Subgroups — #31797682

Order 17536.

396 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11764047	C₂	1	8	yes	maximal, complemented
2	#22578419	C₂	1	6	yes	maximal
3	C₂×Q₈×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11764047	C₂²	1	12	yes	—
5	C₂²⋊#11764047	C₂²	1	4	yes	—
6	C₄×#11764047	C₂²	1	12	yes	—
7	Q₈×Dic₁₃₇	C₄	1	4	yes	—
8	Q₈×Dic₁₃₇	C₂²	1	4	yes	—
9	C₂×C₄⋊Dic₁₃₇	C₄	1	3	yes	—
10	C₂×C₄×Dic₁₃₇	C₂²	1	3	yes	—
11	C₂²×Q₈×C₁₃₇	C₄	1	1	yes	characteristic
12	C₂×#11764047	C₂³	1	8	yes	—
13	C₂²×Dic₁₃₇	C₂×C₄	1	3	yes	—
14	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
15	C₄⋊Dic₁₃₇	C₂×C₄	1	12	yes	—
16	C₄×Dic₁₃₇	C₂×C₄	1	6	yes	—
17	C₄×Dic₁₃₇	C₂³	1	6	yes	—
18	C₂×Q₈×C₁₃₇	C₈	1	8	yes	complemented
19	C₂×Q₈×C₁₃₇	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₁₃₇	C₂×C₄	1	3	yes	—
21	#11764047	—	2	8	no	—
22	C₂×Dic₁₃₇	—	2	2	no	—
23	C₂×Dic₁₃₇	—	4	3	no	—
24	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
25	C₂×Dic₁₃₇	C₂²×C₄	1	6	yes	—
26	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
27	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
28	Q₈×C₁₃₇	C₂×C₈	1	16	yes	—
29	C₂×C₄×C₁₃₇	C₂×C₈	1	12	yes	—
30	C₂×C₄×C₁₃₇	C₂²×C₄	1	6	yes	—
31	C₂²×C₁₃₇	—	2	1	no	—
32	C₂²×C₁₃₇	—	2	1	no	—
33	C₂²×C₁₃₇	C₂²×C₈	1	2	yes	—
34	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
35	C₄×C₁₃₇	C₂²×C₈	1	12	yes	—
36	Dic₁₃₇	—	2	1	no	—
37	Dic₁₃₇	—	2	6	no	—
38	Dic₁₃₇	2₋¹⁺⁴	1	2	yes	—
39	C₂×C₁₃₇	—	2	2	no	—
40	C₂×C₁₃₇	#519	1	1	yes	characteristic
41	C₂×C₁₃₇	C₂³×C₈	1	1	yes	characteristic
42	C₂×C₁₃₇	#568	1	1	yes	characteristic
43	C₁₃₇	#2982	1	1	yes	characteristic, complemented, Sylow
44	#2982	—	137	1	no	maximal, Sylow
45	C₂×C₄×Q₈	—	137	1	no	—
46	C₈×Q₈	—	137	8	no	—
47	#431	—	137	6	no	—
48	C₂²×Q₈	#10587739	1	1	yes	characteristic
49	C₄×Q₈	—	137	4	no	—
50	C₄×Q₈	—	137	4	no	—
51	C₂×C₄⋊C₄	—	137	3	no	—
52	C₂×C₄²	—	137	3	no	—
53	C₄⋊C₈	—	137	12	no	—
54	C₂²⋊C₈	—	137	4	no	—
55	C₄×C₈	—	137	12	no	—
56	C₂×Q₈	#11764047	1	8	yes	complemented
57	C₂×Q₈	C₂×#10587739	1	4	yes	—
58	C₂²×C₄	—	137	1	no	—
59	C₂²×C₄	—	137	3	no	—
60	C₂²×C₄	C₂×#10587739	1	3	yes	—
61	C₂×C₈	—	137	8	no	—
62	C₄⋊C₄	—	137	12	no	—
63	C₄²	—	137	6	no	—
64	C₄²	—	137	6	no	—
65	C₂³	C₂²×#10587739	1	1	yes	characteristic
66	Q₈	C₂×#11764047	1	16	yes	—
67	C₂×C₄	—	274	2	no	—
68	C₂×C₄	—	548	3	no	—
69	C₂×C₄	—	137	1	no	—
70	C₂×C₄	—	137	6	no	—
71	C₂×C₄	—	137	1	no	—
72	C₂×C₄	C₂×#11764047	1	12	yes	—
73	C₂×C₄	C₂²×#10587739	1	6	yes	—
74	C₈	—	274	8	no	—
75	C₂²	—	2	1	no	—
76	C₂²	—	2	1	no	—
77	C₂²	C₂²×#11764047	1	2	yes	—
78	C₂²	C₂³×#10587739	1	1	yes	characteristic
79	C₄	—	274	1	no	—
80	C₄	—	274	6	no	—
81	C₄	—	137	2	no	—
82	C₄	C₂²×#11764047	1	12	yes	—
83	C₂	—	2	2	no	—
84	C₂	#22578471	1	1	yes	characteristic
85	C₂	C₂³×#11764047	1	1	yes	characteristic
86	C₂	#22578483	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10587739
commutator	C₂×C₁₃₇	C₂³×C₈
Fitting	C₂²×Q₈×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂³×C₄
prime radical	C₂³×C₁₃₇	C₂²×C₄
cosocle	C₂×Dic₁₃₇	C₂⁴
hypercenter	C₂²×Q₈	#10587739
hypocenter	C₁₃₇	#2982
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×#11764047	C₂	semidirect
18. C₂×Q₈×C₁₃₇	C₈	semidirect
43. C₁₃₇	#2982	semidirect
56. C₂×Q₈	#11764047	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11764047	2
#22578419	2
C₂×Q₈×Dic₁₃₇	2
#2982	137