Subgroups — C₃₁×#33410 #21490169

Order 7936.

658 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#504	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2913	C₂	1	2	yes	maximal
3	C₃₁×#2159	C₂	1	6	yes	maximal, complemented
4	C₃₁×#2158	C₂	1	3	yes	maximal, complemented
5	C₂×C₃₁×#366	C₂	1	3	yes	maximal, complemented
6	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
7	C₃₁×#520	—	2	2	no	—
8	C₃₁×#504	C₄	1	8	yes	complemented
9	C₂×C₃₁×M₃₂	C₂²	1	3	yes	—
10	C₂²×C₁₆×C₃₁	C₂²	1	3	yes	—
11	C₃₁×#428	—	2	6	no	—
12	C₃₁×#427	C₂²	1	6	yes	—
13	C₂×C₃₁×#107	C₂²	1	6	yes	—
14	C₃₁×#367	C₂²	1	6	yes	—
15	C₃₁×#366	C₂²	1	3	yes	—
16	C₃₁×#366	C₂²	1	6	yes	—
17	C₃₁×#366	C₂²	1	1	yes	characteristic
18	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
19	C₃₁×#131	C₂×C₄	1	4	yes	—
20	C₃₁×#131	Q₈	1	4	yes	—
21	C₂×M₁₆×C₃₁	D₄	1	3	yes	—
22	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
23	C₄×Q₈×C₃₁	—	4	2	no	—
24	C₄×D₄×C₃₁	—	4	6	no	—
25	C₃₁×#117	—	2	6	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	6	no	—
27	C₃₁×M₃₂	C₂×C₄	1	12	yes	complemented
28	C₂×C₁₆×C₃₁	C₂×C₄	1	12	yes	complemented
29	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₃₁	C₂³	1	3	yes	—
31	C₃₁×#107	—	2	6	no	—
32	C₃₁×#107	C₂³	1	6	yes	—
33	C₃₁×#107	C₂³	1	2	yes	—
34	C₃₁×#106	—	2	6	no	—
35	#41×C₃₁	C₄⋊C₄	1	4	yes	—
36	#41×C₃₁	Q₁₆	1	4	yes	—
37	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
38	C₂×D₄×C₃₁	D₈	1	3	yes	—
39	C₂²×C₄×C₃₁	—	4	6	no	—
40	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
41	M₁₆×C₃₁	C₄⋊C₄	1	12	yes	—
42	C₂×C₈×C₃₁	C₂²×C₄	1	6	yes	—
43	C₂×C₈×C₃₁	C₂×D₄	1	3	yes	—
44	C₂×C₈×C₃₁	C₂×Q₈	1	6	yes	—
45	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
46	C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₄⋊C₄×C₃₁	—	2	6	no	—
48	C₄⋊C₄×C₃₁	—	4	6	no	—
49	C₂²⋊C₄×C₃₁	—	4	6	no	—
50	C₄²×C₃₁	—	4	6	no	—
51	C₁₆×C₃₁	C₂²×C₄	1	2	yes	—
52	C₁₆×C₃₁	C₂²×C₄	1	6	yes	—
53	#33410	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂³×C₃₁	C₂×D₈	1	3	yes	—
55	Q₈×C₃₁	#107	1	4	yes	—
56	D₄×C₃₁	#107	1	12	yes	—
57	C₂×C₄×C₃₁	—	4	2	no	—
58	C₂×C₄×C₃₁	—	4	6	no	—
59	C₂×C₄×C₃₁	—	16	6	no	—
60	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	6	yes	—
61	C₂×C₄×C₃₁	C₂×D₈	1	3	yes	—
62	C₂×C₄×C₃₁	C₂×Q₁₆	1	6	yes	—
63	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
64	C₈×C₃₁	C₂×C₄⋊C₄	1	6	yes	—
65	C₈×C₃₁	C₂³×C₄	1	1	yes	characteristic
66	C₈×C₃₁	C₂²×Q₈	1	1	yes	characteristic
67	C₂×#504	C₂×C₃₁	1	1	yes	characteristic
68	#2913	C₂×C₃₁	1	2	yes	—
69	#2159	C₂×C₃₁	1	6	yes	complemented
70	#2158	C₂×C₃₁	1	3	yes	complemented
71	C₂×#366	C₂×C₃₁	1	3	yes	complemented
72	C₂²×C₃₁	—	2	3	no	—
73	C₂²×C₃₁	—	2	3	no	—
74	C₂²×C₃₁	C₂×#107	1	6	yes	—
75	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
76	C₄×C₃₁	—	8	6	no	—
77	C₄×C₃₁	—	8	2	no	—
78	C₄×C₃₁	C₂×#107	1	6	yes	—
79	C₄×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
80	C₄×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
81	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
82	#520	—	2	2	no	—
83	#504	C₄×C₃₁	1	8	yes	complemented
84	C₂×M₃₂	C₂²×C₃₁	1	3	yes	—
85	C₂²×C₁₆	C₂²×C₃₁	1	3	yes	—
86	#428	—	2	6	no	—
87	#427	C₂²×C₃₁	1	6	yes	—
88	C₂×#107	C₂²×C₃₁	1	6	yes	—
89	#367	C₂²×C₃₁	1	6	yes	—
90	#366	C₂²×C₃₁	1	3	yes	—
91	#366	C₂²×C₃₁	1	6	yes	—
92	#366	C₂²×C₃₁	1	1	yes	characteristic
93	C₂×C₃₁	—	2	6	no	—
94	C₂×C₃₁	C₂²×#107	1	1	yes	characteristic
95	C₂×C₃₁	#3415	1	1	yes	characteristic
96	C₂×C₃₁	#3417	1	1	yes	characteristic
97	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
98	#131	C₂×C₄×C₃₁	1	4	yes	—
99	#131	Q₈×C₃₁	1	4	yes	—
100	C₂×M₁₆	D₄×C₃₁	1	3	yes	—
101	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
102	C₄×Q₈	—	4	2	no	—
103	C₄×D₄	—	4	6	no	—
104	#117	—	2	6	no	—
105	C₂×C₄⋊C₄	—	2	6	no	—
106	M₃₂	C₂×C₄×C₃₁	1	12	yes	complemented
107	C₂×C₁₆	C₂×C₄×C₃₁	1	12	yes	complemented
108	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
109	C₂×C₁₆	C₂³×C₃₁	1	3	yes	—
110	#107	—	2	6	no	—
111	#107	C₂³×C₃₁	1	6	yes	—
112	#107	C₂³×C₃₁	1	2	yes	—
113	#106	—	2	6	no	—
114	C₃₁	#33410	1	1	yes	characteristic, direct product factor, Sylow
115	#41	C₄⋊C₄×C₃₁	1	4	yes	—
116	#41	Q₁₆×C₃₁	1	4	yes	—
117	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
118	C₂×D₄	D₈×C₃₁	1	3	yes	—
119	C₂²×C₄	—	4	6	no	—
120	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
121	M₁₆	C₄⋊C₄×C₃₁	1	12	yes	—
122	C₂×C₈	C₂²×C₄×C₃₁	1	6	yes	—
123	C₂×C₈	C₂×D₄×C₃₁	1	3	yes	—
124	C₂×C₈	C₂×Q₈×C₃₁	1	6	yes	—
125	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	2	6	no	—
128	C₄⋊C₄	—	4	6	no	—
129	C₂²⋊C₄	—	4	6	no	—
130	C₄²	—	4	6	no	—
131	C₁₆	C₂²×C₄×C₃₁	1	2	yes	—
132	C₁₆	C₂²×C₄×C₃₁	1	6	yes	—
133	C₂³	C₂×D₈×C₃₁	1	3	yes	—
134	Q₈	C₃₁×#107	1	4	yes	—
135	D₄	C₃₁×#107	1	12	yes	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	4	6	no	—
138	C₂×C₄	—	16	6	no	—
139	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	6	yes	—
140	C₂×C₄	C₂×D₈×C₃₁	1	3	yes	—
141	C₂×C₄	C₂×Q₁₆×C₃₁	1	6	yes	—
142	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
143	C₈	C₂×C₄⋊C₄×C₃₁	1	6	yes	—
144	C₈	C₂³×C₄×C₃₁	1	1	yes	characteristic
145	C₈	C₂²×Q₈×C₃₁	1	1	yes	characteristic
146	C₂²	—	2	3	no	—
147	C₂²	—	2	3	no	—
148	C₂²	C₂×C₃₁×#107	1	6	yes	—
149	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
150	C₄	—	8	6	no	—
151	C₄	—	8	2	no	—
152	C₄	C₂×C₃₁×#107	1	6	yes	—
153	C₄	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
154	C₄	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
155	C₂	—	2	6	no	—
156	C₂	C₂²×C₃₁×#107	1	1	yes	characteristic
157	C₂	C₃₁×#3415	1	1	yes	characteristic
158	C₂	C₃₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator	C₈	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×#41×C₃₁	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×#41×C₃₁ ◁ C₂×C₃₁×#131 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2159	C₂	semidirect
4. C₃₁×#2158	C₂	semidirect
5. C₂×C₃₁×#366	C₂	semidirect
8. C₃₁×#504	C₄	semidirect
27. C₃₁×M₃₂	C₂×C₄	semidirect
28. C₂×C₁₆×C₃₁	C₂×C₄	semidirect
53. #33410	C₃₁	direct (N × H)
69. #2159	C₂×C₃₁	semidirect
70. #2158	C₂×C₃₁	semidirect
71. C₂×#366	C₂×C₃₁	semidirect
83. #504	C₄×C₃₁	semidirect
106. M₃₂	C₂×C₄×C₃₁	semidirect
107. C₂×C₁₆	C₂×C₄×C₃₁	semidirect
114. C₃₁	#33410	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×#504	2
C₃₁×#2913	2
C₃₁×#2159	2
C₃₁×#2158	2
C₂×C₃₁×#366	2
#33410	31

Subgroups — C31×#33410 #21490169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33410 #21490169