Subgroups — #14543727

Order 2176.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763654	C₂	1	2	yes	maximal
2	#11762678	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×C₁₇	C₄	1	1	yes	characteristic
4	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
5	C₂×#63225	—	2	2	no	—
6	C₂×C₄×Dic₁₇	C₄	1	1	yes	characteristic
7	C₂²⋊#3624	—	2	2	no	—
8	C₂²×C₄×C₁₇	—	2	1	no	—
9	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
10	C₂²×Dic₁₇	—	2	1	no	—
11	C₂²×Dic₁₇	D₄	1	2	yes	—
12	C₂×#3624	—	4	2	no	—
13	C₄⋊C₄×C₁₇	—	4	1	no	—
14	C₄⋊Dic₁₇	—	4	1	no	—
15	#63225	—	4	4	no	—
16	C₄×Dic₁₇	—	2	1	no	—
17	C₄×Dic₁₇	C₈	1	2	yes	complemented
18	C₂³×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₇	—	2	1	no	—
20	C₂×C₄×C₁₇	—	2	1	no	—
21	C₂×C₄×C₁₇	—	4	2	no	—
22	C₂×C₄×C₁₇	—	4	1	no	—
23	C₂×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
24	C₂×C₄×C₁₇	M₁₆	1	1	yes	characteristic
25	C₂×Dic₁₇	—	2	2	no	—
26	C₂×Dic₁₇	—	2	2	no	—
27	C₂×Dic₁₇	—	2	2	no	—
28	C₂×Dic₁₇	—	2	2	no	—
29	C₂×Dic₁₇	—	4	2	no	—
30	#3624	—	8	2	no	—
31	#1325	—	17	1	no	maximal, Sylow
32	C₂²×C₁₇	—	2	1	no	—
33	C₂²×C₁₇	—	2	1	no	—
34	C₂²×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
35	C₂²×C₁₇	#99	1	1	yes	characteristic
36	C₂²×C₁₇	#101	1	1	yes	characteristic
37	C₄×C₁₇	—	2	1	no	—
38	C₄×C₁₇	—	2	1	no	—
39	C₄×C₁₇	—	8	1	no	—
40	Dic₁₇	—	2	4	no	—
41	Dic₁₇	—	4	2	no	—
42	#384	—	17	1	no	—
43	#324	—	17	2	no	—
44	C₂×C₁₇	—	2	1	no	—
45	C₂×C₁₇	—	2	1	no	—
46	C₂×C₁₇	#324	1	1	yes	characteristic
47	C₂×C₁₇	#353	1	1	yes	characteristic
48	C₂×C₁₇	#356	1	1	yes	characteristic
49	C₂×C₄⋊C₄	—	17	1	no	—
50	C₂×C₄⋊C₄	—	34	2	no	—
51	C₂×C₄⋊C₄	#595	1	1	yes	characteristic
52	C₂×C₄²	—	17	1	no	—
53	C₂²⋊C₈	—	34	2	no	—
54	C₁₇	#1325	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₄	—	2	1	no	—
56	C₂²×C₄	—	17	2	no	—
57	C₂²×C₄	—	34	1	no	—
58	C₂²×C₄	C₂×#595	1	1	yes	characteristic
59	C₂×C₈	—	68	2	no	—
60	C₄⋊C₄	—	4	1	no	—
61	C₄⋊C₄	—	68	1	no	—
62	C₄⋊C₄	—	68	4	no	—
63	C₄²	—	17	2	no	—
64	C₄²	—	34	1	no	—
65	C₂³	C₂²⋊#595	1	1	yes	characteristic
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	4	1	no	—
70	C₂×C₄	—	34	2	no	—
71	C₂×C₄	—	34	2	no	—
72	C₂×C₄	—	34	2	no	—
73	C₂×C₄	—	34	2	no	—
74	C₂×C₄	—	68	2	no	—
75	C₂×C₄	#63242	1	1	yes	characteristic
76	C₂×C₄	#63243	1	1	yes	characteristic
77	C₈	—	136	2	no	—
78	C₂²	—	2	1	no	—
79	C₂²	—	2	1	no	—
80	C₂²	#10587543	1	1	yes	characteristic
81	C₂²	#10587545	1	1	yes	characteristic
82	C₂²	#10587551	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	2	1	no	—
85	C₄	—	8	1	no	—
86	C₄	—	34	4	no	—
87	C₄	—	68	2	no	—
88	C₂	—	2	1	no	—
89	C₂	—	2	1	no	—
90	C₂	#11763655	1	1	yes	characteristic
91	C₂	#11763713	1	1	yes	characteristic
92	C₂	#11763719	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10587551
commutator	C₂×C₄×C₁₇	C₂×C₈
Frattini	C₂²×C₄	C₂×#595
Fitting	C₂×C₄⋊C₄×C₁₇	C₄
socle	C₂²×C₁₇	#99
prime radical	C₂³×C₁₇	C₂²⋊C₄
cosocle	C₂×C₄⋊Dic₁₇	C₂²
hypercenter	C₂×C₄⋊C₄	#595
hypocenter	C₁₇	#1325
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
17. C₄×Dic₁₇	C₈	semidirect
54. C₁₇	#1325	semidirect

Maximal subgroups

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

subgroup	index
#11763654	2
#11762678	2
#1325	17