Subgroups — #31797844

Order 17536.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14555522	C₂	1	1	yes	maximal, characteristic
2	C₄×#14555522	C₂	1	1	yes	maximal, characteristic
3	#22578452	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555522	—	2	1	no	—
5	C₂×#14555522	—	2	1	no	—
6	#17984155	—	2	1	no	—
7	C₄⋊#11764047	C₄	1	1	yes	characteristic
8	C₄×#11764047	—	2	1	no	—
9	C₄×#11764047	C₂²	1	1	yes	characteristic
10	#17984053	C₄	1	1	yes	characteristic
11	#14555522	—	2	2	no	—
12	#14555522	—	4	1	no	—
13	C₂×#11764047	—	2	1	no	—
14	C₂×#11764047	D₄	1	1	yes	characteristic
15	C₂×#11764047	D₄	1	1	yes	characteristic
16	C₂×Dic₂₇₄	C₈	1	1	yes	characteristic
17	#14555489	—	2	1	no	—
18	C₄×Dic₁₃₇	—	2	1	no	—
19	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
20	C₄⋊C₄×C₁₃₇	C₈	1	1	yes	characteristic
21	#11764047	—	2	1	no	—
22	#11764047	—	4	1	no	—
23	#11764047	SD₁₆	1	1	yes	characteristic
24	#11764047	Q₁₆	1	1	yes	characteristic
25	Dic₂₇₄	—	2	1	no	—
26	Dic₂₇₄	C₁₆	1	2	yes	complemented
27	C₂×Dic₁₃₇	—	2	1	no	—
28	C₂×Dic₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×Dic₁₃₇	M₁₆	1	1	yes	characteristic
30	C₂×C₄×C₁₃₇	—	2	1	no	—
31	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
33	C₄×C₁₃₇	—	4	1	no	—
34	C₄×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
35	C₄×C₁₃₇	M₃₂	1	1	yes	characteristic
36	Dic₁₃₇	—	2	2	no	—
37	Dic₁₃₇	—	2	1	no	—
38	Dic₁₃₇	#103	1	1	yes	characteristic
39	Dic₁₃₇	#104	1	1	yes	characteristic
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	—	137	1	no	—
48	C₄×Q₈	—	137	1	no	—
49	C₂×C₁₆	—	274	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	—	137	1	no	—
55	C₂×Q₈	—	137	1	no	—
56	C₂×C₈	—	274	1	no	—
57	C₂×C₈	—	137	1	no	—
58	C₂×C₈	—	137	1	no	—
59	C₄⋊C₄	—	274	1	no	—
60	C₄⋊C₄	#11764049	1	1	yes	characteristic
61	C₄²	—	274	1	no	—
62	C₄²	—	137	1	no	—
63	C₁₆	—	274	2	no	—
64	C₁₆	—	548	1	no	—
65	Q₈	—	274	1	no	—
66	Q₈	—	137	2	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	274	1	no	—
69	C₂×C₄	—	137	1	no	—
70	C₂×C₄	—	137	1	no	—
71	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
72	C₈	—	274	1	no	—
73	C₈	—	548	1	no	—
74	C₈	—	137	1	no	—
75	C₈	—	137	1	no	—
76	C₂²	C₂²⋊#11764049	1	1	yes	characteristic
77	C₄	—	4	1	no	—
78	C₄	—	274	2	no	—
79	C₄	—	274	1	no	—
80	C₄	—	137	1	no	—
81	C₄	—	137	1	no	—
82	C₄	#17984183	1	1	yes	characteristic
83	C₄	#17984185	1	1	yes	characteristic
84	C₂	Q₈⋊#11764049	1	1	yes	characteristic
85	C₂	#22578500	1	1	yes	characteristic
86	C₂	#22578503	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11764049
commutator	C₄×C₁₃₇	C₂×C₁₆
Frattini	C₂×C₄	C₂×#11764049
Fitting	C₄⋊C₄×C₁₃₇	C₈
socle, prime radical	C₂²×C₁₃₇	C₂²⋊C₈
cosocle	C₄×#11764047	C₂²
hypercenter	C₄⋊C₄	#11764049
hypocenter	C₁₃₇	Q₈⋊C₁₆
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
26. 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
C₄⋊#14555522	2
C₄×#14555522	2
#22578452	2
Q₈⋊C₁₆	137