Subgroups — C₃₁×#3396 #16843329

Order 3968.

384 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#559	C₂	1	2	yes	maximal, complemented
2	C₃₁×#495	C₂	1	2	yes	maximal, complemented
3	C₃₁×#493	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#463	C₂	1	4	yes	maximal, complemented
5	C₃₁×#460	C₂	1	2	yes	maximal, complemented
6	C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₈×C₃₁	C₂	1	2	yes	maximal, complemented
9	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×SD₁₆×C₃₁	C₂²	1	4	yes	—
11	C₂×D₈×C₃₁	C₂²	1	2	yes	—
12	C₂×D₈×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#127	C₂²	1	2	yes	—
14	C₃₁×#124	—	2	2	no	—
15	C₃₁×#124	C₂²	1	4	yes	—
16	C₃₁×#123	—	2	4	no	—
17	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
18	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₃₁	—	2	2	no	—
20	C₄×D₄×C₃₁	C₂²	1	4	yes	—
21	C₃₁×#107	C₂²	1	1	yes	characteristic
22	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
24	C₃₁×#103	C₂²	1	2	yes	—
25	C₃₁×#102	C₂²	1	4	yes	—
26	C₄×C₈×C₃₁	C₂²	1	2	yes	—
27	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
29	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
30	C₂×D₄×C₃₁	—	2	2	no	—
31	C₂×D₄×C₃₁	—	2	2	no	—
32	C₂×D₄×C₃₁	C₂³	1	4	yes	—
33	C₂²×C₄×C₃₁	—	2	4	no	—
34	Q₁₆×C₃₁	—	2	2	no	—
35	SD₁₆×C₃₁	—	4	4	no	—
36	D₈×C₃₁	—	2	4	no	—
37	D₈×C₃₁	—	4	2	no	—
38	C₂×C₈×C₃₁	C₂³	1	2	yes	—
39	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₃₁	—	2	2	no	—
42	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
43	C₄⋊C₄×C₃₁	D₄	1	2	yes	—
44	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
45	C₂²⋊C₄×C₃₁	—	2	4	no	—
46	C₂²⋊C₄×C₃₁	—	2	4	no	—
47	C₂²⋊C₄×C₃₁	—	2	4	no	—
48	C₄²×C₃₁	—	2	2	no	—
49	C₄²×C₃₁	C₂³	1	2	yes	—
50	C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₂³×C₃₁	—	2	4	no	—
52	Q₈×C₃₁	—	2	2	no	—
53	Q₈×C₃₁	—	2	2	no	—
54	Q₈×C₃₁	—	2	2	no	—
55	D₄×C₃₁	—	2	4	no	—
56	D₄×C₃₁	—	2	4	no	—
57	D₄×C₃₁	—	4	4	no	—
58	D₄×C₃₁	—	8	2	no	—
59	C₂×C₄×C₃₁	—	2	2	no	—
60	C₂×C₄×C₃₁	—	2	2	no	—
61	C₂×C₄×C₃₁	—	8	4	no	—
62	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
67	C₈×C₃₁	—	2	1	no	—
68	C₈×C₃₁	—	2	2	no	—
69	C₈×C₃₁	#41	1	2	yes	—
70	#3396	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂²×C₃₁	—	4	4	no	—
72	C₂²×C₃₁	—	4	4	no	—
73	C₂²×C₃₁	—	4	4	no	—
74	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
75	C₄×C₃₁	—	2	2	no	—
76	C₄×C₃₁	—	2	2	no	—
77	C₄×C₃₁	—	2	1	no	—
78	C₄×C₃₁	—	4	2	no	—
79	C₄×C₃₁	—	4	2	no	—
80	C₄×C₃₁	#135	1	2	yes	—
81	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
82	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
83	#559	C₂×C₃₁	1	2	yes	complemented
84	#495	C₂×C₃₁	1	2	yes	complemented
85	#493	C₂×C₃₁	1	1	yes	characteristic
86	#463	C₂×C₃₁	1	4	yes	complemented
87	#460	C₂×C₃₁	1	2	yes	complemented
88	C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
89	C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
90	C₄×D₈	C₂×C₃₁	1	2	yes	complemented
91	C₂×C₃₁	—	8	4	no	—
92	C₂×C₃₁	#550	1	1	yes	characteristic
93	C₂×C₃₁	C₂×#135	1	1	yes	characteristic
94	C₂×C₃₁	#576	1	1	yes	characteristic
95	C₂×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
96	C₂×SD₁₆	C₂²×C₃₁	1	4	yes	—
97	C₂×D₈	C₂²×C₃₁	1	2	yes	—
98	C₂×D₈	C₂²×C₃₁	1	2	yes	—
99	#127	C₂²×C₃₁	1	2	yes	—
100	#124	—	2	2	no	—
101	#124	C₂²×C₃₁	1	4	yes	—
102	#123	—	2	4	no	—
103	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
104	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
105	C₄×D₄	—	2	2	no	—
106	C₄×D₄	C₂²×C₃₁	1	4	yes	—
107	#107	C₂²×C₃₁	1	1	yes	characteristic
108	C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
109	C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
110	#103	C₂²×C₃₁	1	2	yes	—
111	#102	C₂²×C₃₁	1	4	yes	—
112	C₄×C₈	C₂²×C₃₁	1	2	yes	—
113	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
114	C₃₁	#3396	1	1	yes	characteristic, direct product factor, Sylow
115	C₂×Q₈	D₄×C₃₁	1	1	yes	characteristic
116	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
117	C₂×D₄	—	2	2	no	—
118	C₂×D₄	—	2	2	no	—
119	C₂×D₄	C₂³×C₃₁	1	4	yes	—
120	C₂²×C₄	—	2	4	no	—
121	Q₁₆	—	2	2	no	—
122	SD₁₆	—	4	4	no	—
123	D₈	—	2	4	no	—
124	D₈	—	4	2	no	—
125	C₂×C₈	C₂³×C₃₁	1	2	yes	—
126	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
127	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
130	C₄⋊C₄	D₄×C₃₁	1	2	yes	—
131	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₂²⋊C₄	—	2	4	no	—
134	C₂²⋊C₄	—	2	4	no	—
135	C₄²	—	2	2	no	—
136	C₄²	C₂³×C₃₁	1	2	yes	—
137	C₄²	C₂³×C₃₁	1	1	yes	characteristic
138	C₂³	—	2	4	no	—
139	Q₈	—	2	2	no	—
140	Q₈	—	2	2	no	—
141	Q₈	—	2	2	no	—
142	D₄	—	2	4	no	—
143	D₄	—	2	4	no	—
144	D₄	—	4	4	no	—
145	D₄	—	8	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	8	4	no	—
149	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
150	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
151	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
152	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
153	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
154	C₈	—	2	1	no	—
155	C₈	—	2	2	no	—
156	C₈	#41×C₃₁	1	2	yes	—
157	C₂²	—	4	4	no	—
158	C₂²	—	4	4	no	—
159	C₂²	—	4	4	no	—
160	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	2	2	no	—
163	C₄	—	2	1	no	—
164	C₄	—	4	2	no	—
165	C₄	—	4	2	no	—
166	C₄	C₃₁×#135	1	2	yes	—
167	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
168	C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
169	C₂	—	8	4	no	—
170	C₂	C₃₁×#550	1	1	yes	characteristic
171	C₂	C₂×C₃₁×#135	1	1	yes	characteristic
172	C₂	C₃₁×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#493	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#559	C₂	semidirect
2. C₃₁×#495	C₂	semidirect
4. C₃₁×#463	C₂	semidirect
5. C₃₁×#460	C₂	semidirect
6. C₈×Q₈×C₃₁	C₂	semidirect
7. C₄×Q₁₆×C₃₁	C₂	semidirect
8. C₄×D₈×C₃₁	C₂	semidirect
70. #3396	C₃₁	direct (N × H)
83. #559	C₂×C₃₁	semidirect
84. #495	C₂×C₃₁	semidirect
86. #463	C₂×C₃₁	semidirect
87. #460	C₂×C₃₁	semidirect
88. C₈×Q₈	C₂×C₃₁	semidirect
89. C₄×Q₁₆	C₂×C₃₁	semidirect
90. C₄×D₈	C₂×C₃₁	semidirect
114. C₃₁	#3396	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×#559	2
C₃₁×#495	2
C₃₁×#493	2
C₃₁×#463	2
C₃₁×#460	2
C₈×Q₈×C₃₁	2
C₄×Q₁₆×C₃₁	2
C₄×D₈×C₃₁	2
#3396	31

Subgroups — C31×#3396 #16843329

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3396 #16843329