Subgroups — #31797825

Order 17536.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578489	C₂	1	2	yes	maximal
2	#22578347	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#11764049	—	2	2	no	—
4	C₂×C₄⋊#10587739	C₄	1	1	yes	characteristic
5	C₂×C₄⋊#10587739	C₂²	1	1	yes	characteristic
6	(C₂×#10587739)⋊C₄	—	2	2	no	—
7	C₄²×D₁₃₇	C₄	1	1	yes	characteristic
8	C₂×#11764049	—	4	2	no	—
9	C₂²×#10587739	—	2	1	no	—
10	C₂²×#10587739	D₄	1	2	yes	—
11	C₄⋊#10587739	—	4	1	no	—
12	C₄⋊#10587739	—	4	1	no	—
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, complemented
17	C₄²×C₁₃₇	C₈	1	1	yes	characteristic, complemented
18	#11764049	—	8	2	no	—
19	C₂×#10587739	—	2	2	no	—
20	C₂×#10587739	—	4	2	no	—
21	C₂×#10587739	—	4	2	no	—
22	C₂×#10587739	—	4	1	no	—
23	C₂²×D₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×Dic₁₃₇	—	2	1	no	—
25	C₂×Dic₁₃₇	M₁₆	1	1	yes	characteristic
26	C₄×D₁₃₇	—	2	1	no	—
27	C₄×D₁₃₇	—	2	1	no	—
28	C₄×D₁₃₇	—	4	1	no	—
29	C₄×D₁₃₇	—	4	1	no	—
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	C₂×D₁₃₇	—	2	1	no	—
34	C₂×D₁₃₇	—	2	1	no	—
35	C₂×D₁₃₇	#99	1	1	yes	characteristic
36	C₂×D₁₃₇	#101	1	1	yes	characteristic
37	#10587739	—	4	2	no	—
38	#10587739	—	8	1	no	—
39	C₄×C₁₃₇	—	2	1	no	—
40	C₄×C₁₃₇	—	4	1	no	—
41	Dic₁₃₇	—	2	1	no	—
42	Dic₁₃₇	—	4	1	no	—
43	C₂×C₁₃₇	—	2	1	no	—
44	C₂×C₁₃₇	#324	1	1	yes	characteristic
45	D₁₃₇	—	2	1	no	—
46	D₁₃₇	#354	1	1	yes	characteristic
47	D₁₃₇	#355	1	1	yes	characteristic
48	C₁₃₇	#1324	1	1	yes	characteristic, complemented, Sylow
49	#1324	—	137	1	no	maximal, Sylow
50	#382	—	137	1	no	—
51	#324	—	137	2	no	—
52	C₂×C₄⋊C₄	—	137	1	no	—
53	C₂×C₄⋊C₄	—	137	1	no	—
54	C₂×C₄²	—	137	1	no	—
55	C₂²⋊C₈	—	274	2	no	—
56	#95	—	274	2	no	—
57	C₂²×C₄	—	274	1	no	—
58	C₂²×C₄	—	274	1	no	—
59	C₂²×C₄	—	137	1	no	—
60	C₂²×C₄	—	137	2	no	—
61	C₂×C₈	—	548	2	no	—
62	C₄⋊C₄	—	548	1	no	—
63	C₄⋊C₄	—	548	1	no	—
64	C₄²	—	274	1	no	—
65	C₄²	—	137	1	no	—
66	C₄²	#11764049	1	1	yes	characteristic, complemented
67	C₂³	—	137	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	274	1	no	—
70	C₂×C₄	—	274	1	no	—
71	C₂×C₄	—	274	2	no	—
72	C₂×C₄	—	274	1	no	—
73	C₂×C₄	—	548	1	no	—
74	C₂×C₄	—	548	1	no	—
75	C₂×C₄	—	548	2	no	—
76	C₂×C₄	—	548	2	no	—
77	C₂×C₄	—	548	1	no	—
78	C₂×C₄	—	137	1	no	—
79	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
80	C₈	—	1096	2	no	—
81	C₂²	—	274	1	no	—
82	C₂²	—	274	1	no	—
83	C₂²	—	137	1	no	—
84	C₂²	—	137	1	no	—
85	C₂²	C₂²⋊#11764049	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	4	1	no	—
88	C₄	—	274	1	no	—
89	C₄	—	548	1	no	—
90	C₄	—	548	2	no	—
91	C₄	—	1096	1	no	—
92	C₂	—	2	1	no	—
93	C₂	—	274	1	no	—
94	C₂	—	137	1	no	—
95	C₂	—	137	1	no	—
96	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	C₂²⋊C₄
cosocle	C₂×C₄⋊#10587739	C₂²
hypercenter	C₄²	#11764049
hypocenter	C₁₃₇	#1324
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
16. C₄×Dic₁₃₇	C₈	semidirect
17. C₄²×C₁₃₇	C₈	semidirect
48. C₁₃₇	#1324	semidirect
66. C₄²	#11764049	semidirect

Maximal subgroups

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

subgroup	index
#22578489	2
#22578347	2
#1324	137