Subgroups — #31797833

Order 17536.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578506	C₂	1	1	yes	maximal, characteristic, complemented
2	#22578497	C₂	1	1	yes	maximal, characteristic, complemented
3	#22578438	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555522	—	2	1	no	—
5	C₂×#14555522	—	2	1	no	—
6	C₂×#14555519	—	2	1	no	—
7	C₄⋊#11764047	C₄	1	1	yes	characteristic
8	C₂²⋊#11764047	—	2	1	no	—
9	C₄×#11764047	C₂²	1	1	yes	characteristic
10	D₄×Dic₁₃₇	C₄	1	1	yes	characteristic
11	#14555522	—	2	2	no	—
12	#14555522	—	4	1	no	—
13	#14555519	—	4	1	no	—
14	#14555519	—	4	1	no	—
15	C₂×#11764047	—	2	1	no	—
16	C₂×#11764047	D₄	1	1	yes	characteristic
17	C₂×#11764047	D₄	1	1	yes	characteristic
18	C₂²×Dic₁₃₇	—	2	1	no	—
19	C₄⋊Dic₁₃₇	C₈	1	1	yes	characteristic
20	C₂²⋊Dic₁₃₇	—	2	1	no	—
21	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂×D₄×C₁₃₇	C₈	1	1	yes	characteristic
23	#11764047	—	2	1	no	—
24	#11764047	—	4	1	no	—
25	#11764047	D₈	1	1	yes	characteristic
26	#11764047	SD₁₆	1	1	yes	characteristic
27	C₂×Dic₁₃₇	—	2	1	no	—
28	C₂×Dic₁₃₇	—	4	1	no	—
29	C₂×Dic₁₃₇	—	4	1	no	—
30	C₂×Dic₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×Dic₁₃₇	M₁₆	1	1	yes	characteristic
32	C₂³×C₁₃₇	—	2	1	no	—
33	D₄×C₁₃₇	—	4	1	no	—
34	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₁₃₇	—	4	1	no	—
36	C₂²×C₁₃₇	—	4	1	no	—
37	C₂²×C₁₃₇	—	4	1	no	—
38	C₂²×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₁₃₇	—	2	1	no	—
40	Dic₁₃₇	—	2	1	no	—
41	Dic₁₃₇	—	4	1	no	—
42	Dic₁₃₇	#102	1	1	yes	characteristic
43	Dic₁₃₇	#104	1	1	yes	characteristic
44	C₂×C₁₃₇	—	8	1	no	—
45	C₂×C₁₃₇	D₄⋊C₈	1	1	yes	characteristic
46	C₂×C₁₃₇	#349	1	1	yes	characteristic
47	C₂×C₁₃₇	#350	1	1	yes	characteristic
48	C₁₃₇	#1330	1	1	yes	characteristic, complemented, Sylow
49	#1330	—	137	1	no	maximal, Sylow
50	#436	—	137	1	no	—
51	C₄⋊C₁₆	—	137	1	no	—
52	#346	—	137	1	no	—
53	C₂×M₁₆	—	274	1	no	—
54	C₄×D₄	—	137	1	no	—
55	C₂×C₁₆	—	274	1	no	—
56	C₂×C₁₆	—	274	1	no	—
57	C₄⋊C₈	—	137	1	no	—
58	C₂²⋊C₈	—	274	1	no	—
59	C₄×C₈	—	137	1	no	—
60	C₂×D₄	#11764049	1	1	yes	characteristic
61	C₂²×C₄	—	274	1	no	—
62	M₁₆	—	548	1	no	—
63	M₁₆	—	548	1	no	—
64	C₂×C₈	—	274	1	no	—
65	C₂×C₈	—	137	1	no	—
66	C₂×C₈	—	137	1	no	—
67	C₄⋊C₄	—	137	1	no	—
68	C₂²⋊C₄	—	274	1	no	—
69	C₄²	—	137	1	no	—
70	C₁₆	—	274	2	no	—
71	C₁₆	—	548	1	no	—
72	C₂³	—	2	1	no	—
73	D₄	—	4	1	no	—
74	C₂×C₄	—	274	1	no	—
75	C₂×C₄	—	548	1	no	—
76	C₂×C₄	—	548	1	no	—
77	C₂×C₄	—	137	1	no	—
78	C₂×C₄	—	137	1	no	—
79	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
80	C₈	—	274	1	no	—
81	C₈	—	548	1	no	—
82	C₈	—	137	1	no	—
83	C₈	—	137	1	no	—
84	C₂²	—	4	1	no	—
85	C₂²	—	4	1	no	—
86	C₂²	—	4	1	no	—
87	C₂²	C₂²⋊#11764049	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	274	1	no	—
90	C₄	—	548	1	no	—
91	C₄	—	137	1	no	—
92	C₄	—	137	1	no	—
93	C₂	—	8	1	no	—
94	C₂	D₄⋊#11764049	1	1	yes	characteristic
95	C₂	#22578501	1	1	yes	characteristic
96	C₂	#22578503	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11764049
commutator	C₂×C₄×C₁₃₇	C₂×C₈
Frattini	C₂×C₄	C₂×#11764049
Fitting, prime radical	C₂×D₄×C₁₃₇	C₈
socle	C₂²×C₁₃₇	C₂²⋊C₈
cosocle	C₄×#11764047	C₂²
hypercenter	C₂×D₄	#11764049
hypocenter	C₁₃₇	#1330
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. #22578506	C₂	semidirect
2. #22578497	C₂	semidirect
48. C₁₃₇	#1330	semidirect

Maximal subgroups

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

subgroup	index
#22578506	2
#22578497	2
#22578438	2
#1330	137