Subgroups — #31797840

Order 17536.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578508	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×#11764049	C₂	1	1	yes	maximal, characteristic, complemented
3	#22578447	C₂	1	1	yes	maximal, characteristic
4	#17984185	—	2	1	no	—
5	C₄×#11764049	—	2	1	no	—
6	#17984176	—	2	1	no	—
7	#17984159	C₄	1	1	yes	characteristic
8	#17984153	—	2	1	no	—
9	C₈×#10587739	C₂²	1	1	yes	characteristic
10	#17984111	C₄	1	1	yes	characteristic
11	C₂×#11764049	—	2	2	no	—
12	#14555522	—	4	1	no	—
13	#14555520	—	2	1	no	—
14	#14555520	—	4	1	no	—
15	#14555518	—	4	1	no	—
16	#14555518	D₄	1	1	yes	characteristic
17	C₄×#10587739	D₄	1	1	yes	characteristic
18	#14555511	—	2	1	no	—
19	#14555497	—	2	1	no	—
20	D₁₀₉₆	C₈	1	1	yes	characteristic, complemented
21	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
22	Q₁₆×C₁₃₇	C₈	1	1	yes	characteristic, complemented
23	#11764049	—	2	4	no	—
24	C₂×#10587739	—	2	1	no	—
25	#11764047	—	4	1	no	—
26	#11764047	D₈	1	1	yes	characteristic
27	#11764047	SD₁₆	1	1	yes	characteristic
28	D₅₄₈	—	2	1	no	—
29	D₅₄₈	—	4	1	no	—
30	C₄×D₁₃₇	—	4	1	no	—
31	C₄×D₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
32	#11764040	M₁₆	1	1	yes	characteristic
33	Q₈×C₁₃₇	—	2	1	no	—
34	C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
35	C₂×D₁₃₇	—	4	1	no	—
36	C₂×D₁₃₇	#104	1	1	yes	characteristic
37	#10587739	—	2	2	no	—
38	C₄×C₁₃₇	—	4	1	no	—
39	C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
40	Dic₁₃₇	#102	1	1	yes	characteristic
41	C₂×C₁₃₇	D₄⋊C₈	1	1	yes	characteristic
42	D₁₃₇	—	2	1	no	—
43	D₁₃₇	—	8	1	no	—
44	C₁₃₇	#1335	1	1	yes	characteristic, complemented, Sylow
45	#1335	—	137	1	no	maximal, Sylow
46	#443	—	137	1	no	—
47	#364	—	137	1	no	—
48	C₈²	—	137	1	no	—
49	#135	—	137	1	no	—
50	#131	—	274	1	no	—
51	M₃₂	—	274	1	no	—
52	#108	—	137	1	no	—
53	#104	—	274	1	no	—
54	C₄×C₈	—	274	1	no	—
55	C₄×C₈	—	137	1	no	—
56	#41	—	274	1	no	—
57	Q₁₆	#11764049	1	1	yes	characteristic, complemented
58	SD₁₆	—	274	1	no	—
59	D₈	—	137	1	no	—
60	M₁₆	—	274	1	no	—
61	M₁₆	—	548	1	no	—
62	C₂×C₈	—	274	2	no	—
63	C₂×C₈	—	548	1	no	—
64	C₂×C₈	—	137	1	no	—
65	C₂×C₈	—	137	1	no	—
66	C₄²	—	137	1	no	—
67	C₁₆	—	548	1	no	—
68	Q₈	—	2	1	no	—
69	D₄	—	274	1	no	—
70	D₄	—	548	1	no	—
71	C₂×C₄	—	274	1	no	—
72	C₂×C₄	—	548	1	no	—
73	C₂×C₄	—	137	1	no	—
74	C₈	—	274	4	no	—
75	C₈	—	548	1	no	—
76	C₈	—	137	1	no	—
77	C₈	—	137	1	no	—
78	C₈	—	137	1	no	—
79	C₈	C₂×#11764049	1	1	yes	characteristic
80	C₂²	—	548	1	no	—
81	C₂²	—	137	1	no	—
82	C₄	—	4	1	no	—
83	C₄	—	274	2	no	—
84	C₄	—	137	1	no	—
85	C₄	C₂²⋊#11764049	1	1	yes	characteristic
86	C₂	—	274	1	no	—
87	C₂	—	1096	1	no	—
88	C₂	D₄⋊#11764049	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄⋊#11764049
commutator	C₈×C₁₃₇	C₂×C₈
Frattini	C₈	C₂×#11764049
Fitting	Q₁₆×C₁₃₇	C₈
socle	C₂×C₁₃₇	D₄⋊C₈
prime radical	#17984111	C₄
cosocle	C₈×#10587739	C₂²
hypercenter	Q₁₆	#11764049
hypocenter	C₁₃₇	#1335
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
1. #22578508	C₂	semidirect
2. C₈×#11764049	C₂	semidirect
20. D₁₀₉₆	C₈	semidirect
22. Q₁₆×C₁₃₇	C₈	semidirect
44. C₁₃₇	#1335	semidirect
57. Q₁₆	#11764049	semidirect

Maximal subgroups

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

subgroup	index
#22578508	2
C₈×#11764049	2
#22578447	2
#1335	137