Subgroups — C₃₁×#9727 #21466486

Order 7936.

392 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3170	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×#108	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1526	C₂	1	2	yes	maximal
4	C₃₁×#1485	C₂	1	2	yes	maximal
5	C₃₁×#1451	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#533	C₂²	1	2	yes	—
7	C₃₁×#500	C₄	1	1	yes	characteristic
8	C₃₁×#494	C₄	1	1	yes	characteristic
9	C₃₁×#487	C₄	1	2	yes	—
10	C₃₁×#467	—	2	1	no	—
11	C₃₁×#467	—	2	1	no	—
12	C₄×Q₁₆×C₃₁	—	2	1	no	—
13	C₄×Q₁₆×C₃₁	—	2	1	no	—
14	C₃₁×#431	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#108	—	2	1	no	—
16	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
17	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#341	—	2	1	no	—
20	C₃₁×#330	—	2	4	no	—
21	C₃₁×Q₈⋊C₈	—	2	4	no	—
22	C₃₁×#322	—	2	1	no	—
23	C₃₁×#322	—	2	1	no	—
24	C₂×Q₁₆×C₃₁	—	2	1	no	—
25	C₂×Q₁₆×C₃₁	—	2	1	no	—
26	C₂×M₁₆×C₃₁	—	2	1	no	—
27	C₂×M₁₆×C₃₁	—	2	2	no	—
28	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
30	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
31	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
32	C₃₁×#125	—	2	2	no	—
33	C₃₁×C₂²⋊Q₈	—	2	2	no	—
34	C₃₁×C₂²⋊Q₈	—	2	2	no	—
35	C₄×Q₈×C₃₁	—	2	2	no	—
36	C₄×Q₈×C₃₁	—	2	2	no	—
37	C₃₁×#117	—	2	2	no	—
38	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
39	C₃₁×#108	—	2	4	no	—
40	C₃₁×#107	—	2	1	no	—
41	C₃₁×#107	—	2	1	no	—
42	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
43	C₃₁×#103	—	2	2	no	—
44	C₃₁×#103	—	2	2	no	—
45	C₃₁×C₂²⋊C₈	—	2	1	no	—
46	C₃₁×#97	—	2	2	no	—
47	C₃₁×#97	D₄	1	2	yes	—
48	C₄×C₈×C₃₁	—	2	1	no	—
49	C₄×C₈×C₃₁	—	2	1	no	—
50	C₄×C₈×C₃₁	—	2	2	no	—
51	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
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	C₂×Q₈×C₃₁	—	2	2	no	—
56	C₂×Q₈×C₃₁	—	2	2	no	—
57	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
60	Q₁₆×C₃₁	—	4	2	no	—
61	Q₁₆×C₃₁	—	4	2	no	—
62	M₁₆×C₃₁	—	2	4	no	—
63	M₁₆×C₃₁	—	4	2	no	—
64	M₁₆×C₃₁	—	4	2	no	—
65	C₂×C₈×C₃₁	—	2	2	no	—
66	C₂×C₈×C₃₁	—	2	2	no	—
67	C₂×C₈×C₃₁	—	2	2	no	—
68	C₂×C₈×C₃₁	—	4	1	no	—
69	C₂×C₈×C₃₁	—	4	2	no	—
70	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₄⋊C₄×C₃₁	—	2	2	no	—
75	C₄⋊C₄×C₃₁	—	2	2	no	—
76	C₄⋊C₄×C₃₁	—	2	2	no	—
77	C₄⋊C₄×C₃₁	—	2	2	no	—
78	C₄⋊C₄×C₃₁	—	4	2	no	—
79	C₄⋊C₄×C₃₁	—	4	2	no	—
80	C₂²⋊C₄×C₃₁	—	4	2	no	—
81	C₄²×C₃₁	—	4	2	no	—
82	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
86	#9727	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₃₁	#124	1	1	yes	characteristic
88	Q₈×C₃₁	—	4	2	no	—
89	Q₈×C₃₁	—	4	2	no	—
90	Q₈×C₃₁	—	4	4	no	—
91	C₂×C₄×C₃₁	—	2	1	no	—
92	C₂×C₄×C₃₁	—	2	1	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	4	2	no	—
96	C₂×C₄×C₃₁	—	4	2	no	—
97	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
101	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
102	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
103	C₈×C₃₁	—	2	1	no	—
104	C₈×C₃₁	—	2	1	no	—
105	C₈×C₃₁	—	2	1	no	—
106	C₈×C₃₁	—	4	2	no	—
107	C₈×C₃₁	—	4	4	no	—
108	C₈×C₃₁	—	4	2	no	—
109	C₈×C₃₁	#104	1	2	yes	—
110	#3170	C₂×C₃₁	1	1	yes	characteristic
111	C₄×#108	C₂×C₃₁	1	1	yes	characteristic
112	#1526	C₂×C₃₁	1	2	yes	—
113	#1485	C₂×C₃₁	1	2	yes	—
114	#1451	C₂×C₃₁	1	1	yes	characteristic
115	C₂²×C₃₁	—	2	1	no	—
116	C₂²×C₃₁	—	2	1	no	—
117	C₂²×C₃₁	—	2	1	no	—
118	C₂²×C₃₁	#390	1	1	yes	characteristic
119	C₄×C₃₁	—	2	1	no	—
120	C₄×C₃₁	—	2	1	no	—
121	C₄×C₃₁	—	2	1	no	—
122	C₄×C₃₁	—	8	2	no	—
123	C₄×C₃₁	—	8	2	no	—
124	C₄×C₃₁	C₂×#104	1	1	yes	characteristic
125	C₄×C₃₁	#421	1	1	yes	characteristic
126	C₄×C₃₁	#444	1	2	yes	—
127	C₄×C₃₁	#473	1	2	yes	—
128	#533	C₂²×C₃₁	1	2	yes	—
129	#500	C₄×C₃₁	1	1	yes	characteristic
130	#494	C₄×C₃₁	1	1	yes	characteristic
131	#487	C₄×C₃₁	1	2	yes	—
132	#467	—	2	1	no	—
133	#467	—	2	1	no	—
134	C₄×Q₁₆	—	2	1	no	—
135	C₄×Q₁₆	—	2	1	no	—
136	#431	C₂²×C₃₁	1	1	yes	characteristic
137	C₂×#108	—	2	1	no	—
138	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
139	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
141	#341	—	2	1	no	—
142	#330	—	2	4	no	—
143	Q₈⋊C₈	—	2	4	no	—
144	#322	—	2	1	no	—
145	#322	—	2	1	no	—
146	C₂×C₃₁	—	4	1	no	—
147	C₂×C₃₁	#1975	1	1	yes	characteristic
148	C₂×C₃₁	#1979	1	1	yes	characteristic
149	C₂×C₃₁	#1980	1	1	yes	characteristic
150	C₂×Q₁₆	—	2	1	no	—
151	C₂×Q₁₆	—	2	1	no	—
152	C₂×M₁₆	—	2	1	no	—
153	C₂×M₁₆	—	2	2	no	—
154	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
155	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
156	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
157	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
158	#125	—	2	2	no	—
159	C₂²⋊Q₈	—	2	2	no	—
160	C₂²⋊Q₈	—	2	2	no	—
161	C₄×Q₈	—	2	2	no	—
162	C₄×Q₈	—	2	2	no	—
163	#117	—	2	2	no	—
164	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
165	#108	—	2	4	no	—
166	#107	—	2	1	no	—
167	#107	—	2	1	no	—
168	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
169	#103	—	2	2	no	—
170	#103	—	2	2	no	—
171	C₂²⋊C₈	—	2	1	no	—
172	#97	—	2	2	no	—
173	#97	D₄×C₃₁	1	2	yes	—
174	C₄×C₈	—	2	1	no	—
175	C₄×C₈	—	2	1	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
178	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
179	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
180	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
181	C₃₁	#9727	1	1	yes	characteristic, direct product factor, Sylow
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
185	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
186	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
187	Q₁₆	—	4	2	no	—
188	Q₁₆	—	4	2	no	—
189	M₁₆	—	2	4	no	—
190	M₁₆	—	4	2	no	—
191	M₁₆	—	4	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
198	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
199	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
200	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₂²⋊C₄	—	4	2	no	—
208	C₄²	—	4	2	no	—
209	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
210	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
211	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
212	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
213	C₂³	C₃₁×#124	1	1	yes	characteristic
214	Q₈	—	4	2	no	—
215	Q₈	—	4	2	no	—
216	Q₈	—	4	4	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
224	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
225	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
226	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
227	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
228	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
229	C₈	—	2	1	no	—
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	4	2	no	—
233	C₈	—	4	4	no	—
234	C₈	—	4	2	no	—
235	C₈	C₃₁×#104	1	2	yes	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	C₃₁×#390	1	1	yes	characteristic
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	8	2	no	—
244	C₄	—	8	2	no	—
245	C₄	C₂×C₃₁×#104	1	1	yes	characteristic
246	C₄	C₃₁×#421	1	1	yes	characteristic
247	C₄	C₃₁×#444	1	2	yes	—
248	C₄	C₃₁×#473	1	2	yes	—
249	C₂	—	4	1	no	—
250	C₂	C₃₁×#1975	1	1	yes	characteristic
251	C₂	C₃₁×#1979	1	1	yes	characteristic
252	C₂	C₃₁×#1980	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#390
prime radical	C₂³×C₃₁	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
86. #9727	C₃₁	direct (N × H)
181. C₃₁	#9727	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×#3170	2
C₄×C₃₁×#108	2
C₃₁×#1526	2
C₃₁×#1485	2
C₃₁×#1451	2
#9727	31

Subgroups — C31×#9727 #21466486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9727 #21466486