Subgroups — C₃₁×#3486 #16843419

Order 3968.

1492 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Q₈×C₃₁	C₂	1	8	yes	maximal, complemented
4	C₃₁×#548	C₂	1	8	yes	maximal, complemented
5	C₃₁×#536	C₂	1	6	yes	maximal, complemented
6	C₃₁×#525	C₂	1	6	yes	maximal, complemented
7	C₃₁×#519	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×2₋¹⁺⁴	—	2	8	no	—
9	C₂×#41×C₃₁	C₂²	1	6	yes	—
10	C₂×#41×C₃₁	C₂²	1	4	yes	complemented
11	C₂²×Q₈×C₃₁	—	2	8	no	—
12	C₂²×Q₈×C₃₁	C₂²	1	3	yes	—
13	C₂²×Q₈×C₃₁	C₂²	1	8	yes	—
14	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×Q₈×C₃₁	C₂²	1	4	yes	complemented
16	C₂³×C₄×C₃₁	C₂²	1	3	yes	—
17	C₃₁×C₄⋊Q₈	C₂²	1	12	yes	complemented
18	C₃₁×#124	C₂²	1	12	yes	complemented
19	C₃₁×#123	C₂²	1	12	yes	—
20	C₃₁×C₂²⋊Q₈	C₂²	1	12	yes	—
21	C₃₁×C₂²⋊Q₈	C₂²	1	24	yes	complemented
22	C₃₁×#121	C₂²	1	12	yes	—
23	C₃₁×#120	C₂²	1	4	yes	—
24	C₄×Q₈×C₃₁	C₂²	1	8	yes	complemented
25	C₄×D₄×C₃₁	C₂²	1	24	yes	—
26	C₃₁×#117	C₂²	1	6	yes	—
27	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
28	#41×C₃₁	—	2	16	no	—
29	#41×C₃₁	—	2	24	no	—
30	C₂×Q₈×C₃₁	—	2	32	no	—
31	C₂×Q₈×C₃₁	—	2	16	no	—
32	C₂×Q₈×C₃₁	—	2	12	no	—
33	C₂×Q₈×C₃₁	D₄	1	8	yes	complemented
34	C₂×Q₈×C₃₁	C₂³	1	6	yes	—
35	C₂×Q₈×C₃₁	C₂³	1	12	yes	—
36	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
37	C₂×D₄×C₃₁	C₂³	1	8	yes	—
38	C₂×D₄×C₃₁	C₂³	1	4	yes	—
39	C₂×D₄×C₃₁	C₂³	1	6	yes	—
40	C₂²×C₄×C₃₁	—	2	12	no	—
41	C₂²×C₄×C₃₁	C₂³	1	12	yes	—
42	C₂²×C₄×C₃₁	C₂³	1	12	yes	—
43	C₂²×C₄×C₃₁	C₂³	1	3	yes	—
44	C₂²×C₄×C₃₁	C₂³	1	3	yes	—
45	C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
46	C₄⋊C₄×C₃₁	C₂³	1	24	yes	—
47	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
48	C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
49	C₂²⋊C₄×C₃₁	C₂³	1	24	yes	—
50	C₄²×C₃₁	C₂³	1	12	yes	—
51	C₂³×C₃₁	—	2	2	no	—
52	C₂³×C₃₁	—	2	4	no	—
53	C₂³×C₃₁	C₂⁴	1	4	yes	—
54	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
55	C₂³×C₃₁	C₂⁴	1	2	yes	—
56	Q₈×C₃₁	—	2	24	no	—
57	Q₈×C₃₁	—	2	12	no	—
58	Q₈×C₃₁	C₂×D₄	1	16	yes	—
59	D₄×C₃₁	—	2	8	no	—
60	D₄×C₃₁	—	2	16	no	—
61	D₄×C₃₁	—	2	12	no	—
62	C₂×C₄×C₃₁	—	2	6	no	—
63	C₂×C₄×C₃₁	—	2	24	no	—
64	C₂×C₄×C₃₁	—	2	24	no	—
65	C₂×C₄×C₃₁	C₂×D₄	1	12	yes	—
66	C₂×C₄×C₃₁	C₂⁴	1	8	yes	—
67	C₂×C₄×C₃₁	C₂⁴	1	6	yes	—
68	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
69	C₂×C₄×C₃₁	C₂⁴	1	6	yes	—
70	#3486	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂²×C₃₁	—	2	8	no	—
72	C₂²×C₃₁	—	2	2	no	—
73	C₂²×C₃₁	—	2	2	no	—
74	C₂²×C₃₁	—	2	2	no	—
75	C₂²×C₃₁	—	2	4	no	—
76	C₂²×C₃₁	—	2	8	no	—
77	C₂²×C₃₁	C₂²×D₄	1	2	yes	—
78	C₂²×C₃₁	2₋¹⁺⁴	1	4	yes	complemented
79	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
80	C₄×C₃₁	—	2	4	no	—
81	C₄×C₃₁	—	2	6	no	—
82	C₄×C₃₁	—	2	8	no	—
83	C₄×C₃₁	C₂²×D₄	1	12	yes	—
84	C₂×2₋¹⁺⁴	C₂×C₃₁	1	1	yes	characteristic, complemented
85	C₂³×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
86	D₄×Q₈	C₂×C₃₁	1	8	yes	complemented
87	#548	C₂×C₃₁	1	8	yes	complemented
88	#536	C₂×C₃₁	1	6	yes	complemented
89	#525	C₂×C₃₁	1	6	yes	complemented
90	#519	C₂×C₃₁	1	1	yes	characteristic, complemented
91	C₂×C₃₁	—	2	2	no	—
92	C₂×C₃₁	—	2	4	no	—
93	C₂×C₃₁	—	4	4	no	—
94	C₂×C₃₁	C₂³×D₄	1	1	yes	characteristic
95	C₂×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
96	2₋¹⁺⁴	—	2	8	no	—
97	C₂×#41	C₂²×C₃₁	1	6	yes	—
98	C₂×#41	C₂²×C₃₁	1	4	yes	complemented
99	C₂²×Q₈	—	2	8	no	—
100	C₂²×Q₈	C₂²×C₃₁	1	3	yes	—
101	C₂²×Q₈	C₂²×C₃₁	1	8	yes	—
102	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
103	C₂²×Q₈	C₂²×C₃₁	1	4	yes	complemented
104	C₂³×C₄	C₂²×C₃₁	1	3	yes	—
105	C₄⋊Q₈	C₂²×C₃₁	1	12	yes	complemented
106	#124	C₂²×C₃₁	1	12	yes	complemented
107	#123	C₂²×C₃₁	1	12	yes	—
108	C₂²⋊Q₈	C₂²×C₃₁	1	12	yes	—
109	C₂²⋊Q₈	C₂²×C₃₁	1	24	yes	complemented
110	#121	C₂²×C₃₁	1	12	yes	—
111	#120	C₂²×C₃₁	1	4	yes	—
112	C₄×Q₈	C₂²×C₃₁	1	8	yes	complemented
113	C₄×D₄	C₂²×C₃₁	1	24	yes	—
114	#117	C₂²×C₃₁	1	6	yes	—
115	C₃₁	#3486	1	1	yes	characteristic, direct product factor, Sylow
116	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
117	#41	—	2	16	no	—
118	#41	—	2	24	no	—
119	C₂×Q₈	—	2	32	no	—
120	C₂×Q₈	—	2	16	no	—
121	C₂×Q₈	—	2	12	no	—
122	C₂×Q₈	D₄×C₃₁	1	8	yes	complemented
123	C₂×Q₈	C₂³×C₃₁	1	6	yes	—
124	C₂×Q₈	C₂³×C₃₁	1	12	yes	—
125	C₂×Q₈	C₂³×C₃₁	1	4	yes	—
126	C₂×D₄	C₂³×C₃₁	1	8	yes	—
127	C₂×D₄	C₂³×C₃₁	1	4	yes	—
128	C₂×D₄	C₂³×C₃₁	1	6	yes	—
129	C₂²×C₄	—	2	12	no	—
130	C₂²×C₄	C₂³×C₃₁	1	12	yes	—
131	C₂²×C₄	C₂³×C₃₁	1	12	yes	—
132	C₂²×C₄	C₂³×C₃₁	1	3	yes	—
133	C₂²×C₄	C₂³×C₃₁	1	3	yes	—
134	C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
135	C₄⋊C₄	C₂³×C₃₁	1	24	yes	—
136	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
137	C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
138	C₂²⋊C₄	C₂³×C₃₁	1	24	yes	—
139	C₄²	C₂³×C₃₁	1	12	yes	—
140	C₂³	—	2	2	no	—
141	C₂³	—	2	4	no	—
142	C₂³	C₂⁴×C₃₁	1	4	yes	—
143	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
144	C₂³	C₂⁴×C₃₁	1	2	yes	—
145	Q₈	—	2	24	no	—
146	Q₈	—	2	12	no	—
147	Q₈	C₂×D₄×C₃₁	1	16	yes	—
148	D₄	—	2	8	no	—
149	D₄	—	2	16	no	—
150	D₄	—	2	12	no	—
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	—	2	24	no	—
153	C₂×C₄	—	2	24	no	—
154	C₂×C₄	C₂×D₄×C₃₁	1	12	yes	—
155	C₂×C₄	C₂⁴×C₃₁	1	8	yes	—
156	C₂×C₄	C₂⁴×C₃₁	1	6	yes	—
157	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
158	C₂×C₄	C₂⁴×C₃₁	1	6	yes	—
159	C₂²	—	2	8	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	4	no	—
164	C₂²	—	2	8	no	—
165	C₂²	C₂²×D₄×C₃₁	1	2	yes	—
166	C₂²	C₃₁×2₋¹⁺⁴	1	4	yes	complemented
167	C₂²	C₂⁵×C₃₁	1	1	yes	characteristic
168	C₄	—	2	4	no	—
169	C₄	—	2	6	no	—
170	C₄	—	2	8	no	—
171	C₄	C₂²×D₄×C₃₁	1	12	yes	—
172	C₂	—	2	2	no	—
173	C₂	—	2	4	no	—
174	C₂	—	4	4	no	—
175	C₂	C₂³×D₄×C₃₁	1	1	yes	characteristic
176	C₂	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×2₋¹⁺⁴	C₂	semidirect
2. C₂³×Q₈×C₃₁	C₂	semidirect
3. D₄×Q₈×C₃₁	C₂	semidirect
4. C₃₁×#548	C₂	semidirect
5. C₃₁×#536	C₂	semidirect
6. C₃₁×#525	C₂	semidirect
7. C₃₁×#519	C₂	semidirect
10. C₂×#41×C₃₁	C₂²	semidirect
15. C₂²×Q₈×C₃₁	C₂²	semidirect
17. C₃₁×C₄⋊Q₈	C₂²	semidirect
18. C₃₁×#124	C₂²	semidirect
21. C₃₁×C₂²⋊Q₈	C₂²	semidirect
24. C₄×Q₈×C₃₁	C₂²	semidirect
33. C₂×Q₈×C₃₁	D₄	semidirect
70. #3486	C₃₁	direct (N × H)
78. C₂²×C₃₁	2₋¹⁺⁴	semidirect
84. C₂×2₋¹⁺⁴	C₂×C₃₁	semidirect
85. C₂³×Q₈	C₂×C₃₁	semidirect
86. D₄×Q₈	C₂×C₃₁	semidirect
87. #548	C₂×C₃₁	semidirect
88. #536	C₂×C₃₁	semidirect
89. #525	C₂×C₃₁	semidirect
90. #519	C₂×C₃₁	semidirect
98. C₂×#41	C₂²×C₃₁	semidirect
103. C₂²×Q₈	C₂²×C₃₁	semidirect
105. C₄⋊Q₈	C₂²×C₃₁	semidirect
106. #124	C₂²×C₃₁	semidirect
109. C₂²⋊Q₈	C₂²×C₃₁	semidirect
112. C₄×Q₈	C₂²×C₃₁	semidirect
115. C₃₁	#3486	direct (N × H)
122. C₂×Q₈	D₄×C₃₁	semidirect
166. C₂²	C₃₁×2₋¹⁺⁴	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×2₋¹⁺⁴	2
C₂³×Q₈×C₃₁	2
D₄×Q₈×C₃₁	2
C₃₁×#548	2
C₃₁×#536	2
C₃₁×#525	2
C₃₁×#519	2
#3486	31

Subgroups — C31×#3486 #16843419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3486 #16843419