Subgroups — C₂×C₃₁×#2408 #21491430

Order 7936.

8622 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×#121	C₂	1	2	yes	maximal, complemented
3	C₂²×C₃₁×#120	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2408	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₃₁×#394	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#392	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#390	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#380	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#379	C₂	1	2	yes	maximal, complemented
11	C₂⁶×C₃₁	C₂²	1	2	yes	—
12	C₂³×D₄×C₃₁	—	2	16	no	—
13	C₂³×D₄×C₃₁	—	2	8	no	—
14	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
15	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
16	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂³×D₄×C₃₁	C₂²	1	8	yes	complemented
18	C₂³×D₄×C₃₁	C₂²	1	4	yes	complemented
19	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₃₁×#121	—	2	16	no	—
22	C₂×C₃₁×#121	C₂²	1	16	yes	complemented
23	C₂×C₃₁×#120	—	2	16	no	—
24	C₂×C₃₁×#120	C₂²	1	16	yes	complemented
25	C₂×C₄×D₄×C₃₁	—	2	8	no	—
26	C₂×C₄×D₄×C₃₁	C₂²	1	8	yes	complemented
27	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
28	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
29	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
31	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
32	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
33	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
34	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
35	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic, complemented
36	C₃₁×#394	C₂²	1	16	yes	complemented
37	C₃₁×#392	C₂²	1	16	yes	complemented
38	C₃₁×#390	C₂²	1	16	yes	complemented
39	C₃₁×#380	C₂²	1	8	yes	complemented
40	C₃₁×#379	C₂²	1	16	yes	complemented
41	C₂×C₃₁×#95	C₂²	1	2	yes	complemented
42	C₂⁵×C₃₁	—	2	16	no	—
43	C₂⁵×C₃₁	—	2	8	no	—
44	C₂⁵×C₃₁	—	2	8	no	—
45	C₂⁵×C₃₁	—	2	8	no	—
46	C₂⁵×C₃₁	—	4	8	no	—
47	C₂⁵×C₃₁	C₂³	1	4	yes	—
48	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
49	C₂⁵×C₃₁	C₂³	1	8	yes	—
50	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×D₄×C₃₁	—	2	128	no	—
52	C₂²×D₄×C₃₁	—	2	4	no	—
53	C₂²×D₄×C₃₁	—	2	16	no	—
54	C₂²×D₄×C₃₁	—	2	4	no	—
55	C₂²×D₄×C₃₁	—	2	64	no	—
56	C₂²×D₄×C₃₁	—	2	16	no	—
57	C₂²×D₄×C₃₁	—	2	16	no	—
58	C₂²×D₄×C₃₁	—	2	8	no	—
59	C₂²×D₄×C₃₁	—	2	8	no	—
60	C₂²×D₄×C₃₁	—	2	8	no	—
61	C₂²×D₄×C₃₁	—	2	32	no	—
62	C₂²×D₄×C₃₁	—	4	8	no	—
63	C₂²×D₄×C₃₁	D₄	1	16	yes	complemented
64	C₂²×D₄×C₃₁	D₄	1	8	yes	complemented
65	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
66	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
67	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
68	C₂²×D₄×C₃₁	C₂³	1	16	yes	complemented
69	C₂³×C₄×C₃₁	—	2	8	no	—
70	C₂³×C₄×C₃₁	—	2	4	no	—
71	C₂³×C₄×C₃₁	—	2	8	no	—
72	C₂³×C₄×C₃₁	D₄	1	8	yes	complemented
73	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
76	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
78	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₃₁	C₂³	1	4	yes	complemented
81	C₃₁×#121	—	2	64	no	—
82	C₃₁×#120	—	2	64	no	—
83	C₄×D₄×C₃₁	—	2	32	no	—
84	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
85	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	complemented
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	complemented
88	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
89	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
90	C₂×C₂²⋊C₄×C₃₁	—	2	16	no	—
91	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
92	C₂×C₂²⋊C₄×C₃₁	—	2	16	no	—
93	C₂×C₂²⋊C₄×C₃₁	—	4	8	no	—
94	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
96	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	complemented
97	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	complemented
98	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	complemented
99	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	complemented
100	C₂×C₄²×C₃₁	—	2	4	no	—
101	C₂×C₄²×C₃₁	C₂³	1	4	yes	complemented
102	C₃₁×#95	C₂³	1	8	yes	complemented
103	C₂⁴×C₃₁	—	2	4	no	—
104	C₂⁴×C₃₁	—	2	4	no	—
105	C₂⁴×C₃₁	—	2	64	no	—
106	C₂⁴×C₃₁	—	2	2	no	—
107	C₂⁴×C₃₁	—	2	8	no	—
108	C₂⁴×C₃₁	—	2	4	no	—
109	C₂⁴×C₃₁	—	2	32	no	—
110	C₂⁴×C₃₁	—	2	16	no	—
111	C₂⁴×C₃₁	—	2	8	no	—
112	C₂⁴×C₃₁	—	2	8	no	—
113	C₂⁴×C₃₁	—	2	2	no	—
114	C₂⁴×C₃₁	—	2	8	no	—
115	C₂⁴×C₃₁	—	2	8	no	—
116	C₂⁴×C₃₁	—	2	4	no	—
117	C₂⁴×C₃₁	—	2	8	no	—
118	C₂⁴×C₃₁	—	2	16	no	—
119	C₂⁴×C₃₁	—	2	16	no	—
120	C₂⁴×C₃₁	—	4	16	no	—
121	C₂⁴×C₃₁	—	4	64	no	—
122	C₂⁴×C₃₁	—	4	64	no	—
123	C₂⁴×C₃₁	—	4	32	no	—
124	C₂⁴×C₃₁	—	4	32	no	—
125	C₂⁴×C₃₁	C₂×D₄	1	16	yes	—
126	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
127	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
128	C₂⁴×C₃₁	C₂×D₄	1	4	yes	complemented
129	C₂⁴×C₃₁	#41	1	4	yes	—
130	C₂⁴×C₃₁	C₂⁴	1	8	yes	—
131	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
132	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
133	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
134	C₂×D₄×C₃₁	—	2	64	no	—
135	C₂×D₄×C₃₁	—	2	256	no	—
136	C₂×D₄×C₃₁	—	2	32	no	—
137	C₂×D₄×C₃₁	—	2	8	no	—
138	C₂×D₄×C₃₁	—	2	32	no	—
139	C₂×D₄×C₃₁	—	2	8	no	—
140	C₂×D₄×C₃₁	—	2	32	no	—
141	C₂×D₄×C₃₁	—	2	32	no	—
142	C₂×D₄×C₃₁	—	2	64	no	—
143	C₂×D₄×C₃₁	—	2	16	no	—
144	C₂×D₄×C₃₁	—	2	16	no	—
145	C₂×D₄×C₃₁	—	2	16	no	—
146	C₂×D₄×C₃₁	—	4	16	no	—
147	C₂×D₄×C₃₁	—	4	64	no	—
148	C₂×D₄×C₃₁	C₂×D₄	1	32	yes	complemented
149	C₂×D₄×C₃₁	C₂×D₄	1	16	yes	complemented
150	C₂²×C₄×C₃₁	—	2	2	no	—
151	C₂²×C₄×C₃₁	—	2	4	no	—
152	C₂²×C₄×C₃₁	—	2	2	no	—
153	C₂²×C₄×C₃₁	—	2	32	no	—
154	C₂²×C₄×C₃₁	—	2	16	no	—
155	C₂²×C₄×C₃₁	—	2	2	no	—
156	C₂²×C₄×C₃₁	—	2	4	no	—
157	C₂²×C₄×C₃₁	—	2	8	no	—
158	C₂²×C₄×C₃₁	—	2	2	no	—
159	C₂²×C₄×C₃₁	—	2	8	no	—
160	C₂²×C₄×C₃₁	—	2	4	no	—
161	C₂²×C₄×C₃₁	—	2	8	no	—
162	C₂²×C₄×C₃₁	—	2	8	no	—
163	C₂²×C₄×C₃₁	—	2	16	no	—
164	C₂²×C₄×C₃₁	—	4	16	no	—
165	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
166	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
167	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
168	C₂²×C₄×C₃₁	C₂×D₄	1	16	yes	complemented
169	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	complemented
170	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
171	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
173	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
174	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
176	C₄⋊C₄×C₃₁	—	2	8	no	—
177	C₄⋊C₄×C₃₁	—	2	16	no	—
178	C₂²⋊C₄×C₃₁	—	2	8	no	—
179	C₂²⋊C₄×C₃₁	—	2	16	no	—
180	C₂²⋊C₄×C₃₁	—	2	32	no	—
181	C₂²⋊C₄×C₃₁	—	2	8	no	—
182	C₂²⋊C₄×C₃₁	—	2	32	no	—
183	C₂²⋊C₄×C₃₁	—	4	16	no	—
184	C₄²×C₃₁	—	2	8	no	—
185	C₂×#2408	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
186	C₂³×C₃₁	—	2	8	no	—
187	C₂³×C₃₁	—	2	8	no	—
188	C₂³×C₃₁	—	2	16	no	—
189	C₂³×C₃₁	—	2	16	no	—
190	C₂³×C₃₁	—	2	4	no	—
191	C₂³×C₃₁	—	2	4	no	—
192	C₂³×C₃₁	—	2	64	no	—
193	C₂³×C₃₁	—	2	32	no	—
194	C₂³×C₃₁	—	2	8	no	—
195	C₂³×C₃₁	—	2	16	no	—
196	C₂³×C₃₁	—	2	32	no	—
197	C₂³×C₃₁	—	2	32	no	—
198	C₂³×C₃₁	—	2	2	no	—
199	C₂³×C₃₁	—	2	8	no	—
200	C₂³×C₃₁	—	2	4	no	—
201	C₂³×C₃₁	—	2	8	no	—
202	C₂³×C₃₁	—	2	8	no	—
203	C₂³×C₃₁	—	2	2	no	—
204	C₂³×C₃₁	—	2	8	no	—
205	C₂³×C₃₁	—	2	8	no	—
206	C₂³×C₃₁	—	2	4	no	—
207	C₂³×C₃₁	—	2	8	no	—
208	C₂³×C₃₁	—	4	64	no	—
209	C₂³×C₃₁	—	4	64	no	—
210	C₂³×C₃₁	—	4	256	no	—
211	C₂³×C₃₁	—	4	32	no	—
212	C₂³×C₃₁	—	4	4	no	—
213	C₂³×C₃₁	—	4	16	no	—
214	C₂³×C₃₁	—	4	32	no	—
215	C₂³×C₃₁	—	4	4	no	—
216	C₂³×C₃₁	—	4	16	no	—
217	C₂³×C₃₁	—	4	16	no	—
218	C₂³×C₃₁	—	4	8	no	—
219	C₂³×C₃₁	—	4	16	no	—
220	C₂³×C₃₁	#121	1	16	yes	complemented
221	C₂³×C₃₁	C₂²×D₄	1	16	yes	—
222	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
223	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
224	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
225	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
226	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
227	C₂³×C₃₁	C₂²×D₄	1	4	yes	complemented
228	C₂³×C₃₁	C₂×#41	1	4	yes	—
229	C₂³×C₃₁	C₂×#41	1	2	yes	—
230	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
231	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
232	D₄×C₃₁	—	2	64	no	—
233	D₄×C₃₁	—	2	32	no	—
234	D₄×C₃₁	—	2	64	no	—
235	D₄×C₃₁	—	4	64	no	—
236	C₂×C₄×C₃₁	—	2	8	no	—
237	C₂×C₄×C₃₁	—	2	8	no	—
238	C₂×C₄×C₃₁	—	2	2	no	—
239	C₂×C₄×C₃₁	—	2	4	no	—
240	C₂×C₄×C₃₁	—	2	2	no	—
241	C₂×C₄×C₃₁	—	2	32	no	—
242	C₂×C₄×C₃₁	—	2	2	no	—
243	C₂×C₄×C₃₁	—	2	4	no	—
244	C₂×C₄×C₃₁	—	2	8	no	—
245	C₂×C₄×C₃₁	—	2	2	no	—
246	C₂×C₄×C₃₁	—	2	8	no	—
247	C₂×C₄×C₃₁	—	2	4	no	—
248	C₂×C₄×C₃₁	—	2	8	no	—
249	C₂×C₄×C₃₁	—	4	4	no	—
250	C₂×C₄×C₃₁	—	4	4	no	—
251	C₂×C₄×C₃₁	—	4	16	no	—
252	C₂×C₄×C₃₁	—	4	16	no	—
253	C₂×C₄×C₃₁	#120	1	16	yes	complemented
254	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
255	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
256	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
257	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	complemented
258	C₂⁴×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
259	C₂²×#121	C₂×C₃₁	1	2	yes	complemented
260	C₂²×#120	C₂×C₃₁	1	2	yes	complemented
261	C₂²×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
262	#2408	C₂×C₃₁	1	16	yes	direct product factor
263	C₂×#394	C₂×C₃₁	1	2	yes	complemented
264	C₂×#392	C₂×C₃₁	1	2	yes	complemented
265	C₂×#390	C₂×C₃₁	1	2	yes	complemented
266	C₂×#380	C₂×C₃₁	1	1	yes	characteristic, complemented
267	C₂×#379	C₂×C₃₁	1	2	yes	complemented
268	C₂²×C₃₁	—	2	4	no	—
269	C₂²×C₃₁	—	2	32	no	—
270	C₂²×C₃₁	—	2	4	no	—
271	C₂²×C₃₁	—	2	8	no	—
272	C₂²×C₃₁	—	2	8	no	—
273	C₂²×C₃₁	—	2	16	no	—
274	C₂²×C₃₁	—	2	4	no	—
275	C₂²×C₃₁	—	2	8	no	—
276	C₂²×C₃₁	—	2	16	no	—
277	C₂²×C₃₁	—	2	16	no	—
278	C₂²×C₃₁	—	4	128	no	—
279	C₂²×C₃₁	—	4	2	no	—
280	C₂²×C₃₁	—	4	16	no	—
281	C₂²×C₃₁	—	4	64	no	—
282	C₂²×C₃₁	—	4	8	no	—
283	C₂²×C₃₁	—	4	2	no	—
284	C₂²×C₃₁	—	4	8	no	—
285	C₂²×C₃₁	—	4	8	no	—
286	C₂²×C₃₁	—	4	4	no	—
287	C₂²×C₃₁	—	4	8	no	—
288	C₂²×C₃₁	C₂×#120	1	4	yes	—
289	C₂²×C₃₁	C₂×#121	1	4	yes	—
290	C₂²×C₃₁	C₂×#121	1	8	yes	complemented
291	C₂²×C₃₁	#535	1	4	yes	—
292	C₂²×C₃₁	D₄²	1	8	yes	—
293	C₂²×C₃₁	#546	1	8	yes	—
294	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
295	C₂²×C₃₁	C₂³×D₄	1	2	yes	—
296	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
297	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
298	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
299	C₄×C₃₁	—	2	4	no	—
300	C₄×C₃₁	—	2	4	no	—
301	C₄×C₃₁	—	4	2	no	—
302	C₄×C₃₁	—	4	2	no	—
303	C₄×C₃₁	—	4	8	no	—
304	C₄×C₃₁	C₂×#120	1	8	yes	complemented
305	C₂⁶	C₂²×C₃₁	1	2	yes	—
306	C₂³×D₄	—	2	16	no	—
307	C₂³×D₄	—	2	8	no	—
308	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
309	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
310	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
311	C₂³×D₄	C₂²×C₃₁	1	8	yes	complemented
312	C₂³×D₄	C₂²×C₃₁	1	4	yes	complemented
313	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
314	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
315	C₂×#121	—	2	16	no	—
316	C₂×#121	C₂²×C₃₁	1	16	yes	complemented
317	C₂×#120	—	2	16	no	—
318	C₂×#120	C₂²×C₃₁	1	16	yes	complemented
319	C₂×C₄×D₄	—	2	8	no	—
320	C₂×C₄×D₄	C₂²×C₃₁	1	8	yes	complemented
321	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
322	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
323	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
324	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
325	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
326	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
327	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
328	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
329	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic, complemented
330	#394	C₂²×C₃₁	1	16	yes	complemented
331	#392	C₂²×C₃₁	1	16	yes	complemented
332	#390	C₂²×C₃₁	1	16	yes	complemented
333	#380	C₂²×C₃₁	1	8	yes	complemented
334	#379	C₂²×C₃₁	1	16	yes	complemented
335	C₂×#95	C₂²×C₃₁	1	2	yes	complemented
336	C₂×C₃₁	—	2	8	no	—
337	C₂×C₃₁	—	4	4	no	—
338	C₂×C₃₁	—	4	16	no	—
339	C₂×C₃₁	#2408	1	8	yes	direct product factor
340	C₂×C₃₁	C₂²×#120	1	1	yes	characteristic
341	C₂×C₃₁	C₂²×#121	1	1	yes	characteristic
342	C₂×C₃₁	C₂×#535	1	1	yes	characteristic
343	C₂×C₃₁	C₂×D₄²	1	2	yes	—
344	C₂×C₃₁	C₂×#546	1	2	yes	—
345	C₂⁵	—	2	16	no	—
346	C₂⁵	—	2	8	no	—
347	C₂⁵	—	2	8	no	—
348	C₂⁵	—	2	8	no	—
349	C₂⁵	—	4	8	no	—
350	C₂⁵	C₂³×C₃₁	1	4	yes	—
351	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
352	C₂⁵	C₂³×C₃₁	1	8	yes	—
353	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
354	C₂²×D₄	—	2	128	no	—
355	C₂²×D₄	—	2	4	no	—
356	C₂²×D₄	—	2	16	no	—
357	C₂²×D₄	—	2	4	no	—
358	C₂²×D₄	—	2	64	no	—
359	C₂²×D₄	—	2	16	no	—
360	C₂²×D₄	—	2	16	no	—
361	C₂²×D₄	—	2	8	no	—
362	C₂²×D₄	—	2	8	no	—
363	C₂²×D₄	—	2	8	no	—
364	C₂²×D₄	—	2	32	no	—
365	C₂²×D₄	—	4	8	no	—
366	C₂²×D₄	D₄×C₃₁	1	16	yes	complemented
367	C₂²×D₄	D₄×C₃₁	1	8	yes	complemented
368	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
369	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
370	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
371	C₂²×D₄	C₂³×C₃₁	1	16	yes	complemented
372	C₂³×C₄	—	2	8	no	—
373	C₂³×C₄	—	2	4	no	—
374	C₂³×C₄	—	2	8	no	—
375	C₂³×C₄	D₄×C₃₁	1	8	yes	complemented
376	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
377	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
378	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
379	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
380	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
381	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
382	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
383	C₂³×C₄	C₂³×C₃₁	1	4	yes	complemented
384	#121	—	2	64	no	—
385	#120	—	2	64	no	—
386	C₄×D₄	—	2	32	no	—
387	C₂×C₄⋊C₄	—	2	4	no	—
388	C₂×C₄⋊C₄	—	2	8	no	—
389	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	complemented
390	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	complemented
391	C₂×C₂²⋊C₄	—	2	4	no	—
392	C₂×C₂²⋊C₄	—	2	8	no	—
393	C₂×C₂²⋊C₄	—	2	16	no	—
394	C₂×C₂²⋊C₄	—	2	4	no	—
395	C₂×C₂²⋊C₄	—	2	16	no	—
396	C₂×C₂²⋊C₄	—	4	8	no	—
397	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
398	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	complemented
400	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	complemented
401	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	complemented
402	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	complemented
403	C₂×C₄²	—	2	4	no	—
404	C₂×C₄²	C₂³×C₃₁	1	4	yes	complemented
405	#95	C₂³×C₃₁	1	8	yes	complemented
406	C₃₁	C₂×#2408	1	1	yes	characteristic, direct product factor, Sylow
407	C₂⁴	—	2	4	no	—
408	C₂⁴	—	2	4	no	—
409	C₂⁴	—	2	64	no	—
410	C₂⁴	—	2	2	no	—
411	C₂⁴	—	2	8	no	—
412	C₂⁴	—	2	4	no	—
413	C₂⁴	—	2	32	no	—
414	C₂⁴	—	2	16	no	—
415	C₂⁴	—	2	8	no	—
416	C₂⁴	—	2	8	no	—
417	C₂⁴	—	2	2	no	—
418	C₂⁴	—	2	8	no	—
419	C₂⁴	—	2	8	no	—
420	C₂⁴	—	2	4	no	—
421	C₂⁴	—	2	8	no	—
422	C₂⁴	—	2	16	no	—
423	C₂⁴	—	2	16	no	—
424	C₂⁴	—	4	16	no	—
425	C₂⁴	—	4	64	no	—
426	C₂⁴	—	4	64	no	—
427	C₂⁴	—	4	32	no	—
428	C₂⁴	—	4	32	no	—
429	C₂⁴	C₂×D₄×C₃₁	1	16	yes	—
430	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
431	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
432	C₂⁴	C₂×D₄×C₃₁	1	4	yes	complemented
433	C₂⁴	#41×C₃₁	1	4	yes	—
434	C₂⁴	C₂⁴×C₃₁	1	8	yes	—
435	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
436	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
437	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
438	C₂×D₄	—	2	64	no	—
439	C₂×D₄	—	2	256	no	—
440	C₂×D₄	—	2	32	no	—
441	C₂×D₄	—	2	8	no	—
442	C₂×D₄	—	2	32	no	—
443	C₂×D₄	—	2	8	no	—
444	C₂×D₄	—	2	32	no	—
445	C₂×D₄	—	2	32	no	—
446	C₂×D₄	—	2	64	no	—
447	C₂×D₄	—	2	16	no	—
448	C₂×D₄	—	2	16	no	—
449	C₂×D₄	—	2	16	no	—
450	C₂×D₄	—	4	16	no	—
451	C₂×D₄	—	4	64	no	—
452	C₂×D₄	C₂×D₄×C₃₁	1	32	yes	complemented
453	C₂×D₄	C₂×D₄×C₃₁	1	16	yes	complemented
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	4	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	32	no	—
458	C₂²×C₄	—	2	16	no	—
459	C₂²×C₄	—	2	2	no	—
460	C₂²×C₄	—	2	4	no	—
461	C₂²×C₄	—	2	8	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	8	no	—
464	C₂²×C₄	—	2	4	no	—
465	C₂²×C₄	—	2	8	no	—
466	C₂²×C₄	—	2	8	no	—
467	C₂²×C₄	—	2	16	no	—
468	C₂²×C₄	—	4	16	no	—
469	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
470	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
471	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
472	C₂²×C₄	C₂×D₄×C₃₁	1	16	yes	complemented
473	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	complemented
474	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
475	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
476	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
477	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
478	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
479	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
480	C₄⋊C₄	—	2	8	no	—
481	C₄⋊C₄	—	2	16	no	—
482	C₂²⋊C₄	—	2	8	no	—
483	C₂²⋊C₄	—	2	16	no	—
484	C₂²⋊C₄	—	2	32	no	—
485	C₂²⋊C₄	—	2	8	no	—
486	C₂²⋊C₄	—	2	32	no	—
487	C₂²⋊C₄	—	4	16	no	—
488	C₄²	—	2	8	no	—
489	C₂³	—	2	8	no	—
490	C₂³	—	2	8	no	—
491	C₂³	—	2	16	no	—
492	C₂³	—	2	16	no	—
493	C₂³	—	2	4	no	—
494	C₂³	—	2	4	no	—
495	C₂³	—	2	64	no	—
496	C₂³	—	2	32	no	—
497	C₂³	—	2	8	no	—
498	C₂³	—	2	16	no	—
499	C₂³	—	2	32	no	—
500	C₂³	—	2	32	no	—
501	C₂³	—	2	2	no	—
502	C₂³	—	2	8	no	—
503	C₂³	—	2	4	no	—
504	C₂³	—	2	8	no	—
505	C₂³	—	2	8	no	—
506	C₂³	—	2	2	no	—
507	C₂³	—	2	8	no	—
508	C₂³	—	2	8	no	—
509	C₂³	—	2	4	no	—
510	C₂³	—	2	8	no	—
511	C₂³	—	4	64	no	—
512	C₂³	—	4	64	no	—
513	C₂³	—	4	256	no	—
514	C₂³	—	4	32	no	—
515	C₂³	—	4	4	no	—
516	C₂³	—	4	16	no	—
517	C₂³	—	4	32	no	—
518	C₂³	—	4	4	no	—
519	C₂³	—	4	16	no	—
520	C₂³	—	4	16	no	—
521	C₂³	—	4	8	no	—
522	C₂³	—	4	16	no	—
523	C₂³	C₃₁×#121	1	16	yes	complemented
524	C₂³	C₂²×D₄×C₃₁	1	16	yes	—
525	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
526	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
527	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
528	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
529	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
530	C₂³	C₂²×D₄×C₃₁	1	4	yes	complemented
531	C₂³	C₂×#41×C₃₁	1	4	yes	—
532	C₂³	C₂×#41×C₃₁	1	2	yes	—
533	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
534	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
535	D₄	—	2	64	no	—
536	D₄	—	2	32	no	—
537	D₄	—	2	64	no	—
538	D₄	—	4	64	no	—
539	C₂×C₄	—	2	8	no	—
540	C₂×C₄	—	2	8	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	32	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	8	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	8	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	2	8	no	—
552	C₂×C₄	—	4	4	no	—
553	C₂×C₄	—	4	4	no	—
554	C₂×C₄	—	4	16	no	—
555	C₂×C₄	—	4	16	no	—
556	C₂×C₄	C₃₁×#120	1	16	yes	complemented
557	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
558	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
559	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
560	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	complemented
561	C₂²	—	2	4	no	—
562	C₂²	—	2	32	no	—
563	C₂²	—	2	4	no	—
564	C₂²	—	2	8	no	—
565	C₂²	—	2	8	no	—
566	C₂²	—	2	16	no	—
567	C₂²	—	2	4	no	—
568	C₂²	—	2	8	no	—
569	C₂²	—	2	16	no	—
570	C₂²	—	2	16	no	—
571	C₂²	—	4	128	no	—
572	C₂²	—	4	2	no	—
573	C₂²	—	4	8	no	—
574	C₂²	—	4	64	no	—
575	C₂²	—	4	16	no	—
576	C₂²	—	4	8	no	—
577	C₂²	—	4	2	no	—
578	C₂²	—	4	8	no	—
579	C₂²	—	4	8	no	—
580	C₂²	—	4	4	no	—
581	C₂²	C₂×C₃₁×#120	1	4	yes	—
582	C₂²	C₂×C₃₁×#121	1	4	yes	—
583	C₂²	C₂×C₃₁×#121	1	8	yes	complemented
584	C₂²	C₃₁×#535	1	4	yes	—
585	C₂²	D₄²×C₃₁	1	8	yes	—
586	C₂²	C₃₁×#546	1	8	yes	—
587	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
588	C₂²	C₂³×D₄×C₃₁	1	2	yes	—
589	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
590	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
591	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
592	C₄	—	2	4	no	—
593	C₄	—	2	4	no	—
594	C₄	—	4	2	no	—
595	C₄	—	4	2	no	—
596	C₄	—	4	8	no	—
597	C₄	C₂×C₃₁×#120	1	8	yes	complemented
598	C₂	—	2	8	no	—
599	C₂	—	4	16	no	—
600	C₂	—	4	4	no	—
601	C₂	C₃₁×#2408	1	8	yes	direct product factor
602	C₂	C₂²×C₃₁×#120	1	1	yes	characteristic
603	C₂	C₂²×C₃₁×#121	1	1	yes	characteristic
604	C₂	C₂×C₃₁×#535	1	1	yes	characteristic
605	C₂	C₂×D₄²×C₃₁	1	2	yes	—
606	C₂	C₂×C₃₁×#546	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂⁴×D₄×C₃₁	C₂	semidirect
2. C₂²×C₃₁×#121	C₂	semidirect
3. C₂²×C₃₁×#120	C₂	semidirect
4. C₂²×C₄×D₄×C₃₁	C₂	semidirect
5. C₃₁×#2408	C₂	direct (N × H)
6. C₂×C₃₁×#394	C₂	semidirect
7. C₂×C₃₁×#392	C₂	semidirect
8. C₂×C₃₁×#390	C₂	semidirect
9. C₂×C₃₁×#380	C₂	semidirect
10. C₂×C₃₁×#379	C₂	semidirect
17. C₂³×D₄×C₃₁	C₂²	semidirect
18. C₂³×D₄×C₃₁	C₂²	semidirect
20. C₂⁴×C₄×C₃₁	C₂²	semidirect
22. C₂×C₃₁×#121	C₂²	semidirect
24. C₂×C₃₁×#120	C₂²	semidirect
26. C₂×C₄×D₄×C₃₁	C₂²	semidirect
27. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
28. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
31. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
32. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
33. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
34. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
35. C₂²×C₄²×C₃₁	C₂²	semidirect
36. C₃₁×#394	C₂²	semidirect
37. C₃₁×#392	C₂²	semidirect
38. C₃₁×#390	C₂²	semidirect
39. C₃₁×#380	C₂²	semidirect
40. C₃₁×#379	C₂²	semidirect
41. C₂×C₃₁×#95	C₂²	semidirect
63. C₂²×D₄×C₃₁	D₄	semidirect
64. C₂²×D₄×C₃₁	D₄	semidirect
68. C₂²×D₄×C₃₁	C₂³	semidirect
72. C₂³×C₄×C₃₁	D₄	semidirect
80. C₂³×C₄×C₃₁	C₂³	semidirect
86. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
87. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
96. C₂×C₂²⋊C₄×C₃₁	C₂³	semidirect
97. C₂×C₂²⋊C₄×C₃₁	C₂³	semidirect
98. C₂×C₂²⋊C₄×C₃₁	C₂³	semidirect
99. C₂×C₂²⋊C₄×C₃₁	C₂³	semidirect
101. C₂×C₄²×C₃₁	C₂³	semidirect
102. C₃₁×#95	C₂³	semidirect
128. C₂⁴×C₃₁	C₂×D₄	semidirect
148. C₂×D₄×C₃₁	C₂×D₄	semidirect
149. C₂×D₄×C₃₁	C₂×D₄	semidirect
168. C₂²×C₄×C₃₁	C₂×D₄	semidirect
169. C₂²×C₄×C₃₁	C₂×D₄	semidirect
185. C₂×#2408	C₃₁	direct (N × H)
220. C₂³×C₃₁	#121	semidirect
227. C₂³×C₃₁	C₂²×D₄	semidirect
253. C₂×C₄×C₃₁	#120	semidirect
257. C₂×C₄×C₃₁	C₂²×D₄	semidirect
258. C₂⁴×D₄	C₂×C₃₁	semidirect
259. C₂²×#121	C₂×C₃₁	semidirect
260. C₂²×#120	C₂×C₃₁	semidirect
261. C₂²×C₄×D₄	C₂×C₃₁	semidirect
262. #2408	C₂×C₃₁	direct (N × H)
263. C₂×#394	C₂×C₃₁	semidirect
264. C₂×#392	C₂×C₃₁	semidirect
265. C₂×#390	C₂×C₃₁	semidirect
266. C₂×#380	C₂×C₃₁	semidirect
267. C₂×#379	C₂×C₃₁	semidirect
290. C₂²×C₃₁	C₂×#121	semidirect
304. C₄×C₃₁	C₂×#120	semidirect
311. C₂³×D₄	C₂²×C₃₁	semidirect
312. C₂³×D₄	C₂²×C₃₁	semidirect
314. C₂⁴×C₄	C₂²×C₃₁	semidirect
316. C₂×#121	C₂²×C₃₁	semidirect
318. C₂×#120	C₂²×C₃₁	semidirect
320. C₂×C₄×D₄	C₂²×C₃₁	semidirect
321. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
322. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
325. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
326. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
327. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
328. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
329. C₂²×C₄²	C₂²×C₃₁	semidirect
330. #394	C₂²×C₃₁	semidirect
331. #392	C₂²×C₃₁	semidirect
332. #390	C₂²×C₃₁	semidirect
333. #380	C₂²×C₃₁	semidirect
334. #379	C₂²×C₃₁	semidirect
335. C₂×#95	C₂²×C₃₁	semidirect
339. C₂×C₃₁	#2408	direct (N × H)
366. C₂²×D₄	D₄×C₃₁	semidirect
367. C₂²×D₄	D₄×C₃₁	semidirect
371. C₂²×D₄	C₂³×C₃₁	semidirect
375. C₂³×C₄	D₄×C₃₁	semidirect
383. C₂³×C₄	C₂³×C₃₁	semidirect
389. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
390. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
399. C₂×C₂²⋊C₄	C₂³×C₃₁	semidirect
400. C₂×C₂²⋊C₄	C₂³×C₃₁	semidirect
401. C₂×C₂²⋊C₄	C₂³×C₃₁	semidirect
402. C₂×C₂²⋊C₄	C₂³×C₃₁	semidirect
404. C₂×C₄²	C₂³×C₃₁	semidirect
405. #95	C₂³×C₃₁	semidirect
406. C₃₁	C₂×#2408	direct (N × H)
432. C₂⁴	C₂×D₄×C₃₁	semidirect
452. C₂×D₄	C₂×D₄×C₃₁	semidirect
453. C₂×D₄	C₂×D₄×C₃₁	semidirect
472. C₂²×C₄	C₂×D₄×C₃₁	semidirect
473. C₂²×C₄	C₂×D₄×C₃₁	semidirect
523. C₂³	C₃₁×#121	semidirect
530. C₂³	C₂²×D₄×C₃₁	semidirect
556. C₂×C₄	C₃₁×#120	semidirect
560. C₂×C₄	C₂²×D₄×C₃₁	semidirect
583. C₂²	C₂×C₃₁×#121	semidirect
597. C₄	C₂×C₃₁×#120	semidirect
601. C₂	C₃₁×#2408	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×D₄×C₃₁	2
C₂²×C₃₁×#121	2
C₂²×C₃₁×#120	2
C₂²×C₄×D₄×C₃₁	2
C₃₁×#2408	2
C₂×C₃₁×#394	2
C₂×C₃₁×#392	2
C₂×C₃₁×#390	2
C₂×C₃₁×#380	2
C₂×C₃₁×#379	2
C₂×#2408	31

Subgroups — C2×C31×#2408 #21491430

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2408 #21491430