Subgroups — #31795812

Order 17536.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14555513	C₂	1	2	yes	maximal
2	C₈×#11764040	C₂	1	1	yes	maximal, characteristic
3	C₂×#14555513	—	2	2	no	—
4	C₄×#11764040	C₄	1	1	yes	characteristic
5	C₄×#11764040	C₂²	1	1	yes	characteristic
6	C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic
7	#14555513	—	4	2	no	—
8	C₂×#11764040	C₈	1	1	yes	characteristic
9	C₂×#11764040	C₈	1	1	yes	characteristic
10	C₂×#11764040	D₄	1	1	yes	characteristic
11	C₂×#11764040	Q₈	1	1	yes	characteristic
12	C₂×C₈×C₁₃₇	—	2	1	no	—
13	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
14	#11764040	—	2	1	no	—
15	#11764040	—	2	1	no	—
16	#11764040	C₁₆	1	2	yes	complemented
17	#11764040	SD₁₆	1	2	yes	—
18	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
20	C₂×C₄×C₁₃₇	M₁₆	1	1	yes	characteristic
21	C₈×C₁₃₇	—	2	2	no	—
22	C₂²×C₁₃₇	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₁₃₇	—	2	1	no	—
24	C₄×C₁₃₇	#106	1	1	yes	characteristic
25	C₄×C₁₃₇	#108	1	1	yes	characteristic
26	C₄×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
27	C₄×C₁₃₇	M₃₂	1	1	yes	characteristic
28	C₂×C₁₃₇	#334	1	1	yes	characteristic
29	C₂×C₁₃₇	C₄⋊C₁₆	1	1	yes	characteristic
30	C₂×C₁₃₇	#364	1	1	yes	characteristic
31	C₁₃₇	#1367	1	1	yes	characteristic, complemented, Sylow
32	#1367	—	137	1	no	maximal, Sylow
33	C₄⋊C₁₆	—	137	2	no	—
34	C₈²	—	137	1	no	—
35	C₂×C₁₆	—	274	2	no	—
36	C₄×C₈	—	137	1	no	—
37	C₄×C₈	—	137	1	no	—
38	C₄×C₈	#10587739	1	1	yes	characteristic
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	—	137	1	no	—
41	C₂×C₈	—	137	1	no	—
42	C₂×C₈	—	137	1	no	—
43	C₂×C₈	—	137	1	no	—
44	C₄²	C₂×#10587739	1	1	yes	characteristic
45	C₁₆	—	548	2	no	—
46	C₂×C₄	C₄⋊#10587739	1	1	yes	characteristic
47	C₂×C₄	#14555518	1	1	yes	characteristic
48	C₂×C₄	#14555519	1	1	yes	characteristic
49	C₈	—	2	2	no	—
50	C₈	—	274	1	no	—
51	C₈	—	274	1	no	—
52	C₈	—	137	2	no	—
53	C₈	—	137	2	no	—
54	C₂²	#17984155	1	1	yes	characteristic
55	C₄	—	2	1	no	—
56	C₄	#17984156	1	1	yes	characteristic
57	C₄	#17984159	1	1	yes	characteristic
58	C₄	#17984161	1	1	yes	characteristic
59	C₄	#17984163	1	1	yes	characteristic
60	C₂	#22578265	1	1	yes	characteristic
61	C₂	#22578324	1	1	yes	characteristic
62	C₂	#22578326	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#10587739
commutator	C₄×C₁₃₇	C₂×C₁₆
Frattini	C₄²	C₂×#10587739
Fitting	C₄×C₈×C₁₃₇	C₄
socle, prime radical	C₂²×C₁₃₇	C₄⋊C₈
cosocle	C₄×#11764040	C₂²
hypercenter	C₄×C₈	#10587739
hypocenter	C₁₃₇	#1367
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
16. #11764040	C₁₆	semidirect
31. C₁₃₇	#1367	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#14555513	2
C₈×#11764040	2
#1367	137