Subgroups — C₂²×C₃₁×#525 #21519418

Order 7936.

14334 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁵×C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#525	C₂	1	48	yes	maximal, direct product factor
4	C₂²×C₃₁×#123	C₂	1	2	yes	maximal, complemented
5	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
6	C₂²×C₃₁×#121	C₂	1	2	yes	maximal, complemented
7	C₂²×C₃₁×#120	C₂	1	2	yes	maximal, complemented
8	C₂²×C₄×D₄×C₃₁	C₂	1	4	yes	maximal, complemented
9	C₂²×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁶×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×#41×C₃₁	—	2	16	no	—
12	C₂²×#41×C₃₁	C₂²	1	24	yes	complemented
13	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
14	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
15	C₂³×D₄×C₃₁	C₂²	1	4	yes	—
16	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
17	C₂⁴×C₄×C₃₁	—	2	16	no	—
18	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
19	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
21	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₂⁴×C₄×C₃₁	C₂²	1	24	yes	complemented
23	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	complemented
24	C₃₁×#525	C₂²	1	256	yes	direct product factor
25	C₂×C₃₁×#123	C₂²	1	48	yes	complemented
26	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	48	yes	complemented
27	C₂×C₃₁×#121	C₂²	1	48	yes	complemented
28	C₂×C₃₁×#120	C₂²	1	48	yes	complemented
29	C₂×C₄×D₄×C₃₁	C₂²	1	96	yes	complemented
30	C₂×C₃₁×#117	C₂²	1	24	yes	complemented
31	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
32	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
33	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
34	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
35	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
36	C₂²×C₄²×C₃₁	C₂²	1	2	yes	complemented
37	C₂⁵×C₃₁	—	2	8	no	—
38	C₂⁵×C₃₁	—	2	16	no	—
39	C₂⁵×C₃₁	C₂³	1	2	yes	—
40	C₂⁵×C₃₁	C₂³	1	12	yes	—
41	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
42	C₂⁵×C₃₁	C₂³	1	2	yes	—
43	C₂×#41×C₃₁	—	2	192	no	—
44	C₂×#41×C₃₁	C₂³	1	64	yes	complemented
45	C₂²×Q₈×C₃₁	—	2	8	no	—
46	C₂²×Q₈×C₃₁	C₂³	1	12	yes	complemented
47	C₂²×D₄×C₃₁	—	2	16	no	—
48	C₂²×D₄×C₃₁	—	2	32	no	—
49	C₂²×D₄×C₃₁	—	2	8	no	—
50	C₂²×D₄×C₃₁	C₂³	1	48	yes	—
51	C₂²×D₄×C₃₁	C₂³	1	24	yes	—
52	C₂²×D₄×C₃₁	C₂³	1	12	yes	complemented
53	C₂³×C₄×C₃₁	—	2	8	no	—
54	C₂³×C₄×C₃₁	—	2	16	no	—
55	C₂³×C₄×C₃₁	—	2	16	no	—
56	C₂³×C₄×C₃₁	—	2	16	no	—
57	C₂³×C₄×C₃₁	—	2	192	no	—
58	C₂³×C₄×C₃₁	D₄	1	16	yes	complemented
59	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
60	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
61	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
62	C₂³×C₄×C₃₁	C₂³	1	24	yes	—
63	C₂³×C₄×C₃₁	C₂³	1	24	yes	—
64	C₂³×C₄×C₃₁	C₂³	1	12	yes	—
65	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₃₁	C₂³	1	12	yes	—
67	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₃₁	C₂³	1	64	yes	complemented
69	C₂³×C₄×C₃₁	C₂³	1	24	yes	complemented
70	C₃₁×#123	C₂³	1	128	yes	complemented
71	C₃₁×C₂²⋊Q₈	C₂³	1	128	yes	complemented
72	C₃₁×#121	C₂³	1	128	yes	complemented
73	C₃₁×#120	C₂³	1	128	yes	complemented
74	C₄×D₄×C₃₁	C₂³	1	256	yes	complemented
75	C₃₁×#117	C₂³	1	64	yes	complemented
76	C₂×C₄⋊C₄×C₃₁	C₂³	1	24	yes	complemented
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	48	yes	complemented
78	C₂×C₂²⋊C₄×C₃₁	C₂³	1	24	yes	—
79	C₂×C₂²⋊C₄×C₃₁	C₂³	1	48	yes	—
80	C₂×C₂²⋊C₄×C₃₁	C₂³	1	48	yes	complemented
81	C₂×C₄²×C₃₁	C₂³	1	24	yes	complemented
82	C₂⁴×C₃₁	—	2	128	no	—
83	C₂⁴×C₃₁	—	2	48	no	—
84	C₂⁴×C₃₁	—	2	8	no	—
85	C₂⁴×C₃₁	—	2	8	no	—
86	C₂⁴×C₃₁	—	2	8	no	—
87	C₂⁴×C₃₁	—	2	96	no	—
88	C₂⁴×C₃₁	—	2	8	no	—
89	C₂⁴×C₃₁	—	2	16	no	—
90	C₂⁴×C₃₁	C₂×D₄	1	8	yes	—
91	C₂⁴×C₃₁	#41	1	16	yes	complemented
92	C₂⁴×C₃₁	C₂⁴	1	12	yes	—
93	C₂⁴×C₃₁	C₂⁴	1	16	yes	—
94	C₂⁴×C₃₁	C₂⁴	1	6	yes	—
95	C₂⁴×C₃₁	C₂⁴	1	12	yes	—
96	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
97	#41×C₃₁	—	2	256	no	—
98	C₂×Q₈×C₃₁	—	2	48	no	—
99	C₂×Q₈×C₃₁	C₂⁴	1	16	yes	complemented
100	C₂×D₄×C₃₁	—	2	96	no	—
101	C₂×D₄×C₃₁	—	2	192	no	—
102	C₂×D₄×C₃₁	—	2	48	no	—
103	C₂×D₄×C₃₁	C₂⁴	1	64	yes	—
104	C₂×D₄×C₃₁	C₂⁴	1	32	yes	—
105	C₂×D₄×C₃₁	C₂⁴	1	16	yes	complemented
106	C₂²×C₄×C₃₁	—	2	48	no	—
107	C₂²×C₄×C₃₁	—	2	8	no	—
108	C₂²×C₄×C₃₁	—	2	8	no	—
109	C₂²×C₄×C₃₁	—	2	16	no	—
110	C₂²×C₄×C₃₁	—	2	96	no	—
111	C₂²×C₄×C₃₁	—	2	96	no	—
112	C₂²×C₄×C₃₁	—	2	96	no	—
113	C₂²×C₄×C₃₁	—	2	256	no	—
114	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
115	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
116	C₂²×C₄×C₃₁	C₂×D₄	1	96	yes	complemented
117	C₂²×C₄×C₃₁	C₂⁴	1	24	yes	—
118	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
119	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
120	C₂²×C₄×C₃₁	C₂⁴	1	32	yes	—
121	C₂²×C₄×C₃₁	C₂⁴	1	32	yes	—
122	C₂²×C₄×C₃₁	C₂⁴	1	16	yes	—
123	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
124	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
125	C₂²×C₄×C₃₁	C₂⁴	1	16	yes	—
126	C₂²×C₄×C₃₁	C₂⁴	1	32	yes	complemented
127	C₄⋊C₄×C₃₁	C₂⁴	1	32	yes	complemented
128	C₄⋊C₄×C₃₁	C₂⁴	1	64	yes	complemented
129	C₂²⋊C₄×C₃₁	C₂⁴	1	32	yes	—
130	C₂²⋊C₄×C₃₁	C₂⁴	1	64	yes	—
131	C₂²⋊C₄×C₃₁	C₂⁴	1	64	yes	complemented
132	C₄²×C₃₁	C₂⁴	1	32	yes	complemented
133	C₂²×#525	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂³×C₃₁	—	2	384	no	—
135	C₂³×C₃₁	—	2	32	no	—
136	C₂³×C₃₁	—	2	64	no	—
137	C₂³×C₃₁	—	2	32	no	—
138	C₂³×C₃₁	—	2	24	no	—
139	C₂³×C₃₁	—	2	24	no	—
140	C₂³×C₃₁	—	2	24	no	—
141	C₂³×C₃₁	—	2	64	no	—
142	C₂³×C₃₁	—	2	24	no	—
143	C₂³×C₃₁	—	2	48	no	—
144	C₂³×C₃₁	—	4	32	no	—
145	C₂³×C₃₁	C₂²×D₄	1	24	yes	—
146	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
147	C₂³×C₃₁	C₂×#41	1	8	yes	—
148	C₂³×C₃₁	C₂×#41	1	48	yes	complemented
149	C₂³×C₃₁	C₂⁵	1	8	yes	—
150	C₂³×C₃₁	C₂⁵	1	4	yes	—
151	C₂³×C₃₁	C₂⁵	1	8	yes	—
152	C₂³×C₃₁	C₂⁵	1	3	yes	—
153	Q₈×C₃₁	—	2	32	no	—
154	D₄×C₃₁	—	2	64	no	—
155	D₄×C₃₁	—	2	128	no	—
156	D₄×C₃₁	—	2	32	no	—
157	C₂×C₄×C₃₁	—	2	32	no	—
158	C₂×C₄×C₃₁	—	2	24	no	—
159	C₂×C₄×C₃₁	—	2	24	no	—
160	C₂×C₄×C₃₁	—	2	48	no	—
161	C₂×C₄×C₃₁	—	2	64	no	—
162	C₂×C₄×C₃₁	—	2	64	no	—
163	C₂×C₄×C₃₁	—	2	64	no	—
164	C₂×C₄×C₃₁	C₂²×D₄	1	24	yes	—
165	C₂×C₄×C₃₁	C₂²×D₄	1	24	yes	—
166	C₂×C₄×C₃₁	C₂²×D₄	1	64	yes	complemented
167	C₂×C₄×C₃₁	C₂⁵	1	16	yes	—
168	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
169	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
170	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
171	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
172	C₂³×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
173	C₂⁵×C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
174	C₂×#525	C₂×C₃₁	1	48	yes	direct product factor
175	C₂²×#123	C₂×C₃₁	1	2	yes	complemented
176	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	2	yes	complemented
177	C₂²×#121	C₂×C₃₁	1	2	yes	complemented
178	C₂²×#120	C₂×C₃₁	1	2	yes	complemented
179	C₂²×C₄×D₄	C₂×C₃₁	1	4	yes	complemented
180	C₂²×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
181	C₂²×C₃₁	—	2	128	no	—
182	C₂²×C₃₁	—	2	8	no	—
183	C₂²×C₃₁	—	2	8	no	—
184	C₂²×C₃₁	—	2	8	no	—
185	C₂²×C₃₁	—	2	48	no	—
186	C₂²×C₃₁	—	2	96	no	—
187	C₂²×C₃₁	—	2	8	no	—
188	C₂²×C₃₁	—	2	16	no	—
189	C₂²×C₃₁	—	4	48	no	—
190	C₂²×C₃₁	#525	1	16	yes	direct product factor
191	C₂²×C₃₁	C₂³×D₄	1	8	yes	—
192	C₂²×C₃₁	C₂³×D₄	1	6	yes	—
193	C₂²×C₃₁	C₂²×#41	1	12	yes	—
194	C₂²×C₃₁	C₂²×#41	1	16	yes	complemented
195	C₂²×C₃₁	C₂⁶	1	1	yes	characteristic
196	C₄×C₃₁	—	2	8	no	—
197	C₄×C₃₁	—	2	8	no	—
198	C₄×C₃₁	—	2	16	no	—
199	C₄×C₃₁	C₂³×D₄	1	8	yes	—
200	C₄×C₃₁	C₂³×D₄	1	8	yes	—
201	C₂⁶	C₂²×C₃₁	1	1	yes	characteristic
202	C₂²×#41	—	2	16	no	—
203	C₂²×#41	C₂²×C₃₁	1	24	yes	complemented
204	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
205	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
206	C₂³×D₄	C₂²×C₃₁	1	4	yes	—
207	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
208	C₂⁴×C₄	—	2	16	no	—
209	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
210	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
211	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
212	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
213	C₂⁴×C₄	C₂²×C₃₁	1	24	yes	complemented
214	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	complemented
215	#525	C₂²×C₃₁	1	256	yes	direct product factor
216	C₂×#123	C₂²×C₃₁	1	48	yes	complemented
217	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	48	yes	complemented
218	C₂×#121	C₂²×C₃₁	1	48	yes	complemented
219	C₂×#120	C₂²×C₃₁	1	48	yes	complemented
220	C₂×C₄×D₄	C₂²×C₃₁	1	96	yes	complemented
221	C₂×#117	C₂²×C₃₁	1	24	yes	complemented
222	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
223	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
224	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
225	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
226	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
227	C₂²×C₄²	C₂²×C₃₁	1	2	yes	complemented
228	C₂×C₃₁	—	2	8	no	—
229	C₂×C₃₁	—	2	16	no	—
230	C₂×C₃₁	—	4	8	no	—
231	C₂×C₃₁	C₂×#525	1	12	yes	direct product factor
232	C₂×C₃₁	C₂⁴×D₄	1	1	yes	characteristic
233	C₂×C₃₁	C₂³×#41	1	2	yes	—
234	C₂⁵	—	2	8	no	—
235	C₂⁵	—	2	16	no	—
236	C₂⁵	C₂³×C₃₁	1	2	yes	—
237	C₂⁵	C₂³×C₃₁	1	12	yes	—
238	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
239	C₂⁵	C₂³×C₃₁	1	2	yes	—
240	C₂×#41	—	2	192	no	—
241	C₂×#41	C₂³×C₃₁	1	64	yes	complemented
242	C₂²×Q₈	—	2	8	no	—
243	C₂²×Q₈	C₂³×C₃₁	1	12	yes	complemented
244	C₂²×D₄	—	2	16	no	—
245	C₂²×D₄	—	2	32	no	—
246	C₂²×D₄	—	2	8	no	—
247	C₂²×D₄	C₂³×C₃₁	1	48	yes	—
248	C₂²×D₄	C₂³×C₃₁	1	24	yes	—
249	C₂²×D₄	C₂³×C₃₁	1	12	yes	complemented
250	C₂³×C₄	—	2	8	no	—
251	C₂³×C₄	—	2	16	no	—
252	C₂³×C₄	—	2	16	no	—
253	C₂³×C₄	—	2	16	no	—
254	C₂³×C₄	—	2	192	no	—
255	C₂³×C₄	D₄×C₃₁	1	16	yes	complemented
256	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
257	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
258	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
259	C₂³×C₄	C₂³×C₃₁	1	24	yes	—
260	C₂³×C₄	C₂³×C₃₁	1	24	yes	—
261	C₂³×C₄	C₂³×C₃₁	1	12	yes	—
262	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
263	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
264	C₂³×C₄	C₂³×C₃₁	1	12	yes	—
265	C₂³×C₄	C₂³×C₃₁	1	64	yes	complemented
266	C₂³×C₄	C₂³×C₃₁	1	24	yes	complemented
267	#123	C₂³×C₃₁	1	128	yes	complemented
268	C₂²⋊Q₈	C₂³×C₃₁	1	128	yes	complemented
269	#121	C₂³×C₃₁	1	128	yes	complemented
270	#120	C₂³×C₃₁	1	128	yes	complemented
271	C₄×D₄	C₂³×C₃₁	1	256	yes	complemented
272	#117	C₂³×C₃₁	1	64	yes	complemented
273	C₂×C₄⋊C₄	C₂³×C₃₁	1	24	yes	complemented
274	C₂×C₄⋊C₄	C₂³×C₃₁	1	48	yes	complemented
275	C₂×C₂²⋊C₄	C₂³×C₃₁	1	24	yes	—
276	C₂×C₂²⋊C₄	C₂³×C₃₁	1	48	yes	—
277	C₂×C₂²⋊C₄	C₂³×C₃₁	1	48	yes	complemented
278	C₂×C₄²	C₂³×C₃₁	1	24	yes	complemented
279	C₃₁	C₂²×#525	1	1	yes	characteristic, direct product factor, Sylow
280	C₂⁴	—	2	128	no	—
281	C₂⁴	—	2	48	no	—
282	C₂⁴	—	2	8	no	—
283	C₂⁴	—	2	8	no	—
284	C₂⁴	—	2	8	no	—
285	C₂⁴	—	2	96	no	—
286	C₂⁴	—	2	8	no	—
287	C₂⁴	—	2	16	no	—
288	C₂⁴	C₂×D₄×C₃₁	1	8	yes	—
289	C₂⁴	#41×C₃₁	1	16	yes	complemented
290	C₂⁴	C₂⁴×C₃₁	1	12	yes	—
291	C₂⁴	C₂⁴×C₃₁	1	16	yes	—
292	C₂⁴	C₂⁴×C₃₁	1	6	yes	—
293	C₂⁴	C₂⁴×C₃₁	1	12	yes	—
294	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
295	#41	—	2	256	no	—
296	C₂×Q₈	—	2	48	no	—
297	C₂×Q₈	C₂⁴×C₃₁	1	16	yes	complemented
298	C₂×D₄	—	2	96	no	—
299	C₂×D₄	—	2	192	no	—
300	C₂×D₄	—	2	48	no	—
301	C₂×D₄	C₂⁴×C₃₁	1	64	yes	—
302	C₂×D₄	C₂⁴×C₃₁	1	32	yes	—
303	C₂×D₄	C₂⁴×C₃₁	1	16	yes	complemented
304	C₂²×C₄	—	2	48	no	—
305	C₂²×C₄	—	2	8	no	—
306	C₂²×C₄	—	2	8	no	—
307	C₂²×C₄	—	2	16	no	—
308	C₂²×C₄	—	2	96	no	—
309	C₂²×C₄	—	2	96	no	—
310	C₂²×C₄	—	2	96	no	—
311	C₂²×C₄	—	2	256	no	—
312	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
313	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
314	C₂²×C₄	C₂×D₄×C₃₁	1	96	yes	complemented
315	C₂²×C₄	C₂⁴×C₃₁	1	24	yes	—
316	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
317	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
318	C₂²×C₄	C₂⁴×C₃₁	1	32	yes	—
319	C₂²×C₄	C₂⁴×C₃₁	1	32	yes	—
320	C₂²×C₄	C₂⁴×C₃₁	1	16	yes	—
321	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
322	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
323	C₂²×C₄	C₂⁴×C₃₁	1	16	yes	—
324	C₂²×C₄	C₂⁴×C₃₁	1	32	yes	complemented
325	C₄⋊C₄	C₂⁴×C₃₁	1	32	yes	complemented
326	C₄⋊C₄	C₂⁴×C₃₁	1	64	yes	complemented
327	C₂²⋊C₄	C₂⁴×C₃₁	1	32	yes	—
328	C₂²⋊C₄	C₂⁴×C₃₁	1	64	yes	—
329	C₂²⋊C₄	C₂⁴×C₃₁	1	64	yes	complemented
330	C₄²	C₂⁴×C₃₁	1	32	yes	complemented
331	C₂³	—	2	384	no	—
332	C₂³	—	2	32	no	—
333	C₂³	—	2	64	no	—
334	C₂³	—	2	32	no	—
335	C₂³	—	2	24	no	—
336	C₂³	—	2	24	no	—
337	C₂³	—	2	24	no	—
338	C₂³	—	2	64	no	—
339	C₂³	—	2	24	no	—
340	C₂³	—	2	48	no	—
341	C₂³	—	4	32	no	—
342	C₂³	C₂²×D₄×C₃₁	1	24	yes	—
343	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
344	C₂³	C₂×#41×C₃₁	1	8	yes	—
345	C₂³	C₂×#41×C₃₁	1	48	yes	complemented
346	C₂³	C₂⁵×C₃₁	1	8	yes	—
347	C₂³	C₂⁵×C₃₁	1	4	yes	—
348	C₂³	C₂⁵×C₃₁	1	8	yes	—
349	C₂³	C₂⁵×C₃₁	1	3	yes	—
350	Q₈	—	2	32	no	—
351	D₄	—	2	64	no	—
352	D₄	—	2	128	no	—
353	D₄	—	2	32	no	—
354	C₂×C₄	—	2	32	no	—
355	C₂×C₄	—	2	24	no	—
356	C₂×C₄	—	2	24	no	—
357	C₂×C₄	—	2	48	no	—
358	C₂×C₄	—	2	64	no	—
359	C₂×C₄	—	2	64	no	—
360	C₂×C₄	—	2	64	no	—
361	C₂×C₄	C₂²×D₄×C₃₁	1	24	yes	—
362	C₂×C₄	C₂²×D₄×C₃₁	1	24	yes	—
363	C₂×C₄	C₂²×D₄×C₃₁	1	64	yes	complemented
364	C₂×C₄	C₂⁵×C₃₁	1	16	yes	—
365	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
366	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
367	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
368	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
369	C₂²	—	2	128	no	—
370	C₂²	—	2	8	no	—
371	C₂²	—	2	8	no	—
372	C₂²	—	2	8	no	—
373	C₂²	—	2	48	no	—
374	C₂²	—	2	96	no	—
375	C₂²	—	2	8	no	—
376	C₂²	—	2	16	no	—
377	C₂²	—	4	48	no	—
378	C₂²	C₃₁×#525	1	16	yes	direct product factor
379	C₂²	C₂³×D₄×C₃₁	1	8	yes	—
380	C₂²	C₂³×D₄×C₃₁	1	6	yes	—
381	C₂²	C₂²×#41×C₃₁	1	12	yes	—
382	C₂²	C₂²×#41×C₃₁	1	16	yes	complemented
383	C₂²	C₂⁶×C₃₁	1	1	yes	characteristic
384	C₄	—	2	8	no	—
385	C₄	—	2	8	no	—
386	C₄	—	2	16	no	—
387	C₄	C₂³×D₄×C₃₁	1	8	yes	—
388	C₄	C₂³×D₄×C₃₁	1	8	yes	—
389	C₂	—	2	8	no	—
390	C₂	—	2	16	no	—
391	C₂	—	4	8	no	—
392	C₂	C₂×C₃₁×#525	1	12	yes	direct product factor
393	C₂	C₂⁴×D₄×C₃₁	1	1	yes	characteristic
394	C₂	C₂³×#41×C₃₁	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₃₁	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶×C₃₁
socle	C₂⁴×C₃₁	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×#41×C₃₁	C₂	semidirect
2. C₂⁵×C₄×C₃₁	C₂	semidirect
3. C₂×C₃₁×#525	C₂	direct (N × H)
4. C₂²×C₃₁×#123	C₂	semidirect
5. C₂²×C₃₁×C₂²⋊Q₈	C₂	semidirect
6. C₂²×C₃₁×#121	C₂	semidirect
7. C₂²×C₃₁×#120	C₂	semidirect
8. C₂²×C₄×D₄×C₃₁	C₂	semidirect
9. C₂²×C₃₁×#117	C₂	semidirect
12. C₂²×#41×C₃₁	C₂²	semidirect
13. C₂³×Q₈×C₃₁	C₂²	semidirect
16. C₂³×D₄×C₃₁	C₂²	semidirect
22. C₂⁴×C₄×C₃₁	C₂²	semidirect
23. C₂⁴×C₄×C₃₁	C₂²	semidirect
24. C₃₁×#525	C₂²	direct (N × H)
25. C₂×C₃₁×#123	C₂²	semidirect
26. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
27. C₂×C₃₁×#121	C₂²	semidirect
28. C₂×C₃₁×#120	C₂²	semidirect
29. C₂×C₄×D₄×C₃₁	C₂²	semidirect
30. C₂×C₃₁×#117	C₂²	semidirect
31. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
32. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
35. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
36. C₂²×C₄²×C₃₁	C₂²	semidirect
44. C₂×#41×C₃₁	C₂³	semidirect
46. C₂²×Q₈×C₃₁	C₂³	semidirect
52. C₂²×D₄×C₃₁	C₂³	semidirect
58. C₂³×C₄×C₃₁	D₄	semidirect
68. C₂³×C₄×C₃₁	C₂³	semidirect
69. C₂³×C₄×C₃₁	C₂³	semidirect
70. C₃₁×#123	C₂³	semidirect
71. C₃₁×C₂²⋊Q₈	C₂³	semidirect
72. C₃₁×#121	C₂³	semidirect
73. C₃₁×#120	C₂³	semidirect
74. C₄×D₄×C₃₁	C₂³	semidirect
75. C₃₁×#117	C₂³	semidirect
76. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
77. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
80. C₂×C₂²⋊C₄×C₃₁	C₂³	semidirect
81. C₂×C₄²×C₃₁	C₂³	semidirect
91. C₂⁴×C₃₁	#41	semidirect
99. C₂×Q₈×C₃₁	C₂⁴	semidirect
105. C₂×D₄×C₃₁	C₂⁴	semidirect
116. C₂²×C₄×C₃₁	C₂×D₄	semidirect
126. C₂²×C₄×C₃₁	C₂⁴	semidirect
127. C₄⋊C₄×C₃₁	C₂⁴	semidirect
128. C₄⋊C₄×C₃₁	C₂⁴	semidirect
131. C₂²⋊C₄×C₃₁	C₂⁴	semidirect
132. C₄²×C₃₁	C₂⁴	semidirect
133. C₂²×#525	C₃₁	direct (N × H)
148. C₂³×C₃₁	C₂×#41	semidirect
166. C₂×C₄×C₃₁	C₂²×D₄	semidirect
172. C₂³×#41	C₂×C₃₁	semidirect
173. C₂⁵×C₄	C₂×C₃₁	semidirect
174. C₂×#525	C₂×C₃₁	direct (N × H)
175. C₂²×#123	C₂×C₃₁	semidirect
176. C₂²×C₂²⋊Q₈	C₂×C₃₁	semidirect
177. C₂²×#121	C₂×C₃₁	semidirect
178. C₂²×#120	C₂×C₃₁	semidirect
179. C₂²×C₄×D₄	C₂×C₃₁	semidirect
180. C₂²×#117	C₂×C₃₁	semidirect
190. C₂²×C₃₁	#525	direct (N × H)
194. C₂²×C₃₁	C₂²×#41	semidirect
203. C₂²×#41	C₂²×C₃₁	semidirect
204. C₂³×Q₈	C₂²×C₃₁	semidirect
207. C₂³×D₄	C₂²×C₃₁	semidirect
213. C₂⁴×C₄	C₂²×C₃₁	semidirect
214. C₂⁴×C₄	C₂²×C₃₁	semidirect
215. #525	C₂²×C₃₁	direct (N × H)
216. C₂×#123	C₂²×C₃₁	semidirect
217. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
218. C₂×#121	C₂²×C₃₁	semidirect
219. C₂×#120	C₂²×C₃₁	semidirect
220. C₂×C₄×D₄	C₂²×C₃₁	semidirect
221. C₂×#117	C₂²×C₃₁	semidirect
222. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
223. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
226. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
227. C₂²×C₄²	C₂²×C₃₁	semidirect
231. C₂×C₃₁	C₂×#525	direct (N × H)
241. C₂×#41	C₂³×C₃₁	semidirect
243. C₂²×Q₈	C₂³×C₃₁	semidirect
249. C₂²×D₄	C₂³×C₃₁	semidirect
255. C₂³×C₄	D₄×C₃₁	semidirect
265. C₂³×C₄	C₂³×C₃₁	semidirect
266. C₂³×C₄	C₂³×C₃₁	semidirect
267. #123	C₂³×C₃₁	semidirect
268. C₂²⋊Q₈	C₂³×C₃₁	semidirect
269. #121	C₂³×C₃₁	semidirect
270. #120	C₂³×C₃₁	semidirect
271. C₄×D₄	C₂³×C₃₁	semidirect
272. #117	C₂³×C₃₁	semidirect
273. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
274. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
277. C₂×C₂²⋊C₄	C₂³×C₃₁	semidirect
278. C₂×C₄²	C₂³×C₃₁	semidirect
279. C₃₁	C₂²×#525	direct (N × H)
289. C₂⁴	#41×C₃₁	semidirect
297. C₂×Q₈	C₂⁴×C₃₁	semidirect
303. C₂×D₄	C₂⁴×C₃₁	semidirect
314. C₂²×C₄	C₂×D₄×C₃₁	semidirect
324. C₂²×C₄	C₂⁴×C₃₁	semidirect
325. C₄⋊C₄	C₂⁴×C₃₁	semidirect
326. C₄⋊C₄	C₂⁴×C₃₁	semidirect
329. C₂²⋊C₄	C₂⁴×C₃₁	semidirect
330. C₄²	C₂⁴×C₃₁	semidirect
345. C₂³	C₂×#41×C₃₁	semidirect
363. C₂×C₄	C₂²×D₄×C₃₁	semidirect
378. C₂²	C₃₁×#525	direct (N × H)
382. C₂²	C₂²×#41×C₃₁	semidirect
392. C₂	C₂×C₃₁×#525	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#41×C₃₁	2
C₂⁵×C₄×C₃₁	2
C₂×C₃₁×#525	2
C₂²×C₃₁×#123	2
C₂²×C₃₁×C₂²⋊Q₈	2
C₂²×C₃₁×#121	2
C₂²×C₃₁×#120	2
C₂²×C₄×D₄×C₃₁	2
C₂²×C₃₁×#117	2
C₂²×#525	31

Subgroups — C22×C31×#525 #21519418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₁×#525 #21519418