Subgroups — C₃₁×#7180 #21463939

Order 7936.

240 conjugacy classes of subgroups.

The subgroup lattice has 196 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	C₃₁×#1772	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1450	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×M₃₂	—	2	1	no	—
5	C₃₁×#431	—	2	1	no	—
6	C₂×C₃₁×#108	C₄	1	1	yes	characteristic
7	C₃₁×#427	C₄	1	1	yes	characteristic
8	C₃₁×#405	—	2	1	no	—
9	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₃₁×#364	—	2	1	no	—
11	C₃₁×#364	C₄	1	2	yes	complemented
12	C₃₁×#339	—	2	1	no	—
13	C₃₁×#322	—	2	1	no	—
14	C₈²×C₃₁	C₄	1	2	yes	complemented
15	C₂×M₁₆×C₃₁	—	2	1	no	—
16	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
18	C₃₁×#117	—	2	1	no	—
19	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
20	C₃₁×M₃₂	—	2	2	no	—
21	C₃₁×M₃₂	—	4	1	no	—
22	C₂×C₁₆×C₃₁	—	4	1	no	—
23	C₃₁×#108	—	2	1	no	—
24	C₃₁×#108	C₈	1	2	yes	complemented
25	C₃₁×#107	—	2	1	no	—
26	C₃₁×#106	C₈	1	2	yes	complemented
27	C₃₁×C₄⋊C₈	—	2	2	no	—
28	C₃₁×C₂²⋊C₈	—	2	2	no	—
29	C₃₁×#97	—	4	1	no	—
30	C₄×C₈×C₃₁	—	2	2	no	—
31	C₄×C₈×C₃₁	—	4	1	no	—
32	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
35	C₄×C₈×C₃₁	Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₃₁	—	2	1	no	—
37	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
38	M₁₆×C₃₁	—	2	2	no	—
39	M₁₆×C₃₁	—	4	1	no	—
40	C₂×C₈×C₃₁	—	2	4	no	—
41	C₂×C₈×C₃₁	—	4	1	no	—
42	C₂×C₈×C₃₁	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₃₁	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈×C₃₁	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₃₁	M₁₆	1	1	yes	characteristic
49	C₂×C₈×C₃₁	M₁₆	1	1	yes	characteristic
50	C₂×C₈×C₃₁	D₈	1	1	yes	characteristic
51	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₃₁	Q₁₆	1	1	yes	characteristic
54	C₄⋊C₄×C₃₁	—	2	2	no	—
55	C₂²⋊C₄×C₃₁	—	4	1	no	—
56	C₄²×C₃₁	—	2	1	no	—
57	C₄²×C₃₁	—	4	1	no	—
58	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
60	C₁₆×C₃₁	—	4	2	no	—
61	#7180	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂³×C₃₁	#104	1	1	yes	characteristic
63	C₂×C₄×C₃₁	—	2	1	no	—
64	C₂×C₄×C₃₁	—	2	1	no	—
65	C₂×C₄×C₃₁	—	4	2	no	—
66	C₂×C₄×C₃₁	—	4	2	no	—
67	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
68	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
70	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
71	C₂×C₄×C₃₁	#103	1	1	yes	characteristic
72	C₂×C₄×C₃₁	#104	1	1	yes	characteristic
73	C₈×C₃₁	—	4	4	no	—
74	C₈×C₃₁	—	4	2	no	—
75	C₈×C₃₁	C₄×C₈	1	1	yes	characteristic
76	C₈×C₃₁	#97	1	1	yes	characteristic
77	C₈×C₃₁	#102	1	1	yes	characteristic
78	C₈×C₃₁	#103	1	1	yes	characteristic
79	C₈×C₃₁	C₄⋊C₈	1	1	yes	characteristic
80	C₈×C₃₁	C₄⋊C₈	1	1	yes	characteristic
81	C₈×C₃₁	#106	1	1	yes	characteristic
82	C₈×C₃₁	#107	1	1	yes	characteristic
83	C₂×#364	C₂×C₃₁	1	1	yes	characteristic
84	#1772	C₂×C₃₁	1	1	yes	characteristic
85	#1450	C₂×C₃₁	1	1	yes	characteristic
86	C₂²×C₃₁	—	4	1	no	—
87	C₂²×C₃₁	D₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₃₁	Q₈⋊C₈	1	1	yes	characteristic
89	C₂²×C₃₁	#339	1	1	yes	characteristic
90	C₄×C₃₁	—	2	2	no	—
91	C₄×C₃₁	—	4	1	no	—
92	C₄×C₃₁	—	8	2	no	—
93	C₄×C₃₁	D₄⋊C₈	1	1	yes	characteristic
94	C₄×C₃₁	Q₈⋊C₈	1	1	yes	characteristic
95	C₄×C₃₁	#336	1	1	yes	characteristic
96	C₄×C₃₁	#340	1	1	yes	characteristic
97	C₂×M₃₂	—	2	1	no	—
98	#431	—	2	1	no	—
99	C₂×#108	C₄×C₃₁	1	1	yes	characteristic
100	#427	C₄×C₃₁	1	1	yes	characteristic
101	#405	—	2	1	no	—
102	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
103	#364	—	2	1	no	—
104	#364	C₄×C₃₁	1	2	yes	complemented
105	#339	—	2	1	no	—
106	#322	—	2	1	no	—
107	C₈²	C₄×C₃₁	1	2	yes	complemented
108	C₂×C₃₁	—	2	1	no	—
109	C₂×C₃₁	—	2	1	no	—
110	C₂×C₃₁	—	2	1	no	—
111	C₂×C₃₁	#1279	1	1	yes	characteristic
112	C₂×M₁₆	—	2	1	no	—
113	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
114	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
115	#117	—	2	1	no	—
116	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
117	M₃₂	—	2	2	no	—
118	M₃₂	—	4	1	no	—
119	C₂×C₁₆	—	4	1	no	—
120	#108	—	2	1	no	—
121	#108	C₈×C₃₁	1	2	yes	complemented
122	#107	—	2	1	no	—
123	#106	C₈×C₃₁	1	2	yes	complemented
124	C₄⋊C₈	—	2	2	no	—
125	C₂²⋊C₈	—	2	2	no	—
126	#97	—	4	1	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	4	1	no	—
129	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
130	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
131	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
132	C₄×C₈	Q₈×C₃₁	1	1	yes	characteristic
133	C₃₁	#7180	1	1	yes	characteristic, direct product factor, Sylow
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
136	M₁₆	—	2	2	no	—
137	M₁₆	—	4	1	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	C₄²×C₃₁	1	1	yes	characteristic
141	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
142	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
143	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
144	C₂×C₈	C₂×C₈×C₃₁	1	1	yes	characteristic
145	C₂×C₈	C₂×C₈×C₃₁	1	1	yes	characteristic
146	C₂×C₈	M₁₆×C₃₁	1	1	yes	characteristic
147	C₂×C₈	M₁₆×C₃₁	1	1	yes	characteristic
148	C₂×C₈	D₈×C₃₁	1	1	yes	characteristic
149	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
150	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
151	C₂×C₈	Q₁₆×C₃₁	1	1	yes	characteristic
152	C₄⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	4	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	4	1	no	—
156	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
157	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
158	C₁₆	—	4	2	no	—
159	C₂³	C₃₁×#104	1	1	yes	characteristic
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
165	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
166	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
167	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
168	C₂×C₄	C₃₁×#103	1	1	yes	characteristic
169	C₂×C₄	C₃₁×#104	1	1	yes	characteristic
170	C₈	—	4	4	no	—
171	C₈	—	4	2	no	—
172	C₈	C₄×C₈×C₃₁	1	1	yes	characteristic
173	C₈	C₃₁×#97	1	1	yes	characteristic
174	C₈	C₃₁×#102	1	1	yes	characteristic
175	C₈	C₃₁×#103	1	1	yes	characteristic
176	C₈	C₃₁×C₄⋊C₈	1	1	yes	characteristic
177	C₈	C₃₁×C₄⋊C₈	1	1	yes	characteristic
178	C₈	C₃₁×#106	1	1	yes	characteristic
179	C₈	C₃₁×#107	1	1	yes	characteristic
180	C₂²	—	4	1	no	—
181	C₂²	C₃₁×D₄⋊C₈	1	1	yes	characteristic
182	C₂²	C₃₁×Q₈⋊C₈	1	1	yes	characteristic
183	C₂²	C₃₁×#339	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	4	1	no	—
186	C₄	—	8	2	no	—
187	C₄	C₃₁×D₄⋊C₈	1	1	yes	characteristic
188	C₄	C₃₁×Q₈⋊C₈	1	1	yes	characteristic
189	C₄	C₃₁×#336	1	1	yes	characteristic
190	C₄	C₃₁×#340	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	—	2	1	no	—
194	C₂	C₃₁×#1279	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₃₁	#106
commutator	C₈	C₄×C₈×C₃₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₃₁
socle	C₂×C₃₁	#1279
prime radical	C₂³×C₃₁	#104
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₃₁ ◁ C₂²×C₈×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₃₁×#364	C₄	semidirect
14. C₈²×C₃₁	C₄	semidirect
24. C₃₁×#108	C₈	semidirect
26. C₃₁×#106	C₈	semidirect
61. #7180	C₃₁	direct (N × H)
104. #364	C₄×C₃₁	semidirect
107. C₈²	C₄×C₃₁	semidirect
121. #108	C₈×C₃₁	semidirect
123. #106	C₈×C₃₁	semidirect
133. C₃₁	#7180	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×#1772	2
C₃₁×#1450	2
#7180	31

Subgroups — C31×#7180 #21463939

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7180 #21463939