Subgroups — #31798023

Order 17536.

106 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	1	yes	maximal, characteristic, complemented
3	#22578510	C₂	1	2	yes	maximal, complemented
4	C₂×#17984178	C₂	1	1	yes	maximal, characteristic, complemented
5	#22578462	C₂	1	1	yes	maximal, characteristic
6	#17984178	C₂²	1	1	yes	characteristic
7	#17984178	C₂²	1	1	yes	characteristic
8	#17984178	C₂²	1	2	yes	—
9	#17984163	C₄	1	2	yes	—
10	#17984162	C₄	1	1	yes	characteristic
11	#17984161	C₂²	1	2	yes	—
12	C₂×#14555513	C₂²	1	1	yes	characteristic
13	#17984096	C₄	1	1	yes	characteristic
14	#14555513	C₂×C₄	1	1	yes	characteristic
15	#14555513	C₂×C₄	1	2	yes	—
16	#14555513	C₂³	1	1	yes	characteristic
17	#14555510	C₈	1	1	yes	characteristic
18	#14555493	C₈	1	2	yes	—
19	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
20	C₈×D₁₃₇	C₂×C₄	1	2	yes	—
21	M₁₆×C₁₃₇	C₈	1	1	yes	characteristic
22	Dic₂₇₄	C₁₆	1	1	yes	characteristic
23	D₅₄₈	C₁₆	1	1	yes	characteristic
24	C₂²⋊D₁₃₇	C₁₆	1	2	yes	—
25	C₄×D₁₃₇	C₂×C₈	1	2	yes	—
26	#11764040	C₂×C₈	1	1	yes	characteristic
27	#11764040	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
29	C₈×C₁₃₇	C₂×C₈	1	2	yes	—
30	C₂²×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
31	C₂×D₁₃₇	C₂×C₁₆	1	2	yes	—
32	C₄×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
33	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
34	Dic₁₃₇	C₂×C₁₆	1	2	yes	—
35	C₂×C₁₃₇	—	2	1	no	—
36	C₂×C₁₃₇	C₂²×C₁₆	1	1	yes	characteristic
37	D₁₃₇	—	2	2	no	—
38	C₁₃₇	#2258	1	1	yes	characteristic, complemented, Sylow
39	#2258	—	137	1	no	maximal, Sylow
40	#504	—	137	1	no	—
41	M₆₄	—	137	1	no	—
42	M₆₄	—	137	2	no	—
43	C₂×C₃₂	—	137	1	no	—
44	C₂×C₃₂	—	137	2	no	—
45	#131	—	137	1	no	—
46	M₃₂	—	137	1	no	—
47	M₃₂	—	137	2	no	—
48	C₂×C₁₆	—	137	1	no	—
49	C₂×C₁₆	—	137	2	no	—
50	C₃₂	—	137	1	no	—
51	C₃₂	—	137	1	no	—
52	C₃₂	—	137	2	no	—
53	#41	—	137	1	no	—
54	M₁₆	—	137	2	no	—
55	M₁₆	#11764049	1	1	yes	characteristic
56	C₂×C₈	—	137	1	no	—
57	C₂×C₈	—	137	2	no	—
58	C₁₆	—	137	1	no	—
59	C₁₆	—	137	1	no	—
60	C₁₆	—	137	2	no	—
61	Q₈	—	137	1	no	—
62	D₄	—	137	2	no	—
63	D₄	—	137	1	no	—
64	C₂×C₄	—	137	2	no	—
65	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
66	C₈	—	137	1	no	—
67	C₈	—	137	1	no	—
68	C₈	C₂×#11764049	1	2	yes	—
69	C₂²	—	137	2	no	—
70	C₂²	#17984183	1	1	yes	characteristic
71	C₄	—	137	2	no	—
72	C₄	#17984183	1	1	yes	characteristic
73	C₄	C₂²×#11764049	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	274	2	no	—
76	C₂	C₂×#17984183	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×#11764049
commutator, socle	C₂×C₁₃₇	C₂²×C₁₆
Fitting	M₁₆×C₁₃₇	C₈
prime radical	#14555510	C₈
cosocle	#14555513	C₂³
hypercenter	M₁₆	#11764049
hypocenter	C₁₃₇	#2258
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ M₁₆

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
4. C₂×#17984178	C₂	semidirect
38. C₁₃₇	#2258	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×#17984178	2
#22578462	2
#2258	137