Subgroups — #31797821

Order 17536.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578502	C₂	1	2	yes	maximal, complemented
2	#22578378	C₂	1	1	yes	maximal, characteristic
3	#17984184	—	2	2	no	—
4	C₂×#14555520	C₂²	1	1	yes	characteristic
5	C₂×#14555518	C₄	1	1	yes	characteristic
6	#17984145	—	2	2	no	—
7	C₂×#14555511	C₄	1	1	yes	characteristic
8	#14555522	—	4	2	no	—
9	#14555520	—	4	1	no	—
10	#14555518	—	4	1	no	—
11	C₂×#11764047	—	2	1	no	—
12	C₂×#11764047	D₄	1	2	yes	—
13	#14555511	—	4	1	no	—
14	#14555511	—	4	1	no	—
15	C₂×D₅₄₈	—	2	1	no	—
16	C₂×D₅₄₈	C₈	1	1	yes	characteristic
17	C₂×C₄×D₁₃₇	—	2	1	no	—
18	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂×Q₈×C₁₃₇	C₈	1	1	yes	characteristic
20	#11764047	—	2	2	no	—
21	#11764047	—	4	1	no	—
22	C₂²×D₁₃₇	—	2	1	no	—
23	C₂²×D₁₃₇	M₁₆	1	1	yes	characteristic
24	D₅₄₈	—	4	1	no	—
25	D₅₄₈	—	4	2	no	—
26	C₂×Dic₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₄×D₁₃₇	—	2	1	no	—
28	C₄×D₁₃₇	—	2	1	no	—
29	C₄×D₁₃₇	—	4	1	no	—
30	C₄×D₁₃₇	—	4	1	no	—
31	Q₈×C₁₃₇	—	4	1	no	—
32	C₂×C₄×C₁₃₇	—	2	1	no	—
33	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
35	C₂×D₁₃₇	—	2	1	no	—
36	C₂×D₁₃₇	—	4	1	no	—
37	C₂×D₁₃₇	—	4	2	no	—
38	C₂×D₁₃₇	—	4	1	no	—
39	C₄×C₁₃₇	—	2	1	no	—
40	C₄×C₁₃₇	—	4	1	no	—
41	Dic₁₃₇	#99	1	1	yes	characteristic
42	Dic₁₃₇	#100	1	1	yes	characteristic
43	C₂×C₁₃₇	—	2	1	no	—
44	C₂×C₁₃₇	#323	1	1	yes	characteristic
45	D₁₃₇	—	4	1	no	—
46	D₁₃₇	—	8	1	no	—
47	C₁₃₇	#1321	1	1	yes	characteristic, complemented, Sylow
48	#1321	—	137	1	no	maximal, Sylow
49	#408	—	137	1	no	—
50	#349	—	137	2	no	—
51	C₂×#41	—	137	1	no	—
52	C₂×M₁₆	—	137	1	no	—
53	C₂²×C₈	—	137	1	no	—
54	M₃₂	—	274	2	no	—
55	C₂²⋊C₈	—	274	2	no	—
56	#41	—	548	1	no	—
57	#41	—	548	1	no	—
58	C₂×Q₈	#11764049	1	1	yes	characteristic
59	C₂×D₄	—	274	1	no	—
60	C₂×D₄	—	137	1	no	—
61	C₂²×C₄	—	274	1	no	—
62	C₂²×C₄	—	137	1	no	—
63	M₁₆	—	548	1	no	—
64	C₂×C₈	—	274	1	no	—
65	C₂×C₈	—	548	1	no	—
66	C₂×C₈	—	137	2	no	—
67	C₁₆	—	548	2	no	—
68	C₂³	—	274	1	no	—
69	C₂³	—	137	1	no	—
70	Q₈	—	4	1	no	—
71	D₄	—	548	1	no	—
72	D₄	—	548	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	274	1	no	—
75	C₂×C₄	—	274	1	no	—
76	C₂×C₄	—	548	1	no	—
77	C₂×C₄	—	548	1	no	—
78	C₂×C₄	—	137	1	no	—
79	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
80	C₈	—	274	2	no	—
81	C₈	—	548	1	no	—
82	C₂²	—	274	1	no	—
83	C₂²	—	548	1	no	—
84	C₂²	—	548	2	no	—
85	C₂²	—	548	1	no	—
86	C₂²	C₂²⋊#11764049	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	4	1	no	—
89	C₄	—	137	1	no	—
90	C₄	—	137	1	no	—
91	C₂	—	2	1	no	—
92	C₂	—	548	1	no	—
93	C₂	—	1096	1	no	—
94	C₂	#22578488	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22578488
commutator	C₂×C₄×C₁₃₇	C₂×C₈
Frattini	C₂×C₄	C₂×#11764049
Fitting	C₂×Q₈×C₁₃₇	C₈
socle	C₂×C₁₃₇	#323
prime radical	C₂×#14555511	C₄
cosocle	C₂×#14555520	C₂²
hypercenter	C₂×Q₈	#11764049
hypocenter	C₁₃₇	#1321
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #22578502	C₂	semidirect
47. C₁₃₇	#1321	semidirect

Maximal subgroups

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

subgroup	index
#22578502	2
#22578378	2
#1321	137