Subgroups — C₂²×#10587500 #14541896

Order 2176.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₆₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587500	C₂	1	12	yes	maximal, direct product factor
3	C₂²×#63214	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₁₇	—	2	1	no	—
6	C₂²×Dic₃₄	—	2	1	no	—
7	C₂×Q₁₆×C₁₇	—	2	3	no	—
8	C₂×Q₁₆×C₁₇	C₂²	1	6	yes	—
9	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₆₈	—	2	3	no	—
11	C₂×Dic₆₈	C₂²	1	6	yes	—
12	#10587500	C₂²	1	16	yes	direct product factor
13	C₂×#63214	C₂²	1	6	yes	—
14	C₂×Q₈×C₁₇	—	2	6	no	—
15	C₂×Q₈×C₁₇	—	4	3	no	—
16	C₂²×C₄×C₁₇	—	4	1	no	—
17	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
18	C₂²×Dic₁₇	—	4	1	no	—
19	C₂×Dic₃₄	—	2	6	no	—
20	C₂×Dic₃₄	—	4	3	no	—
21	Q₁₆×C₁₇	—	2	6	no	—
22	Q₁₆×C₁₇	C₂³	1	4	yes	—
23	C₂×C₈×C₁₇	D₄	1	3	yes	—
24	C₂×C₈×C₁₇	C₂³	1	3	yes	—
25	Dic₆₈	—	2	6	no	—
26	Dic₆₈	C₂³	1	4	yes	—
27	#63214	C₂³	1	4	yes	—
28	C₂³×C₁₇	D₈	1	1	yes	characteristic
29	Q₈×C₁₇	—	2	4	no	—
30	Q₈×C₁₇	—	4	6	no	—
31	C₂×C₄×C₁₇	—	4	6	no	—
32	C₂×C₄×C₁₇	D₈	1	3	yes	—
33	C₂×C₄×C₁₇	C₂×D₄	1	3	yes	—
34	C₂×Dic₁₇	—	4	6	no	—
35	Dic₃₄	—	2	4	no	—
36	Dic₃₄	—	4	6	no	—
37	C₈×C₁₇	C₂×D₄	1	3	yes	—
38	C₈×C₁₇	C₂⁴	1	1	yes	characteristic
39	C₂²×Q₃₂	—	17	1	no	maximal, Sylow
40	C₂²×C₁₇	Q₃₂	1	4	yes	complemented
41	C₂²×C₁₇	C₂×D₈	1	3	yes	—
42	C₄×C₁₇	—	4	4	no	—
43	C₄×C₁₇	C₂×D₈	1	3	yes	—
44	C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
45	Dic₁₇	—	4	4	no	—
46	C₂²×Q₁₆	—	17	1	no	—
47	C₂²×Q₁₆	D₁₇	1	1	yes	characteristic
48	C₂×Q₃₂	—	17	12	no	—
49	C₂²×C₁₆	—	17	1	no	—
50	C₂×C₁₇	C₂×Q₃₂	1	6	yes	complemented
51	C₂×C₁₇	C₂²×D₈	1	1	yes	characteristic
52	C₂²×Q₈	—	2	1	no	—
53	C₂²×Q₈	—	34	1	no	—
54	C₂×Q₁₆	—	2	3	no	—
55	C₂×Q₁₆	—	17	6	no	—
56	C₂×Q₁₆	—	34	3	no	—
57	C₂×Q₁₆	C₂×D₁₇	1	6	yes	—
58	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
59	Q₃₂	—	17	16	no	—
60	C₂×C₁₆	—	17	6	no	—
61	C₁₇	C₂²×Q₃₂	1	1	yes	characteristic, complemented, Sylow
62	C₂×Q₈	—	2	6	no	—
63	C₂×Q₈	—	4	3	no	—
64	C₂×Q₈	—	34	6	no	—
65	C₂×Q₈	—	68	3	no	—
66	C₂²×C₄	—	4	1	no	—
67	C₂²×C₄	—	68	1	no	—
68	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
69	Q₁₆	—	2	6	no	—
70	Q₁₆	—	17	4	no	—
71	Q₁₆	—	34	6	no	—
72	Q₁₆	C₂²×D₁₇	1	4	yes	—
73	C₂×C₈	C₂²⋊D₁₇	1	3	yes	—
74	C₂×C₈	C₂²×D₁₇	1	3	yes	—
75	C₁₆	—	17	4	no	—
76	C₂³	#63228	1	1	yes	characteristic
77	Q₈	—	2	4	no	—
78	Q₈	—	4	6	no	—
79	Q₈	—	34	4	no	—
80	Q₈	—	68	6	no	—
81	C₂×C₄	—	4	6	no	—
82	C₂×C₄	—	68	6	no	—
83	C₂×C₄	#63228	1	3	yes	—
84	C₂×C₄	C₂×C₂²⋊D₁₇	1	3	yes	—
85	C₈	C₂×C₂²⋊D₁₇	1	3	yes	—
86	C₈	C₂³×D₁₇	1	1	yes	characteristic
87	C₂²	#10587500	1	4	yes	direct product factor
88	C₂²	C₂×#63228	1	3	yes	—
89	C₄	—	4	4	no	—
90	C₄	—	68	4	no	—
91	C₄	C₂×#63228	1	3	yes	—
92	C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
93	C₂	C₂×#10587500	1	6	yes	direct product factor
94	C₂	C₂²×#63228	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#63228
commutator, cosocle	C₈×C₁₇	C₂⁴
Frattini	C₈	C₂³×D₁₇
Fitting	C₂²×Q₁₆×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	D₈
hypercenter	C₂²×Q₁₆	D₁₇
hypocenter	C₁₇	C₂²×Q₃₂
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#10587500	C₂	direct (N × H)
12. #10587500	C₂²	direct (N × H)
40. C₂²×C₁₇	Q₃₂	semidirect
50. C₂×C₁₇	C₂×Q₃₂	semidirect
61. C₁₇	C₂²×Q₃₂	semidirect
87. C₂²	#10587500	direct (N × H)
93. C₂	C₂×#10587500	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₆₈	2
C₂×#10587500	2
C₂²×#63214	2
C₂²×Q₁₆×C₁₇	2
C₂²×Q₃₂	17

Subgroups — C22×#10587500 #14541896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#10587500 #14541896