Subgroups — C₃₁×#7280 #21464039

Order 7936.

226 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2150	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1848	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×M₃₂	—	2	1	no	—
5	C₂×C₃₁×#108	—	2	1	no	—
6	C₂×C₃₁×#108	C₄	1	2	yes	—
7	C₃₁×#423	—	2	1	no	—
8	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
9	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₃₁×C₄⋊C₁₆	—	2	1	no	—
11	C₃₁×C₄⋊C₁₆	C₄	1	2	yes	—
12	C₃₁×#334	—	2	1	no	—
13	C₃₁×#334	C₄	1	2	yes	—
14	C₂×M₁₆×C₃₁	—	2	2	no	—
15	C₂²×C₈×C₃₁	—	4	1	no	—
16	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
19	C₃₁×M₃₂	—	4	1	no	—
20	C₃₁×M₃₂	—	4	1	no	—
21	C₂×C₁₆×C₃₁	—	2	2	no	—
22	C₃₁×#108	—	4	2	no	—
23	C₃₁×#108	—	4	1	no	—
24	C₃₁×#108	—	4	1	no	—
25	C₃₁×C₄⋊C₈	—	2	2	no	—
26	C₃₁×C₄⋊C₈	—	4	1	no	—
27	C₃₁×C₄⋊C₈	—	4	1	no	—
28	C₃₁×C₄⋊C₈	—	4	1	no	—
29	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
32	C₄×C₈×C₃₁	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₃₁	Q₁₆	1	1	yes	characteristic
36	M₁₆×C₃₁	—	4	2	no	—
37	M₁₆×C₃₁	—	4	2	no	—
38	C₂×C₈×C₃₁	—	2	1	no	—
39	C₂×C₈×C₃₁	—	2	1	no	—
40	C₂×C₈×C₃₁	—	4	2	no	—
41	C₂×C₈×C₃₁	—	4	2	no	—
42	C₂×C₈×C₃₁	—	8	2	no	—
43	C₂×C₈×C₃₁	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₃₁	D₈	1	1	yes	characteristic
48	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₃₁	Q₁₆	1	1	yes	characteristic
51	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₃₁	D₈	1	1	yes	characteristic
54	C₄²×C₃₁	SD₁₆	1	1	yes	characteristic
55	C₁₆×C₃₁	—	4	2	no	—
56	#7280	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂³×C₃₁	#103	1	1	yes	characteristic
58	C₂×C₄×C₃₁	—	2	2	no	—
59	C₂×C₄×C₃₁	—	4	1	no	—
60	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
61	C₂×C₄×C₃₁	#100	1	1	yes	characteristic
62	C₂×C₄×C₃₁	#101	1	1	yes	characteristic
63	C₂×C₄×C₃₁	#102	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	C₂×C₄×C₃₁	#106	1	1	yes	characteristic
69	C₂×C₄×C₃₁	#107	1	1	yes	characteristic
70	C₈×C₃₁	—	2	1	no	—
71	C₈×C₃₁	—	2	1	no	—
72	C₈×C₃₁	—	8	2	no	—
73	C₈×C₃₁	—	8	2	no	—
74	C₈×C₃₁	#102	1	1	yes	characteristic
75	C₈×C₃₁	#103	1	1	yes	characteristic
76	C₈×C₃₁	#106	1	1	yes	characteristic
77	C₈×C₃₁	#107	1	1	yes	characteristic
78	#2150	C₂×C₃₁	1	1	yes	characteristic
79	#1848	C₂×C₃₁	1	1	yes	characteristic
80	C₂×#334	C₂×C₃₁	1	1	yes	characteristic
81	C₂²×C₃₁	—	2	1	no	—
82	C₂²×C₃₁	—	2	1	no	—
83	C₂²×C₃₁	#332	1	1	yes	characteristic
84	C₂²×C₃₁	#333	1	1	yes	characteristic
85	C₂²×C₃₁	#340	1	1	yes	characteristic
86	C₄×C₃₁	—	2	1	no	—
87	C₄×C₃₁	—	2	1	no	—
88	C₄×C₃₁	—	2	1	no	—
89	C₄×C₃₁	#331	1	1	yes	characteristic
90	C₄×C₃₁	#332	1	1	yes	characteristic
91	C₄×C₃₁	#340	1	1	yes	characteristic
92	C₄×C₃₁	#343	1	1	yes	characteristic
93	C₄×C₃₁	#365	1	2	yes	—
94	C₂×M₃₂	—	2	1	no	—
95	C₂×#108	—	2	1	no	—
96	C₂×#108	C₄×C₃₁	1	2	yes	—
97	#423	—	2	1	no	—
98	C₂×C₄⋊C₈	—	2	1	no	—
99	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
100	C₄⋊C₁₆	—	2	1	no	—
101	C₄⋊C₁₆	C₄×C₃₁	1	2	yes	—
102	#334	—	2	1	no	—
103	#334	C₄×C₃₁	1	2	yes	—
104	C₂×C₃₁	—	2	1	no	—
105	C₂×C₃₁	—	2	1	no	—
106	C₂×C₃₁	#1296	1	1	yes	characteristic
107	C₂×C₃₁	#1383	1	1	yes	characteristic
108	C₂×C₃₁	#1388	1	1	yes	characteristic
109	C₂×M₁₆	—	2	2	no	—
110	C₂²×C₈	—	4	1	no	—
111	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
112	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
113	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
114	M₃₂	—	4	1	no	—
115	M₃₂	—	4	1	no	—
116	C₂×C₁₆	—	2	2	no	—
117	#108	—	4	2	no	—
118	#108	—	4	1	no	—
119	#108	—	4	1	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	4	1	no	—
122	C₄⋊C₈	—	4	1	no	—
123	C₄⋊C₈	—	4	1	no	—
124	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
125	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
126	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
127	C₄×C₈	Q₈×C₃₁	1	1	yes	characteristic
128	C₃₁	#7280	1	1	yes	characteristic, direct product factor, Sylow
129	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
130	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
131	C₂²×C₄	Q₁₆×C₃₁	1	1	yes	characteristic
132	M₁₆	—	4	2	no	—
133	M₁₆	—	4	2	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	8	2	no	—
139	C₂×C₈	C₄²×C₃₁	1	1	yes	characteristic
140	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
141	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
142	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
143	C₂×C₈	D₈×C₃₁	1	1	yes	characteristic
144	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
145	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
146	C₂×C₈	Q₁₆×C₃₁	1	1	yes	characteristic
147	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
148	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
149	C₄²	D₈×C₃₁	1	1	yes	characteristic
150	C₄²	SD₁₆×C₃₁	1	1	yes	characteristic
151	C₁₆	—	4	2	no	—
152	C₂³	C₃₁×#103	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
156	C₂×C₄	C₃₁×#100	1	1	yes	characteristic
157	C₂×C₄	C₃₁×#101	1	1	yes	characteristic
158	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
159	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
160	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
161	C₂×C₄	C₃₁×#103	1	1	yes	characteristic
162	C₂×C₄	C₃₁×#103	1	1	yes	characteristic
163	C₂×C₄	C₃₁×#106	1	1	yes	characteristic
164	C₂×C₄	C₃₁×#107	1	1	yes	characteristic
165	C₈	—	2	1	no	—
166	C₈	—	2	1	no	—
167	C₈	—	8	2	no	—
168	C₈	—	8	2	no	—
169	C₈	C₃₁×#102	1	1	yes	characteristic
170	C₈	C₃₁×#103	1	1	yes	characteristic
171	C₈	C₃₁×#106	1	1	yes	characteristic
172	C₈	C₃₁×#107	1	1	yes	characteristic
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	C₃₁×#332	1	1	yes	characteristic
176	C₂²	C₃₁×#333	1	1	yes	characteristic
177	C₂²	C₃₁×#340	1	1	yes	characteristic
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	C₃₁×#331	1	1	yes	characteristic
182	C₄	C₃₁×#332	1	1	yes	characteristic
183	C₄	C₃₁×#340	1	1	yes	characteristic
184	C₄	C₃₁×#343	1	1	yes	characteristic
185	C₄	C₃₁×#365	1	2	yes	—
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	C₃₁×#1296	1	1	yes	characteristic
189	C₂	C₃₁×#1383	1	1	yes	characteristic
190	C₂	C₃₁×#1388	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#340
commutator	C₂×C₈	C₄²×C₃₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₃₁
prime radical	C₂³×C₃₁	#103
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
56. #7280	C₃₁	direct (N × H)
128. C₃₁	#7280	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×#2150	2
C₃₁×#1848	2
C₂×C₃₁×#334	2
#7280	31

Subgroups — C31×#7280 #21464039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7280 #21464039