Subgroups — C₃₁×#12953 #21469712

Order 7936.

416 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2948	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2155	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#361	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#361	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1953	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1378	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₃₁×#518	—	2	1	no	—
10	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
12	C₃₁×#442	—	2	2	no	—
13	C₄×D₈×C₃₁	—	2	2	no	—
14	C₃₁×#430	—	2	1	no	—
15	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
17	C₃₁×#427	C₂²	1	1	yes	characteristic
18	C₃₁×#418	—	2	1	no	—
19	C₃₁×#405	C₂²	1	1	yes	characteristic
20	C₃₁×#365	—	2	2	no	—
21	C₃₁×#361	—	2	2	no	—
22	C₃₁×#361	C₄	1	4	yes	complemented
23	C₃₁×#344	—	2	1	no	—
24	C₂²×D₄×C₃₁	—	2	1	no	—
25	C₂×D₈×C₃₁	—	2	1	no	—
26	C₂×D₈×C₃₁	—	2	1	no	—
27	C₂×D₈×C₃₁	—	2	1	no	—
28	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
29	C₂×D₈×C₃₁	D₄	1	2	yes	—
30	C₂×D₈×C₃₁	Q₈	1	2	yes	—
31	C₂×M₁₆×C₃₁	—	2	1	no	—
32	C₂×M₁₆×C₃₁	—	2	1	no	—
33	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
35	C₄×D₄×C₃₁	—	2	4	no	—
36	C₄×D₄×C₃₁	—	4	2	no	—
37	C₃₁×#117	—	2	1	no	—
38	C₃₁×#117	D₄	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
40	C₃₁×M₃₂	—	2	1	no	—
41	C₃₁×M₃₂	—	2	1	no	—
42	C₃₁×M₃₂	—	2	1	no	—
43	C₃₁×M₃₂	C₂×C₄	1	2	yes	complemented
44	C₂×C₁₆×C₃₁	—	2	1	no	—
45	C₂×C₁₆×C₃₁	—	2	1	no	—
46	C₃₁×#108	—	2	1	no	—
47	C₃₁×#108	—	2	1	no	—
48	C₃₁×#108	—	2	1	no	—
49	C₃₁×#108	—	4	2	no	—
50	C₃₁×#108	C₂×C₄	1	2	yes	complemented
51	C₃₁×#107	—	2	1	no	—
52	C₃₁×#106	D₄	1	2	yes	—
53	C₃₁×#102	—	2	4	no	—
54	C₃₁×#97	—	2	1	no	—
55	C₄×C₈×C₃₁	—	2	1	no	—
56	C₂⁴×C₃₁	—	4	1	no	—
57	C₂×D₄×C₃₁	—	2	2	no	—
58	C₂×D₄×C₃₁	—	2	2	no	—
59	C₂×D₄×C₃₁	—	2	2	no	—
60	C₂×D₄×C₃₁	—	4	1	no	—
61	C₂×D₄×C₃₁	—	4	1	no	—
62	C₂×D₄×C₃₁	—	4	1	no	—
63	C₂²×C₄×C₃₁	—	4	4	no	—
64	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
65	D₈×C₃₁	—	2	2	no	—
66	D₈×C₃₁	—	2	2	no	—
67	D₈×C₃₁	—	2	2	no	—
68	D₈×C₃₁	C₄⋊C₄	1	4	yes	—
69	M₁₆×C₃₁	—	2	2	no	—
70	M₁₆×C₃₁	—	2	1	no	—
71	M₁₆×C₃₁	—	2	1	no	—
72	M₁₆×C₃₁	—	4	1	no	—
73	M₁₆×C₃₁	—	4	1	no	—
74	M₁₆×C₃₁	—	4	1	no	—
75	C₂×C₈×C₃₁	—	2	1	no	—
76	C₂×C₈×C₃₁	—	4	1	no	—
77	C₂×C₈×C₃₁	—	4	1	no	—
78	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
83	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
84	C₄⋊C₄×C₃₁	—	2	1	no	—
85	C₄⋊C₄×C₃₁	—	2	2	no	—
86	C₂²⋊C₄×C₃₁	—	4	4	no	—
87	C₂²⋊C₄×C₃₁	—	4	1	no	—
88	C₂²⋊C₄×C₃₁	SD₁₆	1	2	yes	—
89	C₄²×C₃₁	—	2	1	no	—
90	C₄²×C₃₁	—	4	1	no	—
91	C₁₆×C₃₁	—	2	2	no	—
92	C₁₆×C₃₁	—	4	1	no	—
93	#12953	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₃₁	—	4	2	no	—
95	C₂³×C₃₁	—	4	2	no	—
96	C₂³×C₃₁	—	4	2	no	—
97	C₂³×C₃₁	—	16	2	no	—
98	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
99	D₄×C₃₁	—	2	4	no	—
100	D₄×C₃₁	—	4	2	no	—
101	D₄×C₃₁	—	4	2	no	—
102	D₄×C₃₁	—	4	2	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	4	2	no	—
105	C₂×C₄×C₃₁	—	16	4	no	—
106	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
110	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₈×C₃₁	—	4	2	no	—
113	C₈×C₃₁	—	4	1	no	—
114	C₈×C₃₁	—	8	1	no	—
115	C₈×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
117	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
118	C₈×C₃₁	#123	1	1	yes	characteristic
119	#2948	C₂×C₃₁	1	1	yes	characteristic
120	#2155	C₂×C₃₁	1	1	yes	characteristic, complemented
121	C₂×#361	C₂×C₃₁	1	1	yes	characteristic
122	C₂×#361	C₂×C₃₁	1	1	yes	characteristic
123	#1953	C₂×C₃₁	1	1	yes	characteristic, complemented
124	#1389	C₂×C₃₁	1	1	yes	characteristic, complemented
125	#1378	C₂×C₃₁	1	1	yes	characteristic, complemented
126	C₂²×C₃₁	—	4	1	no	—
127	C₂²×C₃₁	—	4	4	no	—
128	C₂²×C₃₁	—	8	4	no	—
129	C₂²×C₃₁	—	16	2	no	—
130	C₂²×C₃₁	—	16	2	no	—
131	C₂²×C₃₁	C₄×SD₁₆	1	1	yes	characteristic
132	C₂²×C₃₁	#450	1	1	yes	characteristic
133	C₂²×C₃₁	#476	1	1	yes	characteristic
134	C₄×C₃₁	—	4	2	no	—
135	C₄×C₃₁	—	8	2	no	—
136	C₄×C₃₁	#385	1	1	yes	characteristic
137	C₄×C₃₁	#442	1	1	yes	characteristic
138	C₄×C₃₁	#449	1	1	yes	characteristic
139	C₄×C₃₁	#478	1	1	yes	characteristic
140	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
141	#518	—	2	1	no	—
142	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
143	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
144	#442	—	2	2	no	—
145	C₄×D₈	—	2	2	no	—
146	#430	—	2	1	no	—
147	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
148	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
149	#427	C₂²×C₃₁	1	1	yes	characteristic
150	#418	—	2	1	no	—
151	#405	C₂²×C₃₁	1	1	yes	characteristic
152	#365	—	2	2	no	—
153	#361	—	2	2	no	—
154	#361	C₄×C₃₁	1	4	yes	complemented
155	#344	—	2	1	no	—
156	C₂×C₃₁	—	2	1	no	—
157	C₂×C₃₁	—	2	1	no	—
158	C₂×C₃₁	—	2	1	no	—
159	C₂×C₃₁	—	8	4	no	—
160	C₂×C₃₁	#1864	1	1	yes	characteristic
161	C₂²×D₄	—	2	1	no	—
162	C₂×D₈	—	2	1	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
166	C₂×D₈	D₄×C₃₁	1	2	yes	—
167	C₂×D₈	Q₈×C₃₁	1	2	yes	—
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	—	2	1	no	—
170	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
172	C₄×D₄	—	2	4	no	—
173	C₄×D₄	—	4	2	no	—
174	#117	—	2	1	no	—
175	#117	D₄×C₃₁	1	1	yes	characteristic
176	C₂×C₂²⋊C₄	—	4	2	no	—
177	M₃₂	—	2	1	no	—
178	M₃₂	—	2	1	no	—
179	M₃₂	—	2	1	no	—
180	M₃₂	C₂×C₄×C₃₁	1	2	yes	complemented
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	—	2	1	no	—
183	#108	—	2	1	no	—
184	#108	—	2	1	no	—
185	#108	—	2	1	no	—
186	#108	—	4	2	no	—
187	#108	C₂×C₄×C₃₁	1	2	yes	complemented
188	#107	—	2	1	no	—
189	#106	D₄×C₃₁	1	2	yes	—
190	#102	—	2	4	no	—
191	#97	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	C₃₁	#12953	1	1	yes	characteristic, direct product factor, Sylow
194	C₂⁴	—	4	1	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂×D₄	—	4	1	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
203	D₈	—	2	2	no	—
204	D₈	—	2	2	no	—
205	D₈	—	2	2	no	—
206	D₈	C₄⋊C₄×C₃₁	1	4	yes	—
207	M₁₆	—	2	2	no	—
208	M₁₆	—	2	1	no	—
209	M₁₆	—	2	1	no	—
210	M₁₆	—	4	1	no	—
211	M₁₆	—	4	1	no	—
212	M₁₆	—	4	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
217	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
218	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
219	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
220	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
221	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	4	4	no	—
225	C₂²⋊C₄	—	4	1	no	—
226	C₂²⋊C₄	SD₁₆×C₃₁	1	2	yes	—
227	C₄²	—	2	1	no	—
228	C₄²	—	4	1	no	—
229	C₁₆	—	2	2	no	—
230	C₁₆	—	4	1	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	4	2	no	—
233	C₂³	—	4	2	no	—
234	C₂³	—	16	2	no	—
235	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
236	D₄	—	2	4	no	—
237	D₄	—	4	2	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	16	4	no	—
243	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
244	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
245	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
246	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
247	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
248	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
249	C₈	—	4	2	no	—
250	C₈	—	4	1	no	—
251	C₈	—	8	1	no	—
252	C₈	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
253	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
254	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
255	C₈	C₃₁×#123	1	1	yes	characteristic
256	C₂²	—	4	1	no	—
257	C₂²	—	4	4	no	—
258	C₂²	—	8	4	no	—
259	C₂²	—	16	2	no	—
260	C₂²	—	16	2	no	—
261	C₂²	C₄×SD₁₆×C₃₁	1	1	yes	characteristic
262	C₂²	C₃₁×#450	1	1	yes	characteristic
263	C₂²	C₃₁×#476	1	1	yes	characteristic
264	C₄	—	4	2	no	—
265	C₄	—	8	2	no	—
266	C₄	C₃₁×#385	1	1	yes	characteristic
267	C₄	C₃₁×#442	1	1	yes	characteristic
268	C₄	C₃₁×#449	1	1	yes	characteristic
269	C₄	C₃₁×#478	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	2	1	no	—
273	C₂	—	8	4	no	—
274	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	C₂²×D₈×C₃₁	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. C₃₁×#2155	C₂	semidirect
5. C₃₁×#1953	C₂	semidirect
6. C₃₁×#1389	C₂	semidirect
7. C₃₁×#1378	C₂	semidirect
22. C₃₁×#361	C₄	semidirect
43. C₃₁×M₃₂	C₂×C₄	semidirect
50. C₃₁×#108	C₂×C₄	semidirect
93. #12953	C₃₁	direct (N × H)
120. #2155	C₂×C₃₁	semidirect
123. #1953	C₂×C₃₁	semidirect
124. #1389	C₂×C₃₁	semidirect
125. #1378	C₂×C₃₁	semidirect
154. #361	C₄×C₃₁	semidirect
180. M₃₂	C₂×C₄×C₃₁	semidirect
187. #108	C₂×C₄×C₃₁	semidirect
193. C₃₁	#12953	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×#2948	2
C₃₁×#2155	2
C₂×C₃₁×#361	2
C₃₁×#1953	2
C₃₁×#1389	2
C₃₁×#1378	2
#12953	31

Subgroups — C31×#12953 #21469712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12953 #21469712