Subgroups — C₃₁×#8294 #21465053

Order 7936.

358 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#108	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆×C₃₁	C₂	1	2	yes	maximal
4	C₂×C₃₁×#322	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1278	C₂	1	2	yes	maximal
6	C₃₁×#431	C₂²	1	2	yes	—
7	C₂×C₃₁×#108	C₄	1	2	yes	—
8	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#341	C₄	1	2	yes	—
13	C₃₁×#335	C₄	1	2	yes	—
14	C₃₁×#334	C₄	1	2	yes	—
15	C₃₁×#322	—	2	2	no	—
16	C₃₁×#322	C₄	1	2	yes	—
17	C₃₁×#322	C₄	1	2	yes	—
18	C₈²×C₃₁	—	2	2	no	—
19	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₃₁	—	2	1	no	—
21	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₃₁	Q₈	1	2	yes	—
24	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
25	C₃₁×#108	C₈	1	8	yes	complemented
26	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
27	C₃₁×C₂²⋊C₈	—	2	2	no	—
28	C₃₁×#97	—	2	2	no	—
29	C₄×C₈×C₃₁	—	2	2	no	—
30	C₄×C₈×C₃₁	—	2	2	no	—
31	C₄×C₈×C₃₁	—	2	1	no	—
32	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₃₁	D₄	1	2	yes	—
39	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
42	M₁₆×C₃₁	—	2	4	no	—
43	M₁₆×C₃₁	C₂×C₈	1	8	yes	—
44	C₂×C₈×C₃₁	—	2	4	no	—
45	C₂×C₈×C₃₁	—	2	2	no	—
46	C₂×C₈×C₃₁	—	2	2	no	—
47	C₂×C₈×C₃₁	—	2	4	no	—
48	C₂×C₈×C₃₁	—	4	2	no	—
49	C₂×C₈×C₃₁	—	4	2	no	—
50	C₂×C₈×C₃₁	C₄²	1	1	yes	characteristic
51	C₂×C₈×C₃₁	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₃₁	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₃₁	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₃₁	C₄⋊C₄	1	4	yes	—
55	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	—
56	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₃₁	#41	1	1	yes	characteristic
60	#8294	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₃₁	C₄×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₃₁	—	2	2	no	—
63	C₂×C₄×C₃₁	—	2	2	no	—
64	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
67	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
71	C₂×C₄×C₃₁	#131	1	2	yes	—
72	C₈×C₃₁	—	2	1	no	—
73	C₈×C₃₁	—	2	1	no	—
74	C₈×C₃₁	—	2	4	no	—
75	C₈×C₃₁	—	2	8	no	—
76	C₈×C₃₁	—	4	2	no	—
77	C₈×C₃₁	—	4	4	no	—
78	C₈×C₃₁	C₄×C₈	1	2	yes	—
79	C₈×C₃₁	C₄×C₈	1	2	yes	—
80	C₄×#108	C₂×C₃₁	1	1	yes	characteristic
81	#1564	C₂×C₃₁	1	1	yes	characteristic
82	C₈×M₁₆	C₂×C₃₁	1	2	yes	—
83	C₂×#322	C₂×C₃₁	1	1	yes	characteristic
84	#1278	C₂×C₃₁	1	2	yes	—
85	C₂²×C₃₁	—	2	1	no	—
86	C₂²×C₃₁	—	2	1	no	—
87	C₂²×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₃₁	C₈×Q₈	1	2	yes	—
89	C₄×C₃₁	—	2	1	no	—
90	C₄×C₃₁	—	2	1	no	—
91	C₄×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
92	C₄×C₃₁	#405	1	1	yes	characteristic
93	C₄×C₃₁	#430	1	2	yes	—
94	C₄×C₃₁	C₈×D₄	1	2	yes	—
95	C₄×C₃₁	#444	1	2	yes	—
96	#431	C₂²×C₃₁	1	2	yes	—
97	C₂×#108	C₄×C₃₁	1	2	yes	—
98	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
99	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
100	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
102	#341	C₄×C₃₁	1	2	yes	—
103	#335	C₄×C₃₁	1	2	yes	—
104	#334	C₄×C₃₁	1	2	yes	—
105	#322	—	2	2	no	—
106	#322	C₄×C₃₁	1	2	yes	—
107	#322	C₄×C₃₁	1	2	yes	—
108	C₈²	—	2	2	no	—
109	C₂×C₃₁	—	2	2	no	—
110	C₂×C₃₁	C₈×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₃₁	#1773	1	1	yes	characteristic
112	C₂×C₃₁	#1778	1	1	yes	characteristic
113	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
114	C₂²×C₈	—	2	1	no	—
115	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
116	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
117	C₂²×C₈	Q₈×C₃₁	1	2	yes	—
118	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
119	#108	C₈×C₃₁	1	8	yes	complemented
120	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
121	C₂²⋊C₈	—	2	2	no	—
122	#97	—	2	2	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	2	1	no	—
126	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
127	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
128	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
129	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
130	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
131	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
132	C₄×C₈	D₄×C₃₁	1	2	yes	—
133	C₃₁	#8294	1	1	yes	characteristic, direct product factor, Sylow
134	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
135	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
136	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
137	M₁₆	—	2	4	no	—
138	M₁₆	C₂×C₈×C₃₁	1	8	yes	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	C₄²×C₃₁	1	1	yes	characteristic
146	C₂×C₈	C₄²×C₃₁	1	1	yes	characteristic
147	C₂×C₈	C₄²×C₃₁	1	1	yes	characteristic
148	C₂×C₈	C₄²×C₃₁	1	1	yes	characteristic
149	C₂×C₈	C₄⋊C₄×C₃₁	1	4	yes	—
150	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
151	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
152	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
153	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
154	C₄²	#41×C₃₁	1	1	yes	characteristic
155	C₂³	C₄×Q₈×C₃₁	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
160	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
161	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
162	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
163	C₂×C₄	C₄×Q₈×C₃₁	1	1	yes	characteristic
164	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
165	C₂×C₄	C₃₁×#131	1	2	yes	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	4	no	—
169	C₈	—	2	8	no	—
170	C₈	—	4	2	no	—
171	C₈	—	4	4	no	—
172	C₈	C₄×C₈×C₃₁	1	2	yes	—
173	C₈	C₄×C₈×C₃₁	1	2	yes	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
177	C₂²	C₈×Q₈×C₃₁	1	2	yes	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	C₂×C₄×C₈×C₃₁	1	1	yes	characteristic
181	C₄	C₃₁×#405	1	1	yes	characteristic
182	C₄	C₃₁×#430	1	2	yes	—
183	C₄	C₈×D₄×C₃₁	1	2	yes	—
184	C₄	C₃₁×#444	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	C₈×C₄⋊C₄×C₃₁	1	1	yes	characteristic
187	C₂	C₃₁×#1773	1	1	yes	characteristic
188	C₂	C₃₁×#1778	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₃₁	C₂×D₄
commutator	C₄	C₂×C₄×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	C₄×C₄⋊C₄
prime radical	C₂³×C₃₁	C₄×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
25. C₃₁×#108	C₈	semidirect
60. #8294	C₃₁	direct (N × H)
119. #108	C₈×C₃₁	semidirect
133. C₃₁	#8294	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₃₁×#108	2
C₃₁×#1564	2
C₈×M₁₆×C₃₁	2
C₂×C₃₁×#322	2
C₃₁×#1278	2
#8294	31

Subgroups — C31×#8294 #21465053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8294 #21465053