Subgroups — C₂×Q₈×#63214 #17187806

Order 4352.

582 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#63214	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊#63214	C₂	1	3	yes	maximal
3	C₂×C₄×#63214	C₂	1	3	yes	maximal
4	C₂×C₈×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63214	C₂²	1	3	yes	—
6	C₂²×#63214	C₂²	1	1	yes	characteristic
7	C₄⋊#63214	C₂²	1	12	yes	—
8	C₄×#63214	C₂²	1	12	yes	—
9	C₂×C₄×Q₈×C₁₇	C₄	1	1	yes	characteristic
10	C₈×Q₈×C₁₇	C₄	1	4	yes	—
11	C₈×Q₈×C₁₇	C₂²	1	4	yes	—
12	C₂×C₁₇×C₄⋊C₈	C₄	1	3	yes	—
13	C₂×C₄×C₈×C₁₇	C₂²	1	3	yes	—
14	C₂²×Q₈×C₁₇	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₁₇	C₂×C₄	1	3	yes	—
16	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
17	C₄×Q₈×C₁₇	C₈	1	4	yes	—
18	C₄×Q₈×C₁₇	C₂×C₄	1	4	yes	—
19	C₂×C₄⋊C₄×C₁₇	C₈	1	3	yes	—
20	C₂×C₄²×C₁₇	C₂×C₄	1	3	yes	—
21	C₁₇×C₄⋊C₈	C₂×C₄	1	12	yes	—
22	C₄×C₈×C₁₇	C₂×C₄	1	6	yes	—
23	C₄×C₈×C₁₇	C₂³	1	6	yes	—
24	C₂×#63214	—	2	6	no	—
25	C₂×#63214	Q₈	1	4	yes	direct product factor
26	C₂×#63214	C₂³	1	6	yes	—
27	C₂×#63214	C₂³	1	2	yes	—
28	C₂×Q₈×C₁₇	C₁₆	1	8	yes	complemented
29	C₂×Q₈×C₁₇	C₂×C₈	1	4	yes	—
30	C₂²×C₄×C₁₇	C₂×C₈	1	3	yes	—
31	C₂²×C₄×C₁₇	C₂×C₈	1	3	yes	—
32	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₇	—	2	6	no	—
34	C₂×C₈×C₁₇	C₂²×C₄	1	6	yes	—
35	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
37	C₂×C₈×C₁₇	#41	1	2	yes	—
38	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
39	C₄⋊C₄×C₁₇	C₂×C₈	1	12	yes	—
40	C₄²×C₁₇	C₂×C₈	1	6	yes	—
41	C₄²×C₁₇	C₂²×C₄	1	6	yes	—
42	#63214	—	2	6	no	—
43	#63214	C₂×Q₈	1	4	yes	direct product factor
44	C₂×Q₈×C₁₆	—	17	1	no	maximal, Sylow
45	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
46	Q₈×C₁₇	C₂×C₁₆	1	16	yes	complemented
47	C₂×C₄×C₁₇	—	2	6	no	—
48	C₂×C₄×C₁₇	C₂×C₁₆	1	12	yes	—
49	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
50	C₂×C₄×C₁₇	C₂²×C₈	1	6	yes	—
51	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂²×C₈	1	6	yes	—
53	C₂×C₄×C₁₇	#131	1	2	yes	—
54	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
55	C₈×C₁₇	—	2	6	no	—
56	C₈×C₁₇	C₄×Q₈	1	2	yes	—
57	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
58	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
59	C₂×C₈×Q₈	D₁₇	1	1	yes	characteristic
60	Q₈×C₁₆	—	17	8	no	—
61	C₂×C₄⋊C₁₆	—	17	3	no	—
62	C₂×C₄×C₁₆	—	17	3	no	—
63	C₂²×C₁₇	C₈×Q₈	1	2	yes	—
64	C₂²×C₁₇	C₂²×C₁₆	1	2	yes	—
65	C₂²×C₁₇	#504	1	2	yes	—
66	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
67	C₄×C₁₇	—	2	6	no	—
68	C₄×C₁₇	C₈×Q₈	1	2	yes	—
69	C₄×C₁₇	C₂²×C₁₆	1	12	yes	—
70	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
71	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
72	C₂×C₄×Q₈	Dic₁₇	1	1	yes	characteristic
73	C₂²×C₁₆	—	17	1	no	—
74	C₂²×C₁₆	—	17	3	no	—
75	C₈×Q₈	Dic₁₇	1	4	yes	—
76	C₈×Q₈	C₂×D₁₇	1	4	yes	—
77	C₂×C₄⋊C₈	Dic₁₇	1	3	yes	—
78	C₂×C₄×C₈	C₂×D₁₇	1	3	yes	—
79	C₄⋊C₁₆	—	17	12	no	—
80	C₄×C₁₆	—	17	12	no	—
81	C₂×C₁₇	Q₈×C₁₆	1	4	yes	complemented
82	C₂×C₁₇	C₂×C₈×Q₈	1	1	yes	characteristic
83	C₂×C₁₇	C₂³×C₁₆	1	1	yes	characteristic
84	C₂×C₁₇	C₂×#504	1	1	yes	characteristic
85	C₂²×Q₈	#3622	1	1	yes	characteristic
86	C₂²×C₈	C₂×Dic₁₇	1	3	yes	—
87	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
88	C₄×Q₈	#3622	1	4	yes	—
89	C₄×Q₈	C₂×Dic₁₇	1	4	yes	—
90	C₂×C₄⋊C₄	#3622	1	3	yes	—
91	C₂×C₄²	C₂×Dic₁₇	1	3	yes	—
92	C₂×C₁₆	—	17	4	no	—
93	C₂×C₁₆	—	17	2	no	—
94	C₂×C₁₆	—	17	6	no	—
95	C₂×C₁₆	—	34	6	no	—
96	C₄⋊C₈	C₂×Dic₁₇	1	12	yes	—
97	C₄×C₈	C₂×Dic₁₇	1	6	yes	—
98	C₄×C₈	C₂²×D₁₇	1	6	yes	—
99	C₁₇	C₂×Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
100	C₂×Q₈	#63214	1	8	yes	direct product factor
101	C₂×Q₈	C₂×#3622	1	4	yes	—
102	C₂²×C₄	C₂×#3622	1	3	yes	—
103	C₂²×C₄	C₂×#3622	1	3	yes	—
104	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
105	C₂×C₈	—	2	6	no	—
106	C₂×C₈	Q₈×D₁₇	1	2	yes	—
107	C₂×C₈	#63256	1	2	yes	—
108	C₂×C₈	C₂²×Dic₁₇	1	6	yes	—
109	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
110	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
111	C₄⋊C₄	C₂×#3622	1	12	yes	—
112	C₄²	C₂×#3622	1	6	yes	—
113	C₄²	C₂²×Dic₁₇	1	6	yes	—
114	C₁₆	—	17	4	no	—
115	C₁₆	—	34	6	no	—
116	C₂³	C₂²×#3622	1	1	yes	characteristic
117	Q₈	C₂×#63214	1	16	yes	direct product factor
118	C₂×C₄	—	2	6	no	—
119	C₂×C₄	C₂×#63214	1	12	yes	—
120	C₂×C₄	C₂²×#3622	1	6	yes	—
121	C₂×C₄	C₂²×#3622	1	6	yes	—
122	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
123	C₂×C₄	Q₈×Dic₁₇	1	2	yes	—
124	C₂×C₄	#10587634	1	2	yes	—
125	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
126	C₈	—	2	6	no	—
127	C₈	Q₈×Dic₁₇	1	2	yes	—
128	C₈	C₂×Q₈×D₁₇	1	1	yes	characteristic
129	C₈	C₂×#63256	1	1	yes	characteristic
130	C₂²	Q₈×#3622	1	2	yes	—
131	C₂²	C₂²×#63214	1	2	yes	—
132	C₂²	#11763226	1	2	yes	—
133	C₂²	C₂³×#3622	1	1	yes	characteristic
134	C₄	—	2	6	no	—
135	C₄	Q₈×#3622	1	2	yes	—
136	C₄	C₂²×#63214	1	12	yes	—
137	C₄	C₂×Q₈×Dic₁₇	1	1	yes	characteristic
138	C₄	C₂×#10587634	1	1	yes	characteristic
139	C₂	Q₈×#63214	1	4	yes	direct product factor
140	C₂	C₂×Q₈×#3622	1	1	yes	characteristic
141	C₂	C₂³×#63214	1	1	yes	characteristic
142	C₂	C₂×#11763226	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₁₇
commutator	C₂×C₁₇	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₈×Q₈×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₇
hypocenter	C₁₇	C₂×Q₈×C₁₆
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×#63214	C₂	direct (N × H)
25. C₂×#63214	Q₈	direct (N × H)
28. C₂×Q₈×C₁₇	C₁₆	semidirect
43. #63214	C₂×Q₈	direct (N × H)
46. Q₈×C₁₇	C₂×C₁₆	semidirect
81. C₂×C₁₇	Q₈×C₁₆	semidirect
99. C₁₇	C₂×Q₈×C₁₆	semidirect
100. C₂×Q₈	#63214	direct (N × H)
117. Q₈	C₂×#63214	direct (N × H)
139. C₂	Q₈×#63214	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#63214	2
C₂×C₄⋊#63214	2
C₂×C₄×#63214	2
C₂×C₈×Q₈×C₁₇	2
C₂×Q₈×C₁₆	17

Subgroups — C2×Q8×#63214 #17187806

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×#63214 #17187806