Subgroups — C₃₁×#7880 #21464639

Order 7936.

618 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	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1277	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₃₁	—	2	1	no	—
6	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
7	C₃₁×#431	—	2	2	no	—
8	C₃₁×#431	—	2	2	no	—
9	C₃₁×#431	—	2	2	no	—
10	C₃₁×#431	C₄	1	4	yes	—
11	C₃₁×#423	—	2	2	no	—
12	C₃₁×#423	C₄	1	4	yes	—
13	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
14	C₂×C₃₁×C₄⋊C₈	C₄	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
18	C₂×C₃₁×C₂²⋊C₈	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₃₁	—	2	1	no	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
21	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₃₁	Q₈	1	1	yes	characteristic
24	C₂×M₁₆×C₃₁	—	2	4	no	—
25	C₂×M₁₆×C₃₁	—	4	4	no	—
26	C₂²×C₈×C₃₁	—	2	2	no	—
27	C₂²×C₈×C₃₁	—	2	2	no	—
28	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₃₁	D₄	1	2	yes	—
34	C₂×C₄²×C₃₁	D₄	1	2	yes	—
35	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
37	C₃₁×C₄⋊C₈	—	2	4	no	—
38	C₃₁×C₄⋊C₈	—	2	4	no	—
39	C₃₁×C₄⋊C₈	—	2	2	no	—
40	C₃₁×C₄⋊C₈	—	2	2	no	—
41	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
42	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
43	C₃₁×C₂²⋊C₈	—	2	4	no	—
44	C₃₁×C₂²⋊C₈	—	2	4	no	—
45	C₄×C₈×C₃₁	—	2	4	no	—
46	C₄×C₈×C₃₁	—	2	2	no	—
47	C₄×C₈×C₃₁	—	2	2	no	—
48	C₂⁴×C₃₁	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	—	2	4	no	—
50	C₂²×C₄×C₃₁	—	2	4	no	—
51	C₂²×C₄×C₃₁	—	2	2	no	—
52	C₂²×C₄×C₃₁	—	2	2	no	—
53	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
54	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
65	M₁₆×C₃₁	—	4	8	no	—
66	C₂×C₈×C₃₁	—	2	4	no	—
67	C₂×C₈×C₃₁	—	2	4	no	—
68	C₂×C₈×C₃₁	—	2	4	no	—
69	C₂×C₈×C₃₁	—	2	8	no	—
70	C₂×C₈×C₃₁	—	4	2	no	—
71	C₂×C₈×C₃₁	—	4	4	no	—
72	C₂×C₈×C₃₁	—	4	2	no	—
73	C₄²×C₃₁	C₄²	1	2	yes	—
74	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
78	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
79	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
83	#7880	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₃₁	—	2	2	no	—
85	C₂³×C₃₁	—	2	2	no	—
86	C₂³×C₃₁	—	2	2	no	—
87	C₂³×C₃₁	#95	1	2	yes	—
88	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₃₁	—	2	2	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	4	no	—
93	C₂×C₄×C₃₁	—	2	4	no	—
94	C₂×C₄×C₃₁	—	2	4	no	—
95	C₂×C₄×C₃₁	—	2	4	no	—
96	C₂×C₄×C₃₁	—	2	4	no	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	#95	1	2	yes	—
100	C₂×C₄×C₃₁	#95	1	2	yes	—
101	C₂×C₄×C₃₁	#95	1	2	yes	—
102	C₂×C₄×C₃₁	#108	1	4	yes	—
103	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	#131	1	4	yes	—
110	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
111	C₂×C₄×C₃₁	#137	1	2	yes	—
112	C₈×C₃₁	—	4	4	no	—
113	C₈×C₃₁	—	4	4	no	—
114	C₈×C₃₁	—	4	4	no	—
115	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
116	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
117	C₂×#423	C₂×C₃₁	1	1	yes	characteristic
118	#1277	C₂×C₃₁	1	4	yes	complemented
119	C₂²×C₃₁	—	2	2	no	—
120	C₂²×C₃₁	—	2	2	no	—
121	C₂²×C₃₁	—	2	2	no	—
122	C₂²×C₃₁	—	4	4	no	—
123	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
124	C₂²×C₃₁	#424	1	2	yes	—
125	C₂²×C₃₁	#428	1	2	yes	—
126	C₂²×C₃₁	C₂×#108	1	2	yes	—
127	C₄×C₃₁	—	2	2	no	—
128	C₄×C₃₁	—	2	2	no	—
129	C₄×C₃₁	—	2	1	no	—
130	C₄×C₃₁	—	2	1	no	—
131	C₄×C₃₁	#341	1	4	yes	—
132	C₄×C₃₁	#405	1	2	yes	—
133	C₄×C₃₁	#408	1	2	yes	—
134	C₄×C₃₁	#417	1	2	yes	—
135	C₄×C₃₁	#418	1	1	yes	characteristic
136	C₄×C₃₁	#419	1	1	yes	characteristic
137	C₂²×M₁₆	—	2	1	no	—
138	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
139	#431	—	2	2	no	—
140	#431	—	2	2	no	—
141	#431	—	2	2	no	—
142	#431	C₄×C₃₁	1	4	yes	—
143	#423	—	2	2	no	—
144	#423	C₄×C₃₁	1	4	yes	—
145	C₂×C₄⋊C₈	—	2	1	no	—
146	C₂×C₄⋊C₈	C₄×C₃₁	1	2	yes	—
147	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
148	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
149	C₂×C₂²⋊C₈	—	2	1	no	—
150	C₂×C₂²⋊C₈	C₄×C₃₁	1	2	yes	—
151	C₂×C₄×C₈	—	2	1	no	—
152	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
153	C₂×C₃₁	—	4	2	no	—
154	C₂×C₃₁	#1574	1	4	yes	—
155	C₂×C₃₁	#1729	1	1	yes	characteristic
156	C₂×C₃₁	#1736	1	1	yes	characteristic
157	C₂×C₃₁	C₂×#341	1	1	yes	characteristic
158	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
159	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
160	C₂³×C₄	Q₈×C₃₁	1	1	yes	characteristic
161	C₂×M₁₆	—	2	4	no	—
162	C₂×M₁₆	—	4	4	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
166	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
167	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
168	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
169	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
170	C₂×C₄²	D₄×C₃₁	1	2	yes	—
171	C₂×C₄²	D₄×C₃₁	1	2	yes	—
172	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
173	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
174	C₄⋊C₈	—	2	4	no	—
175	C₄⋊C₈	—	2	4	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
179	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
180	C₂²⋊C₈	—	2	4	no	—
181	C₂²⋊C₈	—	2	4	no	—
182	C₄×C₈	—	2	4	no	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	—	2	2	no	—
185	C₃₁	#7880	1	1	yes	characteristic, direct product factor, Sylow
186	C₂⁴	C₄⋊C₄×C₃₁	1	1	yes	characteristic
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
192	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
193	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
194	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
195	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
196	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
197	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
198	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
199	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
202	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
203	M₁₆	—	4	8	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	8	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	4	4	no	—
210	C₂×C₈	—	4	2	no	—
211	C₄²	C₄²×C₃₁	1	2	yes	—
212	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
213	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
214	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
215	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
216	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
217	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
218	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
219	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
220	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	C₃₁×#95	1	2	yes	—
225	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	C₃₁×#95	1	2	yes	—
237	C₂×C₄	C₃₁×#95	1	2	yes	—
238	C₂×C₄	C₃₁×#95	1	2	yes	—
239	C₂×C₄	C₃₁×#108	1	4	yes	—
240	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
241	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
242	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
243	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
244	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
245	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
246	C₂×C₄	C₃₁×#131	1	4	yes	—
247	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
248	C₂×C₄	C₃₁×#137	1	2	yes	—
249	C₈	—	4	4	no	—
250	C₈	—	4	4	no	—
251	C₈	—	4	4	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	4	4	no	—
256	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
257	C₂²	C₃₁×#424	1	2	yes	—
258	C₂²	C₃₁×#428	1	2	yes	—
259	C₂²	C₂×C₃₁×#108	1	2	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	C₃₁×#341	1	4	yes	—
265	C₄	C₃₁×#405	1	2	yes	—
266	C₄	C₃₁×#408	1	2	yes	—
267	C₄	C₃₁×#417	1	2	yes	—
268	C₄	C₃₁×#418	1	1	yes	characteristic
269	C₄	C₃₁×#419	1	1	yes	characteristic
270	C₂	—	4	2	no	—
271	C₂	C₃₁×#1574	1	4	yes	—
272	C₂	C₃₁×#1729	1	1	yes	characteristic
273	C₂	C₃₁×#1736	1	1	yes	characteristic
274	C₂	C₂×C₃₁×#341	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1277	C₂	semidirect
83. #7880	C₃₁	direct (N × H)
118. #1277	C₂×C₃₁	semidirect
185. C₃₁	#7880	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#431	2
C₂×C₃₁×#423	2
C₃₁×#1277	2
#7880	31

Subgroups — C31×#7880 #21464639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7880 #21464639