Subgroups — #31795904

Order 17536.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578335	C₂	1	2	yes	maximal, complemented
2	C₂×#17984028	C₂	1	1	yes	maximal, characteristic
3	#17984138	—	2	2	no	—
4	#17984028	—	2	2	no	—
5	C₂×#14555485	C₄	1	1	yes	characteristic
6	C₂×#14555485	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₃₇	C₄	1	1	yes	characteristic
8	#14555485	—	2	1	no	—
9	#14555485	D₄	1	2	yes	—
10	C₂²×C₄×C₁₃₇	C₈	1	1	yes	characteristic
11	C₂×C₈×C₁₃₇	—	2	1	no	—
12	C₂×C₈×C₁₃₇	—	2	1	no	—
13	C₂×C₈×C₁₃₇	C₈	1	1	yes	characteristic
14	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₁₃₇	C₁₆	1	1	yes	characteristic
16	C₂×C₄×C₁₃₇	—	2	1	no	—
17	C₂×C₄×C₁₃₇	—	2	1	no	—
18	C₂×C₄×C₁₃₇	C₁₆	1	1	yes	characteristic
19	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
20	C₈×C₁₃₇	—	2	1	no	—
21	C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₈×C₁₃₇	M₁₆	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	#10587739	1	1	yes	characteristic
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	#11764047	1	1	yes	characteristic
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	#11764047	1	1	yes	characteristic
45	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
46	C₁₆	—	274	1	no	—
47	C₁₆	—	137	2	no	—
48	C₂³	#14555513	1	1	yes	characteristic
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	#14555513	1	1	yes	characteristic
52	C₂×C₄	C₂×#11764047	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	C₂²⋊#10587739	1	1	yes	characteristic
55	C₈	#14555519	1	1	yes	characteristic
56	C₂²	—	2	1	no	—
57	C₂²	—	4	1	no	—
58	C₂²	C₂×#14555513	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	C₂²⋊#11764047	1	1	yes	characteristic
61	C₄	#17984162	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	4	1	no	—
64	C₂	C₂²⋊#14555513	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊#10587739
commutator	C₂²×C₁₃₇	C₂×C₁₆
Frattini	C₂×C₈	C₂×#10587739
Fitting	C₂²×C₈×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₂²⋊C₁₆
prime radical	C₂³×C₁₃₇	C₁₆
cosocle	C₂×#14555485	C₂²
hypercenter	C₂²×C₈	#10587739
hypocenter	C₁₃₇	#1400
LCS	G ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
1. #22578335	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
#22578335	2
C₂×#17984028	2
#1400	137