Subgroups — C₃₁×#8277 #21465036

Order 7936.

394 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#334	C₂	1	2	yes	maximal
5	C₃₁×#1277	C₂	1	2	yes	maximal
6	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₃₁×#97	—	2	1	no	—
9	C₂×C₄×C₈×C₃₁	—	2	1	no	—
10	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₄³×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#336	—	2	1	no	—
14	C₃₁×#336	—	2	1	no	—
15	C₃₁×#336	C₄	1	4	yes	—
16	C₃₁×#334	C₄	1	8	yes	complemented
17	C₂²×C₈×C₃₁	—	2	2	no	—
18	C₂²×C₈×C₃₁	—	2	2	no	—
19	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
25	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
26	C₃₁×C₄⋊C₈	—	2	2	no	—
27	C₃₁×C₄⋊C₈	—	2	2	no	—
28	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
29	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
30	C₃₁×#97	—	4	2	no	—
31	C₄×C₈×C₃₁	—	2	2	no	—
32	C₄×C₈×C₃₁	—	4	2	no	—
33	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
36	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₃₁	—	2	4	no	—
40	C₂×C₈×C₃₁	—	2	2	no	—
41	C₂×C₈×C₃₁	—	2	4	no	—
42	C₂×C₈×C₃₁	—	2	4	no	—
43	C₂×C₈×C₃₁	—	2	4	no	—
44	C₂×C₈×C₃₁	—	4	2	no	—
45	C₂×C₈×C₃₁	—	4	2	no	—
46	C₂×C₈×C₃₁	—	4	2	no	—
47	C₂×C₈×C₃₁	—	4	2	no	—
48	C₂×C₈×C₃₁	C₄²	1	4	yes	—
49	C₄²×C₃₁	—	2	4	no	—
50	C₄²×C₃₁	—	2	2	no	—
51	C₄²×C₃₁	—	4	2	no	—
52	C₄²×C₃₁	SD₁₆	1	2	yes	—
53	C₄²×C₃₁	SD₁₆	1	2	yes	—
54	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
55	C₄²×C₃₁	#41	1	2	yes	—
56	#8277	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₃₁	—	2	2	no	—
59	C₂×C₄×C₃₁	—	2	2	no	—
60	C₂×C₄×C₃₁	—	2	2	no	—
61	C₂×C₄×C₃₁	—	2	2	no	—
62	C₂×C₄×C₃₁	—	2	2	no	—
63	C₂×C₄×C₃₁	—	2	2	no	—
64	C₂×C₄×C₃₁	#106	1	4	yes	—
65	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
67	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
68	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
69	C₂×C₄×C₃₁	#131	1	4	yes	—
70	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₃₁	#135	1	2	yes	—
73	C₈×C₃₁	—	2	4	no	—
74	C₈×C₃₁	—	4	2	no	—
75	C₈×C₃₁	—	4	4	no	—
76	C₈×C₃₁	—	4	4	no	—
77	#1767	C₂×C₃₁	1	1	yes	characteristic
78	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
79	#1744	C₂×C₃₁	1	1	yes	characteristic
80	C₂×#334	C₂×C₃₁	1	2	yes	—
81	#1277	C₂×C₃₁	1	2	yes	—
82	C₂²×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₃₁	#424	1	2	yes	—
84	C₂²×C₃₁	C₂×#106	1	1	yes	characteristic
85	C₂²×C₃₁	#427	1	1	yes	characteristic
86	C₂²×C₃₁	#430	1	2	yes	—
87	C₄×C₃₁	—	2	4	no	—
88	C₄×C₃₁	—	2	2	no	—
89	C₄×C₃₁	—	4	2	no	—
90	C₄×C₃₁	#405	1	2	yes	—
91	C₄×C₃₁	#433	1	2	yes	—
92	C₄×C₃₁	C₄×SD₁₆	1	2	yes	—
93	C₄×C₃₁	C₄×SD₁₆	1	2	yes	—
94	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
95	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
96	C₂×#97	—	2	1	no	—
97	C₂×C₄×C₈	—	2	1	no	—
98	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
99	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
100	C₄³	C₂²×C₃₁	1	1	yes	characteristic
101	#336	—	2	1	no	—
102	#336	—	2	1	no	—
103	#336	C₄×C₃₁	1	4	yes	—
104	#334	C₄×C₃₁	1	8	yes	complemented
105	C₂×C₃₁	#1570	1	2	yes	—
106	C₂×C₃₁	#1572	1	2	yes	—
107	C₂×C₃₁	#1768	1	1	yes	characteristic
108	C₂×C₃₁	C₄×#106	1	1	yes	characteristic
109	C₂×C₃₁	#1778	1	1	yes	characteristic
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
113	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
114	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
115	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
116	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
117	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
118	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
122	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
123	#97	—	4	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	4	2	no	—
126	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
127	C₃₁	#8277	1	1	yes	characteristic, direct product factor, Sylow
128	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
129	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
130	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
131	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
132	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	C₄²×C₃₁	1	4	yes	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	SD₁₆×C₃₁	1	2	yes	—
147	C₄²	SD₁₆×C₃₁	1	2	yes	—
148	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
149	C₄²	#41×C₃₁	1	2	yes	—
150	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	C₃₁×#106	1	4	yes	—
158	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
159	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
160	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
161	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
162	C₂×C₄	C₃₁×#131	1	4	yes	—
163	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
164	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
165	C₂×C₄	C₃₁×#135	1	2	yes	—
166	C₈	—	2	4	no	—
167	C₈	—	4	2	no	—
168	C₈	—	4	4	no	—
169	C₈	—	4	4	no	—
170	C₂²	C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
171	C₂²	C₃₁×#424	1	2	yes	—
172	C₂²	C₂×C₃₁×#106	1	1	yes	characteristic
173	C₂²	C₃₁×#427	1	1	yes	characteristic
174	C₂²	C₃₁×#430	1	2	yes	—
175	C₄	—	2	4	no	—
176	C₄	—	2	2	no	—
177	C₄	—	4	2	no	—
178	C₄	C₃₁×#405	1	2	yes	—
179	C₄	C₃₁×#433	1	2	yes	—
180	C₄	C₄×SD₁₆×C₃₁	1	2	yes	—
181	C₄	C₄×SD₁₆×C₃₁	1	2	yes	—
182	C₂	C₃₁×#1570	1	2	yes	—
183	C₂	C₃₁×#1572	1	2	yes	—
184	C₂	C₃₁×#1768	1	1	yes	characteristic
185	C₂	C₄×C₃₁×#106	1	1	yes	characteristic
186	C₂	C₃₁×#1778	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, prime radical	C₂³×C₃₁	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
16. C₃₁×#334	C₄	semidirect
56. #8277	C₃₁	direct (N × H)
104. #334	C₄×C₃₁	semidirect
127. C₃₁	#8277	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×#1767	2
C₄×C₃₁×C₄⋊C₈	2
C₃₁×#1744	2
C₂×C₃₁×#334	2
C₃₁×#1277	2
#8277	31

Subgroups — C31×#8277 #21465036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8277 #21465036