Subgroups — C₂³×C₇×C₄⋊Q₈ #13239881

Order 1792.

12190 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₇×Q₈	C₂	1	2	yes	maximal
2	C₂²×C₇×C₄⋊Q₈	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₇×C₄⋊C₄	C₂	1	4	yes	maximal
4	C₂³×C₄²×C₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₇×Q₈	—	2	32	no	—
6	C₂³×C₇×Q₈	C₂²	1	56	yes	—
7	C₂⁴×C₄×C₇	C₂²	1	4	yes	—
8	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂⁴×C₄×C₇	C₂²	1	2	yes	—
10	C₂×C₇×C₄⋊Q₈	C₂²	1	448	yes	direct product factor
11	C₂²×C₇×C₄⋊C₄	C₂²	1	112	yes	—
12	C₂²×C₄²×C₇	C₂²	1	28	yes	—
13	C₂³×C₄⋊Q₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂⁵×C₇	C₂³	1	1	yes	characteristic
15	C₂²×C₇×Q₈	—	2	448	no	—
16	C₂²×C₇×Q₈	C₂³	1	224	yes	—
17	C₂³×C₄×C₇	—	2	32	no	—
18	C₂³×C₄×C₇	D₄	1	16	yes	—
19	C₂³×C₄×C₇	Q₈	1	32	yes	complemented
20	C₂³×C₄×C₇	C₂³	1	56	yes	—
21	C₂³×C₄×C₇	C₂³	1	14	yes	—
22	C₂³×C₄×C₇	C₂³	1	28	yes	—
23	C₇×C₄⋊Q₈	C₂³	1	512	yes	direct product factor
24	C₂×C₇×C₄⋊C₄	C₂³	1	448	yes	—
25	C₂×C₄²×C₇	C₂³	1	112	yes	—
26	C₂⁴×Q₈	C₂×C₇	1	2	yes	—
27	C₂²×C₄⋊Q₈	C₂×C₇	1	56	yes	direct product factor
28	C₂³×C₄⋊C₄	C₂×C₇	1	4	yes	—
29	C₂³×C₄²	C₂×C₇	1	1	yes	characteristic
30	C₂⁴×C₇	C₂×D₄	1	8	yes	—
31	C₂⁴×C₇	C₂×Q₈	1	16	yes	—
32	C₂⁴×C₇	C₂⁴	1	7	yes	—
33	C₂×C₇×Q₈	—	2	896	no	—
34	C₂×C₇×Q₈	C₂⁴	1	128	yes	—
35	C₂²×C₄×C₇	—	2	224	no	—
36	C₂²×C₄×C₇	C₂×D₄	1	112	yes	—
37	C₂²×C₄×C₇	C₂×Q₈	1	224	yes	complemented
38	C₂²×C₄×C₇	C₂⁴	1	112	yes	—
39	C₂²×C₄×C₇	C₂⁴	1	28	yes	—
40	C₂²×C₄×C₇	C₂⁴	1	56	yes	—
41	C₇×C₄⋊C₄	C₂⁴	1	256	yes	—
42	C₄²×C₇	C₂⁴	1	64	yes	—
43	C₂³×Q₈	—	2	32	no	—
44	C₂³×Q₈	C₂²×C₇	1	56	yes	—
45	C₂⁴×C₄	C₂²×C₇	1	4	yes	—
46	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
47	C₂⁴×C₄	C₂²×C₇	1	2	yes	—
48	C₂×C₄⋊Q₈	C₂²×C₇	1	448	yes	direct product factor
49	C₂²×C₄⋊C₄	C₂²×C₇	1	112	yes	—
50	C₂²×C₄²	C₂²×C₇	1	28	yes	—
51	C₂³×C₇	C₄⋊Q₈	1	64	yes	direct product factor
52	C₂³×C₇	C₂²×D₄	1	28	yes	—
53	C₂³×C₇	C₂²×Q₈	1	56	yes	—
54	C₂³×C₇	C₂⁵	1	7	yes	—
55	C₇×Q₈	—	2	256	no	—
56	C₂×C₄×C₇	—	2	224	no	—
57	C₂×C₄×C₇	C₂²×D₄	1	112	yes	—
58	C₂×C₄×C₇	C₂²×Q₈	1	224	yes	complemented
59	C₂×C₄×C₇	C₂⁵	1	32	yes	—
60	C₂×C₄×C₇	C₂⁵	1	8	yes	—
61	C₂×C₄×C₇	C₂⁵	1	16	yes	—
62	C₂⁵	C₂³×C₇	1	1	yes	characteristic
63	C₂²×Q₈	—	2	448	no	—
64	C₂²×Q₈	C₂³×C₇	1	224	yes	—
65	C₂³×C₄	—	2	32	no	—
66	C₂³×C₄	C₇×D₄	1	16	yes	—
67	C₂³×C₄	C₇×Q₈	1	32	yes	complemented
68	C₂³×C₄	C₂³×C₇	1	56	yes	—
69	C₂³×C₄	C₂³×C₇	1	14	yes	—
70	C₂³×C₄	C₂³×C₇	1	28	yes	—
71	C₄⋊Q₈	C₂³×C₇	1	512	yes	direct product factor
72	C₂×C₄⋊C₄	C₂³×C₇	1	448	yes	—
73	C₂×C₄²	C₂³×C₇	1	112	yes	—
74	C₂²×C₇	C₂×C₄⋊Q₈	1	112	yes	direct product factor
75	C₂²×C₇	C₂³×D₄	1	14	yes	—
76	C₂²×C₇	C₂³×Q₈	1	28	yes	—
77	C₂²×C₇	C₂⁶	1	1	yes	characteristic
78	C₄×C₇	—	2	32	no	—
79	C₄×C₇	C₂³×D₄	1	16	yes	—
80	C₄×C₇	C₂³×Q₈	1	32	yes	complemented
81	C₂⁴	C₂×C₇×D₄	1	8	yes	—
82	C₂⁴	C₂×C₇×Q₈	1	16	yes	—
83	C₂⁴	C₂⁴×C₇	1	7	yes	—
84	C₂×Q₈	—	2	896	no	—
85	C₂×Q₈	C₂⁴×C₇	1	128	yes	—
86	C₂²×C₄	—	2	224	no	—
87	C₂²×C₄	C₂×C₇×D₄	1	112	yes	—
88	C₂²×C₄	C₂×C₇×Q₈	1	224	yes	complemented
89	C₂²×C₄	C₂⁴×C₇	1	112	yes	—
90	C₂²×C₄	C₂⁴×C₇	1	28	yes	—
91	C₂²×C₄	C₂⁴×C₇	1	56	yes	—
92	C₄⋊C₄	C₂⁴×C₇	1	256	yes	—
93	C₄²	C₂⁴×C₇	1	64	yes	—
94	C₂×C₇	C₂²×C₄⋊Q₈	1	28	yes	direct product factor
95	C₂×C₇	C₂⁴×D₄	1	1	yes	characteristic
96	C₂×C₇	C₂⁴×Q₈	1	2	yes	—
97	C₂³	C₇×C₄⋊Q₈	1	64	yes	direct product factor
98	C₂³	C₂²×C₇×D₄	1	28	yes	—
99	C₂³	C₂²×C₇×Q₈	1	56	yes	—
100	C₂³	C₂⁵×C₇	1	7	yes	—
101	Q₈	—	2	256	no	—
102	C₂×C₄	—	2	224	no	—
103	C₂×C₄	C₂²×C₇×D₄	1	112	yes	—
104	C₂×C₄	C₂²×C₇×Q₈	1	224	yes	complemented
105	C₂×C₄	C₂⁵×C₇	1	32	yes	—
106	C₂×C₄	C₂⁵×C₇	1	8	yes	—
107	C₂×C₄	C₂⁵×C₇	1	16	yes	—
108	C₇	C₂³×C₄⋊Q₈	1	1	yes	characteristic, direct product factor, Sylow
109	C₂²	C₂×C₇×C₄⋊Q₈	1	112	yes	direct product factor
110	C₂²	C₂³×C₇×D₄	1	14	yes	—
111	C₂²	C₂³×C₇×Q₈	1	28	yes	—
112	C₂²	C₂⁶×C₇	1	1	yes	characteristic
113	C₄	—	2	32	no	—
114	C₄	C₂³×C₇×D₄	1	16	yes	—
115	C₄	C₂³×C₇×Q₈	1	32	yes	complemented
116	C₂	C₂²×C₇×C₄⋊Q₈	1	28	yes	direct product factor
117	C₂	C₂⁴×C₇×D₄	1	1	yes	characteristic
118	C₂	C₂⁴×C₇×Q₈	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶×C₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇×C₄⋊Q₈	C₂	direct (N × H)
10. C₂×C₇×C₄⋊Q₈	C₂²	direct (N × H)
13. C₂³×C₄⋊Q₈	C₇	direct (N × H)
19. C₂³×C₄×C₇	Q₈	semidirect
23. C₇×C₄⋊Q₈	C₂³	direct (N × H)
27. C₂²×C₄⋊Q₈	C₂×C₇	direct (N × H)
37. C₂²×C₄×C₇	C₂×Q₈	semidirect
48. C₂×C₄⋊Q₈	C₂²×C₇	direct (N × H)
51. C₂³×C₇	C₄⋊Q₈	direct (N × H)
58. C₂×C₄×C₇	C₂²×Q₈	semidirect
67. C₂³×C₄	C₇×Q₈	semidirect
71. C₄⋊Q₈	C₂³×C₇	direct (N × H)
74. C₂²×C₇	C₂×C₄⋊Q₈	direct (N × H)
80. C₄×C₇	C₂³×Q₈	semidirect
88. C₂²×C₄	C₂×C₇×Q₈	semidirect
94. C₂×C₇	C₂²×C₄⋊Q₈	direct (N × H)
97. C₂³	C₇×C₄⋊Q₈	direct (N × H)
104. C₂×C₄	C₂²×C₇×Q₈	semidirect
108. C₇	C₂³×C₄⋊Q₈	direct (N × H)
109. C₂²	C₂×C₇×C₄⋊Q₈	direct (N × H)
115. C₄	C₂³×C₇×Q₈	semidirect
116. C₂	C₂²×C₇×C₄⋊Q₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₇×Q₈	2
C₂²×C₇×C₄⋊Q₈	2
C₂³×C₇×C₄⋊C₄	2
C₂³×C₄²×C₇	2
C₂³×C₄⋊Q₈	7

Subgroups — C23×C7×C4⋊Q8 #13239881

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₇×C₄⋊Q₈ #13239881