Subgroups — #31778769

Order 17536.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₂₇₄	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#14555485	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₄×Dic₂₇₄	C₄	1	1	yes	characteristic
5	C₂×#14555485	—	2	1	no	—
6	C₄⋊#11764040	C₄	1	1	yes	characteristic
7	#17984017	—	2	1	no	—
8	C₈×Dic₁₃₇	—	2	1	no	—
9	C₂×C₁₆×C₁₃₇	—	2	1	no	—
10	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₇₄	C₈	1	1	yes	characteristic
12	C₂×#11764040	—	2	1	no	—
13	C₄⋊Dic₁₃₇	C₈	1	1	yes	characteristic
14	#14555489	—	2	1	no	—
15	C₄×Dic₁₃₇	—	2	1	no	—
16	#14555485	—	4	1	no	—
17	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
18	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
19	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₁₃₇	—	2	2	no	—
21	Dic₂₇₄	—	2	1	no	—
22	Dic₂₇₄	C₁₆	1	2	yes	complemented
23	C₂×Dic₁₃₇	—	2	1	no	—
24	C₂×Dic₁₃₇	—	2	1	no	—
25	#11764040	—	4	1	no	—
26	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₁₃₇	M₁₆	1	1	yes	characteristic
29	C₈×C₁₃₇	—	2	1	no	—
30	C₈×C₁₃₇	SD₁₆	1	1	yes	characteristic
31	C₈×C₁₃₇	Q₁₆	1	1	yes	characteristic
32	C₂²×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
33	C₄×C₁₃₇	—	2	1	no	—
34	C₄×C₁₃₇	#103	1	1	yes	characteristic
35	C₄×C₁₃₇	#104	1	1	yes	characteristic
36	C₄×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
37	C₄×C₁₃₇	M₃₂	1	1	yes	characteristic
38	Dic₁₃₇	—	2	2	no	—
39	Dic₁₃₇	—	4	1	no	—
40	C₂×C₁₃₇	Q₈⋊C₈	1	1	yes	characteristic
41	C₂×C₁₃₇	C₂²⋊C₁₆	1	1	yes	characteristic
42	C₂×C₁₃₇	#350	1	1	yes	characteristic
43	C₁₃₇	Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
44	Q₈⋊C₁₆	—	137	1	no	maximal, Sylow
45	C₈×Q₈	—	137	1	no	—
46	C₄⋊C₁₆	—	137	1	no	—
47	C₄×C₁₆	D₁₃₇	1	1	yes	characteristic
48	C₄×Q₈	—	137	1	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	274	1	no	—
51	C₄⋊C₈	—	274	1	no	—
52	C₄⋊C₈	—	137	1	no	—
53	C₄×C₈	—	274	1	no	—
54	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
55	C₂×Q₈	—	137	1	no	—
56	C₂×C₈	—	274	1	no	—
57	C₂×C₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
58	C₂×C₈	D₅₄₈	1	1	yes	characteristic
59	C₄⋊C₄	—	274	1	no	—
60	C₄⋊C₄	—	137	1	no	—
61	C₄²	—	274	1	no	—
62	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
63	C₁₆	—	2	2	no	—
64	C₁₆	—	548	1	no	—
65	Q₈	—	274	1	no	—
66	Q₈	—	137	2	no	—
67	C₂×C₄	—	274	1	no	—
68	C₂×C₄	—	274	1	no	—
69	C₂×C₄	#14555487	1	1	yes	characteristic
70	C₂×C₄	C₈×D₁₃₇	1	1	yes	characteristic
71	C₂×C₄	#14555493	1	1	yes	characteristic
72	C₈	—	2	1	no	—
73	C₈	—	548	1	no	—
74	C₈	#14555499	1	1	yes	characteristic
75	C₈	Dic₅₄₈	1	1	yes	characteristic
76	C₂²	#17984000	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	274	2	no	—
79	C₄	—	548	1	no	—
80	C₄	#17984013	1	1	yes	characteristic
81	C₄	#17984016	1	1	yes	characteristic
82	C₄	C₁₆×D₁₃₇	1	1	yes	characteristic
83	C₄	#17984027	1	1	yes	characteristic
84	C₂	#22577135	1	1	yes	characteristic
85	C₂	#22577182	1	1	yes	characteristic
86	C₂	#22577186	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₅₄₈
commutator	C₄×C₁₃₇	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₁₃₇
Fitting	C₄×C₁₆×C₁₃₇	C₂
socle, prime radical	C₂²×C₁₃₇	C₂²⋊C₈
cosocle	C₄×C₈×C₁₃₇	C₂²
hypercenter	C₄×C₁₆	D₁₃₇
hypocenter	C₁₃₇	Q₈⋊C₁₆
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
22. Dic₂₇₄	C₁₆	semidirect
43. C₁₃₇	Q₈⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×Dic₂₇₄	2
C₄⋊#14555485	2
C₄×C₁₆×C₁₃₇	2
Q₈⋊C₁₆	137