Subgroups — C₃₁×#3176 #16843109

Order 3968.

386 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#540	C₂	1	2	yes	maximal, complemented
2	C₃₁×#499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#490	C₂	1	2	yes	maximal, complemented
5	C₃₁×#469	C₂	1	2	yes	maximal, complemented
6	C₃₁×#466	C₂	1	2	yes	maximal, complemented
7	C₃₁×#442	C₂	1	2	yes	maximal, complemented
8	C₄×D₈×C₃₁	C₂	1	2	yes	maximal, complemented
9	C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₈×C₃₁	C₂²	1	2	yes	—
11	C₂×D₈×C₃₁	C₂²	1	2	yes	—
12	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#127	C₂²	1	2	yes	—
15	C₃₁×#125	C₂²	1	2	yes	—
16	C₃₁×#123	—	2	4	no	—
17	C₃₁×#121	—	2	4	no	—
18	C₃₁×#121	C₂²	1	4	yes	—
19	C₄×D₄×C₃₁	C₂²	1	4	yes	—
20	C₃₁×#117	C₂²	1	1	yes	characteristic
21	C₃₁×#107	C₂²	1	2	yes	—
22	C₃₁×#106	C₂²	1	2	yes	—
23	C₃₁×#102	C₂²	1	4	yes	—
24	C₃₁×#102	C₂²	1	4	yes	—
25	C₃₁×#97	C₂²	1	1	yes	characteristic
26	C₃₁×#97	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×D₄×C₃₁	—	2	4	no	—
30	C₂×D₄×C₃₁	—	2	4	no	—
31	C₂×D₄×C₃₁	C₂³	1	4	yes	—
32	C₂²×C₄×C₃₁	—	2	4	no	—
33	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
34	D₈×C₃₁	—	2	4	no	—
35	D₈×C₃₁	—	2	4	no	—
36	M₁₆×C₃₁	—	2	2	no	—
37	C₂×C₈×C₃₁	—	2	2	no	—
38	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₃₁	—	2	4	no	—
43	C₄⋊C₄×C₃₁	D₄	1	2	yes	—
44	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
45	C₂²⋊C₄×C₃₁	—	2	4	no	—
46	C₂²⋊C₄×C₃₁	—	2	4	no	—
47	C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
48	C₄²×C₃₁	C₂³	1	1	yes	characteristic
49	C₄²×C₃₁	C₂³	1	1	yes	characteristic
50	C₂³×C₃₁	—	2	4	no	—
51	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
52	D₄×C₃₁	—	2	4	no	—
53	D₄×C₃₁	—	2	4	no	—
54	D₄×C₃₁	—	8	4	no	—
55	D₄×C₃₁	—	8	4	no	—
56	C₂×C₄×C₃₁	—	2	1	no	—
57	C₂×C₄×C₃₁	—	2	1	no	—
58	C₂×C₄×C₃₁	—	2	4	no	—
59	C₂×C₄×C₃₁	—	8	4	no	—
60	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
64	C₈×C₃₁	—	2	1	no	—
65	C₈×C₃₁	—	2	1	no	—
66	C₈×C₃₁	#41	1	2	yes	—
67	C₈×C₃₁	#41	1	2	yes	—
68	#3176	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂²×C₃₁	—	2	1	no	—
70	C₂²×C₃₁	—	2	2	no	—
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	1	no	—
76	C₄×C₃₁	—	2	2	no	—
77	C₄×C₃₁	—	2	2	no	—
78	C₄×C₃₁	—	4	4	no	—
79	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
80	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
81	#540	C₂×C₃₁	1	2	yes	complemented
82	#499	C₂×C₃₁	1	1	yes	characteristic, complemented
83	#493	C₂×C₃₁	1	1	yes	characteristic, complemented
84	#490	C₂×C₃₁	1	2	yes	complemented
85	#469	C₂×C₃₁	1	2	yes	complemented
86	#466	C₂×C₃₁	1	2	yes	complemented
87	#442	C₂×C₃₁	1	2	yes	complemented
88	C₄×D₈	C₂×C₃₁	1	2	yes	complemented
89	#405	C₂×C₃₁	1	1	yes	characteristic, complemented
90	C₂×C₃₁	—	4	1	no	—
91	C₂×C₃₁	—	8	4	no	—
92	C₂×C₃₁	#532	1	1	yes	characteristic
93	C₂×C₃₁	#576	1	2	yes	—
94	C₂×D₈	C₂²×C₃₁	1	2	yes	—
95	C₂×D₈	C₂²×C₃₁	1	2	yes	—
96	C₂×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
97	C₂²×C₈	C₂²×C₃₁	1	1	yes	characteristic
98	#127	C₂²×C₃₁	1	2	yes	—
99	#125	C₂²×C₃₁	1	2	yes	—
100	#123	—	2	4	no	—
101	#121	—	2	4	no	—
102	#121	C₂²×C₃₁	1	4	yes	—
103	C₄×D₄	C₂²×C₃₁	1	4	yes	—
104	#117	C₂²×C₃₁	1	1	yes	characteristic
105	#107	C₂²×C₃₁	1	2	yes	—
106	#106	C₂²×C₃₁	1	2	yes	—
107	#102	C₂²×C₃₁	1	4	yes	—
108	#102	C₂²×C₃₁	1	4	yes	—
109	#97	C₂²×C₃₁	1	1	yes	characteristic
110	#97	C₂²×C₃₁	1	1	yes	characteristic
111	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
112	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
113	C₃₁	#3176	1	1	yes	characteristic, direct product factor, Sylow
114	C₂×D₄	—	2	4	no	—
115	C₂×D₄	—	2	4	no	—
116	C₂×D₄	C₂³×C₃₁	1	4	yes	—
117	C₂²×C₄	—	2	4	no	—
118	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
119	D₈	—	2	4	no	—
120	D₈	—	2	4	no	—
121	M₁₆	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
124	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
125	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
126	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
127	C₄⋊C₄	—	2	4	no	—
128	C₄⋊C₄	D₄×C₃₁	1	2	yes	—
129	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
130	C₂²⋊C₄	—	2	4	no	—
131	C₂²⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
133	C₄²	C₂³×C₃₁	1	1	yes	characteristic
134	C₄²	C₂³×C₃₁	1	1	yes	characteristic
135	C₂³	—	2	4	no	—
136	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
137	D₄	—	2	4	no	—
138	D₄	—	2	4	no	—
139	D₄	—	8	4	no	—
140	D₄	—	8	4	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	8	4	no	—
145	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
146	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
147	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
148	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	#41×C₃₁	1	2	yes	—
152	C₈	#41×C₃₁	1	2	yes	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	2	no	—
155	C₂²	—	4	4	no	—
156	C₂²	—	4	4	no	—
157	C₂²	—	4	4	no	—
158	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	4	4	no	—
163	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
164	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
165	C₂	—	4	1	no	—
166	C₂	—	8	4	no	—
167	C₂	C₃₁×#532	1	1	yes	characteristic
168	C₂	C₃₁×#576	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#540	C₂	semidirect
2. C₃₁×#499	C₂	semidirect
3. C₃₁×#493	C₂	semidirect
4. C₃₁×#490	C₂	semidirect
5. C₃₁×#469	C₂	semidirect
6. C₃₁×#466	C₂	semidirect
7. C₃₁×#442	C₂	semidirect
8. C₄×D₈×C₃₁	C₂	semidirect
9. C₃₁×#405	C₂	semidirect
68. #3176	C₃₁	direct (N × H)
81. #540	C₂×C₃₁	semidirect
82. #499	C₂×C₃₁	semidirect
83. #493	C₂×C₃₁	semidirect
84. #490	C₂×C₃₁	semidirect
85. #469	C₂×C₃₁	semidirect
86. #466	C₂×C₃₁	semidirect
87. #442	C₂×C₃₁	semidirect
88. C₄×D₈	C₂×C₃₁	semidirect
89. #405	C₂×C₃₁	semidirect
113. C₃₁	#3176	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×#540	2
C₃₁×#499	2
C₃₁×#493	2
C₃₁×#490	2
C₃₁×#469	2
C₃₁×#466	2
C₃₁×#442	2
C₄×D₈×C₃₁	2
C₃₁×#405	2
#3176	31

Subgroups — C31×#3176 #16843109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3176 #16843109