Subgroups — #22578009

Order 8768.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984085	C₂	1	1	yes	maximal, characteristic, complemented
2	#17984073	C₂	1	4	yes	maximal, complemented
3	#17984063	C₂	1	2	yes	maximal, complemented
4	#17984063	C₂	1	4	yes	maximal, complemented
5	#17984050	C₂	1	2	yes	maximal, complemented
6	#17984045	C₂	1	1	yes	maximal, characteristic, complemented
7	#125×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₅₄₈	C₂²	1	2	yes	—
9	C₂×D₅₄₈	C₂²	1	4	yes	—
10	C₂×D₅₄₈	C₂²	1	4	yes	—
11	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₃₇	C₂²	1	4	yes	—
13	#14555489	C₂²	1	2	yes	—
14	#14555487	C₂²	1	2	yes	—
15	#14555487	C₂²	1	4	yes	—
16	#14555487	C₂²	1	4	yes	—
17	C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
19	C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
20	C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂²×D₁₃₇	C₂³	1	1	yes	characteristic
22	C₂²×D₁₃₇	C₂³	1	4	yes	—
23	D₅₄₈	—	2	4	no	—
24	D₅₄₈	—	4	4	no	—
25	D₅₄₈	—	4	4	no	—
26	C₂×Dic₁₃₇	C₂³	1	2	yes	—
27	C₂×Dic₁₃₇	C₂³	1	1	yes	characteristic
28	C₄×D₁₃₇	—	2	2	no	—
29	C₄×D₁₃₇	—	4	4	no	—
30	C₂×C₄×C₁₃₇	C₂³	1	2	yes	—
31	C₂×C₄×C₁₃₇	C₂³	1	4	yes	—
32	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
33	C₂²×C₁₃₇	C₂⁴	1	1	yes	characteristic
34	C₂×D₁₃₇	—	2	1	no	—
35	C₂×D₁₃₇	—	2	4	no	—
36	C₂×D₁₃₇	—	2	4	no	—
37	C₂×D₁₃₇	—	2	2	no	—
38	C₂×D₁₃₇	—	2	4	no	—
39	C₄×C₁₃₇	—	2	4	no	—
40	C₄×C₁₃₇	—	2	2	no	—
41	C₄×C₁₃₇	#41	1	2	yes	—
42	Dic₁₃₇	—	2	1	no	—
43	Dic₁₃₇	—	2	2	no	—
44	C₂×C₁₃₇	C₂×#41	1	1	yes	characteristic
45	C₂×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
46	D₁₃₇	—	4	1	no	—
47	D₁₃₇	—	4	4	no	—
48	C₁₃₇	#540	1	1	yes	characteristic, complemented, Sylow
49	#540	—	137	1	no	maximal, Sylow
50	#127	—	137	1	no	—
51	#125	D₁₃₇	1	1	yes	characteristic, complemented
52	#123	—	137	4	no	—
53	#121	—	137	4	no	—
54	#121	—	137	2	no	—
55	C₄×D₄	—	137	2	no	—
56	#117	—	137	1	no	—
57	C₂×D₄	—	137	2	no	—
58	C₂×D₄	—	137	4	no	—
59	C₂×D₄	—	137	4	no	—
60	C₂²×C₄	—	137	1	no	—
61	C₂²×C₄	—	137	4	no	—
62	C₄⋊C₄	—	137	2	no	—
63	C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
64	C₄⋊C₄	C₂×D₁₃₇	1	4	yes	—
65	C₂²⋊C₄	—	137	2	no	—
66	C₂²⋊C₄	—	137	4	no	—
67	C₂²⋊C₄	—	137	4	no	—
68	C₄²	—	137	1	no	—
69	C₄²	C₂×D₁₃₇	1	1	yes	characteristic
70	C₂³	—	137	1	no	—
71	C₂³	—	137	4	no	—
72	D₄	—	274	4	no	—
73	D₄	—	548	4	no	—
74	D₄	—	548	4	no	—
75	C₂×C₄	—	274	2	no	—
76	C₂×C₄	—	548	4	no	—
77	C₂×C₄	—	137	2	no	—
78	C₂×C₄	—	137	1	no	—
79	C₂×C₄	C₂²×D₁₃₇	1	2	yes	—
80	C₂×C₄	C₂²×D₁₃₇	1	4	yes	—
81	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
82	C₂²	—	274	1	no	—
83	C₂²	—	274	4	no	—
84	C₂²	—	274	4	no	—
85	C₂²	—	274	2	no	—
86	C₂²	—	274	4	no	—
87	C₂²	C₂³×D₁₃₇	1	1	yes	characteristic
88	C₄	—	2	4	no	—
89	C₄	—	2	2	no	—
90	C₄	—	274	1	no	—
91	C₄	—	274	2	no	—
92	C₄	#14555511	1	2	yes	—
93	C₂	—	548	1	no	—
94	C₂	—	548	4	no	—
95	C₂	C₂×#14555511	1	1	yes	characteristic
96	C₂	#17984134	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂⁴
Fitting	#125×C₁₃₇	C₂
hypercenter	#125	D₁₃₇
hypocenter	C₁₃₇	#540
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #17984085	C₂	semidirect
2. #17984073	C₂	semidirect
3. #17984063	C₂	semidirect
4. #17984063	C₂	semidirect
5. #17984050	C₂	semidirect
6. #17984045	C₂	semidirect
7. #125×C₁₃₇	C₂	semidirect
48. C₁₃₇	#540	semidirect
51. #125	D₁₃₇	semidirect