Subgroups — C₃₁×#13162 #21469921

Order 7936.

290 conjugacy classes of subgroups.

The subgroup lattice has 186 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×#2151	C₂	1	2	yes	maximal
3	C₂×C₃₁×#346	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1772	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1379	C₂	1	2	yes	maximal
6	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
7	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#108	C₄	1	1	yes	characteristic
10	C₃₁×#427	C₄	1	1	yes	characteristic
11	C₃₁×#405	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#364	—	2	2	no	—
14	C₃₁×C₄⋊C₁₆	—	2	4	no	—
15	C₃₁×#346	—	2	2	no	—
16	C₃₁×#339	C₄	1	2	yes	—
17	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
20	C₃₁×#117	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₃₁×M₃₂	—	2	2	no	—
23	C₃₁×M₃₂	—	2	2	no	—
24	C₂×C₁₆×C₃₁	—	2	1	no	—
25	C₂×C₁₆×C₃₁	—	2	1	no	—
26	C₂×C₁₆×C₃₁	—	2	1	no	—
27	C₂×C₁₆×C₃₁	—	2	1	no	—
28	C₂×C₁₆×C₃₁	—	2	2	no	—
29	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
30	C₂×C₁₆×C₃₁	Q₈	1	2	yes	—
31	C₃₁×#108	—	2	2	no	—
32	C₃₁×#107	C₈	1	2	yes	—
33	C₃₁×#106	C₈	1	2	yes	—
34	C₃₁×#97	—	2	2	no	—
35	C₄×C₈×C₃₁	—	2	2	no	—
36	C₄×C₈×C₃₁	D₄	1	2	yes	—
37	C₄×C₈×C₃₁	Q₈	1	2	yes	—
38	C₂²×C₄×C₃₁	—	2	1	no	—
39	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
40	M₁₆×C₃₁	—	2	2	no	—
41	M₁₆×C₃₁	—	2	2	no	—
42	C₂×C₈×C₃₁	—	2	2	no	—
43	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₃₁	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₃₁	C₂×C₈	1	1	yes	characteristic
46	C₂×C₈×C₃₁	M₁₆	1	2	yes	—
47	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
52	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
53	C₄⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
54	C₂²⋊C₄×C₃₁	—	2	2	no	—
55	C₄²×C₃₁	—	2	2	no	—
56	C₄²×C₃₁	—	2	1	no	—
57	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
58	C₁₆×C₃₁	—	2	2	no	—
59	C₁₆×C₃₁	—	2	2	no	—
60	C₁₆×C₃₁	—	4	2	no	—
61	#13162	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂³×C₃₁	#131	1	1	yes	characteristic
63	C₂×C₄×C₃₁	—	2	2	no	—
64	C₂×C₄×C₃₁	—	2	4	no	—
65	C₂×C₄×C₃₁	—	4	2	no	—
66	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₃₁	#131	1	1	yes	characteristic
72	C₈×C₃₁	—	4	2	no	—
73	C₈×C₃₁	C₄⋊C₈	1	2	yes	—
74	C₈×C₃₁	C₄⋊C₈	1	2	yes	—
75	C₈×C₃₁	#121	1	1	yes	characteristic
76	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
77	C₈×C₃₁	#125	1	1	yes	characteristic
78	C₈×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×#364	C₂×C₃₁	1	1	yes	characteristic
80	#2151	C₂×C₃₁	1	2	yes	—
81	C₂×#346	C₂×C₃₁	1	1	yes	characteristic
82	#1772	C₂×C₃₁	1	1	yes	characteristic
83	#1379	C₂×C₃₁	1	2	yes	—
84	C₂²×C₃₁	—	2	2	no	—
85	C₂²×C₃₁	#424	1	1	yes	characteristic
86	C₂²×C₃₁	C₈×D₄	1	1	yes	characteristic
87	C₂²×C₃₁	#446	1	1	yes	characteristic
88	C₄×C₃₁	—	4	2	no	—
89	C₄×C₃₁	—	4	4	no	—
90	C₄×C₃₁	#389	1	1	yes	characteristic
91	C₄×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
92	C₄×C₃₁	#436	1	1	yes	characteristic
93	C₄×C₃₁	C₈×Q₈	1	1	yes	characteristic
94	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
95	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
96	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
97	C₂×#108	C₄×C₃₁	1	1	yes	characteristic
98	#427	C₄×C₃₁	1	1	yes	characteristic
99	#405	C₂²×C₃₁	1	2	yes	—
100	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
101	#364	—	2	2	no	—
102	C₄⋊C₁₆	—	2	4	no	—
103	#346	—	2	2	no	—
104	#339	C₄×C₃₁	1	2	yes	—
105	C₂×C₃₁	—	2	1	no	—
106	C₂×C₃₁	—	2	1	no	—
107	C₂×C₃₁	#1939	1	1	yes	characteristic
108	C₂×C₃₁	#2171	1	2	yes	—
109	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
110	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
111	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
112	#117	C₂×C₄×C₃₁	1	2	yes	—
113	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
114	M₃₂	—	2	2	no	—
115	M₃₂	—	2	2	no	—
116	C₂×C₁₆	—	2	1	no	—
117	C₂×C₁₆	—	2	1	no	—
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
122	C₂×C₁₆	Q₈×C₃₁	1	2	yes	—
123	#108	—	2	2	no	—
124	#107	C₈×C₃₁	1	2	yes	—
125	#106	C₈×C₃₁	1	2	yes	—
126	#97	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	D₄×C₃₁	1	2	yes	—
129	C₄×C₈	Q₈×C₃₁	1	2	yes	—
130	C₃₁	#13162	1	1	yes	characteristic, direct product factor, Sylow
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
133	M₁₆	—	2	2	no	—
134	M₁₆	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
137	C₂×C₈	C₂×C₈×C₃₁	1	1	yes	characteristic
138	C₂×C₈	C₂×C₈×C₃₁	1	1	yes	characteristic
139	C₂×C₈	M₁₆×C₃₁	1	2	yes	—
140	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
141	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
142	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
143	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
144	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
145	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
146	C₄⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
147	C₂²⋊C₄	—	2	2	no	—
148	C₄²	—	2	2	no	—
149	C₄²	—	2	1	no	—
150	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
151	C₁₆	—	2	2	no	—
152	C₁₆	—	2	2	no	—
153	C₁₆	—	4	2	no	—
154	C₂³	C₃₁×#131	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
159	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
160	C₂×C₄	C₄×Q₈×C₃₁	1	1	yes	characteristic
161	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
162	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
163	C₂×C₄	C₃₁×#131	1	1	yes	characteristic
164	C₈	—	4	2	no	—
165	C₈	C₃₁×C₄⋊C₈	1	2	yes	—
166	C₈	C₃₁×C₄⋊C₈	1	2	yes	—
167	C₈	C₃₁×#121	1	1	yes	characteristic
168	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
169	C₈	C₃₁×#125	1	1	yes	characteristic
170	C₈	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
171	C₂²	—	2	2	no	—
172	C₂²	C₃₁×#424	1	1	yes	characteristic
173	C₂²	C₈×D₄×C₃₁	1	1	yes	characteristic
174	C₂²	C₃₁×#446	1	1	yes	characteristic
175	C₄	—	4	2	no	—
176	C₄	—	4	4	no	—
177	C₄	C₃₁×#389	1	1	yes	characteristic
178	C₄	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
179	C₄	C₃₁×#436	1	1	yes	characteristic
180	C₄	C₈×Q₈×C₃₁	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	C₃₁×#1939	1	1	yes	characteristic
184	C₂	C₃₁×#2171	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₈×D₄
prime radical	C₂³×C₃₁	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ C₂²×C₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
61. #13162	C₃₁	direct (N × H)
130. C₃₁	#13162	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×#2151	2
C₂×C₃₁×#346	2
C₃₁×#1772	2
C₃₁×#1379	2
#13162	31

Subgroups — C31×#13162 #21469921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13162 #21469921