Subgroups — C₂×C₃₁×#2454 #21491458

Order 7936.

3094 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2454	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#399	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#396	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#390	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃₁×#390	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃₁×#389	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#389	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#385	C₂	1	2	yes	maximal, complemented
11	C₂×C₃₁×#380	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
15	C₂×C₃₁×#125	—	2	8	no	—
16	C₂×C₃₁×#125	C₂²	1	8	yes	complemented
17	C₂×C₄×D₄×C₃₁	—	2	8	no	—
18	C₂×C₄×D₄×C₃₁	C₂²	1	8	yes	—
19	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
30	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
31	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
32	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
33	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
34	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
35	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
36	C₃₁×#399	C₂²	1	16	yes	complemented
37	C₃₁×#396	C₂²	1	16	yes	complemented
38	C₃₁×#390	C₂²	1	8	yes	—
39	C₃₁×#390	C₂²	1	8	yes	—
40	C₃₁×#389	C₂²	1	8	yes	complemented
41	C₃₁×#389	C₂²	1	8	yes	complemented
42	C₃₁×#385	C₂²	1	16	yes	complemented
43	C₃₁×#380	C₂²	1	16	yes	complemented
44	C₄×C₄⋊C₄×C₃₁	C₂²	1	8	yes	complemented
45	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
46	C₂×C₃₁×#95	C₂²	1	2	yes	—
47	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
48	C₂×C₃₁×#95	C₂²	1	2	yes	—
49	C₂⁵×C₃₁	C₂³	1	2	yes	—
50	C₂²×D₄×C₃₁	—	2	4	no	—
51	C₂²×D₄×C₃₁	—	2	4	no	—
52	C₂²×D₄×C₃₁	Q₈	1	8	yes	—
53	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
54	C₂³×C₄×C₃₁	—	2	8	no	—
55	C₂³×C₄×C₃₁	—	2	8	no	—
56	C₂³×C₄×C₃₁	—	4	4	no	—
57	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
62	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
63	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
64	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
65	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₃₁×#125	—	2	32	no	—
69	C₄×D₄×C₃₁	—	2	32	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
74	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
75	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
76	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
77	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
78	C₂×C₄⋊C₄×C₃₁	D₄	1	8	yes	—
79	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
88	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
89	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
90	C₂×C₂²⋊C₄×C₃₁	—	4	4	no	—
91	C₂×C₂²⋊C₄×C₃₁	—	4	4	no	—
92	C₂×C₂²⋊C₄×C₃₁	—	4	4	no	—
93	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
94	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
95	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
96	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
97	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
98	C₂×C₄²×C₃₁	—	2	4	no	—
99	C₂×C₄²×C₃₁	—	2	4	no	—
100	C₂×C₄²×C₃₁	—	2	4	no	—
101	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
102	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
103	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
104	C₃₁×#95	C₂³	1	4	yes	—
105	C₃₁×#95	C₂³	1	8	yes	—
106	C₃₁×#95	C₂³	1	4	yes	—
107	C₃₁×#95	C₂³	1	8	yes	—
108	C₂⁴×C₃₁	—	2	4	no	—
109	C₂⁴×C₃₁	—	2	4	no	—
110	C₂⁴×C₃₁	—	2	4	no	—
111	C₂⁴×C₃₁	—	2	4	no	—
112	C₂⁴×C₃₁	—	2	4	no	—
113	C₂⁴×C₃₁	—	2	4	no	—
114	C₂⁴×C₃₁	C₂×Q₈	1	8	yes	—
115	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
116	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
117	C₂×D₄×C₃₁	—	2	32	no	—
118	C₂×D₄×C₃₁	—	2	8	no	—
119	C₂×D₄×C₃₁	—	2	8	no	—
120	C₂×D₄×C₃₁	C₂×Q₈	1	16	yes	—
121	C₂²×C₄×C₃₁	—	2	2	no	—
122	C₂²×C₄×C₃₁	—	2	2	no	—
123	C₂²×C₄×C₃₁	—	2	2	no	—
124	C₂²×C₄×C₃₁	—	2	4	no	—
125	C₂²×C₄×C₃₁	—	2	4	no	—
126	C₂²×C₄×C₃₁	—	2	2	no	—
127	C₂²×C₄×C₃₁	—	2	2	no	—
128	C₂²×C₄×C₃₁	—	2	2	no	—
129	C₂²×C₄×C₃₁	—	2	2	no	—
130	C₂²×C₄×C₃₁	—	2	4	no	—
131	C₂²×C₄×C₃₁	—	2	2	no	—
132	C₂²×C₄×C₃₁	—	2	4	no	—
133	C₂²×C₄×C₃₁	—	2	4	no	—
134	C₂²×C₄×C₃₁	—	2	2	no	—
135	C₂²×C₄×C₃₁	—	2	2	no	—
136	C₂²×C₄×C₃₁	—	2	2	no	—
137	C₂²×C₄×C₃₁	—	2	2	no	—
138	C₂²×C₄×C₃₁	—	2	2	no	—
139	C₂²×C₄×C₃₁	—	2	2	no	—
140	C₂²×C₄×C₃₁	—	2	4	no	—
141	C₂²×C₄×C₃₁	—	2	4	no	—
142	C₂²×C₄×C₃₁	—	2	4	no	—
143	C₂²×C₄×C₃₁	—	2	2	no	—
144	C₂²×C₄×C₃₁	—	2	2	no	—
145	C₂²×C₄×C₃₁	—	2	16	no	—
146	C₂²×C₄×C₃₁	—	2	16	no	—
147	C₂²×C₄×C₃₁	—	4	8	no	—
148	C₂²×C₄×C₃₁	—	4	32	no	—
149	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
150	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
151	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
152	C₂²×C₄×C₃₁	#41	1	4	yes	—
153	C₂²×C₄×C₃₁	#41	1	4	yes	—
154	C₂²×C₄×C₃₁	#41	1	4	yes	—
155	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
156	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
158	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
161	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
162	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
163	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
165	C₄⋊C₄×C₃₁	—	2	16	no	—
166	C₄⋊C₄×C₃₁	—	2	16	no	—
167	C₄⋊C₄×C₃₁	—	2	8	no	—
168	C₄⋊C₄×C₃₁	—	2	8	no	—
169	C₄⋊C₄×C₃₁	—	2	8	no	—
170	C₄⋊C₄×C₃₁	—	2	8	no	—
171	C₄⋊C₄×C₃₁	—	4	8	no	—
172	C₄⋊C₄×C₃₁	—	4	8	no	—
173	C₄⋊C₄×C₃₁	C₂×D₄	1	16	yes	—
174	C₂²⋊C₄×C₃₁	—	2	16	no	—
175	C₂²⋊C₄×C₃₁	—	2	16	no	—
176	C₂²⋊C₄×C₃₁	—	4	8	no	—
177	C₂²⋊C₄×C₃₁	—	4	8	no	—
178	C₂²⋊C₄×C₃₁	—	4	8	no	—
179	C₄²×C₃₁	—	2	8	no	—
180	C₄²×C₃₁	—	2	8	no	—
181	C₄²×C₃₁	—	2	8	no	—
182	C₂×#2454	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂³×C₃₁	—	2	16	no	—
184	C₂³×C₃₁	—	2	16	no	—
185	C₂³×C₃₁	—	2	16	no	—
186	C₂³×C₃₁	—	2	4	no	—
187	C₂³×C₃₁	—	2	4	no	—
188	C₂³×C₃₁	—	2	4	no	—
189	C₂³×C₃₁	—	2	4	no	—
190	C₂³×C₃₁	—	2	4	no	—
191	C₂³×C₃₁	—	2	4	no	—
192	C₂³×C₃₁	—	4	8	no	—
193	C₂³×C₃₁	—	4	8	no	—
194	C₂³×C₃₁	—	4	8	no	—
195	C₂³×C₃₁	—	4	8	no	—
196	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
197	C₂³×C₃₁	C₂²×Q₈	1	8	yes	—
198	C₂³×C₃₁	C₂²×Q₈	1	2	yes	—
199	C₂³×C₃₁	C₂×#41	1	2	yes	—
200	C₂³×C₃₁	C₂×#41	1	2	yes	—
201	C₂³×C₃₁	C₂×#41	1	2	yes	—
202	C₂³×C₃₁	H(5,𝔽₂)	1	2	yes	—
203	C₂³×C₃₁	H(5,𝔽₂)	1	2	yes	—
204	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
205	D₄×C₃₁	—	2	32	no	—
206	C₂×C₄×C₃₁	—	2	8	no	—
207	C₂×C₄×C₃₁	—	2	8	no	—
208	C₂×C₄×C₃₁	—	2	16	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	4	no	—
213	C₂×C₄×C₃₁	—	2	4	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	4	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	2	4	no	—
221	C₂×C₄×C₃₁	—	2	4	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	2	2	no	—
226	C₂×C₄×C₃₁	—	2	2	no	—
227	C₂×C₄×C₃₁	—	2	2	no	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	2	4	no	—
230	C₂×C₄×C₃₁	—	2	4	no	—
231	C₂×C₄×C₃₁	—	2	2	no	—
232	C₂×C₄×C₃₁	—	2	2	no	—
233	C₂×C₄×C₃₁	—	4	32	no	—
234	C₂×C₄×C₃₁	—	4	4	no	—
235	C₂×C₄×C₃₁	—	4	8	no	—
236	C₂×C₄×C₃₁	—	4	4	no	—
237	C₂×C₄×C₃₁	—	4	4	no	—
238	C₂×C₄×C₃₁	—	4	4	no	—
239	C₂×C₄×C₃₁	—	4	8	no	—
240	C₂×C₄×C₃₁	C₂²⋊Q₈	1	16	yes	—
241	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
242	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
243	C₂×C₄×C₃₁	C₂²×Q₈	1	4	yes	—
244	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
245	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
246	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
247	C₂²×#125	C₂×C₃₁	1	1	yes	characteristic, complemented
248	C₂²×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic
249	#2454	C₂×C₃₁	1	16	yes	direct product factor
250	C₂×#399	C₂×C₃₁	1	2	yes	complemented
251	C₂×#396	C₂×C₃₁	1	2	yes	complemented
252	C₂×#390	C₂×C₃₁	1	1	yes	characteristic
253	C₂×#390	C₂×C₃₁	1	1	yes	characteristic
254	C₂×#389	C₂×C₃₁	1	1	yes	characteristic, complemented
255	C₂×#389	C₂×C₃₁	1	1	yes	characteristic, complemented
256	C₂×#385	C₂×C₃₁	1	2	yes	complemented
257	C₂×#380	C₂×C₃₁	1	2	yes	complemented
258	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
259	C₂²×C₃₁	—	2	8	no	—
260	C₂²×C₃₁	—	2	8	no	—
261	C₂²×C₃₁	—	2	8	no	—
262	C₂²×C₃₁	—	4	4	no	—
263	C₂²×C₃₁	—	4	4	no	—
264	C₂²×C₃₁	—	4	4	no	—
265	C₂²×C₃₁	—	4	4	no	—
266	C₂²×C₃₁	—	4	32	no	—
267	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	4	yes	—
268	C₂²×C₃₁	#529	1	4	yes	—
269	C₂²×C₃₁	#540	1	4	yes	—
270	C₂²×C₃₁	#546	1	4	yes	—
271	C₂²×C₃₁	#550	1	4	yes	—
272	C₂²×C₃₁	#554	1	4	yes	—
273	C₂²×C₃₁	#554	1	4	yes	—
274	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
275	C₂²×C₃₁	C₂³×Q₈	1	1	yes	characteristic
276	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
277	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
278	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
279	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
280	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
281	C₄×C₃₁	—	2	4	no	—
282	C₄×C₃₁	—	2	4	no	—
283	C₄×C₃₁	—	2	8	no	—
284	C₄×C₃₁	—	4	2	no	—
285	C₄×C₃₁	—	4	4	no	—
286	C₄×C₃₁	—	4	2	no	—
287	C₄×C₃₁	—	4	2	no	—
288	C₄×C₃₁	—	4	2	no	—
289	C₄×C₃₁	—	4	4	no	—
290	C₄×C₃₁	C₂×C₂²⋊Q₈	1	8	yes	—
291	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
292	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
293	C₂×#125	—	2	8	no	—
294	C₂×#125	C₂²×C₃₁	1	8	yes	complemented
295	C₂×C₄×D₄	—	2	8	no	—
296	C₂×C₄×D₄	C₂²×C₃₁	1	8	yes	—
297	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
298	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
299	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
300	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
301	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
302	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
303	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
304	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
305	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
306	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
307	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
308	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
309	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
310	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
311	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
312	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
313	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
314	#399	C₂²×C₃₁	1	16	yes	complemented
315	#396	C₂²×C₃₁	1	16	yes	complemented
316	#390	C₂²×C₃₁	1	8	yes	—
317	#390	C₂²×C₃₁	1	8	yes	—
318	#389	C₂²×C₃₁	1	8	yes	complemented
319	#389	C₂²×C₃₁	1	8	yes	complemented
320	#385	C₂²×C₃₁	1	16	yes	complemented
321	#380	C₂²×C₃₁	1	16	yes	complemented
322	C₄×C₄⋊C₄	C₂²×C₃₁	1	8	yes	complemented
323	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
324	C₂×#95	C₂²×C₃₁	1	2	yes	—
325	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
326	C₂×#95	C₂²×C₃₁	1	2	yes	—
327	C₂×C₃₁	—	4	8	no	—
328	C₂×C₃₁	#2454	1	8	yes	direct product factor
329	C₂×C₃₁	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
330	C₂×C₃₁	C₂×#529	1	1	yes	characteristic
331	C₂×C₃₁	C₂×#540	1	1	yes	characteristic
332	C₂×C₃₁	C₂×#546	1	1	yes	characteristic
333	C₂×C₃₁	C₂×#550	1	1	yes	characteristic
334	C₂×C₃₁	C₂×#554	1	1	yes	characteristic
335	C₂×C₃₁	C₂×#554	1	1	yes	characteristic
336	C₂⁵	C₂³×C₃₁	1	2	yes	—
337	C₂²×D₄	—	2	4	no	—
338	C₂²×D₄	—	2	4	no	—
339	C₂²×D₄	Q₈×C₃₁	1	8	yes	—
340	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
341	C₂³×C₄	—	2	8	no	—
342	C₂³×C₄	—	2	8	no	—
343	C₂³×C₄	—	4	4	no	—
344	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
345	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
346	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
347	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
348	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
349	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
350	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
351	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
352	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
353	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
354	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
355	#125	—	2	32	no	—
356	C₄×D₄	—	2	32	no	—
357	C₂×C₄⋊C₄	—	2	8	no	—
358	C₂×C₄⋊C₄	—	2	8	no	—
359	C₂×C₄⋊C₄	—	2	4	no	—
360	C₂×C₄⋊C₄	—	2	4	no	—
361	C₂×C₄⋊C₄	—	2	4	no	—
362	C₂×C₄⋊C₄	—	2	4	no	—
363	C₂×C₄⋊C₄	—	4	4	no	—
364	C₂×C₄⋊C₄	—	4	4	no	—
365	C₂×C₄⋊C₄	D₄×C₃₁	1	8	yes	—
366	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
367	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
368	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
369	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
370	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
371	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
372	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
373	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
374	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
375	C₂×C₂²⋊C₄	—	2	8	no	—
376	C₂×C₂²⋊C₄	—	2	8	no	—
377	C₂×C₂²⋊C₄	—	4	4	no	—
378	C₂×C₂²⋊C₄	—	4	4	no	—
379	C₂×C₂²⋊C₄	—	4	4	no	—
380	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
383	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
384	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
385	C₂×C₄²	—	2	4	no	—
386	C₂×C₄²	—	2	4	no	—
387	C₂×C₄²	—	2	4	no	—
388	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
389	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
390	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
391	#95	C₂³×C₃₁	1	4	yes	—
392	#95	C₂³×C₃₁	1	4	yes	—
393	#95	C₂³×C₃₁	1	8	yes	—
394	#95	C₂³×C₃₁	1	8	yes	—
395	C₃₁	C₂×#2454	1	1	yes	characteristic, direct product factor, Sylow
396	C₂⁴	—	2	4	no	—
397	C₂⁴	—	2	4	no	—
398	C₂⁴	—	2	4	no	—
399	C₂⁴	—	2	4	no	—
400	C₂⁴	—	2	4	no	—
401	C₂⁴	—	2	4	no	—
402	C₂⁴	C₂×Q₈×C₃₁	1	8	yes	—
403	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
404	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
405	C₂×D₄	—	2	32	no	—
406	C₂×D₄	—	2	8	no	—
407	C₂×D₄	—	2	8	no	—
408	C₂×D₄	C₂×Q₈×C₃₁	1	16	yes	—
409	C₂²×C₄	—	2	2	no	—
410	C₂²×C₄	—	2	2	no	—
411	C₂²×C₄	—	2	2	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	2	no	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	2	2	no	—
424	C₂²×C₄	—	2	2	no	—
425	C₂²×C₄	—	2	2	no	—
426	C₂²×C₄	—	2	2	no	—
427	C₂²×C₄	—	2	2	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	2	no	—
433	C₂²×C₄	—	2	16	no	—
434	C₂²×C₄	—	2	16	no	—
435	C₂²×C₄	—	4	8	no	—
436	C₂²×C₄	—	4	32	no	—
437	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
438	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
439	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
440	C₂²×C₄	#41×C₃₁	1	4	yes	—
441	C₂²×C₄	#41×C₃₁	1	4	yes	—
442	C₂²×C₄	#41×C₃₁	1	4	yes	—
443	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
444	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
445	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
446	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
447	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
448	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
449	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
450	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
451	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
452	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
453	C₄⋊C₄	—	2	16	no	—
454	C₄⋊C₄	—	2	16	no	—
455	C₄⋊C₄	—	2	8	no	—
456	C₄⋊C₄	—	2	8	no	—
457	C₄⋊C₄	—	2	8	no	—
458	C₄⋊C₄	—	2	8	no	—
459	C₄⋊C₄	—	4	8	no	—
460	C₄⋊C₄	—	4	8	no	—
461	C₄⋊C₄	C₂×D₄×C₃₁	1	16	yes	—
462	C₂²⋊C₄	—	2	16	no	—
463	C₂²⋊C₄	—	2	16	no	—
464	C₂²⋊C₄	—	4	8	no	—
465	C₂²⋊C₄	—	4	8	no	—
466	C₂²⋊C₄	—	4	8	no	—
467	C₄²	—	2	8	no	—
468	C₄²	—	2	8	no	—
469	C₄²	—	2	8	no	—
470	C₂³	—	2	16	no	—
471	C₂³	—	2	16	no	—
472	C₂³	—	2	16	no	—
473	C₂³	—	2	4	no	—
474	C₂³	—	2	4	no	—
475	C₂³	—	2	4	no	—
476	C₂³	—	2	4	no	—
477	C₂³	—	2	4	no	—
478	C₂³	—	2	4	no	—
479	C₂³	—	4	8	no	—
480	C₂³	—	4	8	no	—
481	C₂³	—	4	8	no	—
482	C₂³	—	4	8	no	—
483	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
484	C₂³	C₂²×Q₈×C₃₁	1	8	yes	—
485	C₂³	C₂²×Q₈×C₃₁	1	2	yes	—
486	C₂³	C₂×#41×C₃₁	1	2	yes	—
487	C₂³	C₂×#41×C₃₁	1	2	yes	—
488	C₂³	C₂×#41×C₃₁	1	2	yes	—
489	C₂³	C₃₁×H(5,𝔽₂)	1	2	yes	—
490	C₂³	C₃₁×H(5,𝔽₂)	1	2	yes	—
491	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
492	D₄	—	2	32	no	—
493	C₂×C₄	—	2	8	no	—
494	C₂×C₄	—	2	8	no	—
495	C₂×C₄	—	2	16	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	4	32	no	—
521	C₂×C₄	—	4	4	no	—
522	C₂×C₄	—	4	8	no	—
523	C₂×C₄	—	4	4	no	—
524	C₂×C₄	—	4	4	no	—
525	C₂×C₄	—	4	4	no	—
526	C₂×C₄	—	4	8	no	—
527	C₂×C₄	C₃₁×C₂²⋊Q₈	1	16	yes	—
528	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
529	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
530	C₂×C₄	C₂²×Q₈×C₃₁	1	4	yes	—
531	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
532	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
533	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
534	C₂²	—	2	8	no	—
535	C₂²	—	2	8	no	—
536	C₂²	—	2	8	no	—
537	C₂²	—	4	4	no	—
538	C₂²	—	4	4	no	—
539	C₂²	—	4	4	no	—
540	C₂²	—	4	4	no	—
541	C₂²	—	4	32	no	—
542	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	4	yes	—
543	C₂²	C₃₁×#529	1	4	yes	—
544	C₂²	C₃₁×#540	1	4	yes	—
545	C₂²	C₃₁×#546	1	4	yes	—
546	C₂²	C₃₁×#550	1	4	yes	—
547	C₂²	C₃₁×#554	1	4	yes	—
548	C₂²	C₃₁×#554	1	4	yes	—
549	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
550	C₂²	C₂³×Q₈×C₃₁	1	1	yes	characteristic
551	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
552	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
553	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
554	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
555	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
556	C₄	—	2	4	no	—
557	C₄	—	2	4	no	—
558	C₄	—	2	8	no	—
559	C₄	—	4	2	no	—
560	C₄	—	4	4	no	—
561	C₄	—	4	2	no	—
562	C₄	—	4	2	no	—
563	C₄	—	4	2	no	—
564	C₄	—	4	4	no	—
565	C₄	C₂×C₃₁×C₂²⋊Q₈	1	8	yes	—
566	C₂	—	4	8	no	—
567	C₂	C₃₁×#2454	1	8	yes	direct product factor
568	C₂	C₂²×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
569	C₂	C₂×C₃₁×#529	1	1	yes	characteristic
570	C₂	C₂×C₃₁×#540	1	1	yes	characteristic
571	C₂	C₂×C₃₁×#546	1	1	yes	characteristic
572	C₂	C₂×C₃₁×#550	1	1	yes	characteristic
573	C₂	C₂×C₃₁×#554	1	1	yes	characteristic
574	C₂	C₂×C₃₁×#554	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂³×D₄×C₃₁	C₂²
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×#125	C₂	semidirect
3. C₃₁×#2454	C₂	direct (N × H)
4. C₂×C₃₁×#399	C₂	semidirect
5. C₂×C₃₁×#396	C₂	semidirect
8. C₂×C₃₁×#389	C₂	semidirect
9. C₂×C₃₁×#389	C₂	semidirect
10. C₂×C₃₁×#385	C₂	semidirect
11. C₂×C₃₁×#380	C₂	semidirect
12. C₂×C₄×C₄⋊C₄×C₃₁	C₂	semidirect
16. C₂×C₃₁×#125	C₂²	semidirect
36. C₃₁×#399	C₂²	semidirect
37. C₃₁×#396	C₂²	semidirect
40. C₃₁×#389	C₂²	semidirect
41. C₃₁×#389	C₂²	semidirect
42. C₃₁×#385	C₂²	semidirect
43. C₃₁×#380	C₂²	semidirect
44. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
182. C₂×#2454	C₃₁	direct (N × H)
247. C₂²×#125	C₂×C₃₁	semidirect
249. #2454	C₂×C₃₁	direct (N × H)
250. C₂×#399	C₂×C₃₁	semidirect
251. C₂×#396	C₂×C₃₁	semidirect
254. C₂×#389	C₂×C₃₁	semidirect
255. C₂×#389	C₂×C₃₁	semidirect
256. C₂×#385	C₂×C₃₁	semidirect
257. C₂×#380	C₂×C₃₁	semidirect
258. C₂×C₄×C₄⋊C₄	C₂×C₃₁	semidirect
294. C₂×#125	C₂²×C₃₁	semidirect
314. #399	C₂²×C₃₁	semidirect
315. #396	C₂²×C₃₁	semidirect
318. #389	C₂²×C₃₁	semidirect
319. #389	C₂²×C₃₁	semidirect
320. #385	C₂²×C₃₁	semidirect
321. #380	C₂²×C₃₁	semidirect
322. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
328. C₂×C₃₁	#2454	direct (N × H)
395. C₃₁	C₂×#2454	direct (N × H)
567. C₂	C₃₁×#2454	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×#125	2
C₂²×C₄×D₄×C₃₁	2
C₃₁×#2454	2
C₂×C₃₁×#399	2
C₂×C₃₁×#396	2
C₂×C₃₁×#390	2
C₂×C₃₁×#389	2
C₂×C₃₁×#385	2
C₂×C₃₁×#380	2
C₂×C₄×C₄⋊C₄×C₃₁	2
C₂×#2454	31

Subgroups — C2×C31×#2454 #21491458

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2454 #21491458