Subgroups — #31798044

Order 17536.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578525	C₂	1	8	yes	maximal, complemented
2	#22578522	C₂	1	6	yes	maximal, complemented
3	C₂×#17984176	C₂	1	1	yes	maximal, characteristic
4	#17984185	C₂²	1	12	yes	complemented
5	#17984184	C₂²	1	4	yes	—
6	#17984183	C₂²	1	12	yes	complemented
7	#17984176	C₄	1	4	yes	—
8	#17984176	C₂²	1	4	yes	—
9	C₂×#14555520	C₄	1	3	yes	—
10	C₂×#14555518	C₂²	1	3	yes	—
11	C₂×#14555511	C₄	1	1	yes	characteristic
12	#14555522	C₂³	1	8	yes	—
13	#14555520	C₂×C₄	1	12	yes	—
14	#14555518	C₂×C₄	1	6	yes	—
15	#14555518	C₂³	1	6	yes	—
16	C₂×#11764047	C₂×C₄	1	3	yes	—
17	C₂×#11764047	C₂³	1	1	yes	characteristic
18	#14555511	C₈	1	4	yes	—
19	#14555511	C₂×C₄	1	4	yes	—
20	C₂×D₅₄₈	C₈	1	3	yes	—
21	C₂×C₄×D₁₃₇	C₂×C₄	1	3	yes	—
22	C₂×Q₈×C₁₃₇	C₈	1	1	yes	characteristic
23	#11764047	C₂²×C₄	1	6	yes	—
24	#11764047	C₂²×C₄	1	1	yes	characteristic
25	#11764047	C₂⁴	1	1	yes	characteristic
26	C₂²×D₁₃₇	C₂×C₈	1	3	yes	—
27	D₅₄₈	C₂×C₈	1	12	yes	—
28	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
29	C₄×D₁₃₇	C₂×C₈	1	6	yes	—
30	C₄×D₁₃₇	C₂²×C₄	1	6	yes	—
31	Q₈×C₁₃₇	C₂×C₈	1	4	yes	—
32	C₂×C₄×C₁₃₇	C₂×C₈	1	3	yes	—
33	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
34	C₂×D₁₃₇	—	4	3	no	—
35	C₂×D₁₃₇	C₂²×C₈	1	6	yes	—
36	C₄×C₁₃₇	C₂²×C₈	1	6	yes	—
37	Dic₁₃₇	C₂²×C₈	1	1	yes	characteristic
38	Dic₁₃₇	C₂³×C₄	1	1	yes	characteristic
39	C₂×C₁₃₇	—	2	1	no	—
40	C₂×C₁₃₇	C₂³×C₈	1	1	yes	characteristic
41	D₁₃₇	—	2	6	no	—
42	C₁₃₇	#3407	1	1	yes	characteristic, complemented, Sylow
43	#3407	—	137	1	no	maximal, Sylow
44	C₂×#131	—	137	1	no	—
45	#504	—	137	8	no	—
46	C₂×M₃₂	—	137	6	no	—
47	C₂×#41	—	137	1	no	—
48	#131	—	137	4	no	—
49	#131	—	137	4	no	—
50	C₂×M₁₆	—	137	3	no	—
51	C₂²×C₈	—	137	3	no	—
52	M₃₂	—	137	4	no	—
53	M₃₂	—	137	12	no	—
54	C₂×C₁₆	—	137	12	no	—
55	#41	—	137	4	no	—
56	#41	—	137	4	no	—
57	C₂×Q₈	#11764049	1	1	yes	characteristic
58	C₂×D₄	—	137	3	no	—
59	C₂²×C₄	—	137	3	no	—
60	M₁₆	—	137	12	no	—
61	C₂×C₈	—	137	6	no	—
62	C₂×C₈	—	137	1	no	—
63	C₂×C₈	—	137	3	no	—
64	C₂×C₈	—	137	6	no	—
65	C₁₆	—	137	8	no	—
66	C₂³	—	137	3	no	—
67	Q₈	C₂×#11764049	1	4	yes	—
68	D₄	—	137	12	no	—
69	C₂×C₄	—	137	1	no	—
70	C₂×C₄	—	137	6	no	—
71	C₂×C₄	—	137	6	no	—
72	C₂×C₄	C₂×#11764049	1	3	yes	—
73	C₈	—	137	1	no	—
74	C₈	—	137	1	no	—
75	C₈	—	137	6	no	—
76	C₂²	—	548	3	no	—
77	C₂²	—	137	6	no	—
78	C₂²	C₂²×#11764049	1	1	yes	characteristic
79	C₄	—	137	1	no	—
80	C₄	—	137	1	no	—
81	C₄	C₂²×#11764049	1	6	yes	—
82	C₂	—	2	1	no	—
83	C₂	—	274	6	no	—
84	C₂	C₂³×#11764049	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#11764049
commutator, socle	C₂×C₁₃₇	C₂³×C₈
Fitting	C₂×Q₈×C₁₃₇	C₈
prime radical	C₂×#14555511	C₄
cosocle	#11764047	C₂⁴
hypercenter	C₂×Q₈	#11764049
hypocenter	C₁₃₇	#3407
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #22578525	C₂	semidirect
2. #22578522	C₂	semidirect
4. #17984185	C₂²	semidirect
6. #17984183	C₂²	semidirect
42. C₁₃₇	#3407	semidirect

Maximal subgroups

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

subgroup	index
#22578525	2
#22578522	2
C₂×#17984176	2
#3407	137