Subgroups — #14530596

Order 2176.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11762630	C₂	1	2	yes	maximal, complemented
2	#11762591	C₂	1	2	yes	maximal
3	#11762587	C₂	1	2	yes	maximal, complemented
4	C₄×M₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×M₁₆×C₁₇	C₄	1	1	yes	characteristic
6	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
7	C₂×C₄²×C₁₇	C₄	1	1	yes	characteristic
8	C₁₇×#97	C₄	1	1	yes	characteristic
9	C₁₇×#97	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
11	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
12	#10587483	C₂²	1	4	yes	—
13	C₂²×C₄×C₁₇	—	2	1	no	—
14	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
15	M₁₆×C₁₇	—	2	2	no	—
16	M₁₆×C₁₇	—	2	2	no	—
17	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
21	C₄²×C₁₇	—	2	1	no	—
22	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
24	#63214	—	2	4	no	—
25	C₂³×C₁₇	M₁₆	1	1	yes	characteristic
26	C₂×C₄×C₁₇	—	2	1	no	—
27	C₂×C₄×C₁₇	—	2	1	no	—
28	C₂×C₄×C₁₇	—	2	1	no	—
29	C₂×C₄×C₁₇	—	2	1	no	—
30	C₂×C₄×C₁₇	—	4	2	no	—
31	C₂×C₄×C₁₇	M₁₆	1	1	yes	characteristic
32	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₁₇	—	2	1	no	—
34	C₈×C₁₇	—	2	1	no	—
35	C₈×C₁₇	M₁₆	1	2	yes	—
36	C₈×C₁₇	#41	1	2	yes	—
37	#2165	—	17	1	no	maximal, Sylow
38	C₂²×C₁₇	—	2	1	no	—
39	C₂²×C₁₇	—	4	1	no	—
40	C₂²×C₁₇	C₂×M₁₆	1	1	yes	characteristic
41	C₄×C₁₇	—	2	2	no	—
42	C₄×C₁₇	—	2	1	no	—
43	C₄×C₁₇	—	4	1	no	—
44	C₄×C₁₇	#117	1	1	yes	characteristic
45	C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
46	C₄×M₁₆	D₁₇	1	1	yes	characteristic
47	#364	—	17	2	no	—
48	#349	—	17	2	no	—
49	#347	—	17	2	no	—
50	C₂×C₁₇	—	2	1	no	—
51	C₂×C₁₇	—	4	1	no	—
52	C₂×C₁₇	#432	1	1	yes	characteristic
53	C₂×M₁₆	Dic₁₇	1	1	yes	characteristic
54	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
55	C₂×C₄²	Dic₁₇	1	1	yes	characteristic
56	M₃₂	—	17	4	no	—
57	#97	Dic₁₇	1	1	yes	characteristic
58	#97	C₂×D₁₇	1	1	yes	characteristic
59	C₄×C₈	Dic₁₇	1	1	yes	characteristic
60	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
61	C₁₇	#2165	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₄	—	2	1	no	—
63	C₂²×C₄	C₂×Dic₁₇	1	1	yes	characteristic
64	M₁₆	—	2	2	no	—
65	M₁₆	—	2	2	no	—
66	C₂×C₈	C₂×Dic₁₇	1	1	yes	characteristic
67	C₂×C₈	C₂×Dic₁₇	1	1	yes	characteristic
68	C₂×C₈	C₂×Dic₁₇	1	1	yes	characteristic
69	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
70	C₄²	—	2	1	no	—
71	C₄²	C₂×Dic₁₇	1	1	yes	characteristic
72	C₄²	C₂×Dic₁₇	1	1	yes	characteristic
73	C₁₆	—	34	4	no	—
74	C₂³	#63223	1	1	yes	characteristic
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	#63223	1	1	yes	characteristic
81	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
82	C₈	—	2	1	no	—
83	C₈	—	2	1	no	—
84	C₈	#63223	1	2	yes	—
85	C₈	#63252	1	2	yes	—
86	C₂²	—	2	1	no	—
87	C₂²	—	4	1	no	—
88	C₂²	C₂×#63223	1	1	yes	characteristic
89	C₄	—	2	2	no	—
90	C₄	—	2	1	no	—
91	C₄	—	4	1	no	—
92	C₄	C₂×#63223	1	1	yes	characteristic
93	C₄	#10587612	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	4	1	no	—
96	C₂	#11762866	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#10587612
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₇
Fitting	C₄×M₁₆×C₁₇	C₂
socle	C₂×C₁₇	#432
prime radical	C₂³×C₁₇	M₁₆
cosocle	C₂×C₈×C₁₇	C₂³
hypercenter	C₄×M₁₆	D₁₇
hypocenter	C₁₇	#2165
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ #97 ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. #11762630	C₂	semidirect
3. #11762587	C₂	semidirect
61. C₁₇	#2165	semidirect

Maximal subgroups

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

subgroup	index
#11762630	2
#11762591	2
#11762587	2
C₄×M₁₆×C₁₇	2
#2165	17