Subgroups — #31797883

Order 17536.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578511	C₂	1	2	yes	maximal, complemented
2	C₂×#17984162	C₂	1	1	yes	maximal, characteristic
3	#17984178	—	2	2	no	—
4	#17984162	—	2	2	no	—
5	C₂×#14555513	C₄	1	1	yes	characteristic
6	C₂×#14555513	C₂²	1	1	yes	characteristic
7	C₂²×#11764040	C₄	1	1	yes	characteristic
8	#14555513	—	2	1	no	—
9	#14555513	D₄	1	2	yes	—
10	C₂×#11764040	—	2	1	no	—
11	C₂×#11764040	—	2	1	no	—
12	C₂×#11764040	C₈	1	1	yes	characteristic
13	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₃₇	C₈	1	1	yes	characteristic
15	#11764040	—	2	1	no	—
16	#11764040	C₂²⋊C₄	1	1	yes	characteristic
17	#11764040	M₁₆	1	1	yes	characteristic
18	C₂³×C₁₃₇	C₁₆	1	1	yes	characteristic
19	C₂×C₄×C₁₃₇	—	2	1	no	—
20	C₂×C₄×C₁₃₇	—	2	1	no	—
21	C₂×C₄×C₁₃₇	C₁₆	1	1	yes	characteristic
22	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
23	C₂²×C₁₃₇	—	2	1	no	—
24	C₂²×C₁₃₇	—	4	1	no	—
25	C₂²×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
26	C₄×C₁₃₇	—	2	1	no	—
27	C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
28	C₄×C₁₃₇	M₃₂	1	1	yes	characteristic
29	C₂×C₁₃₇	—	2	1	no	—
30	C₂×C₁₃₇	—	4	1	no	—
31	C₂×C₁₃₇	C₂²⋊C₁₆	1	1	yes	characteristic
32	C₁₃₇	#1400	1	1	yes	characteristic, complemented, Sylow
33	#1400	—	137	1	no	maximal, Sylow
34	C₂×M₃₂	—	137	1	no	—
35	M₆₄	—	137	2	no	—
36	C₂²×C₈	—	137	1	no	—
37	M₃₂	—	274	2	no	—
38	C₂×C₁₆	—	137	1	no	—
39	C₂×C₁₆	—	137	1	no	—
40	C₃₂	—	274	2	no	—
41	C₂²×C₄	#11764049	1	1	yes	characteristic
42	C₂×C₈	—	274	1	no	—
43	C₂×C₈	—	274	1	no	—
44	C₂×C₈	—	137	1	no	—
45	C₂×C₈	—	137	1	no	—
46	C₁₆	—	274	1	no	—
47	C₁₆	—	137	2	no	—
48	C₂³	#14555522	1	1	yes	characteristic
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	#14555522	1	1	yes	characteristic
52	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
53	C₈	—	274	1	no	—
54	C₈	—	137	1	no	—
55	C₈	—	137	1	no	—
56	C₂²	—	2	1	no	—
57	C₂²	—	4	1	no	—
58	C₂²	C₂×#14555522	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	C₂²⋊#11764049	1	1	yes	characteristic
61	C₄	#17984184	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	4	1	no	—
64	C₂	C₂²⋊#14555522	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#11764049
commutator	C₂²×C₁₃₇	C₂×C₁₆
Frattini	C₂×C₄	C₂×#11764049
Fitting	C₂²×C₄×C₁₃₇	C₈
socle	C₂×C₁₃₇	C₂²⋊C₁₆
prime radical	C₂³×C₁₃₇	C₁₆
cosocle	C₂×#14555513	C₂²
hypercenter	C₂²×C₄	#11764049
hypocenter	C₁₃₇	#1400
LCS	G ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. #22578511	C₂	semidirect
32. C₁₃₇	#1400	semidirect

Maximal subgroups

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

subgroup	index
#22578511	2
C₂×#17984162	2
#1400	137