Subgroups — C₃₁×#8396 #21465155

Order 7936.

738 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈²×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1801	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1549	C₂	1	2	yes	maximal
4	C₃₁×#1487	C₂	1	2	yes	maximal
5	C₃₁×#1461	C₂	1	1	yes	maximal, characteristic
6	Q₈²×C₃₁	—	2	4	no	—
7	Q₈²×C₃₁	—	2	4	no	—
8	C₂×C₃₁×C₄⋊Q₈	—	2	2	no	—
9	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
10	C₂×C₃₁×C₄⋊Q₈	C₄	1	2	yes	—
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
13	C₂×C₄×Q₈×C₃₁	C₄	1	2	yes	—
14	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#431	C₂²	1	2	yes	—
17	C₃₁×#423	C₂²	1	2	yes	—
18	C₂×C₃₁×#101	—	2	1	no	—
19	C₃₁×#343	—	2	1	no	—
20	C₃₁×#332	—	2	4	no	—
21	C₃₁×Q₈⋊C₈	—	2	4	no	—
22	C₃₁×#324	—	2	2	no	—
23	C₂²×Q₈×C₃₁	—	2	2	no	—
24	C₂²×Q₈×C₃₁	—	2	1	no	—
25	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
27	C₂×M₁₆×C₃₁	—	2	2	no	—
28	C₃₁×C₄⋊Q₈	—	2	16	no	—
29	C₃₁×C₄⋊Q₈	—	2	8	no	—
30	C₃₁×C₄⋊Q₈	—	2	4	no	—
31	C₃₁×C₄⋊Q₈	—	2	4	no	—
32	C₃₁×C₄⋊Q₈	—	2	2	no	—
33	C₃₁×C₄⋊Q₈	D₄	1	4	yes	—
34	C₄×Q₈×C₃₁	—	2	8	no	—
35	C₄×Q₈×C₃₁	—	2	4	no	—
36	C₄×Q₈×C₃₁	—	2	4	no	—
37	C₄×Q₈×C₃₁	—	2	2	no	—
38	C₄×Q₈×C₃₁	—	2	2	no	—
39	C₄×Q₈×C₃₁	—	2	2	no	—
40	C₄×Q₈×C₃₁	D₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
45	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₃₁	—	2	2	no	—
50	C₂×C₄²×C₃₁	—	2	1	no	—
51	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×C₄⋊C₈	—	2	2	no	—
54	C₃₁×C₄⋊C₈	—	2	2	no	—
55	C₃₁×C₄⋊C₈	—	2	2	no	—
56	C₃₁×#101	—	4	2	no	—
57	C₃₁×C₂²⋊C₈	—	2	2	no	—
58	C₄×C₈×C₃₁	—	2	2	no	—
59	C₂×Q₈×C₃₁	—	2	4	no	—
60	C₂×Q₈×C₃₁	—	2	8	no	—
61	C₂×Q₈×C₃₁	—	2	1	no	—
62	C₂×Q₈×C₃₁	—	2	2	no	—
63	C₂×Q₈×C₃₁	—	2	4	no	—
64	C₂×Q₈×C₃₁	—	2	2	no	—
65	C₂×Q₈×C₃₁	—	8	2	no	—
66	C₂×Q₈×C₃₁	—	8	4	no	—
67	C₂×Q₈×C₃₁	C₂²⋊C₄	1	2	yes	—
68	C₂×Q₈×C₃₁	SD₁₆	1	2	yes	—
69	C₂×Q₈×C₃₁	Q₁₆	1	2	yes	—
70	C₂²×C₄×C₃₁	—	2	1	no	—
71	C₂²×C₄×C₃₁	—	2	2	no	—
72	C₂²×C₄×C₃₁	—	2	1	no	—
73	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
78	M₁₆×C₃₁	—	4	2	no	—
79	M₁₆×C₃₁	—	4	2	no	—
80	C₂×C₈×C₃₁	—	4	2	no	—
81	C₂×C₈×C₃₁	—	4	2	no	—
82	C₄⋊C₄×C₃₁	—	2	8	no	—
83	C₄⋊C₄×C₃₁	—	2	8	no	—
84	C₄⋊C₄×C₃₁	—	2	1	no	—
85	C₄⋊C₄×C₃₁	—	2	2	no	—
86	C₄⋊C₄×C₃₁	—	2	4	no	—
87	C₄⋊C₄×C₃₁	—	2	4	no	—
88	C₄⋊C₄×C₃₁	—	2	4	no	—
89	C₄⋊C₄×C₃₁	—	2	2	no	—
90	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
91	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₃₁	—	2	8	no	—
93	C₄²×C₃₁	—	2	2	no	—
94	C₄²×C₃₁	—	2	2	no	—
95	C₄²×C₃₁	—	2	4	no	—
96	C₄²×C₃₁	C₂×D₄	1	2	yes	—
97	C₄²×C₃₁	C₂×D₄	1	2	yes	—
98	#8396	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	Q₈×C₃₁	—	2	4	no	—
101	Q₈×C₃₁	—	2	2	no	—
102	Q₈×C₃₁	—	2	4	no	—
103	Q₈×C₃₁	—	2	4	no	—
104	Q₈×C₃₁	—	8	4	no	—
105	Q₈×C₃₁	—	8	8	no	—
106	Q₈×C₃₁	#101	1	4	yes	complemented
107	C₂×C₄×C₃₁	—	2	4	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	2	4	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	4	1	no	—
116	C₂×C₄×C₃₁	—	4	4	no	—
117	C₂×C₄×C₃₁	—	4	2	no	—
118	C₂×C₄×C₃₁	—	4	2	no	—
119	C₂×C₄×C₃₁	#103	1	4	yes	—
120	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₃₁	#120	1	2	yes	—
123	C₂×C₄×C₃₁	#120	1	2	yes	—
124	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×C₄×C₃₁	#137	1	2	yes	—
127	C₈×C₃₁	—	8	2	no	—
128	C₈×C₃₁	—	8	2	no	—
129	C₂×Q₈²	C₂×C₃₁	1	1	yes	characteristic
130	#1801	C₂×C₃₁	1	1	yes	characteristic
131	#1549	C₂×C₃₁	1	2	yes	—
132	#1487	C₂×C₃₁	1	2	yes	—
133	#1461	C₂×C₃₁	1	1	yes	characteristic
134	C₂²×C₃₁	—	2	1	no	—
135	C₂²×C₃₁	—	2	1	no	—
136	C₂²×C₃₁	#379	1	1	yes	characteristic
137	C₂²×C₃₁	C₂×#103	1	1	yes	characteristic
138	C₂²×C₃₁	#419	1	1	yes	characteristic
139	C₄×C₃₁	—	2	2	no	—
140	C₄×C₃₁	—	2	4	no	—
141	C₄×C₃₁	—	2	1	no	—
142	C₄×C₃₁	—	2	4	no	—
143	C₄×C₃₁	—	4	4	no	—
144	C₄×C₃₁	—	4	2	no	—
145	C₄×C₃₁	—	4	2	no	—
146	C₄×C₃₁	C₂×#101	1	2	yes	—
147	C₄×C₃₁	#448	1	2	yes	—
148	C₄×C₃₁	#451	1	2	yes	—
149	Q₈²	—	2	4	no	—
150	Q₈²	—	2	4	no	—
151	C₂×C₄⋊Q₈	—	2	2	no	—
152	C₂×C₄⋊Q₈	—	2	1	no	—
153	C₂×C₄⋊Q₈	C₄×C₃₁	1	2	yes	—
154	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
155	C₂×C₄×Q₈	—	2	1	no	—
156	C₂×C₄×Q₈	C₄×C₃₁	1	2	yes	—
157	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
158	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
159	#431	C₂²×C₃₁	1	2	yes	—
160	#423	C₂²×C₃₁	1	2	yes	—
161	C₂×#101	—	2	1	no	—
162	#343	—	2	1	no	—
163	#332	—	2	4	no	—
164	Q₈⋊C₈	—	2	4	no	—
165	#324	—	2	2	no	—
166	C₂×C₃₁	—	2	1	no	—
167	C₂×C₃₁	—	2	1	no	—
168	C₂×C₃₁	#1784	1	1	yes	characteristic
169	C₂×C₃₁	#1787	1	1	yes	characteristic
170	C₂×C₃₁	#1795	1	1	yes	characteristic
171	C₂²×Q₈	—	2	2	no	—
172	C₂²×Q₈	—	2	1	no	—
173	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
174	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
175	C₂×M₁₆	—	2	2	no	—
176	C₄⋊Q₈	—	2	16	no	—
177	C₄⋊Q₈	—	2	8	no	—
178	C₄⋊Q₈	—	2	4	no	—
179	C₄⋊Q₈	—	2	4	no	—
180	C₄⋊Q₈	—	2	2	no	—
181	C₄⋊Q₈	D₄×C₃₁	1	4	yes	—
182	C₄×Q₈	—	2	8	no	—
183	C₄×Q₈	—	2	4	no	—
184	C₄×Q₈	—	2	4	no	—
185	C₄×Q₈	—	2	2	no	—
186	C₄×Q₈	—	2	2	no	—
187	C₄×Q₈	—	2	2	no	—
188	C₄×Q₈	D₄×C₃₁	1	4	yes	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
194	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
195	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
196	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
200	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	—	2	2	no	—
203	C₄⋊C₈	—	2	2	no	—
204	#101	—	4	2	no	—
205	C₂²⋊C₈	—	2	2	no	—
206	C₄×C₈	—	2	2	no	—
207	C₃₁	#8396	1	1	yes	characteristic, direct product factor, Sylow
208	C₂×Q₈	—	2	4	no	—
209	C₂×Q₈	—	2	8	no	—
210	C₂×Q₈	—	2	1	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	2	4	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	8	2	no	—
215	C₂×Q₈	—	8	4	no	—
216	C₂×Q₈	C₂²⋊C₄×C₃₁	1	2	yes	—
217	C₂×Q₈	SD₁₆×C₃₁	1	2	yes	—
218	C₂×Q₈	Q₁₆×C₃₁	1	2	yes	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
223	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
224	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
227	M₁₆	—	4	2	no	—
228	M₁₆	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₄⋊C₄	—	2	8	no	—
232	C₄⋊C₄	—	2	8	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
240	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
241	C₄²	—	2	8	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	4	no	—
245	C₄²	C₂×D₄×C₃₁	1	2	yes	—
246	C₄²	C₂×D₄×C₃₁	1	2	yes	—
247	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
248	Q₈	—	2	4	no	—
249	Q₈	—	2	2	no	—
250	Q₈	—	2	4	no	—
251	Q₈	—	2	4	no	—
252	Q₈	—	8	4	no	—
253	Q₈	—	8	8	no	—
254	Q₈	C₃₁×#101	1	4	yes	complemented
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	C₃₁×#103	1	4	yes	—
268	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
270	C₂×C₄	C₃₁×#120	1	2	yes	—
271	C₂×C₄	C₃₁×#120	1	2	yes	—
272	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
273	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
274	C₂×C₄	C₃₁×#137	1	2	yes	—
275	C₈	—	8	2	no	—
276	C₈	—	8	2	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	C₃₁×#379	1	1	yes	characteristic
280	C₂²	C₂×C₃₁×#103	1	1	yes	characteristic
281	C₂²	C₃₁×#419	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	4	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	4	no	—
286	C₄	—	4	4	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	2	no	—
289	C₄	C₂×C₃₁×#101	1	2	yes	—
290	C₄	C₃₁×#448	1	2	yes	—
291	C₄	C₃₁×#451	1	2	yes	—
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	C₃₁×#1784	1	1	yes	characteristic
295	C₂	C₃₁×#1787	1	1	yes	characteristic
296	C₂	C₃₁×#1795	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#379
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂³×C₃₁	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
98. #8396	C₃₁	direct (N × H)
106. Q₈×C₃₁	#101	semidirect
207. C₃₁	#8396	direct (N × H)
254. Q₈	C₃₁×#101	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈²×C₃₁	2
C₃₁×#1801	2
C₃₁×#1549	2
C₃₁×#1487	2
C₃₁×#1461	2
#8396	31

Subgroups — C31×#8396 #21465155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8396 #21465155