Subgroups — C₃₁×#12948 #21469707

Order 7936.

384 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2957	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#365	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#360	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2143	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1951	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1385	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1378	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
9	C₃₁×#568	—	2	1	no	—
10	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
12	C₃₁×#444	—	2	2	no	—
13	C₃₁×#443	—	2	2	no	—
14	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
16	C₃₁×#428	—	2	1	no	—
17	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
18	C₃₁×#421	—	2	1	no	—
19	C₃₁×#405	C₂²	1	1	yes	characteristic
20	C₃₁×#365	—	2	2	no	—
21	C₃₁×#360	—	2	2	no	—
22	C₃₁×#359	C₄	1	4	yes	complemented
23	C₃₁×#343	—	2	1	no	—
24	C₂×#41×C₃₁	—	2	1	no	—
25	C₃₁×#135	—	2	1	no	—
26	C₃₁×#135	—	2	1	no	—
27	C₃₁×#135	C₂×C₄	1	2	yes	—
28	C₃₁×#135	D₄	1	2	yes	—
29	C₂×Q₁₆×C₃₁	Q₈	1	1	yes	characteristic
30	C₂×SD₁₆×C₃₁	—	2	1	no	—
31	C₂×D₈×C₃₁	Q₈	1	1	yes	characteristic
32	C₃₁×#131	—	2	4	no	—
33	C₃₁×#131	—	4	2	no	—
34	C₂×M₁₆×C₃₁	—	2	1	no	—
35	C₂×M₁₆×C₃₁	—	2	1	no	—
36	C₂×M₁₆×C₃₁	—	4	2	no	—
37	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₃₁×#117	D₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
41	C₃₁×M₃₂	—	2	1	no	—
42	C₃₁×M₃₂	—	2	1	no	—
43	C₃₁×M₃₂	—	2	1	no	—
44	C₃₁×M₃₂	—	2	1	no	—
45	C₂×C₁₆×C₃₁	—	2	1	no	—
46	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
47	C₃₁×#108	—	2	1	no	—
48	C₃₁×#108	—	2	1	no	—
49	C₃₁×#108	C₂×C₄	1	2	yes	—
50	C₃₁×#108	D₄	1	2	yes	—
51	C₃₁×#107	—	4	1	no	—
52	C₃₁×#106	—	2	1	no	—
53	C₃₁×#106	—	2	1	no	—
54	C₃₁×#106	—	4	1	no	—
55	C₃₁×#104	—	2	4	no	—
56	C₃₁×#97	—	2	1	no	—
57	C₄×C₈×C₃₁	—	2	1	no	—
58	#41×C₃₁	—	2	2	no	—
59	#41×C₃₁	—	2	2	no	—
60	#41×C₃₁	—	4	1	no	—
61	#41×C₃₁	—	4	1	no	—
62	C₂×Q₈×C₃₁	—	2	1	no	—
63	C₂×D₄×C₃₁	—	2	1	no	—
64	C₂×D₄×C₃₁	—	4	1	no	—
65	C₂²×C₄×C₃₁	—	4	1	no	—
66	C₂²×C₄×C₃₁	—	4	1	no	—
67	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
68	Q₁₆×C₃₁	—	2	1	no	—
69	Q₁₆×C₃₁	C₄⋊C₄	1	2	yes	—
70	SD₁₆×C₃₁	—	2	2	no	—
71	SD₁₆×C₃₁	—	2	2	no	—
72	D₈×C₃₁	—	2	1	no	—
73	D₈×C₃₁	C₄⋊C₄	1	2	yes	—
74	M₁₆×C₃₁	—	2	1	no	—
75	M₁₆×C₃₁	—	2	2	no	—
76	M₁₆×C₃₁	—	2	2	no	—
77	M₁₆×C₃₁	—	4	4	no	—
78	M₁₆×C₃₁	—	4	1	no	—
79	M₁₆×C₃₁	—	4	1	no	—
80	M₁₆×C₃₁	SD₁₆	1	2	yes	—
81	C₂×C₈×C₃₁	—	2	1	no	—
82	C₂×C₈×C₃₁	—	4	4	no	—
83	C₂×C₈×C₃₁	—	4	1	no	—
84	C₂×C₈×C₃₁	—	4	1	no	—
85	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
90	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
91	C₄⋊C₄×C₃₁	—	2	1	no	—
92	C₄⋊C₄×C₃₁	—	4	1	no	—
93	C₄⋊C₄×C₃₁	—	4	1	no	—
94	C₂²⋊C₄×C₃₁	—	2	1	no	—
95	C₄²×C₃₁	—	2	1	no	—
96	C₁₆×C₃₁	—	2	2	no	—
97	C₁₆×C₃₁	—	4	1	no	—
98	#12948	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₃₁	—	4	1	no	—
100	C₂³×C₃₁	#135	1	1	yes	characteristic
101	Q₈×C₃₁	—	2	2	no	—
102	Q₈×C₃₁	—	4	1	no	—
103	D₄×C₃₁	—	2	2	no	—
104	D₄×C₃₁	—	4	2	no	—
105	D₄×C₃₁	—	4	2	no	—
106	D₄×C₃₁	—	4	1	no	—
107	C₂×C₄×C₃₁	—	4	1	no	—
108	C₂×C₄×C₃₁	—	4	2	no	—
109	C₂×C₄×C₃₁	—	4	2	no	—
110	C₂×C₄×C₃₁	—	4	1	no	—
111	C₂×C₄×C₃₁	—	8	1	no	—
112	C₂×C₄×C₃₁	—	16	1	no	—
113	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₈×C₃₁	—	2	2	no	—
120	C₈×C₃₁	—	4	2	no	—
121	C₈×C₃₁	—	4	2	no	—
122	C₈×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
124	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
125	C₈×C₃₁	#123	1	1	yes	characteristic
126	#2957	C₂×C₃₁	1	1	yes	characteristic
127	C₂×#365	C₂×C₃₁	1	1	yes	characteristic, complemented
128	C₂×#360	C₂×C₃₁	1	1	yes	characteristic
129	#2143	C₂×C₃₁	1	1	yes	characteristic
130	#1951	C₂×C₃₁	1	1	yes	characteristic, complemented
131	#1385	C₂×C₃₁	1	1	yes	characteristic, complemented
132	#1378	C₂×C₃₁	1	1	yes	characteristic, complemented
133	C₂²×C₃₁	—	4	1	no	—
134	C₂²×C₃₁	—	4	2	no	—
135	C₂²×C₃₁	—	16	1	no	—
136	C₂²×C₃₁	C₄×SD₁₆	1	1	yes	characteristic
137	C₂²×C₃₁	#449	1	1	yes	characteristic
138	C₂²×C₃₁	#478	1	1	yes	characteristic
139	C₄×C₃₁	—	4	2	no	—
140	C₄×C₃₁	—	8	1	no	—
141	C₄×C₃₁	—	16	1	no	—
142	C₄×C₃₁	#385	1	1	yes	characteristic
143	C₄×C₃₁	#442	1	1	yes	characteristic
144	C₄×C₃₁	#450	1	1	yes	characteristic
145	C₄×C₃₁	#476	1	1	yes	characteristic
146	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
147	#568	—	2	1	no	—
148	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
149	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
150	#444	—	2	2	no	—
151	#443	—	2	2	no	—
152	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
153	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
154	#428	—	2	1	no	—
155	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
156	#421	—	2	1	no	—
157	#405	C₂²×C₃₁	1	1	yes	characteristic
158	#365	—	2	2	no	—
159	#360	—	2	2	no	—
160	#359	C₄×C₃₁	1	4	yes	complemented
161	#343	—	2	1	no	—
162	C₂×C₃₁	—	2	1	no	—
163	C₂×C₃₁	—	2	1	no	—
164	C₂×C₃₁	—	2	1	no	—
165	C₂×C₃₁	—	8	2	no	—
166	C₂×C₃₁	#1864	1	1	yes	characteristic
167	C₂×#41	—	2	1	no	—
168	#135	—	2	1	no	—
169	#135	—	2	1	no	—
170	#135	C₂×C₄×C₃₁	1	2	yes	—
171	#135	D₄×C₃₁	1	2	yes	—
172	C₂×Q₁₆	Q₈×C₃₁	1	1	yes	characteristic
173	C₂×SD₁₆	—	2	1	no	—
174	C₂×D₈	Q₈×C₃₁	1	1	yes	characteristic
175	#131	—	2	4	no	—
176	#131	—	4	2	no	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	4	2	no	—
180	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
182	#117	D₄×C₃₁	1	1	yes	characteristic
183	C₂×C₄⋊C₄	—	2	1	no	—
184	M₃₂	—	2	1	no	—
185	M₃₂	—	2	1	no	—
186	M₃₂	—	2	1	no	—
187	M₃₂	—	2	1	no	—
188	C₂×C₁₆	—	2	1	no	—
189	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
190	#108	—	2	1	no	—
191	#108	—	2	1	no	—
192	#108	C₂×C₄×C₃₁	1	2	yes	—
193	#108	D₄×C₃₁	1	2	yes	—
194	#107	—	4	1	no	—
195	#106	—	2	1	no	—
196	#106	—	2	1	no	—
197	#106	—	4	1	no	—
198	#104	—	2	4	no	—
199	#97	—	2	1	no	—
200	C₄×C₈	—	2	1	no	—
201	C₃₁	#12948	1	1	yes	characteristic, direct product factor, Sylow
202	#41	—	2	2	no	—
203	#41	—	2	2	no	—
204	#41	—	4	1	no	—
205	#41	—	4	1	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	C₂×D₄×C₃₁	1	1	yes	characteristic
212	Q₁₆	—	2	1	no	—
213	Q₁₆	C₄⋊C₄×C₃₁	1	2	yes	—
214	SD₁₆	—	2	2	no	—
215	SD₁₆	—	2	2	no	—
216	D₈	—	2	1	no	—
217	D₈	C₄⋊C₄×C₃₁	1	2	yes	—
218	M₁₆	—	2	1	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	4	4	no	—
222	M₁₆	—	4	1	no	—
223	M₁₆	—	4	1	no	—
224	M₁₆	SD₁₆×C₃₁	1	2	yes	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	4	4	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
230	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
231	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
232	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
233	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
234	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₂²⋊C₄	—	2	1	no	—
239	C₄²	—	2	1	no	—
240	C₁₆	—	2	2	no	—
241	C₁₆	—	4	1	no	—
242	C₂³	—	4	1	no	—
243	C₂³	C₃₁×#135	1	1	yes	characteristic
244	Q₈	—	2	2	no	—
245	Q₈	—	4	1	no	—
246	D₄	—	2	2	no	—
247	D₄	—	4	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	—	16	1	no	—
256	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
257	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
258	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
259	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
260	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
261	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
262	C₈	—	2	2	no	—
263	C₈	—	4	2	no	—
264	C₈	—	4	2	no	—
265	C₈	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
266	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
267	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
268	C₈	C₃₁×#123	1	1	yes	characteristic
269	C₂²	—	4	1	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	16	1	no	—
272	C₂²	C₄×SD₁₆×C₃₁	1	1	yes	characteristic
273	C₂²	C₃₁×#449	1	1	yes	characteristic
274	C₂²	C₃₁×#478	1	1	yes	characteristic
275	C₄	—	4	2	no	—
276	C₄	—	8	1	no	—
277	C₄	—	16	1	no	—
278	C₄	C₃₁×#385	1	1	yes	characteristic
279	C₄	C₃₁×#442	1	1	yes	characteristic
280	C₄	C₃₁×#450	1	1	yes	characteristic
281	C₄	C₃₁×#476	1	1	yes	characteristic
282	C₂	—	2	1	no	—
283	C₂	—	2	1	no	—
284	C₂	—	2	1	no	—
285	C₂	—	8	2	no	—
286	C₂	C₃₁×#1864	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#1864
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₃₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#365	C₂	semidirect
5. C₃₁×#1951	C₂	semidirect
6. C₃₁×#1385	C₂	semidirect
7. C₃₁×#1378	C₂	semidirect
22. C₃₁×#359	C₄	semidirect
98. #12948	C₃₁	direct (N × H)
127. C₂×#365	C₂×C₃₁	semidirect
130. #1951	C₂×C₃₁	semidirect
131. #1385	C₂×C₃₁	semidirect
132. #1378	C₂×C₃₁	semidirect
160. #359	C₄×C₃₁	semidirect
201. C₃₁	#12948	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2957	2
C₂×C₃₁×#365	2
C₂×C₃₁×#360	2
C₃₁×#2143	2
C₃₁×#1951	2
C₃₁×#1385	2
C₃₁×#1378	2
#12948	31

Subgroups — C31×#12948 #21469707

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12948 #21469707