Subgroups — #31795745

Order 17536.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578289	C₂	1	1	yes	maximal, characteristic
2	#22578266	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555513	—	2	1	no	—
5	C₄⋊#11764047	—	2	1	no	—
6	Q₈×Dic₁₃₇	—	2	1	no	—
7	#17984022	C₄	1	1	yes	characteristic
8	#17984012	—	2	1	no	—
9	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×Q₁₆×C₁₃₇	C₄	1	1	yes	characteristic
11	C₂×#11764047	—	4	1	no	—
12	#14555513	—	2	2	no	—
13	C₂×#11764040	D₄	1	1	yes	characteristic
14	C₄⋊Dic₁₃₇	—	2	1	no	—
15	C₄⋊Dic₁₃₇	—	4	1	no	—
16	C₄×Dic₁₃₇	—	4	1	no	—
17	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
18	C₂×Q₈×C₁₃₇	—	2	1	no	—
19	Q₁₆×C₁₃₇	—	2	1	no	—
20	Q₁₆×C₁₃₇	C₈	1	2	yes	complemented
21	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
22	#11764047	—	8	1	no	—
23	C₂×Dic₁₃₇	—	4	1	no	—
24	C₂×Dic₁₃₇	D₈	1	1	yes	characteristic
25	C₂×Dic₁₃₇	SD₁₆	1	1	yes	characteristic
26	#11764040	—	2	1	no	—
27	Q₈×C₁₃₇	—	2	2	no	—
28	Q₈×C₁₃₇	—	4	1	no	—
29	C₂×C₄×C₁₃₇	—	4	1	no	—
30	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
32	C₈×C₁₃₇	M₁₆	1	1	yes	characteristic
33	C₂²×C₁₃₇	#102	1	1	yes	characteristic
34	C₄×C₁₃₇	—	4	2	no	—
35	C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
36	C₄×C₁₃₇	#104	1	1	yes	characteristic
37	Dic₁₃₇	—	2	1	no	—
38	Dic₁₃₇	—	8	1	no	—
39	Dic₁₃₇	SD₃₂	1	1	yes	characteristic
40	Dic₁₃₇	Q₃₂	1	1	yes	characteristic
41	C₂×C₁₃₇	D₄⋊C₈	1	1	yes	characteristic
42	C₂×C₁₃₇	#358	1	1	yes	characteristic
43	C₂×C₁₃₇	#359	1	1	yes	characteristic
44	C₁₃₇	#1332	1	1	yes	characteristic, complemented, Sylow
45	#1332	—	137	1	no	maximal, Sylow
46	C₄×Q₁₆	—	137	1	no	—
47	C₄×C₁₆	—	137	1	no	—
48	#335	—	137	1	no	—
49	C₂×Q₁₆	#10587739	1	1	yes	characteristic
50	C₄×Q₈	—	274	1	no	—
51	C₂×C₁₆	—	274	1	no	—
52	#107	—	137	1	no	—
53	C₄⋊C₈	—	274	1	no	—
54	#103	—	274	1	no	—
55	C₄×C₈	—	137	1	no	—
56	C₂×Q₈	—	2	1	no	—
57	Q₁₆	—	2	1	no	—
58	Q₁₆	#11764047	1	2	yes	complemented
59	C₂×C₈	—	548	1	no	—
60	C₂×C₈	—	137	1	no	—
61	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
62	C₄⋊C₄	—	274	1	no	—
63	C₄⋊C₄	—	548	1	no	—
64	C₄²	—	548	1	no	—
65	C₄²	—	137	1	no	—
66	C₁₆	—	274	2	no	—
67	Q₈	—	2	2	no	—
68	Q₈	—	4	1	no	—
69	C₂×C₄	—	4	1	no	—
70	C₂×C₄	—	548	1	no	—
71	C₂×C₄	—	137	1	no	—
72	C₂×C₄	—	137	1	no	—
73	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
74	C₈	—	274	1	no	—
75	C₈	—	1096	1	no	—
76	C₈	C₂×#11764047	1	1	yes	characteristic
77	C₈	#14555519	1	1	yes	characteristic
78	C₂²	#17984150	1	1	yes	characteristic
79	C₄	—	4	2	no	—
80	C₄	—	274	1	no	—
81	C₄	—	1096	1	no	—
82	C₄	—	137	1	no	—
83	C₄	—	137	1	no	—
84	C₄	C₂²⋊#11764047	1	1	yes	characteristic
85	C₄	#17984152	1	1	yes	characteristic
86	C₂	D₄⋊#11764047	1	1	yes	characteristic
87	C₂	#22578316	1	1	yes	characteristic
88	C₂	#22578318	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17984150
commutator	C₈×C₁₃₇	C₂×C₈
Frattini	C₂×C₈	C₂×#10587739
Fitting	C₂×Q₁₆×C₁₃₇	C₄
socle, prime radical	C₂²×C₁₃₇	#102
cosocle	C₈×Dic₁₃₇	C₂²
hypercenter	C₂×Q₁₆	#10587739
hypocenter	C₁₃₇	#1332
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
20. Q₁₆×C₁₃₇	C₈	semidirect
44. C₁₃₇	#1332	semidirect
58. Q₁₆	#11764047	semidirect

Maximal subgroups

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

subgroup	index
#22578289	2
#22578266	2
Q₁₆×Dic₁₃₇	2
#1332	137