Subgroups — #14543707

Order 2176.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#63216	C₂	1	2	yes	maximal, complemented
2	#11762777	C₂	1	1	yes	maximal, characteristic
3	C₂×#63252	C₄	1	1	yes	characteristic
4	C₂×M₁₆×C₁₇	C₄	1	1	yes	characteristic
5	C₂²×#3622	C₂²	1	1	yes	characteristic
6	C₂×#63216	—	2	2	no	—
7	#10587491	—	2	2	no	—
8	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
9	C₂×C₂²⋊D₁₇	—	2	1	no	—
10	#63252	—	4	1	no	—
11	#63252	—	4	1	no	—
12	C₂×D₆₈	C₈	1	1	yes	characteristic
13	C₂×C₄×D₁₇	—	2	1	no	—
14	C₂×Dic₃₄	C₈	1	1	yes	characteristic
15	M₁₆×C₁₇	—	4	1	no	—
16	C₂×C₈×C₁₇	—	2	1	no	—
17	C₂×#3622	—	4	1	no	—
18	C₂×#3622	D₄	1	2	yes	—
19	#63216	—	4	2	no	—
20	C₂³×C₁₇	M₁₆	1	1	yes	characteristic
21	C₂²×D₁₇	—	2	1	no	—
22	C₂×C₄×C₁₇	—	2	1	no	—
23	C₂×C₄×C₁₇	—	2	1	no	—
24	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
26	C₂²⋊D₁₇	—	4	2	no	—
27	C₂×Dic₁₇	—	2	1	no	—
28	D₆₈	—	4	1	no	—
29	C₄×D₁₇	—	4	1	no	—
30	C₄×D₁₇	—	4	1	no	—
31	Dic₃₄	—	4	1	no	—
32	C₈×C₁₇	—	4	1	no	—
33	#3622	—	2	2	no	—
34	#1315	—	17	1	no	maximal, Sylow
35	C₂²×C₁₇	—	2	1	no	—
36	C₂²×C₁₇	—	2	2	no	—
37	C₂²×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
38	C₂×D₁₇	—	4	1	no	—
39	C₂×D₁₇	—	4	2	no	—
40	C₄×C₁₇	—	2	1	no	—
41	C₄×C₁₇	#99	1	1	yes	characteristic
42	C₄×C₁₇	#100	1	1	yes	characteristic
43	Dic₁₇	—	4	1	no	—
44	#408	—	17	1	no	—
45	C₂²⋊C₁₆	—	17	2	no	—
46	C₂×C₁₇	—	4	1	no	—
47	C₂×C₁₇	#323	1	1	yes	characteristic
48	C₂×C₁₇	#350	1	2	yes	—
49	D₁₇	—	8	1	no	—
50	C₂×#41	—	17	1	no	—
51	C₂×M₁₆	#595	1	1	yes	characteristic
52	C₂²×C₈	—	17	1	no	—
53	C₂×C₁₆	—	34	2	no	—
54	C₂²⋊C₈	—	34	2	no	—
55	C₁₇	#1315	1	1	yes	characteristic, complemented, Sylow
56	#41	—	68	1	no	—
57	#41	—	68	1	no	—
58	C₂×Q₈	—	17	1	no	—
59	C₂×D₄	—	17	1	no	—
60	C₂×D₄	—	34	1	no	—
61	C₂²×C₄	—	34	1	no	—
62	C₂²×C₄	C₂×#595	1	1	yes	characteristic
63	M₁₆	—	4	1	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	—	17	2	no	—
66	C₂×C₈	—	68	1	no	—
67	C₁₆	—	68	2	no	—
68	C₂³	—	34	1	no	—
69	C₂³	#63243	1	1	yes	characteristic
70	Q₈	—	68	1	no	—
71	D₄	—	68	1	no	—
72	D₄	—	68	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	34	1	no	—
76	C₂×C₄	—	68	1	no	—
77	C₂×C₄	—	68	1	no	—
78	C₂×C₄	#63242	1	1	yes	characteristic
79	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
80	C₈	—	4	1	no	—
81	C₈	—	34	2	no	—
82	C₂²	—	2	1	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	68	1	no	—
85	C₂²	—	68	2	no	—
86	C₂²	#10587543	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	68	1	no	—
89	C₄	#10587539	1	1	yes	characteristic
90	C₄	#10587553	1	1	yes	characteristic
91	C₂	—	4	1	no	—
92	C₂	—	136	1	no	—
93	C₂	#11763653	1	1	yes	characteristic
94	C₂	#11763708	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊#595
commutator	C₂×C₄×C₁₇	C₂×C₈
Frattini	C₂²×C₄	C₂×#595
Fitting	C₂×M₁₆×C₁₇	C₄
socle	C₂²×C₁₇	C₂²⋊C₈
prime radical	C₂×#63252	C₄
cosocle	C₂²×#3622	C₂²
hypercenter	C₂×M₁₆	#595
hypocenter	C₁₇	#1315
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²⋊#63216	C₂	semidirect
55. C₁₇	#1315	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊#63216	2
#11762777	2
#1315	17