Subgroups — #31797828

Order 17536.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578502	C₂	1	2	yes	maximal
2	#22578408	C₂	1	1	yes	maximal, characteristic
3	#17984184	—	2	2	no	—
4	C₂×#14555520	C₄	1	1	yes	characteristic
5	C₂×#14555520	C₂²	1	1	yes	characteristic
6	#17984155	—	2	2	no	—
7	C₄²×D₁₃₇	C₄	1	1	yes	characteristic
8	#14555522	—	4	2	no	—
9	#14555520	—	4	1	no	—
10	#14555520	—	4	1	no	—
11	C₂×#11764047	—	2	1	no	—
12	C₂×#11764047	D₄	1	2	yes	—
13	C₂×C₄×D₁₃₇	—	2	1	no	—
14	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
15	C₄×Dic₁₃₇	—	2	1	no	—
16	C₄×Dic₁₃₇	C₈	1	1	yes	characteristic
17	C₄²×C₁₃₇	C₈	1	1	yes	characteristic
18	#11764047	—	2	2	no	—
19	#11764047	—	4	1	no	—
20	C₂²×D₁₃₇	M₁₆	1	1	yes	characteristic
21	C₂×Dic₁₃₇	—	2	1	no	—
22	C₂×Dic₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×D₁₃₇	—	2	1	no	—
24	C₄×D₁₃₇	—	2	1	no	—
25	C₄×D₁₃₇	—	4	2	no	—
26	C₂×C₄×C₁₃₇	—	2	1	no	—
27	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
28	C₂²×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
29	C₂×D₁₃₇	—	2	1	no	—
30	C₂×D₁₃₇	—	4	1	no	—
31	C₄×C₁₃₇	—	2	1	no	—
32	C₄×C₁₃₇	—	4	1	no	—
33	Dic₁₃₇	—	2	2	no	—
34	Dic₁₃₇	#99	1	1	yes	characteristic
35	Dic₁₃₇	#101	1	1	yes	characteristic
36	C₂×C₁₃₇	—	2	1	no	—
37	C₂×C₁₃₇	#324	1	1	yes	characteristic
38	D₁₃₇	—	4	1	no	—
39	C₁₃₇	#1327	1	1	yes	characteristic, complemented, Sylow
40	#1327	—	137	1	no	maximal, Sylow
41	#423	—	137	1	no	—
42	#349	—	137	2	no	—
43	C₂×M₁₆	—	137	1	no	—
44	C₂×M₁₆	—	137	1	no	—
45	C₂×C₄²	—	137	1	no	—
46	M₃₂	—	274	2	no	—
47	C₄⋊C₈	—	274	2	no	—
48	C₂²×C₄	—	274	1	no	—
49	C₂²×C₄	—	137	1	no	—
50	M₁₆	—	548	1	no	—
51	M₁₆	—	548	1	no	—
52	C₂×C₈	—	274	1	no	—
53	C₂×C₈	—	137	2	no	—
54	C₄²	—	274	1	no	—
55	C₄²	—	137	1	no	—
56	C₄²	#11764049	1	1	yes	characteristic
57	C₁₆	—	548	2	no	—
58	C₂³	—	137	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	274	1	no	—
61	C₂×C₄	—	274	1	no	—
62	C₂×C₄	—	274	1	no	—
63	C₂×C₄	—	548	2	no	—
64	C₂×C₄	—	137	1	no	—
65	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
66	C₈	—	274	2	no	—
67	C₈	—	548	1	no	—
68	C₂²	—	274	1	no	—
69	C₂²	—	548	1	no	—
70	C₂²	C₂²⋊#11764049	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	274	2	no	—
74	C₄	—	137	1	no	—
75	C₄	—	137	1	no	—
76	C₂	—	2	1	no	—
77	C₂	—	548	1	no	—
78	C₂	#22578489	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22578489
commutator	C₂×C₄×C₁₃₇	C₂×C₈
Frattini	C₂×C₄	C₂×#11764049
Fitting	C₄²×C₁₃₇	C₈
socle	C₂×C₁₃₇	#324
prime radical	C₂²×D₁₃₇	M₁₆
cosocle	C₂×#14555520	C₂²
hypercenter	C₄²	#11764049
hypocenter	C₁₃₇	#1327
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
39. C₁₃₇	#1327	semidirect

Maximal subgroups

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

subgroup	index
#22578502	2
#22578408	2
#1327	137