Subgroups — #31798020

Order 17536.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578512	C₂	1	2	yes	maximal, complemented
2	#22578511	C₂	1	2	yes	maximal, complemented
3	#22578510	C₂	1	2	yes	maximal, complemented
4	C₂×#17984161	C₂	1	1	yes	maximal, characteristic
5	#17984178	C₂²	1	4	yes	complemented
6	#17984161	C₄	1	1	yes	characteristic
7	#17984161	C₄	1	1	yes	characteristic
8	#17984161	C₂²	1	1	yes	characteristic
9	#17984161	C₂²	1	1	yes	characteristic
10	C₂×#14555513	C₄	1	1	yes	characteristic
11	C₂×#14555513	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₁₃₇	C₄	1	1	yes	characteristic
13	#14555513	C₂×C₄	1	1	yes	characteristic
14	#14555513	C₂×C₄	1	1	yes	characteristic
15	#14555513	C₂×C₄	1	1	yes	characteristic
16	#14555513	C₂³	1	1	yes	characteristic
17	C₂×C₄×D₁₃₇	C₈	1	1	yes	characteristic
18	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
19	C₈×D₁₃₇	C₈	1	1	yes	characteristic
20	C₈×D₁₃₇	C₈	1	1	yes	characteristic
21	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
22	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₃₇	C₈	1	1	yes	characteristic
24	C₂²×D₁₃₇	C₁₆	1	1	yes	characteristic
25	C₂×Dic₁₃₇	C₁₆	1	1	yes	characteristic
26	C₄×D₁₃₇	C₁₆	1	1	yes	characteristic
27	C₄×D₁₃₇	C₁₆	1	1	yes	characteristic
28	C₄×D₁₃₇	C₂×C₈	1	1	yes	characteristic
29	C₄×D₁₃₇	C₂×C₈	1	1	yes	characteristic
30	#11764040	C₂×C₈	1	1	yes	characteristic
31	#11764040	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
33	C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
34	C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
36	C₂×D₁₃₇	—	2	2	no	—
37	C₂×D₁₃₇	C₂×C₁₆	1	1	yes	characteristic
38	C₂×D₁₃₇	C₂×C₁₆	1	1	yes	characteristic
39	C₄×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
40	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
41	Dic₁₃₇	C₂×C₁₆	1	1	yes	characteristic
42	Dic₁₃₇	C₂×C₁₆	1	1	yes	characteristic
43	C₂×C₁₃₇	—	2	1	no	—
44	C₂×C₁₃₇	C₂²×C₁₆	1	1	yes	characteristic
45	D₁₃₇	—	2	1	no	—
46	D₁₃₇	M₆₄	1	2	yes	complemented
47	C₁₃₇	C₂×M₆₄	1	1	yes	characteristic, complemented, Sylow
48	C₂×M₆₄	—	137	1	no	maximal, Sylow
49	C₂²×C₁₆	—	137	1	no	—
50	M₆₄	—	137	2	no	—
51	M₆₄	—	137	2	no	—
52	C₂×C₃₂	—	137	2	no	—
53	C₂²×C₈	—	137	1	no	—
54	C₂×C₁₆	—	137	1	no	—
55	C₂×C₁₆	—	137	1	no	—
56	C₂×C₁₆	—	137	1	no	—
57	C₂×C₁₆	—	137	1	no	—
58	C₂×C₁₆	—	137	1	no	—
59	C₂×C₁₆	—	137	1	no	—
60	C₃₂	—	137	4	no	—
61	C₂²×C₄	—	137	1	no	—
62	C₂×C₈	—	137	1	no	—
63	C₂×C₈	—	137	1	no	—
64	C₂×C₈	—	137	1	no	—
65	C₂×C₈	—	137	1	no	—
66	C₂×C₈	—	137	1	no	—
67	C₂×C₈	#11764049	1	1	yes	characteristic
68	C₁₆	—	137	1	no	—
69	C₁₆	—	137	1	no	—
70	C₁₆	—	137	1	no	—
71	C₁₆	—	137	1	no	—
72	C₂³	—	137	1	no	—
73	C₂×C₄	—	137	1	no	—
74	C₂×C₄	—	137	1	no	—
75	C₂×C₄	—	137	1	no	—
76	C₂×C₄	—	137	1	no	—
77	C₂×C₄	—	137	1	no	—
78	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
79	C₈	—	137	1	no	—
80	C₈	—	137	1	no	—
81	C₈	C₂×#11764049	1	1	yes	characteristic
82	C₈	C₂×#11764049	1	1	yes	characteristic
83	C₂²	—	274	2	no	—
84	C₂²	—	137	1	no	—
85	C₂²	—	137	1	no	—
86	C₂²	#17984183	1	1	yes	characteristic
87	C₄	—	137	1	no	—
88	C₄	—	137	1	no	—
89	C₄	#17984183	1	1	yes	characteristic
90	C₄	C₂²×#11764049	1	1	yes	characteristic
91	C₂	—	2	1	no	—
92	C₂	—	274	1	no	—
93	C₂	—	137	2	no	—
94	C₂	C₂×#17984183	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×#11764049
commutator, socle	C₂×C₁₃₇	C₂²×C₁₆
Frattini	C₄	C₂²×#11764049
Fitting	C₂×C₈×C₁₃₇	C₈
prime radical	C₂²×D₁₃₇	C₁₆
cosocle	#14555513	C₂³
hypercenter	C₂×C₈	#11764049
hypocenter	C₁₃₇	C₂×M₆₄
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. #22578512	C₂	semidirect
2. #22578511	C₂	semidirect
3. #22578510	C₂	semidirect
5. #17984178	C₂²	semidirect
46. D₁₃₇	M₆₄	semidirect
47. C₁₃₇	C₂×M₆₄	semidirect

Maximal subgroups

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

subgroup	index
#22578512	2
#22578511	2
#22578510	2
C₂×#17984161	2
C₂×M₆₄	137