Subgroups — C₃₁×#7920 #21464679

Order 7936.

614 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2920	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1281	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₃₁	C₄	1	2	yes	—
5	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
6	C₃₁×#432	—	2	2	no	—
7	C₃₁×#432	C₄	1	4	yes	—
8	C₃₁×#431	—	2	2	no	—
9	C₃₁×#431	C₂²	1	4	yes	—
10	C₃₁×#423	—	2	2	no	—
11	C₃₁×#407	C₄	1	4	yes	—
12	C₄×M₁₆×C₃₁	—	2	2	no	—
13	C₄×M₁₆×C₃₁	—	2	2	no	—
14	C₄×M₁₆×C₃₁	—	2	2	no	—
15	C₂×C₃₁×#97	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
17	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
19	C₂×M₁₆×C₃₁	—	2	2	no	—
20	C₂×M₁₆×C₃₁	—	2	2	no	—
21	C₂×M₁₆×C₃₁	—	2	4	no	—
22	C₂×M₁₆×C₃₁	—	2	8	no	—
23	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₃₁	D₄	1	2	yes	—
27	C₂×C₄²×C₃₁	D₄	1	2	yes	—
28	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
31	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
33	C₃₁×C₄⋊C₈	—	2	4	no	—
34	C₃₁×C₄⋊C₈	—	2	4	no	—
35	C₃₁×C₂²⋊C₈	—	2	4	no	—
36	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	—
37	C₃₁×#97	—	2	4	no	—
38	C₃₁×#97	—	2	4	no	—
39	C₄×C₈×C₃₁	—	2	4	no	—
40	C₄×C₈×C₃₁	—	2	4	no	—
41	C₂⁴×C₃₁	C₄²	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	—	2	4	no	—
43	C₂²×C₄×C₃₁	—	4	4	no	—
44	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
46	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
56	M₁₆×C₃₁	—	2	8	no	—
57	M₁₆×C₃₁	—	2	8	no	—
58	M₁₆×C₃₁	—	4	8	no	—
59	C₂×C₈×C₃₁	—	2	4	no	—
60	C₂×C₈×C₃₁	—	2	4	no	—
61	C₂×C₈×C₃₁	—	2	8	no	—
62	C₂×C₈×C₃₁	M₁₆	1	8	yes	complemented
63	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
67	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
68	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
69	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
70	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₃₁	C₂×D₄	1	2	yes	—
72	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
73	#7920	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₃₁	—	2	2	no	—
75	C₂³×C₃₁	—	4	2	no	—
76	C₂³×C₃₁	#95	1	1	yes	characteristic
77	C₂³×C₃₁	#95	1	1	yes	characteristic
78	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	—	2	4	no	—
81	C₂×C₄×C₃₁	—	2	4	no	—
82	C₂×C₄×C₃₁	—	2	4	no	—
83	C₂×C₄×C₃₁	—	2	2	no	—
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	4	2	no	—
88	C₂×C₄×C₃₁	—	4	2	no	—
89	C₂×C₄×C₃₁	—	4	2	no	—
90	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
91	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
92	C₂×C₄×C₃₁	#95	1	2	yes	—
93	C₂×C₄×C₃₁	#95	1	2	yes	—
94	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
95	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
99	C₈×C₃₁	—	2	8	no	—
100	C₈×C₃₁	—	4	8	no	—
101	#2920	C₂×C₃₁	1	2	yes	—
102	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
103	#1281	C₂×C₃₁	1	4	yes	complemented
104	C₂²×C₃₁	—	2	2	no	—
105	C₂²×C₃₁	—	2	1	no	—
106	C₂²×C₃₁	—	2	1	no	—
107	C₂²×C₃₁	—	2	2	no	—
108	C₂²×C₃₁	—	2	2	no	—
109	C₂²×C₃₁	—	4	4	no	—
110	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
111	C₂²×C₃₁	C₄×M₁₆	1	2	yes	—
112	C₄×C₃₁	—	2	1	no	—
113	C₄×C₃₁	—	2	1	no	—
114	C₄×C₃₁	—	2	2	no	—
115	C₄×C₃₁	—	2	2	no	—
116	C₄×C₃₁	—	2	2	no	—
117	C₄×C₃₁	#344	1	4	yes	—
118	C₄×C₃₁	#407	1	2	yes	—
119	C₄×C₃₁	#423	1	2	yes	—
120	C₂²×M₁₆	C₄×C₃₁	1	2	yes	—
121	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
122	#432	—	2	2	no	—
123	#432	C₄×C₃₁	1	4	yes	—
124	#431	—	2	2	no	—
125	#431	C₂²×C₃₁	1	4	yes	—
126	#423	—	2	2	no	—
127	#407	C₄×C₃₁	1	4	yes	—
128	C₄×M₁₆	—	2	2	no	—
129	C₄×M₁₆	—	2	2	no	—
130	C₄×M₁₆	—	2	2	no	—
131	C₂×#97	C₂²×C₃₁	1	2	yes	—
132	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
133	C₂×C₃₁	—	2	2	no	—
134	C₂×C₃₁	—	4	1	no	—
135	C₂×C₃₁	—	4	1	no	—
136	C₂×C₃₁	#1728	1	1	yes	characteristic
137	C₂×C₃₁	C₂×#344	1	2	yes	—
138	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
139	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	2	4	no	—
143	C₂×M₁₆	—	2	8	no	—
144	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
145	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
146	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
147	C₂×C₄²	D₄×C₃₁	1	2	yes	—
148	C₂×C₄²	D₄×C₃₁	1	2	yes	—
149	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
150	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
151	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
152	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
153	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
154	C₄⋊C₈	—	2	4	no	—
155	C₄⋊C₈	—	2	4	no	—
156	C₂²⋊C₈	—	2	4	no	—
157	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
158	#97	—	2	4	no	—
159	#97	—	2	4	no	—
160	C₄×C₈	—	2	4	no	—
161	C₄×C₈	—	2	4	no	—
162	C₃₁	#7920	1	1	yes	characteristic, direct product factor, Sylow
163	C₂⁴	C₄²×C₃₁	1	1	yes	characteristic
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	4	4	no	—
166	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
167	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
168	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
169	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
170	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
171	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
172	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
173	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
174	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
175	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
176	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
177	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
178	M₁₆	—	2	8	no	—
179	M₁₆	—	2	8	no	—
180	M₁₆	—	4	8	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	8	no	—
184	C₂×C₈	M₁₆×C₃₁	1	8	yes	complemented
185	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
186	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
187	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
188	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
189	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
190	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
191	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
192	C₄²	C₂×D₄×C₃₁	1	2	yes	—
193	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
194	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
195	C₂³	—	2	2	no	—
196	C₂³	—	4	2	no	—
197	C₂³	C₃₁×#95	1	1	yes	characteristic
198	C₂³	C₃₁×#95	1	1	yes	characteristic
199	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
212	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
213	C₂×C₄	C₃₁×#95	1	2	yes	—
214	C₂×C₄	C₃₁×#95	1	2	yes	—
215	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
216	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
217	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
218	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
219	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
220	C₈	—	2	8	no	—
221	C₈	—	4	8	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	4	4	no	—
228	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
229	C₂²	C₄×M₁₆×C₃₁	1	2	yes	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	C₃₁×#344	1	4	yes	—
236	C₄	C₃₁×#407	1	2	yes	—
237	C₄	C₃₁×#423	1	2	yes	—
238	C₂	—	2	2	no	—
239	C₂	—	4	1	no	—
240	C₂	—	4	1	no	—
241	C₂	C₃₁×#1728	1	1	yes	characteristic
242	C₂	C₂×C₃₁×#344	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄²×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#95
prime radical	C₂⁴×C₃₁	C₄²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1281	C₂	semidirect
62. C₂×C₈×C₃₁	M₁₆	semidirect
73. #7920	C₃₁	direct (N × H)
103. #1281	C₂×C₃₁	semidirect
162. C₃₁	#7920	direct (N × H)
184. C₂×C₈	M₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2920	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#1281	2
#7920	31

Subgroups — C31×#7920 #21464679

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7920 #21464679