Subgroups — C₂×C₃₁×#2539 #21491578

Order 7936.

5726 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#123	C₂	1	2	yes	maximal, complemented
2	C₂²×C₃₁×#121	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃₁×#120	C₂	1	2	yes	maximal, complemented
4	C₂³×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2539	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₃₁×#399	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#392	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#390	C₂	1	4	yes	maximal, complemented
9	C₂×C₄×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁶×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
12	C₂³×D₄×C₃₁	C₂²	1	4	yes	—
13	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
14	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#123	—	2	16	no	—
17	C₂×C₃₁×#123	C₂²	1	16	yes	complemented
18	C₂×C₃₁×#121	—	2	8	no	—
19	C₂×C₃₁×#121	C₂²	1	8	yes	complemented
20	C₂×C₃₁×#120	—	2	16	no	—
21	C₂×C₃₁×#120	C₂²	1	16	yes	complemented
22	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
23	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
24	C₂²×C₂²⋊C₄×C₃₁	—	2	8	no	—
25	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
27	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	complemented
30	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
31	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	complemented
32	C₂²×C₄²×C₃₁	C₂²	1	2	yes	complemented
33	C₃₁×#399	C₂²	1	16	yes	complemented
34	C₃₁×#392	C₂²	1	16	yes	complemented
35	C₃₁×#390	C₂²	1	32	yes	complemented
36	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	complemented
37	C₂×C₃₁×#95	C₂²	1	2	yes	complemented
38	C₂⁵×C₃₁	—	2	4	no	—
39	C₂⁵×C₃₁	—	2	8	no	—
40	C₂⁵×C₃₁	—	2	8	no	—
41	C₂⁵×C₃₁	—	4	4	no	—
42	C₂⁵×C₃₁	C₂³	1	2	yes	—
43	C₂⁵×C₃₁	C₂³	1	2	yes	—
44	C₂⁵×C₃₁	C₂³	1	4	yes	—
45	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₃₁	—	2	8	no	—
47	C₂²×D₄×C₃₁	—	2	8	no	—
48	C₂²×D₄×C₃₁	—	2	8	no	—
49	C₂²×D₄×C₃₁	—	2	16	no	—
50	C₂²×D₄×C₃₁	—	2	16	no	—
51	C₂²×D₄×C₃₁	—	2	4	no	—
52	C₂²×D₄×C₃₁	—	4	8	no	—
53	C₂²×D₄×C₃₁	—	4	4	no	—
54	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
55	C₂²×D₄×C₃₁	C₂³	1	16	yes	—
56	C₂²×D₄×C₃₁	C₂³	1	4	yes	complemented
57	C₂³×C₄×C₃₁	—	2	4	no	—
58	C₂³×C₄×C₃₁	—	2	8	no	—
59	C₂³×C₄×C₃₁	—	4	4	no	—
60	C₂³×C₄×C₃₁	D₄	1	8	yes	complemented
61	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
62	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
63	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
64	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
66	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
68	C₃₁×#123	—	2	64	no	—
69	C₃₁×#121	—	2	32	no	—
70	C₃₁×#120	—	2	64	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	16	no	—
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	complemented
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	16	yes	complemented
75	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
76	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
77	C₂×C₂²⋊C₄×C₃₁	—	2	16	no	—
78	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
79	C₂×C₂²⋊C₄×C₃₁	—	2	16	no	—
80	C₂×C₂²⋊C₄×C₃₁	—	2	32	no	—
81	C₂×C₂²⋊C₄×C₃₁	D₄	1	16	yes	complemented
82	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
83	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
84	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
85	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
86	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	complemented
87	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	complemented
88	C₂×C₄²×C₃₁	—	4	4	no	—
89	C₂×C₄²×C₃₁	C₂³	1	8	yes	complemented
90	C₃₁×#95	C₂³	1	8	yes	complemented
91	C₂⁴×C₃₁	—	2	4	no	—
92	C₂⁴×C₃₁	—	2	4	no	—
93	C₂⁴×C₃₁	—	2	32	no	—
94	C₂⁴×C₃₁	—	2	8	no	—
95	C₂⁴×C₃₁	—	2	4	no	—
96	C₂⁴×C₃₁	—	2	8	no	—
97	C₂⁴×C₃₁	—	2	16	no	—
98	C₂⁴×C₃₁	—	2	4	no	—
99	C₂⁴×C₃₁	—	2	8	no	—
100	C₂⁴×C₃₁	—	2	4	no	—
101	C₂⁴×C₃₁	—	2	4	no	—
102	C₂⁴×C₃₁	—	2	16	no	—
103	C₂⁴×C₃₁	—	2	4	no	—
104	C₂⁴×C₃₁	—	2	8	no	—
105	C₂⁴×C₃₁	—	2	8	no	—
106	C₂⁴×C₃₁	—	4	8	no	—
107	C₂⁴×C₃₁	—	4	64	no	—
108	C₂⁴×C₃₁	—	4	16	no	—
109	C₂⁴×C₃₁	—	4	16	no	—
110	C₂⁴×C₃₁	C₂×D₄	1	8	yes	—
111	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
112	C₂⁴×C₃₁	#41	1	8	yes	—
113	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
114	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
115	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
116	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
117	C₂×D₄×C₃₁	—	2	16	no	—
118	C₂×D₄×C₃₁	—	2	16	no	—
119	C₂×D₄×C₃₁	—	2	16	no	—
120	C₂×D₄×C₃₁	—	2	32	no	—
121	C₂×D₄×C₃₁	—	2	32	no	—
122	C₂×D₄×C₃₁	—	2	8	no	—
123	C₂×D₄×C₃₁	—	4	16	no	—
124	C₂×D₄×C₃₁	—	4	8	no	—
125	C₂×D₄×C₃₁	—	4	64	no	—
126	C₂×D₄×C₃₁	—	4	16	no	—
127	C₂×D₄×C₃₁	—	4	32	no	—
128	C₂²×C₄×C₃₁	—	2	8	no	—
129	C₂²×C₄×C₃₁	—	2	4	no	—
130	C₂²×C₄×C₃₁	—	2	8	no	—
131	C₂²×C₄×C₃₁	—	2	4	no	—
132	C₂²×C₄×C₃₁	—	2	8	no	—
133	C₂²×C₄×C₃₁	—	2	4	no	—
134	C₂²×C₄×C₃₁	—	2	4	no	—
135	C₂²×C₄×C₃₁	—	2	8	no	—
136	C₂²×C₄×C₃₁	—	2	4	no	—
137	C₂²×C₄×C₃₁	—	2	4	no	—
138	C₂²×C₄×C₃₁	—	2	16	no	—
139	C₂²×C₄×C₃₁	—	4	16	no	—
140	C₂²×C₄×C₃₁	—	4	8	no	—
141	C₂²×C₄×C₃₁	—	4	16	no	—
142	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
143	C₂²×C₄×C₃₁	C₂×D₄	1	16	yes	—
144	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
145	C₂²×C₄×C₃₁	C₂×D₄	1	16	yes	complemented
146	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
147	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
149	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
150	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
151	C₄⋊C₄×C₃₁	—	2	8	no	—
152	C₄⋊C₄×C₃₁	—	2	32	no	—
153	C₂²⋊C₄×C₃₁	—	2	16	no	—
154	C₂²⋊C₄×C₃₁	—	2	16	no	—
155	C₂²⋊C₄×C₃₁	—	2	32	no	—
156	C₂²⋊C₄×C₃₁	—	2	16	no	—
157	C₂²⋊C₄×C₃₁	—	2	32	no	—
158	C₂²⋊C₄×C₃₁	C₂×D₄	1	32	yes	complemented
159	C₄²×C₃₁	—	4	8	no	—
160	C₂×#2539	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂³×C₃₁	—	2	8	no	—
162	C₂³×C₃₁	—	2	8	no	—
163	C₂³×C₃₁	—	2	32	no	—
164	C₂³×C₃₁	—	2	4	no	—
165	C₂³×C₃₁	—	2	4	no	—
166	C₂³×C₃₁	—	2	32	no	—
167	C₂³×C₃₁	—	2	16	no	—
168	C₂³×C₃₁	—	2	32	no	—
169	C₂³×C₃₁	—	2	4	no	—
170	C₂³×C₃₁	—	2	8	no	—
171	C₂³×C₃₁	—	2	4	no	—
172	C₂³×C₃₁	—	2	8	no	—
173	C₂³×C₃₁	—	2	4	no	—
174	C₂³×C₃₁	—	2	4	no	—
175	C₂³×C₃₁	—	2	4	no	—
176	C₂³×C₃₁	—	2	8	no	—
177	C₂³×C₃₁	—	2	8	no	—
178	C₂³×C₃₁	—	4	64	no	—
179	C₂³×C₃₁	—	4	16	no	—
180	C₂³×C₃₁	—	4	128	no	—
181	C₂³×C₃₁	—	4	16	no	—
182	C₂³×C₃₁	—	4	8	no	—
183	C₂³×C₃₁	—	4	8	no	—
184	C₂³×C₃₁	—	4	16	no	—
185	C₂³×C₃₁	—	4	8	no	—
186	C₂³×C₃₁	—	4	8	no	—
187	C₂³×C₃₁	—	4	8	no	—
188	C₂³×C₃₁	—	4	16	no	—
189	C₂³×C₃₁	—	4	16	no	—
190	C₂³×C₃₁	#127	1	16	yes	—
191	C₂³×C₃₁	C₂²×D₄	1	8	yes	—
192	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
193	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
194	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
195	C₂³×C₃₁	C₂×#41	1	8	yes	—
196	C₂³×C₃₁	C₂×#41	1	4	yes	—
197	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
198	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
199	D₄×C₃₁	—	4	64	no	—
200	D₄×C₃₁	—	4	16	no	—
201	D₄×C₃₁	—	4	32	no	—
202	C₂×C₄×C₃₁	—	2	8	no	—
203	C₂×C₄×C₃₁	—	2	8	no	—
204	C₂×C₄×C₃₁	—	2	32	no	—
205	C₂×C₄×C₃₁	—	2	8	no	—
206	C₂×C₄×C₃₁	—	2	4	no	—
207	C₂×C₄×C₃₁	—	2	8	no	—
208	C₂×C₄×C₃₁	—	2	4	no	—
209	C₂×C₄×C₃₁	—	2	4	no	—
210	C₂×C₄×C₃₁	—	2	4	no	—
211	C₂×C₄×C₃₁	—	2	8	no	—
212	C₂×C₄×C₃₁	—	2	4	no	—
213	C₂×C₄×C₃₁	—	2	4	no	—
214	C₂×C₄×C₃₁	—	4	16	no	—
215	C₂×C₄×C₃₁	—	4	16	no	—
216	C₂×C₄×C₃₁	—	4	8	no	—
217	C₂×C₄×C₃₁	—	4	8	no	—
218	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
219	C₂×C₄×C₃₁	C₂²×D₄	1	16	yes	—
220	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
221	C₂²×#123	C₂×C₃₁	1	2	yes	complemented
222	C₂²×#121	C₂×C₃₁	1	1	yes	characteristic, complemented
223	C₂²×#120	C₂×C₃₁	1	2	yes	complemented
224	C₂³×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#2539	C₂×C₃₁	1	16	yes	direct product factor
226	C₂×#399	C₂×C₃₁	1	2	yes	complemented
227	C₂×#392	C₂×C₃₁	1	2	yes	complemented
228	C₂×#390	C₂×C₃₁	1	4	yes	complemented
229	C₂×C₄×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
230	C₂²×C₃₁	—	2	4	no	—
231	C₂²×C₃₁	—	2	4	no	—
232	C₂²×C₃₁	—	2	16	no	—
233	C₂²×C₃₁	—	2	32	no	—
234	C₂²×C₃₁	—	2	8	no	—
235	C₂²×C₃₁	—	2	16	no	—
236	C₂²×C₃₁	—	4	64	no	—
237	C₂²×C₃₁	—	4	4	no	—
238	C₂²×C₃₁	—	4	4	no	—
239	C₂²×C₃₁	—	4	8	no	—
240	C₂²×C₃₁	—	4	32	no	—
241	C₂²×C₃₁	—	4	4	no	—
242	C₂²×C₃₁	—	4	4	no	—
243	C₂²×C₃₁	—	4	4	no	—
244	C₂²×C₃₁	—	4	8	no	—
245	C₂²×C₃₁	—	4	8	no	—
246	C₂²×C₃₁	—	8	16	no	—
247	C₂²×C₃₁	#525	1	4	yes	—
248	C₂²×C₃₁	C₂×#127	1	8	yes	—
249	C₂²×C₃₁	C₂×#127	1	4	yes	—
250	C₂²×C₃₁	#534	1	4	yes	—
251	C₂²×C₃₁	#546	1	16	yes	—
252	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
253	C₂²×C₃₁	C₂³×D₄	1	2	yes	—
254	C₂²×C₃₁	C₂²×#41	1	2	yes	—
255	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
256	C₄×C₃₁	—	2	4	no	—
257	C₄×C₃₁	—	2	4	no	—
258	C₄×C₃₁	—	2	16	no	—
259	C₄×C₃₁	—	4	4	no	—
260	C₄×C₃₁	—	4	4	no	—
261	C₂⁶	C₂²×C₃₁	1	1	yes	characteristic
262	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
263	C₂³×D₄	C₂²×C₃₁	1	4	yes	—
264	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
265	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
266	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
267	C₂×#123	—	2	16	no	—
268	C₂×#123	C₂²×C₃₁	1	16	yes	complemented
269	C₂×#121	—	2	8	no	—
270	C₂×#121	C₂²×C₃₁	1	8	yes	complemented
271	C₂×#120	—	2	16	no	—
272	C₂×#120	C₂²×C₃₁	1	16	yes	complemented
273	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
274	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
275	C₂²×C₂²⋊C₄	—	2	8	no	—
276	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
277	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
278	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
279	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
280	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	complemented
281	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
282	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	complemented
283	C₂²×C₄²	C₂²×C₃₁	1	2	yes	complemented
284	#399	C₂²×C₃₁	1	16	yes	complemented
285	#392	C₂²×C₃₁	1	16	yes	complemented
286	#390	C₂²×C₃₁	1	32	yes	complemented
287	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	complemented
288	C₂×#95	C₂²×C₃₁	1	2	yes	complemented
289	C₂×C₃₁	—	2	8	no	—
290	C₂×C₃₁	—	4	8	no	—
291	C₂×C₃₁	—	8	4	no	—
292	C₂×C₃₁	#2539	1	8	yes	direct product factor
293	C₂×C₃₁	C₂×#525	1	1	yes	characteristic
294	C₂×C₃₁	C₂²×#127	1	1	yes	characteristic
295	C₂×C₃₁	C₂×#534	1	1	yes	characteristic
296	C₂×C₃₁	C₂×#546	1	4	yes	—
297	C₂⁵	—	2	4	no	—
298	C₂⁵	—	2	8	no	—
299	C₂⁵	—	2	8	no	—
300	C₂⁵	—	4	4	no	—
301	C₂⁵	C₂³×C₃₁	1	2	yes	—
302	C₂⁵	C₂³×C₃₁	1	2	yes	—
303	C₂⁵	C₂³×C₃₁	1	4	yes	—
304	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
305	C₂²×D₄	—	2	8	no	—
306	C₂²×D₄	—	2	8	no	—
307	C₂²×D₄	—	2	8	no	—
308	C₂²×D₄	—	2	16	no	—
309	C₂²×D₄	—	2	16	no	—
310	C₂²×D₄	—	2	4	no	—
311	C₂²×D₄	—	4	8	no	—
312	C₂²×D₄	—	4	4	no	—
313	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
314	C₂²×D₄	C₂³×C₃₁	1	16	yes	—
315	C₂²×D₄	C₂³×C₃₁	1	4	yes	complemented
316	C₂³×C₄	—	2	4	no	—
317	C₂³×C₄	—	2	8	no	—
318	C₂³×C₄	—	4	4	no	—
319	C₂³×C₄	D₄×C₃₁	1	8	yes	complemented
320	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
321	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
322	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
323	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
324	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
325	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
326	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
327	#123	—	2	64	no	—
328	#121	—	2	32	no	—
329	#120	—	2	64	no	—
330	C₂×C₄⋊C₄	—	2	4	no	—
331	C₂×C₄⋊C₄	—	2	16	no	—
332	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	complemented
333	C₂×C₄⋊C₄	C₂³×C₃₁	1	16	yes	complemented
334	C₂×C₂²⋊C₄	—	2	8	no	—
335	C₂×C₂²⋊C₄	—	2	8	no	—
336	C₂×C₂²⋊C₄	—	2	16	no	—
337	C₂×C₂²⋊C₄	—	2	8	no	—
338	C₂×C₂²⋊C₄	—	2	16	no	—
339	C₂×C₂²⋊C₄	—	2	32	no	—
340	C₂×C₂²⋊C₄	D₄×C₃₁	1	16	yes	complemented
341	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
342	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
343	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
344	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
345	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	complemented
346	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	complemented
347	C₂×C₄²	—	4	4	no	—
348	C₂×C₄²	C₂³×C₃₁	1	8	yes	complemented
349	#95	C₂³×C₃₁	1	8	yes	complemented
350	C₃₁	C₂×#2539	1	1	yes	characteristic, direct product factor, Sylow
351	C₂⁴	—	2	4	no	—
352	C₂⁴	—	2	4	no	—
353	C₂⁴	—	2	32	no	—
354	C₂⁴	—	2	8	no	—
355	C₂⁴	—	2	4	no	—
356	C₂⁴	—	2	8	no	—
357	C₂⁴	—	2	16	no	—
358	C₂⁴	—	2	4	no	—
359	C₂⁴	—	2	8	no	—
360	C₂⁴	—	2	4	no	—
361	C₂⁴	—	2	4	no	—
362	C₂⁴	—	2	16	no	—
363	C₂⁴	—	2	4	no	—
364	C₂⁴	—	2	8	no	—
365	C₂⁴	—	2	8	no	—
366	C₂⁴	—	4	8	no	—
367	C₂⁴	—	4	64	no	—
368	C₂⁴	—	4	16	no	—
369	C₂⁴	—	4	16	no	—
370	C₂⁴	C₂×D₄×C₃₁	1	8	yes	—
371	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
372	C₂⁴	#41×C₃₁	1	8	yes	—
373	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
374	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
375	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
376	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
377	C₂×D₄	—	2	16	no	—
378	C₂×D₄	—	2	16	no	—
379	C₂×D₄	—	2	16	no	—
380	C₂×D₄	—	2	32	no	—
381	C₂×D₄	—	2	32	no	—
382	C₂×D₄	—	2	8	no	—
383	C₂×D₄	—	4	16	no	—
384	C₂×D₄	—	4	8	no	—
385	C₂×D₄	—	4	64	no	—
386	C₂×D₄	—	4	16	no	—
387	C₂×D₄	—	4	32	no	—
388	C₂²×C₄	—	2	8	no	—
389	C₂²×C₄	—	2	4	no	—
390	C₂²×C₄	—	2	8	no	—
391	C₂²×C₄	—	2	4	no	—
392	C₂²×C₄	—	2	8	no	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	—	2	4	no	—
395	C₂²×C₄	—	2	8	no	—
396	C₂²×C₄	—	2	4	no	—
397	C₂²×C₄	—	2	4	no	—
398	C₂²×C₄	—	2	16	no	—
399	C₂²×C₄	—	4	16	no	—
400	C₂²×C₄	—	4	8	no	—
401	C₂²×C₄	—	4	16	no	—
402	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
403	C₂²×C₄	C₂×D₄×C₃₁	1	16	yes	—
404	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
405	C₂²×C₄	C₂×D₄×C₃₁	1	16	yes	complemented
406	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
407	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
408	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
409	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
410	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
411	C₄⋊C₄	—	2	8	no	—
412	C₄⋊C₄	—	2	32	no	—
413	C₂²⋊C₄	—	2	16	no	—
414	C₂²⋊C₄	—	2	16	no	—
415	C₂²⋊C₄	—	2	32	no	—
416	C₂²⋊C₄	—	2	16	no	—
417	C₂²⋊C₄	—	2	32	no	—
418	C₂²⋊C₄	C₂×D₄×C₃₁	1	32	yes	complemented
419	C₄²	—	4	8	no	—
420	C₂³	—	2	8	no	—
421	C₂³	—	2	8	no	—
422	C₂³	—	2	32	no	—
423	C₂³	—	2	4	no	—
424	C₂³	—	2	4	no	—
425	C₂³	—	2	32	no	—
426	C₂³	—	2	16	no	—
427	C₂³	—	2	32	no	—
428	C₂³	—	2	4	no	—
429	C₂³	—	2	8	no	—
430	C₂³	—	2	4	no	—
431	C₂³	—	2	8	no	—
432	C₂³	—	2	4	no	—
433	C₂³	—	2	4	no	—
434	C₂³	—	2	4	no	—
435	C₂³	—	2	8	no	—
436	C₂³	—	2	8	no	—
437	C₂³	—	4	64	no	—
438	C₂³	—	4	16	no	—
439	C₂³	—	4	128	no	—
440	C₂³	—	4	16	no	—
441	C₂³	—	4	8	no	—
442	C₂³	—	4	8	no	—
443	C₂³	—	4	16	no	—
444	C₂³	—	4	8	no	—
445	C₂³	—	4	8	no	—
446	C₂³	—	4	8	no	—
447	C₂³	—	4	16	no	—
448	C₂³	—	4	16	no	—
449	C₂³	C₃₁×#127	1	16	yes	—
450	C₂³	C₂²×D₄×C₃₁	1	8	yes	—
451	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
452	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
453	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
454	C₂³	C₂×#41×C₃₁	1	8	yes	—
455	C₂³	C₂×#41×C₃₁	1	4	yes	—
456	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
457	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
458	D₄	—	4	64	no	—
459	D₄	—	4	16	no	—
460	D₄	—	4	32	no	—
461	C₂×C₄	—	2	8	no	—
462	C₂×C₄	—	2	8	no	—
463	C₂×C₄	—	2	32	no	—
464	C₂×C₄	—	2	8	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	8	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	8	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	4	16	no	—
474	C₂×C₄	—	4	16	no	—
475	C₂×C₄	—	4	8	no	—
476	C₂×C₄	—	4	8	no	—
477	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
478	C₂×C₄	C₂²×D₄×C₃₁	1	16	yes	—
479	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
480	C₂²	—	2	4	no	—
481	C₂²	—	2	4	no	—
482	C₂²	—	2	16	no	—
483	C₂²	—	2	32	no	—
484	C₂²	—	2	8	no	—
485	C₂²	—	2	16	no	—
486	C₂²	—	4	64	no	—
487	C₂²	—	4	4	no	—
488	C₂²	—	4	4	no	—
489	C₂²	—	4	8	no	—
490	C₂²	—	4	32	no	—
491	C₂²	—	4	4	no	—
492	C₂²	—	4	4	no	—
493	C₂²	—	4	4	no	—
494	C₂²	—	4	8	no	—
495	C₂²	—	4	8	no	—
496	C₂²	—	8	16	no	—
497	C₂²	C₃₁×#525	1	4	yes	—
498	C₂²	C₂×C₃₁×#127	1	8	yes	—
499	C₂²	C₂×C₃₁×#127	1	4	yes	—
500	C₂²	C₃₁×#534	1	4	yes	—
501	C₂²	C₃₁×#546	1	16	yes	—
502	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
503	C₂²	C₂³×D₄×C₃₁	1	2	yes	—
504	C₂²	C₂²×#41×C₃₁	1	2	yes	—
505	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
506	C₄	—	2	4	no	—
507	C₄	—	2	4	no	—
508	C₄	—	2	16	no	—
509	C₄	—	4	4	no	—
510	C₄	—	4	4	no	—
511	C₂	—	2	8	no	—
512	C₂	—	4	8	no	—
513	C₂	—	8	4	no	—
514	C₂	C₃₁×#2539	1	8	yes	direct product factor
515	C₂	C₂×C₃₁×#525	1	1	yes	characteristic
516	C₂	C₂²×C₃₁×#127	1	1	yes	characteristic
517	C₂	C₂×C₃₁×#534	1	1	yes	characteristic
518	C₂	C₂×C₃₁×#546	1	4	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. C₂²×C₃₁×#123	C₂	semidirect
2. C₂²×C₃₁×#121	C₂	semidirect
3. C₂²×C₃₁×#120	C₂	semidirect
4. C₂³×C₂²⋊C₄×C₃₁	C₂	semidirect
5. C₃₁×#2539	C₂	direct (N × H)
6. C₂×C₃₁×#399	C₂	semidirect
7. C₂×C₃₁×#392	C₂	semidirect
8. C₂×C₃₁×#390	C₂	semidirect
9. C₂×C₄×C₂²⋊C₄×C₃₁	C₂	semidirect
13. C₂³×D₄×C₃₁	C₂²	semidirect
17. C₂×C₃₁×#123	C₂²	semidirect
19. C₂×C₃₁×#121	C₂²	semidirect
21. C₂×C₃₁×#120	C₂²	semidirect
22. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
23. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
29. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
30. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
31. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
32. C₂²×C₄²×C₃₁	C₂²	semidirect
33. C₃₁×#399	C₂²	semidirect
34. C₃₁×#392	C₂²	semidirect
35. C₃₁×#390	C₂²	semidirect
36. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
37. C₂×C₃₁×#95	C₂²	semidirect
56. C₂²×D₄×C₃₁	C₂³	semidirect
60. C₂³×C₄×C₃₁	D₄	semidirect
73. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
74. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
81. C₂×C₂²⋊C₄×C₃₁	D₄	semidirect
86. C₂×C₂²⋊C₄×C₃₁	C₂³	semidirect
87. C₂×C₂²⋊C₄×C₃₁	C₂³	semidirect
89. C₂×C₄²×C₃₁	C₂³	semidirect
90. C₃₁×#95	C₂³	semidirect
145. C₂²×C₄×C₃₁	C₂×D₄	semidirect
158. C₂²⋊C₄×C₃₁	C₂×D₄	semidirect
160. C₂×#2539	C₃₁	direct (N × H)
221. C₂²×#123	C₂×C₃₁	semidirect
222. C₂²×#121	C₂×C₃₁	semidirect
223. C₂²×#120	C₂×C₃₁	semidirect
224. C₂³×C₂²⋊C₄	C₂×C₃₁	semidirect
225. #2539	C₂×C₃₁	direct (N × H)
226. C₂×#399	C₂×C₃₁	semidirect
227. C₂×#392	C₂×C₃₁	semidirect
228. C₂×#390	C₂×C₃₁	semidirect
229. C₂×C₄×C₂²⋊C₄	C₂×C₃₁	semidirect
264. C₂³×D₄	C₂²×C₃₁	semidirect
268. C₂×#123	C₂²×C₃₁	semidirect
270. C₂×#121	C₂²×C₃₁	semidirect
272. C₂×#120	C₂²×C₃₁	semidirect
273. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
274. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
280. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
281. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
282. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
283. C₂²×C₄²	C₂²×C₃₁	semidirect
284. #399	C₂²×C₃₁	semidirect
285. #392	C₂²×C₃₁	semidirect
286. #390	C₂²×C₃₁	semidirect
287. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
288. C₂×#95	C₂²×C₃₁	semidirect
292. C₂×C₃₁	#2539	direct (N × H)
315. C₂²×D₄	C₂³×C₃₁	semidirect
319. C₂³×C₄	D₄×C₃₁	semidirect
332. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
333. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
340. C₂×C₂²⋊C₄	D₄×C₃₁	semidirect
345. C₂×C₂²⋊C₄	C₂³×C₃₁	semidirect
346. C₂×C₂²⋊C₄	C₂³×C₃₁	semidirect
348. C₂×C₄²	C₂³×C₃₁	semidirect
349. #95	C₂³×C₃₁	semidirect
350. C₃₁	C₂×#2539	direct (N × H)
405. C₂²×C₄	C₂×D₄×C₃₁	semidirect
418. C₂²⋊C₄	C₂×D₄×C₃₁	semidirect
514. C₂	C₃₁×#2539	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#123	2
C₂²×C₃₁×#121	2
C₂²×C₃₁×#120	2
C₂³×C₂²⋊C₄×C₃₁	2
C₃₁×#2539	2
C₂×C₃₁×#399	2
C₂×C₃₁×#392	2
C₂×C₃₁×#390	2
C₂×C₄×C₂²⋊C₄×C₃₁	2
C₂×#2539	31

Subgroups — C2×C31×#2539 #21491578

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2539 #21491578