Subgroups — #31795804

Order 17536.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578324	C₂	1	1	yes	maximal, characteristic
2	#22578267	C₂	1	1	yes	maximal, characteristic
3	#22577781	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555513	—	2	1	no	—
5	C₄⋊#11764047	—	2	1	no	—
6	C₂×Dic₅₄₈	C₄	1	1	yes	characteristic
7	C₂×#14555500	—	2	1	no	—
8	#17984086	—	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	—	4	1	no	—
13	C₂×Dic₂₇₄	—	2	1	no	—
14	Dic₅₄₈	—	4	1	no	—
15	#14555500	—	4	1	no	—
16	#14555500	—	4	1	no	—
17	C₂×#11764040	D₄	1	1	yes	characteristic
18	#14555489	—	4	1	no	—
19	#14555489	—	4	1	no	—
20	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
21	C₂×Q₈×C₁₃₇	—	2	1	no	—
22	Q₁₆×C₁₃₇	—	4	1	no	—
23	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
24	#11764047	—	8	1	no	—
25	Dic₂₇₄	—	4	1	no	—
26	Dic₂₇₄	—	4	1	no	—
27	C₂×Dic₁₃₇	—	4	1	no	—
28	C₂×Dic₁₃₇	D₈	1	1	yes	characteristic
29	C₂×Dic₁₃₇	SD₁₆	1	1	yes	characteristic
30	#11764040	D₈	1	1	yes	characteristic
31	#11764040	SD₁₆	1	1	yes	characteristic
32	Q₈×C₁₃₇	—	4	1	no	—
33	Q₈×C₁₃₇	—	4	1	no	—
34	C₂×C₄×C₁₃₇	—	4	1	no	—
35	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₈×C₁₃₇	—	2	1	no	—
37	C₂²×C₁₃₇	#102	1	1	yes	characteristic
38	C₄×C₁₃₇	—	8	1	no	—
39	C₄×C₁₃₇	#100	1	1	yes	characteristic
40	C₄×C₁₃₇	#102	1	1	yes	characteristic
41	Dic₁₃₇	—	2	1	no	—
42	Dic₁₃₇	—	8	1	no	—
43	Dic₁₃₇	SD₃₂	1	1	yes	characteristic
44	Dic₁₃₇	Q₃₂	1	1	yes	characteristic
45	C₂×C₁₃₇	#331	1	1	yes	characteristic
46	C₂×C₁₃₇	#358	1	1	yes	characteristic
47	C₂×C₁₃₇	#362	1	1	yes	characteristic
48	C₁₃₇	#1362	1	1	yes	characteristic, complemented, Sylow
49	#1362	—	137	1	no	maximal, Sylow
50	#494	—	137	1	no	—
51	C₄⋊C₁₆	—	137	1	no	—
52	#335	—	137	1	no	—
53	C₂×Q₁₆	—	274	1	no	—
54	C₂×Q₁₆	—	137	1	no	—
55	C₂×Q₁₆	#10587739	1	1	yes	characteristic
56	C₄⋊Q₈	—	274	1	no	—
57	C₂×C₁₆	—	274	1	no	—
58	C₄⋊C₈	—	274	1	no	—
59	C₄×C₈	—	137	1	no	—
60	C₂×Q₈	—	2	1	no	—
61	C₂×Q₈	—	274	1	no	—
62	Q₁₆	—	4	1	no	—
63	Q₁₆	—	548	1	no	—
64	Q₁₆	—	548	1	no	—
65	Q₁₆	—	548	1	no	—
66	C₂×C₈	—	548	1	no	—
67	C₂×C₈	—	137	1	no	—
68	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
69	C₄⋊C₄	—	548	1	no	—
70	C₄⋊C₄	—	548	1	no	—
71	C₄²	—	137	1	no	—
72	C₁₆	—	548	1	no	—
73	Q₈	—	4	1	no	—
74	Q₈	—	4	1	no	—
75	Q₈	—	548	1	no	—
76	Q₈	—	548	1	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	—	548	1	no	—
79	C₂×C₄	—	137	1	no	—
80	C₂×C₄	—	137	1	no	—
81	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
82	C₈	—	2	1	no	—
83	C₈	—	1096	1	no	—
84	C₈	—	137	1	no	—
85	C₈	—	137	1	no	—
86	C₂²	#17984150	1	1	yes	characteristic
87	C₄	—	8	1	no	—
88	C₄	—	274	1	no	—
89	C₄	—	1096	1	no	—
90	C₄	—	137	1	no	—
91	C₄	—	137	1	no	—
92	C₄	#17984148	1	1	yes	characteristic
93	C₄	#17984150	1	1	yes	characteristic
94	C₂	#22578261	1	1	yes	characteristic
95	C₂	#22578316	1	1	yes	characteristic
96	C₂	#22578322	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17984150
commutator	C₂×C₈×C₁₃₇	C₂×C₄
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	#1362
LCS	G ▷ C₂×C₈×C₁₃₇ ▷ C₂×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
48. C₁₃₇	#1362	semidirect

Maximal subgroups

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

subgroup	index
#22578324	2
#22578267	2
#22577781	2
#1362	137