Subgroups — #31797830

Order 17536.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578501	C₂	1	1	yes	maximal, characteristic
2	#22578501	C₂	1	1	yes	maximal, characteristic
3	#22578409	C₂	1	1	yes	maximal, characteristic
4	#17984184	—	2	1	no	—
5	#17984184	—	2	1	no	—
6	C₂×#14555519	C₂²	1	1	yes	characteristic
7	C₂²×#11764047	C₄	1	1	yes	characteristic
8	#17984155	—	2	1	no	—
9	C₄⋊#11764047	—	2	1	no	—
10	#17984044	C₄	1	1	yes	characteristic
11	#14555522	—	4	1	no	—
12	#14555522	—	4	1	no	—
13	#14555519	—	4	1	no	—
14	C₂×#11764047	—	2	1	no	—
15	C₂×#11764047	—	4	1	no	—
16	C₂×#11764047	D₄	1	1	yes	characteristic
17	C₂×#11764047	D₄	1	1	yes	characteristic
18	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
19	#14555489	—	2	1	no	—
20	C₂²⋊Dic₁₃₇	C₈	1	1	yes	characteristic
21	C₄×Dic₁₃₇	—	2	1	no	—
22	C₂²⋊C₄×C₁₃₇	C₈	1	1	yes	characteristic
23	#11764047	—	2	1	no	—
24	#11764047	—	2	1	no	—
25	#11764047	—	4	1	no	—
26	C₂×Dic₁₃₇	—	2	1	no	—
27	C₂×Dic₁₃₇	—	2	1	no	—
28	C₂×Dic₁₃₇	—	2	1	no	—
29	C₂×Dic₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×Dic₁₃₇	M₁₆	1	1	yes	characteristic
31	C₂³×C₁₃₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₁₃₇	—	2	1	no	—
33	C₂²×C₁₃₇	—	2	1	no	—
34	C₂²×C₁₃₇	—	4	1	no	—
35	C₂²×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
36	C₄×C₁₃₇	—	4	1	no	—
37	Dic₁₃₇	—	2	1	no	—
38	Dic₁₃₇	—	4	1	no	—
39	Dic₁₃₇	#99	1	1	yes	characteristic
40	Dic₁₃₇	#101	1	1	yes	characteristic
41	C₂×C₁₃₇	—	2	1	no	—
42	C₂×C₁₃₇	—	4	1	no	—
43	C₂×C₁₃₇	#324	1	1	yes	characteristic
44	C₁₃₇	#1328	1	1	yes	characteristic, complemented, Sylow
45	#1328	—	137	1	no	maximal, Sylow
46	#424	—	137	1	no	—
47	#349	—	137	1	no	—
48	#349	—	137	1	no	—
49	C₂×M₁₆	—	137	1	no	—
50	C₂²×C₈	—	137	1	no	—
51	#117	—	137	1	no	—
52	M₃₂	—	274	1	no	—
53	M₃₂	—	274	1	no	—
54	C₄⋊C₈	—	274	1	no	—
55	C₄⋊C₈	—	274	1	no	—
56	C₂²×C₄	—	137	1	no	—
57	M₁₆	—	548	1	no	—
58	C₂×C₈	—	274	1	no	—
59	C₂×C₈	—	548	1	no	—
60	C₂×C₈	—	137	1	no	—
61	C₂×C₈	—	137	1	no	—
62	C₄⋊C₄	—	274	1	no	—
63	C₂²⋊C₄	—	137	1	no	—
64	C₂²⋊C₄	#11764049	1	1	yes	characteristic
65	C₄²	—	274	1	no	—
66	C₁₆	—	548	1	no	—
67	C₁₆	—	548	1	no	—
68	C₂³	C₂×#11764049	1	1	yes	characteristic
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	274	1	no	—
71	C₂×C₄	—	274	1	no	—
72	C₂×C₄	—	274	1	no	—
73	C₂×C₄	—	137	1	no	—
74	C₂×C₄	—	137	1	no	—
75	C₈	—	274	1	no	—
76	C₈	—	274	1	no	—
77	C₈	—	548	1	no	—
78	C₂²	—	2	1	no	—
79	C₂²	—	4	1	no	—
80	C₂²	C₂²⋊#11764049	1	1	yes	characteristic
81	C₄	—	4	1	no	—
82	C₄	—	274	1	no	—
83	C₄	—	548	1	no	—
84	C₄	—	137	1	no	—
85	C₄	—	137	1	no	—
86	C₂	—	2	1	no	—
87	C₂	—	4	1	no	—
88	C₂	#22578489	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22578489
commutator, prime radical	C₂³×C₁₃₇	C₂×C₈
Frattini	C₂³	C₂×#11764049
Fitting	C₂²⋊C₄×C₁₃₇	C₈
socle	C₂×C₁₃₇	#324
cosocle	C₂×#14555519	C₂²
hypercenter	C₂²⋊C₄	#11764049
hypocenter	C₁₃₇	#1328
LCS	G ▷ C₂³×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂³ ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
44. C₁₃₇	#1328	semidirect

Maximal subgroups

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

subgroup	index
#22578501	2
#22578409	2
#1328	137