Subgroups — C₂²×Q₈×#3622 #17898676

Order 4352.

2478 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#3622	C₂	1	24	yes	maximal, direct product factor
2	C₂²×C₄⋊#3622	C₂	1	3	yes	maximal
3	C₂²×C₄×#3622	C₂	1	3	yes	maximal
4	C₂²×C₄×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×#3622	C₂²	1	3	yes	—
6	C₂³×#3622	C₂²	1	1	yes	characteristic
7	Q₈×#3622	C₂²	1	64	yes	direct product factor
8	C₂×C₄⋊#3622	C₂²	1	36	yes	—
9	C₂×C₄×#3622	C₂²	1	36	yes	—
10	C₂³×Q₈×C₁₇	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₁₇	C₄	1	12	yes	—
12	C₂×C₄×Q₈×C₁₇	C₂²	1	12	yes	—
13	C₂²×C₄⋊C₄×C₁₇	C₄	1	3	yes	—
14	C₂²×C₄²×C₁₇	C₂²	1	3	yes	—
15	C₂²×Q₈×C₁₇	C₈	1	16	yes	complemented
16	C₂²×Q₈×C₁₇	C₂×C₄	1	12	yes	—
17	C₂³×C₄×C₁₇	C₂×C₄	1	3	yes	—
18	C₂³×C₄×C₁₇	C₂×C₄	1	3	yes	—
19	C₂³×C₄×C₁₇	C₂³	1	1	yes	characteristic
20	C₄×Q₈×C₁₇	C₂×C₄	1	48	yes	—
21	C₄×Q₈×C₁₇	C₂³	1	16	yes	—
22	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	36	yes	—
23	C₂×C₄²×C₁₇	C₂×C₄	1	18	yes	—
24	C₂×C₄²×C₁₇	C₂³	1	18	yes	—
25	C₂²×#3622	—	2	12	no	—
26	C₂²×#3622	Q₈	1	8	yes	direct product factor
27	C₂²×#3622	C₂³	1	18	yes	—
28	C₂²×#3622	C₂³	1	6	yes	—
29	C₄⋊#3622	C₂³	1	48	yes	—
30	C₄×#3622	C₂³	1	48	yes	—
31	C₂⁴×C₁₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×Q₈×C₁₇	C₂×C₈	1	96	yes	complemented
33	C₂×Q₈×C₁₇	C₂²×C₄	1	16	yes	—
34	C₂²×C₄×C₁₇	—	2	12	no	—
35	C₂²×C₄×C₁₇	C₂×C₈	1	24	yes	—
36	C₂²×C₄×C₁₇	C₂²×C₄	1	18	yes	—
37	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
38	C₂²×C₄×C₁₇	C₂²×C₄	1	18	yes	—
39	C₂²×C₄×C₁₇	C₂×Q₈	1	4	yes	—
40	C₂²×C₄×C₁₇	#41	1	4	yes	—
41	C₂²×C₄×C₁₇	C₂⁴	1	3	yes	—
42	C₄⋊C₄×C₁₇	C₂²×C₄	1	48	yes	—
43	C₄²×C₁₇	C₂²×C₄	1	36	yes	—
44	C₄²×C₁₇	C₂⁴	1	12	yes	—
45	C₂×#3622	—	2	36	no	—
46	C₂×#3622	C₂×Q₈	1	24	yes	direct product factor
47	C₂×#3622	C₂⁴	1	12	yes	—
48	C₂×#3622	C₂⁴	1	4	yes	—
49	C₂²×C₈×Q₈	—	17	1	no	maximal, Sylow
50	C₂³×C₁₇	C₄×Q₈	1	4	yes	—
51	C₂³×C₁₇	C₂²×C₈	1	4	yes	—
52	C₂³×C₁₇	#131	1	4	yes	—
53	C₂³×C₁₇	C₂³×C₄	1	3	yes	—
54	Q₈×C₁₇	C₂²×C₈	1	64	yes	complemented
55	C₂×C₄×C₁₇	—	2	36	no	—
56	C₂×C₄×C₁₇	C₄×Q₈	1	12	yes	—
57	C₂×C₄×C₁₇	C₂²×C₈	1	72	yes	—
58	C₂×C₄×C₁₇	C₂³×C₄	1	12	yes	—
59	C₂×C₄×C₁₇	C₂³×C₄	1	3	yes	—
60	C₂×C₄×C₁₇	C₂³×C₄	1	12	yes	—
61	C₂×C₄×C₁₇	C₂²×Q₈	1	6	yes	—
62	C₂×C₄×C₁₇	C₂×#41	1	6	yes	—
63	C₂×C₄×C₁₇	C₂⁵	1	1	yes	characteristic
64	#3622	—	2	12	no	—
65	#3622	C₂²×Q₈	1	8	yes	direct product factor
66	C₂²×C₄×Q₈	D₁₇	1	1	yes	characteristic
67	C₂×C₈×Q₈	—	17	24	no	—
68	C₂²×C₄⋊C₈	—	17	3	no	—
69	C₂²×C₄×C₈	—	17	3	no	—
70	C₂²×C₁₇	C₈×Q₈	1	16	yes	complemented
71	C₂²×C₁₇	C₂×C₄×Q₈	1	6	yes	—
72	C₂²×C₁₇	C₂³×C₈	1	6	yes	—
73	C₂²×C₁₇	C₂×#131	1	6	yes	—
74	C₂²×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
75	C₄×C₁₇	—	2	12	no	—
76	C₄×C₁₇	C₂×C₄×Q₈	1	6	yes	—
77	C₄×C₁₇	C₂³×C₈	1	24	yes	—
78	C₄×C₁₇	C₂³×Q₈	1	1	yes	characteristic
79	C₄×C₁₇	C₂²×#41	1	1	yes	characteristic
80	C₂³×Q₈	Dic₁₇	1	1	yes	characteristic
81	C₂³×C₈	—	17	1	no	—
82	C₂³×C₈	—	17	3	no	—
83	C₂×C₄×Q₈	Dic₁₇	1	12	yes	—
84	C₂×C₄×Q₈	C₂×D₁₇	1	12	yes	—
85	C₂²×C₄⋊C₄	Dic₁₇	1	3	yes	—
86	C₂²×C₄²	C₂×D₁₇	1	3	yes	—
87	C₈×Q₈	—	17	64	no	—
88	C₂×C₄⋊C₈	—	17	36	no	—
89	C₂×C₄×C₈	—	17	36	no	—
90	C₂×C₁₇	C₂×C₈×Q₈	1	12	yes	complemented
91	C₂×C₁₇	C₂²×C₄×Q₈	1	1	yes	characteristic
92	C₂×C₁₇	C₂⁴×C₈	1	1	yes	characteristic
93	C₂×C₁₇	C₂²×#131	1	1	yes	characteristic
94	C₂²×Q₈	#3622	1	16	yes	direct product factor
95	C₂²×Q₈	C₂×Dic₁₇	1	12	yes	—
96	C₂³×C₄	C₂×Dic₁₇	1	3	yes	—
97	C₂³×C₄	C₂×Dic₁₇	1	3	yes	—
98	C₂³×C₄	C₂²×D₁₇	1	1	yes	characteristic
99	C₂²×C₈	—	17	8	no	—
100	C₂²×C₈	—	17	6	no	—
101	C₂²×C₈	—	17	18	no	—
102	C₂²×C₈	—	34	12	no	—
103	C₄×Q₈	C₂×Dic₁₇	1	48	yes	—
104	C₄×Q₈	C₂²×D₁₇	1	16	yes	—
105	C₂×C₄⋊C₄	C₂×Dic₁₇	1	36	yes	—
106	C₂×C₄²	C₂×Dic₁₇	1	18	yes	—
107	C₂×C₄²	C₂²×D₁₇	1	18	yes	—
108	C₄⋊C₈	—	17	48	no	—
109	C₄×C₈	—	17	48	no	—
110	C₁₇	C₂²×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
111	C₂⁴	C₂²×Dic₁₇	1	1	yes	characteristic
112	C₂×Q₈	C₂×#3622	1	96	yes	direct product factor
113	C₂×Q₈	C₂²×Dic₁₇	1	16	yes	—
114	C₂²×C₄	—	2	12	no	—
115	C₂²×C₄	C₂×#3622	1	24	yes	—
116	C₂²×C₄	Q₈×D₁₇	1	4	yes	—
117	C₂²×C₄	#63256	1	4	yes	—
118	C₂²×C₄	C₂²×Dic₁₇	1	18	yes	—
119	C₂²×C₄	C₂²×Dic₁₇	1	18	yes	—
120	C₂²×C₄	C₂²×Dic₁₇	1	3	yes	—
121	C₂²×C₄	C₂³×D₁₇	1	3	yes	—
122	C₂×C₈	—	17	24	no	—
123	C₂×C₈	—	17	4	no	—
124	C₂×C₈	—	17	12	no	—
125	C₂×C₈	—	34	36	no	—
126	C₄⋊C₄	C₂²×Dic₁₇	1	48	yes	—
127	C₄²	C₂²×Dic₁₇	1	36	yes	—
128	C₄²	C₂³×D₁₇	1	12	yes	—
129	C₂³	C₂²×#3622	1	4	yes	—
130	C₂³	Q₈×Dic₁₇	1	4	yes	—
131	C₂³	#10587634	1	4	yes	—
132	C₂³	C₂³×Dic₁₇	1	3	yes	—
133	Q₈	C₂²×#3622	1	64	yes	direct product factor
134	C₂×C₄	—	2	36	no	—
135	C₂×C₄	C₂²×#3622	1	72	yes	—
136	C₂×C₄	Q₈×Dic₁₇	1	12	yes	—
137	C₂×C₄	C₂×Q₈×D₁₇	1	6	yes	—
138	C₂×C₄	C₂×#63256	1	6	yes	—
139	C₂×C₄	C₂³×Dic₁₇	1	12	yes	—
140	C₂×C₄	C₂³×Dic₁₇	1	3	yes	—
141	C₂×C₄	C₂³×Dic₁₇	1	12	yes	—
142	C₂×C₄	C₂⁴×D₁₇	1	1	yes	characteristic
143	C₈	—	17	8	no	—
144	C₈	—	34	12	no	—
145	C₂²	Q₈×#3622	1	16	yes	direct product factor
146	C₂²	C₂×Q₈×Dic₁₇	1	6	yes	—
147	C₂²	C₂³×#3622	1	6	yes	—
148	C₂²	C₂×#10587634	1	6	yes	—
149	C₂²	C₂⁴×Dic₁₇	1	1	yes	characteristic
150	C₄	—	2	12	no	—
151	C₄	C₂×Q₈×Dic₁₇	1	6	yes	—
152	C₄	C₂³×#3622	1	24	yes	—
153	C₄	C₂²×Q₈×D₁₇	1	1	yes	characteristic
154	C₄	C₂²×#63256	1	1	yes	characteristic
155	C₂	C₂×Q₈×#3622	1	12	yes	direct product factor
156	C₂	C₂²×Q₈×Dic₁₇	1	1	yes	characteristic
157	C₂	C₂⁴×#3622	1	1	yes	characteristic
158	C₂	C₂²×#10587634	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₁₇
commutator	C₂×C₁₇	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₁₇
Fitting	C₂²×C₄×Q₈×C₁₇	C₂
socle, prime radical	C₂⁴×C₁₇	C₂²×C₄
cosocle	C₂×C₄×C₁₇	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₁₇
hypocenter	C₁₇	C₂²×C₈×Q₈
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×#3622	C₂	direct (N × H)
7. Q₈×#3622	C₂²	direct (N × H)
15. C₂²×Q₈×C₁₇	C₈	semidirect
26. C₂²×#3622	Q₈	direct (N × H)
32. C₂×Q₈×C₁₇	C₂×C₈	semidirect
46. C₂×#3622	C₂×Q₈	direct (N × H)
54. Q₈×C₁₇	C₂²×C₈	semidirect
65. #3622	C₂²×Q₈	direct (N × H)
70. C₂²×C₁₇	C₈×Q₈	semidirect
90. C₂×C₁₇	C₂×C₈×Q₈	semidirect
94. C₂²×Q₈	#3622	direct (N × H)
110. C₁₇	C₂²×C₈×Q₈	semidirect
112. C₂×Q₈	C₂×#3622	direct (N × H)
133. Q₈	C₂²×#3622	direct (N × H)
145. C₂²	Q₈×#3622	direct (N × H)
155. C₂	C₂×Q₈×#3622	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×Q₈×#3622	2
C₂²×C₄⋊#3622	2
C₂²×C₄×#3622	2
C₂²×C₄×Q₈×C₁₇	2
C₂²×C₈×Q₈	17

Subgroups — C22×Q8×#3622 #17898676

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₈×#3622 #17898676