Subgroups — #16858914

Order 3968.

1540 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521532	C₂	1	8	yes	maximal, complemented
2	#14521527	C₂	1	2	yes	maximal, complemented
3	#14521524	C₂	1	6	yes	maximal, complemented
4	#41×Dic₃₁	C₂	1	8	yes	maximal, complemented
5	C₂×#11758503	C₂	1	6	yes	maximal, complemented
6	C₂²×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
7	C₂×#41×C₃₁	C₄	1	16	yes	complemented
8	C₂×#41×C₃₁	C₂²	1	4	yes	—
9	C₂×#41×C₃₁	C₂²	1	4	yes	—
10	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₃₁	C₂²	1	3	yes	—
12	C₂³×C₄×C₃₁	C₂²	1	3	yes	—
13	C₂×C₂²⋊Dic₃₁	C₂²	1	12	yes	—
14	Q₈×Dic₃₁	C₂²	1	16	yes	complemented
15	D₄×Dic₃₁	C₂²	1	48	yes	complemented
16	#11758503	C₂²	1	4	yes	—
17	#11758503	C₂²	1	12	yes	—
18	#11758503	C₂²	1	12	yes	complemented
19	#11758503	C₂²	1	12	yes	complemented
20	C₂×C₄⋊Dic₃₁	C₂²	1	12	yes	complemented
21	C₂×C₄×Dic₃₁	C₂²	1	12	yes	complemented
22	C₂⁴×C₃₁	C₂³	1	3	yes	—
23	#41×C₃₁	C₂×C₄	1	32	yes	complemented
24	#41×C₃₁	C₂×C₄	1	32	yes	complemented
25	C₂×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
26	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
27	C₂×D₄×C₃₁	C₂×C₄	1	24	yes	complemented
28	C₂×D₄×C₃₁	C₂³	1	12	yes	—
29	C₂²×C₄×C₃₁	C₂×C₄	1	24	yes	complemented
30	C₂²×C₄×C₃₁	C₂³	1	6	yes	—
31	C₂²×C₄×C₃₁	C₂³	1	6	yes	—
32	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₃₁	C₂³	1	3	yes	—
34	C₂²×Dic₃₁	C₂³	1	24	yes	—
35	C₂²⋊Dic₃₁	C₂³	1	8	yes	—
36	C₂²⋊Dic₃₁	C₂³	1	24	yes	—
37	C₄⋊Dic₃₁	C₂³	1	8	yes	—
38	C₄⋊Dic₃₁	C₂³	1	24	yes	—
39	C₄×Dic₃₁	C₂³	1	8	yes	—
40	C₄×Dic₃₁	C₂³	1	24	yes	—
41	C₂³×C₃₁	—	4	6	no	—
42	C₂³×C₃₁	C₂²×C₄	1	12	yes	—
43	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
44	C₂³×C₃₁	C₂⁴	1	6	yes	—
45	Q₈×C₃₁	C₂²×C₄	1	16	yes	—
46	D₄×C₃₁	C₂²×C₄	1	48	yes	—
47	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₃₁	C₂²×C₄	1	12	yes	—
49	C₂×C₄×C₃₁	C₂²×C₄	1	24	yes	—
50	C₂×C₄×C₃₁	C₂²×C₄	1	24	yes	—
51	C₂×C₄×C₃₁	C₂⁴	1	6	yes	—
52	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
53	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₃₁	—	4	24	no	—
55	C₂×Dic₃₁	C₂⁴	1	16	yes	—
56	#3428	—	31	1	no	maximal, Sylow
57	C₂²×C₃₁	—	2	12	no	—
58	C₂²×C₃₁	—	2	2	no	—
59	C₂²×C₃₁	—	4	12	no	—
60	C₂²×C₃₁	C₂³×C₄	1	2	yes	—
61	C₂²×C₃₁	C₂³×C₄	1	12	yes	—
62	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
63	C₄×C₃₁	C₂³×C₄	1	2	yes	—
64	C₄×C₃₁	C₂³×C₄	1	2	yes	—
65	C₄×C₃₁	C₂³×C₄	1	12	yes	—
66	Dic₃₁	—	2	16	no	—
67	C₂²×#41	D₃₁	1	1	yes	characteristic
68	#520	—	31	8	no	—
69	#519	—	31	2	no	—
70	#518	—	31	6	no	—
71	C₄×#41	—	31	8	no	—
72	C₂×#117	—	31	6	no	—
73	C₂×C₃₁	—	2	2	no	—
74	C₂×C₃₁	—	2	12	no	—
75	C₂×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
76	C₂×C₃₁	#585	1	2	yes	—
77	C₂×#41	Dic₃₁	1	16	yes	complemented
78	C₂×#41	C₂×D₃₁	1	4	yes	—
79	C₂×#41	C₂×D₃₁	1	4	yes	—
80	C₂²×Q₈	C₂×D₃₁	1	1	yes	characteristic
81	C₂²×D₄	C₂×D₃₁	1	3	yes	—
82	C₂³×C₄	C₂×D₃₁	1	3	yes	—
83	C₄×Q₈	—	31	16	no	—
84	C₄×D₄	—	31	48	no	—
85	#117	—	31	4	no	—
86	#117	—	31	12	no	—
87	#117	—	31	12	no	—
88	#117	—	31	12	no	—
89	C₂×C₄⋊C₄	—	31	12	no	—
90	C₂×C₂²⋊C₄	—	31	12	no	—
91	C₂×C₄²	—	31	12	no	—
92	C₃₁	#3428	1	1	yes	characteristic, complemented, Sylow
93	C₂⁴	C₂²×D₃₁	1	3	yes	—
94	#41	C₂×Dic₃₁	1	32	yes	complemented
95	#41	C₂×Dic₃₁	1	32	yes	complemented
96	C₂×Q₈	C₂×Dic₃₁	1	8	yes	complemented
97	C₂×Q₈	C₂²×D₃₁	1	4	yes	—
98	C₂×D₄	C₂×Dic₃₁	1	24	yes	complemented
99	C₂×D₄	C₂²×D₃₁	1	12	yes	—
100	C₂²×C₄	—	31	24	no	—
101	C₂²×C₄	C₂×Dic₃₁	1	24	yes	complemented
102	C₂²×C₄	C₂²×D₃₁	1	6	yes	—
103	C₂²×C₄	C₂²×D₃₁	1	3	yes	—
104	C₂²×C₄	C₂²×D₃₁	1	6	yes	—
105	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
106	C₄⋊C₄	—	31	8	no	—
107	C₄⋊C₄	—	31	24	no	—
108	C₂²⋊C₄	—	31	8	no	—
109	C₂²⋊C₄	—	31	24	no	—
110	C₄²	—	31	24	no	—
111	C₄²	—	31	8	no	—
112	C₂³	—	4	6	no	—
113	C₂³	C₂²×Dic₃₁	1	12	yes	—
114	C₂³	C₂³×D₃₁	1	1	yes	characteristic
115	C₂³	C₂³×D₃₁	1	6	yes	—
116	Q₈	C₂²×Dic₃₁	1	16	yes	—
117	D₄	C₂²×Dic₃₁	1	48	yes	—
118	C₂×C₄	—	31	16	no	—
119	C₂×C₄	—	124	24	no	—
120	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
121	C₂×C₄	C₂²×Dic₃₁	1	12	yes	—
122	C₂×C₄	C₂²×Dic₃₁	1	24	yes	—
123	C₂×C₄	C₂²×Dic₃₁	1	24	yes	—
124	C₂×C₄	C₂³×D₃₁	1	6	yes	—
125	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
126	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
127	C₂²	—	2	12	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	4	12	no	—
130	C₂²	C₂³×Dic₃₁	1	2	yes	—
131	C₂²	C₂³×Dic₃₁	1	12	yes	—
132	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
133	C₄	—	62	16	no	—
134	C₄	C₂³×Dic₃₁	1	2	yes	—
135	C₄	C₂³×Dic₃₁	1	2	yes	—
136	C₄	C₂³×Dic₃₁	1	12	yes	—
137	C₂	—	2	2	no	—
138	C₂	—	2	12	no	—
139	C₂	C₂⁴×Dic₃₁	1	1	yes	characteristic
140	C₂	#14521885	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₃₁
commutator	C₂×C₃₁	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₃₁
Fitting, prime radical	C₂²×#41×C₃₁	C₂
socle, cosocle	C₂²×C₃₁	C₂⁵
hypercenter	C₂²×#41	D₃₁
hypocenter	C₃₁	#3428
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. #14521532	C₂	semidirect
2. #14521527	C₂	semidirect
3. #14521524	C₂	semidirect
4. #41×Dic₃₁	C₂	semidirect
5. C₂×#11758503	C₂	semidirect
7. C₂×#41×C₃₁	C₄	semidirect
14. Q₈×Dic₃₁	C₂²	semidirect
15. D₄×Dic₃₁	C₂²	semidirect
18. #11758503	C₂²	semidirect
19. #11758503	C₂²	semidirect
20. C₂×C₄⋊Dic₃₁	C₂²	semidirect
21. C₂×C₄×Dic₃₁	C₂²	semidirect
23. #41×C₃₁	C₂×C₄	semidirect
24. #41×C₃₁	C₂×C₄	semidirect
25. C₂×Q₈×C₃₁	C₂×C₄	semidirect
27. C₂×D₄×C₃₁	C₂×C₄	semidirect
29. C₂²×C₄×C₃₁	C₂×C₄	semidirect
77. C₂×#41	Dic₃₁	semidirect
92. C₃₁	#3428	semidirect
94. #41	C₂×Dic₃₁	semidirect
95. #41	C₂×Dic₃₁	semidirect
96. C₂×Q₈	C₂×Dic₃₁	semidirect
98. C₂×D₄	C₂×Dic₃₁	semidirect
101. C₂²×C₄	C₂×Dic₃₁	semidirect

Maximal subgroups

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

subgroup	index
#14521532	2
#14521527	2
#14521524	2
#41×Dic₃₁	2
C₂×#11758503	2
C₂²×#41×C₃₁	2
#3428	31