Subgroups — C₃₁×#7271 #21464030

Order 7936.

222 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#364	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#364	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1846	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×M₃₂	—	2	1	no	—
5	C₂×C₃₁×M₃₂	—	2	1	no	—
6	C₂×C₃₁×#107	—	2	1	no	—
7	C₂×C₃₁×#106	—	2	1	no	—
8	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₃₁×#382	C₄	1	2	yes	—
10	C₃₁×#364	—	2	1	no	—
11	C₃₁×#364	—	2	1	no	—
12	C₃₁×#364	C₄	1	2	yes	complemented
13	C₃₁×#364	C₄	1	2	yes	complemented
14	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
17	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₃₁×M₃₂	—	2	2	no	—
19	C₃₁×M₃₂	—	2	2	no	—
20	C₃₁×M₃₂	—	4	1	no	—
21	C₃₁×M₃₂	—	4	1	no	—
22	C₂×C₁₆×C₃₁	—	4	1	no	—
23	C₂×C₁₆×C₃₁	—	4	1	no	—
24	C₃₁×#107	—	4	1	no	—
25	C₃₁×#107	—	4	1	no	—
26	C₃₁×#106	—	4	1	no	—
27	C₃₁×#106	—	4	1	no	—
28	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
31	C₄×C₈×C₃₁	Q₈	1	1	yes	characteristic
32	C₃₁×#95	—	4	2	no	—
33	C₂²×C₄×C₃₁	—	2	1	no	—
34	C₂²×C₄×C₃₁	—	4	2	no	—
35	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₃₁	D₈	1	1	yes	characteristic
41	C₂×C₈×C₃₁	D₈	1	1	yes	characteristic
42	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
43	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
44	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₃₁	Q₁₆	1	1	yes	characteristic
47	C₂×C₈×C₃₁	Q₁₆	1	1	yes	characteristic
48	C₄⋊C₄×C₃₁	—	4	2	no	—
49	C₄⋊C₄×C₃₁	—	4	2	no	—
50	C₄²×C₃₁	—	2	1	no	—
51	C₄²×C₃₁	C₄²	1	1	yes	characteristic
52	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
53	C₁₆×C₃₁	—	4	2	no	—
54	C₁₆×C₃₁	—	4	2	no	—
55	#7271	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂³×C₃₁	#100	1	1	yes	characteristic
57	C₂×C₄×C₃₁	—	2	1	no	—
58	C₂×C₄×C₃₁	—	2	1	no	—
59	C₂×C₄×C₃₁	—	4	2	no	—
60	C₂×C₄×C₃₁	—	8	4	no	—
61	C₂×C₄×C₃₁	—	8	2	no	—
62	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
63	C₂×C₄×C₃₁	#101	1	1	yes	characteristic
64	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
65	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
66	C₂×C₄×C₃₁	#103	1	1	yes	characteristic
67	C₂×C₄×C₃₁	#103	1	1	yes	characteristic
68	C₈×C₃₁	#102	1	1	yes	characteristic
69	C₈×C₃₁	#102	1	1	yes	characteristic
70	C₈×C₃₁	#103	1	1	yes	characteristic
71	C₈×C₃₁	#103	1	1	yes	characteristic
72	C₈×C₃₁	#106	1	1	yes	characteristic
73	C₈×C₃₁	#106	1	1	yes	characteristic
74	C₈×C₃₁	#107	1	1	yes	characteristic
75	C₈×C₃₁	#107	1	1	yes	characteristic
76	C₂×#364	C₂×C₃₁	1	1	yes	characteristic
77	C₂×#364	C₂×C₃₁	1	1	yes	characteristic
78	#1846	C₂×C₃₁	1	1	yes	characteristic
79	C₂²×C₃₁	—	2	2	no	—
80	C₂²×C₃₁	#331	1	1	yes	characteristic
81	C₂²×C₃₁	#333	1	1	yes	characteristic
82	C₂²×C₃₁	#343	1	1	yes	characteristic
83	C₄×C₃₁	—	4	1	no	—
84	C₄×C₃₁	—	4	1	no	—
85	C₄×C₃₁	—	16	2	no	—
86	C₄×C₃₁	#332	1	1	yes	characteristic
87	C₄×C₃₁	#332	1	1	yes	characteristic
88	C₄×C₃₁	#340	1	1	yes	characteristic
89	C₄×C₃₁	#340	1	1	yes	characteristic
90	C₂×M₃₂	—	2	1	no	—
91	C₂×M₃₂	—	2	1	no	—
92	C₂×#107	—	2	1	no	—
93	C₂×#106	—	2	1	no	—
94	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
95	#382	C₄×C₃₁	1	2	yes	—
96	#364	—	2	1	no	—
97	#364	—	2	1	no	—
98	#364	C₄×C₃₁	1	2	yes	complemented
99	#364	C₄×C₃₁	1	2	yes	complemented
100	C₂×C₃₁	—	2	1	no	—
101	C₂×C₃₁	—	2	1	no	—
102	C₂×C₃₁	#1296	1	1	yes	characteristic
103	C₂×C₃₁	#1366	1	2	yes	—
104	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
105	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
108	M₃₂	—	2	2	no	—
109	M₃₂	—	2	2	no	—
110	M₃₂	—	4	1	no	—
111	M₃₂	—	4	1	no	—
112	C₂×C₁₆	—	4	1	no	—
113	C₂×C₁₆	—	4	1	no	—
114	#107	—	4	1	no	—
115	#107	—	4	1	no	—
116	#106	—	4	1	no	—
117	#106	—	4	1	no	—
118	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
119	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
120	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
121	C₄×C₈	Q₈×C₃₁	1	1	yes	characteristic
122	#95	—	4	2	no	—
123	C₃₁	#7271	1	1	yes	characteristic, direct product factor, Sylow
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	4	2	no	—
126	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
127	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
128	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
129	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
130	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
131	C₂×C₈	D₈×C₃₁	1	1	yes	characteristic
132	C₂×C₈	D₈×C₃₁	1	1	yes	characteristic
133	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
134	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
135	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
136	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
137	C₂×C₈	Q₁₆×C₃₁	1	1	yes	characteristic
138	C₂×C₈	Q₁₆×C₃₁	1	1	yes	characteristic
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄²	—	2	1	no	—
142	C₄²	C₄²×C₃₁	1	1	yes	characteristic
143	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
144	C₁₆	—	4	2	no	—
145	C₁₆	—	4	2	no	—
146	C₂³	C₃₁×#100	1	1	yes	characteristic
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	8	4	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
153	C₂×C₄	C₃₁×#101	1	1	yes	characteristic
154	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
155	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
156	C₂×C₄	C₃₁×#103	1	1	yes	characteristic
157	C₂×C₄	C₃₁×#103	1	1	yes	characteristic
158	C₈	C₃₁×#102	1	1	yes	characteristic
159	C₈	C₃₁×#102	1	1	yes	characteristic
160	C₈	C₃₁×#103	1	1	yes	characteristic
161	C₈	C₃₁×#103	1	1	yes	characteristic
162	C₈	C₃₁×#106	1	1	yes	characteristic
163	C₈	C₃₁×#106	1	1	yes	characteristic
164	C₈	C₃₁×#107	1	1	yes	characteristic
165	C₈	C₃₁×#107	1	1	yes	characteristic
166	C₂²	—	2	2	no	—
167	C₂²	C₃₁×#331	1	1	yes	characteristic
168	C₂²	C₃₁×#333	1	1	yes	characteristic
169	C₂²	C₃₁×#343	1	1	yes	characteristic
170	C₄	—	4	1	no	—
171	C₄	—	4	1	no	—
172	C₄	—	16	2	no	—
173	C₄	C₃₁×#332	1	1	yes	characteristic
174	C₄	C₃₁×#332	1	1	yes	characteristic
175	C₄	C₃₁×#340	1	1	yes	characteristic
176	C₄	C₃₁×#340	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	C₃₁×#1296	1	1	yes	characteristic
180	C₂	C₃₁×#1366	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	#333
commutator	C₄²	C₄²×C₃₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₃₁
prime radical	C₂³×C₃₁	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₃₁×#364	C₄	semidirect
13. C₃₁×#364	C₄	semidirect
55. #7271	C₃₁	direct (N × H)
98. #364	C₄×C₃₁	semidirect
99. #364	C₄×C₃₁	semidirect
123. C₃₁	#7271	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×#364	2
C₃₁×#1846	2
#7271	31

Subgroups — C31×#7271 #21464030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7271 #21464030