Subgroups — C₂×Q₈×C₇⋊C₁₆ #13507278

Order 1792.

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	C₂×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₇⋊C₁₆	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊(C₇⋊C₁₆)	C₂	1	3	yes	maximal
4	C₂×C₄×C₇⋊C₁₆	C₂	1	3	yes	maximal
5	C₂×C₄×C₇×Q₈	C₄	1	1	yes	characteristic
6	C₇×C₈×Q₈	C₄	1	4	yes	—
7	C₇×C₈×Q₈	C₂²	1	4	yes	—
8	C₂×C₇×C₄⋊C₈	C₄	1	3	yes	—
9	C₂×C₄×C₇×C₈	C₂²	1	3	yes	—
10	C₂²×C₇⋊C₁₆	C₂²	1	3	yes	—
11	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₄⋊(C₇⋊C₁₆)	C₂²	1	12	yes	—
13	C₄×C₇⋊C₁₆	C₂²	1	12	yes	—
14	C₂×Q₈×C₁₆	—	7	1	no	maximal, Sylow
15	C₂²×C₇×Q₈	C₈	1	1	yes	characteristic
16	C₂²×C₇×C₈	C₂×C₄	1	3	yes	—
17	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
18	C₄×C₇×Q₈	C₈	1	4	yes	—
19	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
20	C₂×C₇×C₄⋊C₄	C₈	1	3	yes	—
21	C₂×C₄²×C₇	C₂×C₄	1	3	yes	—
22	C₇×C₄⋊C₈	C₂×C₄	1	12	yes	—
23	C₄×C₇×C₈	C₂×C₄	1	6	yes	—
24	C₄×C₇×C₈	C₂³	1	6	yes	—
25	C₂×C₇⋊C₁₆	—	2	6	no	—
26	C₂×C₇⋊C₁₆	Q₈	1	4	yes	direct product factor
27	C₂×C₇⋊C₁₆	C₂³	1	6	yes	—
28	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
29	C₂×C₈×Q₈	D₇	1	1	yes	characteristic
30	Q₈×C₁₆	—	7	8	no	—
31	C₂×C₄⋊C₁₆	—	7	3	no	—
32	C₂×C₄×C₁₆	—	7	3	no	—
33	C₂×C₇×Q₈	C₁₆	1	8	yes	complemented
34	C₂×C₇×Q₈	C₂×C₈	1	4	yes	—
35	C₂²×C₄×C₇	C₂×C₈	1	3	yes	—
36	C₂²×C₄×C₇	C₂×C₈	1	3	yes	—
37	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₇×C₈	—	2	6	no	—
39	C₂×C₇×C₈	C₂²×C₄	1	6	yes	—
40	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
42	C₂×C₇×C₈	#41	1	2	yes	—
43	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
44	C₇×C₄⋊C₄	C₂×C₈	1	12	yes	—
45	C₄²×C₇	C₂×C₈	1	6	yes	—
46	C₄²×C₇	C₂²×C₄	1	6	yes	—
47	C₇⋊C₁₆	—	2	6	no	—
48	C₇⋊C₁₆	C₂×Q₈	1	4	yes	direct product factor
49	C₂×C₄×Q₈	Dic₇	1	1	yes	characteristic
50	C₂²×C₁₆	—	7	1	no	—
51	C₂²×C₁₆	—	7	3	no	—
52	C₈×Q₈	Dic₇	1	4	yes	—
53	C₈×Q₈	C₂×D₇	1	4	yes	—
54	C₂×C₄⋊C₈	Dic₇	1	3	yes	—
55	C₂×C₄×C₈	C₂×D₇	1	3	yes	—
56	C₄⋊C₁₆	—	7	12	no	—
57	C₄×C₁₆	—	7	12	no	—
58	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
59	C₇×Q₈	C₂×C₁₆	1	16	yes	complemented
60	C₂×C₄×C₇	—	2	6	no	—
61	C₂×C₄×C₇	C₂×C₁₆	1	12	yes	—
62	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
63	C₂×C₄×C₇	C₂²×C₈	1	6	yes	—
64	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₇	C₂²×C₈	1	6	yes	—
66	C₂×C₄×C₇	#131	1	2	yes	—
67	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
68	C₇×C₈	—	2	6	no	—
69	C₇×C₈	C₄×Q₈	1	2	yes	—
70	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
71	C₇×C₈	C₂×#41	1	1	yes	characteristic
72	C₂²×Q₈	C₇⋊C₈	1	1	yes	characteristic
73	C₂²×C₈	C₂×Dic₇	1	3	yes	—
74	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
75	C₄×Q₈	C₇⋊C₈	1	4	yes	—
76	C₄×Q₈	C₂×Dic₇	1	4	yes	—
77	C₂×C₄⋊C₄	C₇⋊C₈	1	3	yes	—
78	C₂×C₄²	C₂×Dic₇	1	3	yes	—
79	C₂×C₁₆	—	7	4	no	—
80	C₂×C₁₆	—	7	2	no	—
81	C₂×C₁₆	—	7	6	no	—
82	C₂×C₁₆	—	14	6	no	—
83	C₄⋊C₈	C₂×Dic₇	1	12	yes	—
84	C₄×C₈	C₂×Dic₇	1	6	yes	—
85	C₄×C₈	C₂²×D₇	1	6	yes	—
86	C₂²×C₇	C₈×Q₈	1	2	yes	—
87	C₂²×C₇	C₂²×C₁₆	1	2	yes	—
88	C₂²×C₇	#504	1	2	yes	—
89	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
90	C₄×C₇	—	2	6	no	—
91	C₄×C₇	C₈×Q₈	1	2	yes	—
92	C₄×C₇	C₂²×C₁₆	1	12	yes	—
93	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
94	C₄×C₇	C₂×#131	1	1	yes	characteristic
95	C₂×Q₈	C₇⋊C₁₆	1	8	yes	direct product factor
96	C₂×Q₈	C₂×C₇⋊C₈	1	4	yes	—
97	C₂²×C₄	C₂×C₇⋊C₈	1	3	yes	—
98	C₂²×C₄	C₂×C₇⋊C₈	1	3	yes	—
99	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
100	C₂×C₈	—	2	6	no	—
101	C₂×C₈	Q₈×D₇	1	2	yes	—
102	C₂×C₈	#1161	1	2	yes	—
103	C₂×C₈	C₂²×Dic₇	1	6	yes	—
104	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
105	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
106	C₄⋊C₄	C₂×C₇⋊C₈	1	12	yes	—
107	C₄²	C₂×C₇⋊C₈	1	6	yes	—
108	C₄²	C₂²×Dic₇	1	6	yes	—
109	C₁₆	—	7	4	no	—
110	C₁₆	—	14	6	no	—
111	C₂×C₇	Q₈×C₁₆	1	4	yes	complemented
112	C₂×C₇	C₂×C₈×Q₈	1	1	yes	characteristic
113	C₂×C₇	C₂³×C₁₆	1	1	yes	characteristic
114	C₂×C₇	C₂×#504	1	1	yes	characteristic
115	C₂³	C₂²×C₇⋊C₈	1	1	yes	characteristic
116	Q₈	C₂×C₇⋊C₁₆	1	16	yes	direct product factor
117	C₂×C₄	—	2	6	no	—
118	C₂×C₄	C₂×C₇⋊C₁₆	1	12	yes	—
119	C₂×C₄	C₂²×C₇⋊C₈	1	6	yes	—
120	C₂×C₄	C₂²×C₇⋊C₈	1	6	yes	—
121	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
122	C₂×C₄	Q₈×Dic₇	1	2	yes	—
123	C₂×C₄	#6511	1	2	yes	—
124	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
125	C₈	—	2	6	no	—
126	C₈	Q₈×Dic₇	1	2	yes	—
127	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
128	C₈	C₂×#1161	1	1	yes	characteristic
129	C₇	C₂×Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
130	C₂²	Q₈×C₇⋊C₈	1	2	yes	—
131	C₂²	C₂²×C₇⋊C₁₆	1	2	yes	—
132	C₂²	#89912	1	2	yes	—
133	C₂²	C₂³×C₇⋊C₈	1	1	yes	characteristic
134	C₄	—	2	6	no	—
135	C₄	Q₈×C₇⋊C₈	1	2	yes	—
136	C₄	C₂²×C₇⋊C₁₆	1	12	yes	—
137	C₄	C₂×Q₈×Dic₇	1	1	yes	characteristic
138	C₄	C₂×#6511	1	1	yes	characteristic
139	C₂	Q₈×C₇⋊C₁₆	1	4	yes	direct product factor
140	C₂	C₂×Q₈×C₇⋊C₈	1	1	yes	characteristic
141	C₂	C₂³×C₇⋊C₁₆	1	1	yes	characteristic
142	C₂	C₂×#89912	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₇
commutator	C₂×C₇	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₇×C₈×Q₈	C₂
socle, prime radical	C₂³×C₇	C₂²×C₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	C₂×C₈×Q₈	D₇
hypocenter	C₇	C₂×Q₈×C₁₆
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×C₇⋊C₁₆	C₂	direct (N × H)
26. C₂×C₇⋊C₁₆	Q₈	direct (N × H)
33. C₂×C₇×Q₈	C₁₆	semidirect
48. C₇⋊C₁₆	C₂×Q₈	direct (N × H)
59. C₇×Q₈	C₂×C₁₆	semidirect
95. C₂×Q₈	C₇⋊C₁₆	direct (N × H)
111. C₂×C₇	Q₈×C₁₆	semidirect
116. Q₈	C₂×C₇⋊C₁₆	direct (N × H)
129. C₇	C₂×Q₈×C₁₆	semidirect
139. C₂	Q₈×C₇⋊C₁₆	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×C₇×C₈×Q₈	2
Q₈×C₇⋊C₁₆	2
C₂×C₄⋊(C₇⋊C₁₆)	2
C₂×C₄×C₇⋊C₁₆	2
C₂×Q₈×C₁₆	7

Subgroups — C2×Q8×C7⋊C16 #13507278

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×C₇⋊C₁₆ #13507278