Subgroups — C₂²×#11762643 #17192430

Order 4352.

502 conjugacy classes of subgroups.

The subgroup lattice has 114 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762643	C₂	1	12	yes	maximal, direct product factor
2	C₂²×#10587465	C₂	1	2	yes	maximal, complemented
3	C₂³×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
4	#11762643	C₂²	1	16	yes	direct product factor
5	C₂×#10587465	C₂²	1	12	yes	complemented
6	C₂³×C₈×C₁₇	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₇	C₄	1	6	yes	—
8	C₂²×C₁₆×C₁₇	C₄	1	1	yes	characteristic
9	C₂²×C₁₆×C₁₇	C₂²	1	6	yes	—
10	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₇	C₈	1	1	yes	characteristic
12	C₂²×C₈×C₁₇	C₈	1	6	yes	—
13	C₂²×C₈×C₁₇	C₈	1	1	yes	characteristic
14	C₂²×C₈×C₁₇	C₂×C₄	1	6	yes	—
15	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₆×C₁₇	C₂×C₄	1	6	yes	—
17	C₂×C₁₆×C₁₇	C₂×C₄	1	12	yes	—
18	C₂×C₁₆×C₁₇	C₂×C₄	1	3	yes	—
19	C₂×C₁₆×C₁₇	C₂³	1	3	yes	—
20	C₂×C₁₆×C₁₇	C₂³	1	4	yes	—
21	#10587465	C₂³	1	8	yes	complemented
22	C₂⁴×C₁₇	C₁₆	1	1	yes	characteristic
23	C₂²×C₄×C₁₇	C₁₆	1	6	yes	—
24	C₂²×C₄×C₁₇	C₁₆	1	1	yes	characteristic
25	C₂²×C₄×C₁₇	C₂×C₈	1	6	yes	—
26	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₁₇	C₂×C₈	1	12	yes	—
28	C₂×C₈×C₁₇	C₂×C₈	1	6	yes	—
29	C₂×C₈×C₁₇	C₂×C₈	1	3	yes	—
30	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
31	C₂×C₈×C₁₇	C₂²×C₄	1	4	yes	—
32	C₁₆×C₁₇	C₂²×C₄	1	3	yes	—
33	C₁₆×C₁₇	C₂²×C₄	1	4	yes	—
34	C₁₆×C₁₇	C₂⁴	1	1	yes	characteristic
35	C₂²×M₆₄	—	17	1	no	maximal, Sylow
36	C₂³×C₁₇	—	2	4	no	—
37	C₂³×C₁₇	C₂×C₁₆	1	6	yes	—
38	C₂³×C₁₇	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄×C₁₇	C₂×C₁₆	1	12	yes	—
40	C₂×C₄×C₁₇	C₂×C₁₆	1	3	yes	—
41	C₂×C₄×C₁₇	C₂×C₁₆	1	6	yes	—
42	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
43	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
44	C₈×C₁₇	C₂²×C₈	1	3	yes	—
45	C₈×C₁₇	C₂²×C₈	1	4	yes	—
46	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
47	C₂³×C₁₆	D₁₇	1	1	yes	characteristic
48	C₂×M₆₄	—	17	12	no	—
49	C₂²×C₃₂	—	17	2	no	—
50	C₂²×C₁₇	—	2	12	no	—
51	C₂²×C₁₇	M₆₄	1	4	yes	complemented
52	C₂²×C₁₇	C₂²×C₁₆	1	4	yes	—
53	C₂²×C₁₇	C₂²×C₁₆	1	3	yes	—
54	C₄×C₁₇	C₂²×C₁₆	1	3	yes	—
55	C₄×C₁₇	C₂²×C₁₆	1	4	yes	—
56	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
57	C₂³×C₈	Dic₁₇	1	1	yes	characteristic
58	C₂²×C₁₆	Dic₁₇	1	1	yes	characteristic
59	C₂²×C₁₆	Dic₁₇	1	6	yes	—
60	C₂²×C₁₆	C₂×D₁₇	1	6	yes	—
61	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
62	M₆₄	—	17	16	no	—
63	C₂×C₃₂	—	17	12	no	—
64	C₂×C₁₇	—	2	4	no	—
65	C₂×C₁₇	C₂×M₆₄	1	6	yes	complemented
66	C₂×C₁₇	C₂³×C₁₆	1	1	yes	characteristic
67	C₂³×C₄	#3622	1	1	yes	characteristic
68	C₂²×C₈	#3622	1	1	yes	characteristic
69	C₂²×C₈	#3622	1	6	yes	—
70	C₂²×C₈	C₂×Dic₁₇	1	6	yes	—
71	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
72	C₂×C₁₆	C₂×Dic₁₇	1	6	yes	—
73	C₂×C₁₆	C₂×Dic₁₇	1	12	yes	—
74	C₂×C₁₆	C₂×Dic₁₇	1	3	yes	—
75	C₂×C₁₆	C₂²×D₁₇	1	3	yes	—
76	C₂×C₁₆	C₂²×D₁₇	1	4	yes	—
77	C₃₂	—	17	8	no	—
78	C₁₇	C₂²×M₆₄	1	1	yes	characteristic, complemented, Sylow
79	C₂⁴	#63214	1	1	yes	characteristic
80	C₂²×C₄	#63214	1	6	yes	—
81	C₂²×C₄	#63214	1	1	yes	characteristic
82	C₂²×C₄	C₂×#3622	1	6	yes	—
83	C₂²×C₄	C₂×#3622	1	1	yes	characteristic
84	C₂×C₈	C₂×#3622	1	12	yes	—
85	C₂×C₈	C₂×#3622	1	6	yes	—
86	C₂×C₈	C₂×#3622	1	3	yes	—
87	C₂×C₈	C₂²×Dic₁₇	1	4	yes	—
88	C₂×C₈	C₂²×Dic₁₇	1	3	yes	—
89	C₁₆	C₂²×Dic₁₇	1	4	yes	—
90	C₁₆	C₂²×Dic₁₇	1	3	yes	—
91	C₁₆	C₂³×D₁₇	1	1	yes	characteristic
92	C₂³	—	2	4	no	—
93	C₂³	C₂×#63214	1	6	yes	—
94	C₂³	C₂×#63214	1	1	yes	characteristic
95	C₂×C₄	C₂×#63214	1	12	yes	—
96	C₂×C₄	C₂×#63214	1	3	yes	—
97	C₂×C₄	C₂×#63214	1	6	yes	—
98	C₂×C₄	C₂²×#3622	1	4	yes	—
99	C₂×C₄	C₂²×#3622	1	3	yes	—
100	C₈	C₂²×#3622	1	4	yes	—
101	C₈	C₂²×#3622	1	3	yes	—
102	C₈	C₂³×Dic₁₇	1	1	yes	characteristic
103	C₂²	—	2	12	no	—
104	C₂²	#11762643	1	4	yes	direct product factor
105	C₂²	C₂²×#63214	1	4	yes	—
106	C₂²	C₂²×#63214	1	3	yes	—
107	C₄	C₂²×#63214	1	3	yes	—
108	C₄	C₂²×#63214	1	4	yes	—
109	C₄	C₂³×#3622	1	1	yes	characteristic
110	C₂	—	2	4	no	—
111	C₂	C₂×#11762643	1	6	yes	direct product factor
112	C₂	C₂³×#63214	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₁₇
commutator	C₂×C₁₇	C₂³×C₁₆
Frattini	C₁₆	C₂³×D₁₇
Fitting	C₂³×C₁₆×C₁₇	C₂
socle	C₂³×C₁₇	C₂×C₁₆
prime radical	C₂⁴×C₁₇	C₁₆
cosocle	C₁₆×C₁₇	C₂⁴
hypercenter	C₂³×C₁₆	D₁₇
hypocenter	C₁₇	C₂²×M₆₄
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#11762643	C₂	direct (N × H)
2. C₂²×#10587465	C₂	semidirect
4. #11762643	C₂²	direct (N × H)
5. C₂×#10587465	C₂²	semidirect
21. #10587465	C₂³	semidirect
51. C₂²×C₁₇	M₆₄	semidirect
65. C₂×C₁₇	C₂×M₆₄	semidirect
78. C₁₇	C₂²×M₆₄	semidirect
104. C₂²	#11762643	direct (N × H)
111. C₂	C₂×#11762643	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11762643	2
C₂²×#10587465	2
C₂³×C₁₆×C₁₇	2
C₂²×M₆₄	17

Subgroups — C22×#11762643 #17192430

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11762643 #17192430