Subgroups — C₃₁×#11666 #21468425

Order 7936.

368 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#135	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2163	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1566	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1334	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1333	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#135	C₄	1	1	yes	characteristic
7	C₄×#41×C₃₁	—	2	1	no	—
8	C₄×Q₁₆×C₃₁	C₄	1	1	yes	characteristic
9	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₄×SD₁₆×C₃₁	—	2	1	no	—
11	C₄×SD₁₆×C₃₁	—	2	1	no	—
12	C₄×D₈×C₃₁	C₄	1	1	yes	characteristic
13	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#427	C₄	1	1	yes	characteristic
15	C₃₁×#423	—	2	1	no	—
16	C₃₁×#416	—	2	1	no	—
17	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×C₄⋊C₁₆	—	2	1	no	—
19	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
20	C₃₁×#346	C₂²	1	2	yes	—
21	C₃₁×#334	—	2	1	no	—
22	C₃₁×#334	C₂²	1	2	yes	—
23	C₂×#41×C₃₁	—	2	1	no	—
24	C₃₁×#135	—	2	2	no	—
25	C₃₁×#135	—	2	2	no	—
26	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂×SD₁₆×C₃₁	—	2	1	no	—
28	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂×M₁₆×C₃₁	—	4	1	no	—
30	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
32	C₄×Q₈×C₃₁	—	2	1	no	—
33	C₄×Q₈×C₃₁	—	2	1	no	—
34	C₄×D₄×C₃₁	—	2	1	no	—
35	C₄×D₄×C₃₁	—	2	1	no	—
36	C₄×D₄×C₃₁	—	4	1	no	—
37	C₄×D₄×C₃₁	—	4	1	no	—
38	C₃₁×#117	—	2	1	no	—
39	C₃₁×#117	—	4	1	no	—
40	C₂×C₄²×C₃₁	—	4	1	no	—
41	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₃₁	—	2	2	no	—
43	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
44	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
45	C₃₁×#106	—	2	1	no	—
46	C₃₁×C₄⋊C₈	—	2	2	no	—
47	C₃₁×C₄⋊C₈	—	4	1	no	—
48	C₃₁×#103	—	2	1	no	—
49	C₃₁×#103	—	2	1	no	—
50	C₃₁×#102	—	2	1	no	—
51	C₃₁×#102	—	2	1	no	—
52	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
54	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
55	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	#41×C₃₁	—	2	4	no	—
57	#41×C₃₁	—	4	2	no	—
58	C₂×Q₈×C₃₁	—	2	1	no	—
59	C₂×D₄×C₃₁	—	2	1	no	—
60	C₂×D₄×C₃₁	—	4	1	no	—
61	C₂²×C₄×C₃₁	—	4	1	no	—
62	C₂²×C₄×C₃₁	—	4	1	no	—
63	C₂²×C₄×C₃₁	—	4	1	no	—
64	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
67	Q₁₆×C₃₁	—	2	1	no	—
68	Q₁₆×C₃₁	M₁₆	1	2	yes	complemented
69	SD₁₆×C₃₁	—	2	4	no	—
70	D₈×C₃₁	—	2	1	no	—
71	D₈×C₃₁	M₁₆	1	2	yes	complemented
72	M₁₆×C₃₁	—	8	2	no	—
73	C₂×C₈×C₃₁	—	2	1	no	—
74	C₂×C₈×C₃₁	—	2	1	no	—
75	C₂×C₈×C₃₁	—	4	1	no	—
76	C₂×C₈×C₃₁	—	4	2	no	—
77	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₃₁	—	2	1	no	—
82	C₄⋊C₄×C₃₁	—	2	1	no	—
83	C₄⋊C₄×C₃₁	—	4	1	no	—
84	C₄⋊C₄×C₃₁	—	4	1	no	—
85	C₂²⋊C₄×C₃₁	—	4	1	no	—
86	C₂²⋊C₄×C₃₁	—	4	1	no	—
87	C₂²⋊C₄×C₃₁	—	4	1	no	—
88	C₄²×C₃₁	—	4	1	no	—
89	C₄²×C₃₁	—	4	1	no	—
90	C₄²×C₃₁	—	4	1	no	—
91	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₃₁	D₈	1	1	yes	characteristic
93	C₄²×C₃₁	SD₁₆	1	1	yes	characteristic
94	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₁₆×C₃₁	—	4	2	no	—
96	#11666	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₃₁	—	4	1	no	—
98	C₂³×C₃₁	#102	1	1	yes	characteristic
99	Q₈×C₃₁	—	2	2	no	—
100	Q₈×C₃₁	—	4	1	no	—
101	D₄×C₃₁	—	2	2	no	—
102	D₄×C₃₁	—	4	4	no	—
103	D₄×C₃₁	—	4	1	no	—
104	C₂×C₄×C₃₁	—	2	2	no	—
105	C₂×C₄×C₃₁	—	2	1	no	—
106	C₂×C₄×C₃₁	—	2	1	no	—
107	C₂×C₄×C₃₁	—	4	1	no	—
108	C₂×C₄×C₃₁	—	4	4	no	—
109	C₂×C₄×C₃₁	—	4	1	no	—
110	C₂×C₄×C₃₁	—	4	1	no	—
111	C₂×C₄×C₃₁	—	4	1	no	—
112	C₂×C₄×C₃₁	—	8	2	no	—
113	C₂×C₄×C₃₁	—	8	2	no	—
114	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
115	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
116	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
120	C₈×C₃₁	—	2	1	no	—
121	C₈×C₃₁	—	2	1	no	—
122	C₈×C₃₁	—	2	1	no	—
123	C₈×C₃₁	—	8	2	no	—
124	C₈×C₃₁	C₂×M₁₆	1	1	yes	characteristic
125	C₈×C₃₁	C₂×M₁₆	1	1	yes	characteristic
126	C₄×#135	C₂×C₃₁	1	1	yes	characteristic
127	#2163	C₂×C₃₁	1	1	yes	characteristic, complemented
128	#1566	C₂×C₃₁	1	1	yes	characteristic, complemented
129	#1334	C₂×C₃₁	1	2	yes	complemented
130	#1333	C₂×C₃₁	1	2	yes	complemented
131	C₂²×C₃₁	—	2	1	no	—
132	C₂²×C₃₁	—	2	1	no	—
133	C₂²×C₃₁	—	2	1	no	—
134	C₂²×C₃₁	—	4	2	no	—
135	C₂²×C₃₁	—	8	1	no	—
136	C₂²×C₃₁	—	8	1	no	—
137	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
138	C₄×C₃₁	—	2	1	no	—
139	C₄×C₃₁	—	2	1	no	—
140	C₄×C₃₁	—	2	1	no	—
141	C₄×C₃₁	—	2	1	no	—
142	C₄×C₃₁	—	4	2	no	—
143	C₄×C₃₁	—	8	1	no	—
144	C₄×C₃₁	—	8	1	no	—
145	C₄×C₃₁	#360	1	2	yes	—
146	C₄×C₃₁	#407	1	1	yes	characteristic
147	C₄×C₃₁	#421	1	1	yes	characteristic
148	C₂×#135	C₄×C₃₁	1	1	yes	characteristic
149	C₄×#41	—	2	1	no	—
150	C₄×Q₁₆	C₄×C₃₁	1	1	yes	characteristic
151	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
152	C₄×SD₁₆	—	2	1	no	—
153	C₄×SD₁₆	—	2	1	no	—
154	C₄×D₈	C₄×C₃₁	1	1	yes	characteristic
155	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
156	#427	C₄×C₃₁	1	1	yes	characteristic
157	#423	—	2	1	no	—
158	#416	—	2	1	no	—
159	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
160	C₄⋊C₁₆	—	2	1	no	—
161	C₂²⋊C₁₆	—	2	1	no	—
162	#346	C₂²×C₃₁	1	2	yes	—
163	#334	—	2	1	no	—
164	#334	C₂²×C₃₁	1	2	yes	—
165	C₂×C₃₁	—	4	1	no	—
166	C₂×C₃₁	—	8	2	no	—
167	C₂×C₃₁	#1486	1	1	yes	characteristic
168	C₂×C₃₁	C₂×#360	1	1	yes	characteristic
169	C₂×C₃₁	#2148	1	1	yes	characteristic
170	C₂×#41	—	2	1	no	—
171	#135	—	2	2	no	—
172	#135	—	2	2	no	—
173	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
174	C₂×SD₁₆	—	2	1	no	—
175	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
176	C₂×M₁₆	—	4	1	no	—
177	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
178	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
179	C₄×Q₈	—	2	1	no	—
180	C₄×Q₈	—	2	1	no	—
181	C₄×D₄	—	2	1	no	—
182	C₄×D₄	—	2	1	no	—
183	C₄×D₄	—	4	1	no	—
184	C₄×D₄	—	4	1	no	—
185	#117	—	2	1	no	—
186	#117	—	4	1	no	—
187	C₂×C₄²	—	4	1	no	—
188	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
189	C₂×C₁₆	—	2	2	no	—
190	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
191	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
192	#106	—	2	1	no	—
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	—	4	1	no	—
195	#103	—	2	1	no	—
196	#103	—	2	1	no	—
197	#102	—	2	1	no	—
198	#102	—	2	1	no	—
199	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
200	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
201	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
202	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
203	C₃₁	#11666	1	1	yes	characteristic, direct product factor, Sylow
204	#41	—	2	4	no	—
205	#41	—	4	2	no	—
206	C₂×Q₈	—	2	1	no	—
207	C₂×D₄	—	2	1	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂²×C₄	—	4	1	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	—	4	1	no	—
212	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
213	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
214	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
215	Q₁₆	—	2	1	no	—
216	Q₁₆	M₁₆×C₃₁	1	2	yes	complemented
217	SD₁₆	—	2	4	no	—
218	D₈	—	2	1	no	—
219	D₈	M₁₆×C₃₁	1	2	yes	complemented
220	M₁₆	—	8	2	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
226	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
227	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₂²⋊C₄	—	4	1	no	—
234	C₂²⋊C₄	—	4	1	no	—
235	C₂²⋊C₄	—	4	1	no	—
236	C₄²	—	4	1	no	—
237	C₄²	—	4	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
240	C₄²	D₈×C₃₁	1	1	yes	characteristic
241	C₄²	SD₁₆×C₃₁	1	1	yes	characteristic
242	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
243	C₁₆	—	4	2	no	—
244	C₂³	—	4	1	no	—
245	C₂³	C₃₁×#102	1	1	yes	characteristic
246	Q₈	—	2	2	no	—
247	Q₈	—	4	1	no	—
248	D₄	—	2	2	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	8	2	no	—
260	C₂×C₄	—	8	2	no	—
261	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
262	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
263	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
264	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
265	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
266	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	—	8	2	no	—
271	C₈	C₂×M₁₆×C₃₁	1	1	yes	characteristic
272	C₈	C₂×M₁₆×C₃₁	1	1	yes	characteristic
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	8	1	no	—
278	C₂²	—	8	1	no	—
279	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	4	2	no	—
285	C₄	—	8	1	no	—
286	C₄	—	8	1	no	—
287	C₄	C₃₁×#360	1	2	yes	—
288	C₄	C₃₁×#407	1	1	yes	characteristic
289	C₄	C₃₁×#421	1	1	yes	characteristic
290	C₂	—	4	1	no	—
291	C₂	—	8	2	no	—
292	C₂	C₃₁×#1486	1	1	yes	characteristic
293	C₂	C₂×C₃₁×#360	1	1	yes	characteristic
294	C₂	C₃₁×#2148	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₄×C₈	C₂³×C₃₁
prime radical	C₂×C₃₁×#135	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2163	C₂	semidirect
3. C₃₁×#1566	C₂	semidirect
4. C₃₁×#1334	C₂	semidirect
5. C₃₁×#1333	C₂	semidirect
68. Q₁₆×C₃₁	M₁₆	semidirect
71. D₈×C₃₁	M₁₆	semidirect
96. #11666	C₃₁	direct (N × H)
127. #2163	C₂×C₃₁	semidirect
128. #1566	C₂×C₃₁	semidirect
129. #1334	C₂×C₃₁	semidirect
130. #1333	C₂×C₃₁	semidirect
203. C₃₁	#11666	direct (N × H)
216. Q₁₆	M₁₆×C₃₁	semidirect
219. D₈	M₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#135	2
C₃₁×#2163	2
C₃₁×#1566	2
C₃₁×#1334	2
C₃₁×#1333	2
#11666	31

Subgroups — C31×#11666 #21468425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11666 #21468425