Subgroups — #31778950

Order 17536.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577798	C₂	1	1	yes	maximal, characteristic
2	#22577198	C₂	1	1	yes	maximal, characteristic
3	#355×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555500	—	2	1	no	—
5	C₂×#14555498	—	2	1	no	—
6	#17984088	C₄	1	1	yes	characteristic, complemented
7	#17984003	—	2	1	no	—
8	#17983999	C₄	1	1	yes	characteristic, complemented
9	#124×C₁₃₇	C₂²	1	1	yes	characteristic
10	#101×C₁₃₇	—	2	1	no	—
11	C₂×Dic₂₇₄	—	2	1	no	—
12	#14555500	—	4	2	no	—
13	#14555498	—	4	2	no	—
14	C₂×#11764040	—	2	1	no	—
15	C₄⋊Dic₁₃₇	—	4	1	no	—
16	C₂²⋊Dic₁₃₇	—	4	1	no	—
17	C₂×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
18	C₂×D₄×C₁₃₇	D₄	1	1	yes	characteristic
19	M₁₆×C₁₃₇	—	4	1	no	—
20	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
21	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
22	Dic₂₇₄	—	4	1	no	—
23	Dic₂₇₄	—	4	1	no	—
24	C₂×Dic₁₃₇	—	4	1	no	—
25	C₂×Dic₁₃₇	—	8	1	no	—
26	#11764040	—	2	2	no	—
27	C₂³×C₁₃₇	—	2	1	no	—
28	Q₈×C₁₃₇	—	4	1	no	—
29	D₄×C₁₃₇	—	4	1	no	—
30	C₂×C₄×C₁₃₇	—	2	1	no	—
31	C₂×C₄×C₁₃₇	—	2	1	no	—
32	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₈×C₁₃₇	—	8	1	no	—
34	C₂²×C₁₃₇	—	4	1	no	—
35	C₂²×C₁₃₇	—	8	1	no	—
36	C₂²×C₁₃₇	#99	1	1	yes	characteristic
37	C₄×C₁₃₇	—	2	1	no	—
38	C₄×C₁₃₇	—	4	1	no	—
39	C₄×C₁₃₇	—	4	1	no	—
40	Dic₁₃₇	—	8	1	no	—
41	Dic₁₃₇	—	16	1	no	—
42	C₂×C₁₃₇	—	2	1	no	—
43	C₂×C₁₃₇	—	8	1	no	—
44	C₂×C₁₃₇	#353	1	1	yes	characteristic
45	C₁₃₇	#1407	1	1	yes	characteristic, complemented, Sylow
46	#1407	—	137	1	no	maximal, Sylow
47	#497	—	137	1	no	—
48	#355	D₁₃₇	1	1	yes	characteristic
49	#354	—	137	1	no	—
50	C₂×Q₁₆	—	274	1	no	—
51	C₂×SD₁₆	—	274	1	no	—
52	C₄⋊Q₈	—	137	1	no	—
53	#124	C₂×D₁₃₇	1	1	yes	characteristic
54	#101	—	2	1	no	—
55	#99	—	274	1	no	—
56	#97	—	137	1	no	—
57	C₂×Q₈	—	274	1	no	—
58	C₂×Q₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
59	C₂×D₄	D₅₄₈	1	1	yes	characteristic
60	Q₁₆	—	548	2	no	—
61	SD₁₆	—	548	2	no	—
62	M₁₆	—	4	1	no	—
63	C₂×C₈	—	274	1	no	—
64	C₄⋊C₄	—	548	1	no	—
65	C₂²⋊C₄	—	4	1	no	—
66	C₂²⋊C₄	—	548	1	no	—
67	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
68	C₂³	—	2	1	no	—
69	Q₈	—	4	1	no	—
70	Q₈	—	548	1	no	—
71	Q₈	—	548	1	no	—
72	D₄	—	4	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	548	1	no	—
76	C₂×C₄	—	1096	1	no	—
77	C₂×C₄	#14555487	1	1	yes	characteristic
78	C₈	—	8	1	no	—
79	C₈	—	274	2	no	—
80	C₂²	—	4	1	no	—
81	C₂²	—	8	1	no	—
82	C₂²	#17984002	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	1096	1	no	—
87	C₄	—	2192	1	no	—
88	C₂	—	2	1	no	—
89	C₂	—	8	1	no	—
90	C₂	#22577195	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22577195
commutator	C₄²×C₁₃₇	C₂×C₄
Frattini	#124	C₂×D₁₃₇
Fitting	#355×C₁₃₇	C₂
socle	C₂×C₁₃₇	#353
prime radical	C₂×D₄×C₁₃₇	D₄
cosocle	#124×C₁₃₇	C₂²
hypercenter	#355	D₁₃₇
hypocenter	C₁₃₇	#1407
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #355

Complements

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

N	H ≅ G/N	type
6. #17984088	C₄	semidirect
8. #17983999	C₄	semidirect
45. C₁₃₇	#1407	semidirect

Maximal subgroups

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

subgroup	index
#22577798	2
#22577198	2
#355×C₁₃₇	2
#1407	137