Subgroups — C₂²×C₃₁×#474 #21517232

Order 7936.

2790 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#474	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₃₁×#107	C₂	1	2	yes	maximal, complemented
5	C₂²×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₃₁×#102	C₂	1	2	yes	maximal
7	C₂³×Q₈×C₃₁	—	2	1	no	—
8	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₂⁴×C₄×C₃₁	—	2	1	no	—
10	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
11	C₂×C₃₁×C₄⋊Q₈	—	2	6	no	—
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	12	yes	complemented
13	C₂×C₄×D₄×C₃₁	—	2	6	no	—
14	C₂×C₄×D₄×C₃₁	C₂²	1	12	yes	—
15	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂²×C₂²⋊C₄×C₃₁	—	2	1	no	—
19	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#474	C₂²	1	64	yes	direct product factor
21	C₂×C₃₁×#107	C₂²	1	24	yes	complemented
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	12	yes	complemented
23	C₂×C₃₁×#102	C₂²	1	24	yes	—
24	C₂⁵×C₃₁	—	2	1	no	—
25	C₂²×Q₈×C₃₁	—	2	12	no	—
26	C₂²×Q₈×C₃₁	—	4	8	no	—
27	C₂²×D₄×C₃₁	—	2	3	no	—
28	C₂²×D₄×C₃₁	—	2	4	no	—
29	C₂²×D₄×C₃₁	Q₈	1	8	yes	complemented
30	C₂²×D₄×C₃₁	C₂³	1	6	yes	—
31	C₂³×C₄×C₃₁	—	2	1	no	—
32	C₂³×C₄×C₃₁	—	2	2	no	—
33	C₂³×C₄×C₃₁	—	2	12	no	—
34	C₂³×C₄×C₃₁	—	4	8	no	—
35	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	—	2	8	no	—
39	C₂²×C₈×C₃₁	C₂³	1	12	yes	—
40	C₃₁×C₄⋊Q₈	—	2	24	no	—
41	C₃₁×C₄⋊Q₈	C₂³	1	16	yes	complemented
42	C₄×D₄×C₃₁	—	2	24	no	—
43	C₄×D₄×C₃₁	C₂³	1	16	yes	—
44	C₂×C₄⋊C₄×C₃₁	—	2	12	no	—
45	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	6	no	—
47	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
48	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
49	C₂×C₂²⋊C₄×C₃₁	—	2	12	no	—
50	C₂×C₄²×C₃₁	D₄	1	6	yes	—
51	C₂×C₄²×C₃₁	C₂³	1	6	yes	—
52	C₃₁×#107	C₂³	1	32	yes	complemented
53	C₃₁×C₄⋊C₈	C₂³	1	16	yes	complemented
54	C₃₁×#102	C₂³	1	32	yes	—
55	C₂⁴×C₃₁	—	2	8	no	—
56	C₂⁴×C₃₁	—	2	6	no	—
57	C₂⁴×C₃₁	—	4	4	no	—
58	C₂⁴×C₃₁	—	4	4	no	—
59	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂×Q₈×C₃₁	—	2	16	no	—
61	C₂×Q₈×C₃₁	—	4	48	no	—
62	C₂×D₄×C₃₁	—	2	24	no	—
63	C₂×D₄×C₃₁	—	2	6	no	—
64	C₂×D₄×C₃₁	—	2	12	no	—
65	C₂×D₄×C₃₁	C₂×Q₈	1	24	yes	complemented
66	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
67	C₂²×C₄×C₃₁	—	2	4	no	—
68	C₂²×C₄×C₃₁	—	2	6	no	—
69	C₂²×C₄×C₃₁	—	2	12	no	—
70	C₂²×C₄×C₃₁	—	2	16	no	—
71	C₂²×C₄×C₃₁	—	4	4	no	—
72	C₂²×C₄×C₃₁	—	4	8	no	—
73	C₂²×C₄×C₃₁	—	4	48	no	—
74	C₂²×C₄×C₃₁	D₈	1	8	yes	—
75	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
76	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
77	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
78	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
79	C₂²×C₄×C₃₁	#41	1	4	yes	—
80	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
81	C₂×C₈×C₃₁	—	2	24	no	—
82	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
83	C₄⋊C₄×C₃₁	—	2	16	no	—
84	C₄⋊C₄×C₃₁	—	2	6	no	—
85	C₄⋊C₄×C₃₁	—	2	12	no	—
86	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
87	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
88	C₂²⋊C₄×C₃₁	—	2	16	no	—
89	C₄²×C₃₁	C₂×D₄	1	12	yes	—
90	C₄²×C₃₁	C₂⁴	1	4	yes	—
91	C₂²×#474	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₃₁	—	2	24	no	—
93	C₂³×C₃₁	—	2	4	no	—
94	C₂³×C₃₁	—	4	12	no	—
95	C₂³×C₃₁	—	4	12	no	—
96	C₂³×C₃₁	—	4	32	no	—
97	C₂³×C₃₁	C₂²⋊Q₈	1	4	yes	—
98	C₂³×C₃₁	C₂×D₈	1	4	yes	—
99	C₂³×C₃₁	#137	1	4	yes	—
100	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
101	Q₈×C₃₁	—	4	32	no	—
102	D₄×C₃₁	—	2	24	no	—
103	D₄×C₃₁	—	2	4	no	—
104	D₄×C₃₁	C₂²×Q₈	1	8	yes	complemented
105	C₂×C₄×C₃₁	—	2	12	no	—
106	C₂×C₄×C₃₁	—	2	4	no	—
107	C₂×C₄×C₃₁	—	2	8	no	—
108	C₂×C₄×C₃₁	—	4	12	no	—
109	C₂×C₄×C₃₁	—	4	24	no	—
110	C₂×C₄×C₃₁	—	4	32	no	—
111	C₂×C₄×C₃₁	C₂²⋊Q₈	1	12	yes	—
112	C₂×C₄×C₃₁	C₂×D₈	1	24	yes	—
113	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
114	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
115	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
116	C₂×C₄×C₃₁	C₂²×Q₈	1	6	yes	—
117	C₂×C₄×C₃₁	C₂×#41	1	6	yes	—
118	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
119	C₈×C₃₁	—	2	8	no	—
120	C₂²×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
121	C₂²×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic
122	C₂×#474	C₂×C₃₁	1	24	yes	direct product factor
123	C₂²×#107	C₂×C₃₁	1	2	yes	complemented
124	C₂²×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
125	C₂²×#102	C₂×C₃₁	1	2	yes	—
126	C₂²×C₃₁	—	2	8	no	—
127	C₂²×C₃₁	—	4	4	no	—
128	C₂²×C₃₁	—	4	4	no	—
129	C₂²×C₃₁	—	4	48	no	—
130	C₂²×C₃₁	#474	1	16	yes	direct product factor
131	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	6	yes	—
132	C₂²×C₃₁	C₂²×D₈	1	6	yes	—
133	C₂²×C₃₁	C₂×#137	1	6	yes	—
134	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
135	C₄×C₃₁	—	2	4	no	—
136	C₄×C₃₁	—	4	4	no	—
137	C₄×C₃₁	—	4	8	no	—
138	C₄×C₃₁	C₂×C₂²⋊Q₈	1	6	yes	—
139	C₄×C₃₁	C₂²×D₈	1	8	yes	—
140	C₄×C₃₁	C₂³×Q₈	1	1	yes	characteristic
141	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
142	C₂³×Q₈	—	2	1	no	—
143	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
144	C₂⁴×C₄	—	2	1	no	—
145	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
146	C₂×C₄⋊Q₈	—	2	6	no	—
147	C₂×C₄⋊Q₈	C₂²×C₃₁	1	12	yes	complemented
148	C₂×C₄×D₄	—	2	6	no	—
149	C₂×C₄×D₄	C₂²×C₃₁	1	12	yes	—
150	C₂²×C₄⋊C₄	—	2	1	no	—
151	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
152	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
153	C₂²×C₂²⋊C₄	—	2	1	no	—
154	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
155	#474	C₂²×C₃₁	1	64	yes	direct product factor
156	C₂×#107	C₂²×C₃₁	1	24	yes	complemented
157	C₂×C₄⋊C₈	C₂²×C₃₁	1	12	yes	complemented
158	C₂×#102	C₂²×C₃₁	1	24	yes	—
159	C₂×C₃₁	—	4	8	no	—
160	C₂×C₃₁	C₂×#474	1	12	yes	direct product factor
161	C₂×C₃₁	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₃₁	C₂³×D₈	1	1	yes	characteristic
163	C₂×C₃₁	C₂²×#137	1	1	yes	characteristic
164	C₂⁵	—	2	1	no	—
165	C₂²×Q₈	—	2	12	no	—
166	C₂²×Q₈	—	4	8	no	—
167	C₂²×D₄	—	2	3	no	—
168	C₂²×D₄	—	2	4	no	—
169	C₂²×D₄	Q₈×C₃₁	1	8	yes	complemented
170	C₂²×D₄	C₂³×C₃₁	1	6	yes	—
171	C₂³×C₄	—	2	1	no	—
172	C₂³×C₄	—	2	2	no	—
173	C₂³×C₄	—	2	12	no	—
174	C₂³×C₄	—	4	8	no	—
175	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
176	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
177	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
178	C₂²×C₈	—	2	8	no	—
179	C₂²×C₈	C₂³×C₃₁	1	12	yes	—
180	C₄⋊Q₈	—	2	24	no	—
181	C₄⋊Q₈	C₂³×C₃₁	1	16	yes	complemented
182	C₄×D₄	—	2	24	no	—
183	C₄×D₄	C₂³×C₃₁	1	16	yes	—
184	C₂×C₄⋊C₄	—	2	12	no	—
185	C₂×C₄⋊C₄	—	2	3	no	—
186	C₂×C₄⋊C₄	—	2	6	no	—
187	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
188	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
189	C₂×C₂²⋊C₄	—	2	12	no	—
190	C₂×C₄²	D₄×C₃₁	1	6	yes	—
191	C₂×C₄²	C₂³×C₃₁	1	6	yes	—
192	#107	C₂³×C₃₁	1	32	yes	complemented
193	C₄⋊C₈	C₂³×C₃₁	1	16	yes	complemented
194	#102	C₂³×C₃₁	1	32	yes	—
195	C₃₁	C₂²×#474	1	1	yes	characteristic, direct product factor, Sylow
196	C₂⁴	—	2	8	no	—
197	C₂⁴	—	2	6	no	—
198	C₂⁴	—	4	4	no	—
199	C₂⁴	—	4	4	no	—
200	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
201	C₂×Q₈	—	2	16	no	—
202	C₂×Q₈	—	4	48	no	—
203	C₂×D₄	—	2	24	no	—
204	C₂×D₄	—	2	6	no	—
205	C₂×D₄	—	2	12	no	—
206	C₂×D₄	C₂×Q₈×C₃₁	1	24	yes	complemented
207	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	6	no	—
210	C₂²×C₄	—	2	12	no	—
211	C₂²×C₄	—	2	16	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	4	8	no	—
214	C₂²×C₄	—	4	48	no	—
215	C₂²×C₄	D₈×C₃₁	1	8	yes	—
216	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
217	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
218	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
219	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
220	C₂²×C₄	#41×C₃₁	1	4	yes	—
221	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
222	C₂×C₈	—	2	24	no	—
223	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
224	C₄⋊C₄	—	2	16	no	—
225	C₄⋊C₄	—	2	6	no	—
226	C₄⋊C₄	—	2	12	no	—
227	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
228	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
229	C₂²⋊C₄	—	2	16	no	—
230	C₄²	C₂×D₄×C₃₁	1	12	yes	—
231	C₄²	C₂⁴×C₃₁	1	4	yes	—
232	C₂³	—	2	24	no	—
233	C₂³	—	2	4	no	—
234	C₂³	—	4	32	no	—
235	C₂³	—	4	12	no	—
236	C₂³	—	4	12	no	—
237	C₂³	C₃₁×C₂²⋊Q₈	1	4	yes	—
238	C₂³	C₂×D₈×C₃₁	1	4	yes	—
239	C₂³	C₃₁×#137	1	4	yes	—
240	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
241	Q₈	—	4	32	no	—
242	D₄	—	2	24	no	—
243	D₄	—	2	4	no	—
244	D₄	C₂²×Q₈×C₃₁	1	8	yes	complemented
245	C₂×C₄	—	2	12	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	4	12	no	—
249	C₂×C₄	—	4	24	no	—
250	C₂×C₄	—	4	32	no	—
251	C₂×C₄	C₃₁×C₂²⋊Q₈	1	12	yes	—
252	C₂×C₄	C₂×D₈×C₃₁	1	24	yes	—
253	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
254	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
255	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
256	C₂×C₄	C₂²×Q₈×C₃₁	1	6	yes	—
257	C₂×C₄	C₂×#41×C₃₁	1	6	yes	—
258	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
259	C₈	—	2	8	no	—
260	C₂²	—	2	8	no	—
261	C₂²	—	4	4	no	—
262	C₂²	—	4	4	no	—
263	C₂²	—	4	48	no	—
264	C₂²	C₃₁×#474	1	16	yes	direct product factor
265	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	6	yes	—
266	C₂²	C₂²×D₈×C₃₁	1	6	yes	—
267	C₂²	C₂×C₃₁×#137	1	6	yes	—
268	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
269	C₄	—	2	4	no	—
270	C₄	—	4	4	no	—
271	C₄	—	4	8	no	—
272	C₄	C₂×C₃₁×C₂²⋊Q₈	1	6	yes	—
273	C₄	C₂²×D₈×C₃₁	1	8	yes	—
274	C₄	C₂³×Q₈×C₃₁	1	1	yes	characteristic
275	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
276	C₂	—	4	8	no	—
277	C₂	C₂×C₃₁×#474	1	12	yes	direct product factor
278	C₂	C₂²×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
279	C₂	C₂³×D₈×C₃₁	1	1	yes	characteristic
280	C₂	C₂²×C₃₁×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂³×D₄×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×C₄⋊Q₈	C₂	semidirect
3. C₂×C₃₁×#474	C₂	direct (N × H)
4. C₂²×C₃₁×#107	C₂	semidirect
5. C₂²×C₃₁×C₄⋊C₈	C₂	semidirect
12. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
20. C₃₁×#474	C₂²	direct (N × H)
21. C₂×C₃₁×#107	C₂²	semidirect
22. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
29. C₂²×D₄×C₃₁	Q₈	semidirect
41. C₃₁×C₄⋊Q₈	C₂³	semidirect
52. C₃₁×#107	C₂³	semidirect
53. C₃₁×C₄⋊C₈	C₂³	semidirect
65. C₂×D₄×C₃₁	C₂×Q₈	semidirect
91. C₂²×#474	C₃₁	direct (N × H)
104. D₄×C₃₁	C₂²×Q₈	semidirect
120. C₂²×C₄⋊Q₈	C₂×C₃₁	semidirect
122. C₂×#474	C₂×C₃₁	direct (N × H)
123. C₂²×#107	C₂×C₃₁	semidirect
124. C₂²×C₄⋊C₈	C₂×C₃₁	semidirect
130. C₂²×C₃₁	#474	direct (N × H)
147. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
155. #474	C₂²×C₃₁	direct (N × H)
156. C₂×#107	C₂²×C₃₁	semidirect
157. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
160. C₂×C₃₁	C₂×#474	direct (N × H)
169. C₂²×D₄	Q₈×C₃₁	semidirect
181. C₄⋊Q₈	C₂³×C₃₁	semidirect
192. #107	C₂³×C₃₁	semidirect
193. C₄⋊C₈	C₂³×C₃₁	semidirect
195. C₃₁	C₂²×#474	direct (N × H)
206. C₂×D₄	C₂×Q₈×C₃₁	semidirect
244. D₄	C₂²×Q₈×C₃₁	semidirect
264. C₂²	C₃₁×#474	direct (N × H)
277. C₂	C₂×C₃₁×#474	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×C₄⋊Q₈	2
C₂²×C₄×D₄×C₃₁	2
C₂×C₃₁×#474	2
C₂²×C₃₁×#107	2
C₂²×C₃₁×C₄⋊C₈	2
C₂²×C₃₁×#102	2
C₂²×#474	31

Subgroups — C22×C31×#474 #21517232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₁×#474 #21517232