Subgroups — C₂³×#6392 #14217441

Order 1792.

3614 conjugacy classes of subgroups.

The subgroup lattice has 130 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊Dic₇	C₂	1	2	yes	maximal
3	C₂²×#6392	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
5	C₂⁴×Dic₇	—	2	2	no	—
6	C₂³×C₇×C₈	C₄	1	16	yes	complemented
7	C₂³×C₇×C₈	C₂²	1	14	yes	—
8	C₂²×C₄⋊Dic₇	—	2	14	no	—
9	C₂²×C₄⋊Dic₇	C₂²	1	28	yes	—
10	C₂×#6392	C₂²	1	112	yes	direct product factor
11	C₂³×#107	—	7	1	no	maximal, Sylow
12	C₂⁵×C₇	D₄	1	1	yes	characteristic
13	C₂³×C₄×C₇	C₂×C₄	1	8	yes	—
14	C₂³×C₄×C₇	D₄	1	7	yes	—
15	C₂³×C₄×C₇	Q₈	1	8	yes	—
16	C₂³×C₄×C₇	C₂³	1	7	yes	—
17	C₂³×Dic₇	—	2	28	no	—
18	C₂³×Dic₇	—	4	16	no	—
19	C₂²×C₇×C₈	C₂×C₄	1	112	yes	complemented
20	C₂²×C₇×C₈	C₂³	1	28	yes	—
21	C₂×C₄⋊Dic₇	—	2	84	no	—
22	C₂×C₄⋊Dic₇	C₂³	1	56	yes	—
23	#6392	C₂³	1	64	yes	direct product factor
24	C₂⁴×C₈	D₇	1	1	yes	characteristic
25	C₂³×C₄⋊C₄	—	7	2	no	—
26	C₂²×#107	—	7	28	no	—
27	C₂⁴×C₇	C₄⋊C₄	1	8	yes	—
28	C₂⁴×C₇	D₈	1	8	yes	—
29	C₂⁴×C₇	Q₁₆	1	8	yes	—
30	C₂⁴×C₇	C₂×D₄	1	7	yes	—
31	C₂²×C₄×C₇	C₄⋊C₄	1	56	yes	—
32	C₂²×C₄×C₇	C₂²×C₄	1	28	yes	—
33	C₂²×C₄×C₇	C₂×D₄	1	21	yes	—
34	C₂²×C₄×C₇	C₂×Q₈	1	28	yes	—
35	C₂²×C₄×C₇	C₂⁴	1	7	yes	—
36	C₂²×Dic₇	—	2	56	no	—
37	C₂²×Dic₇	—	4	112	no	—
38	C₂×C₇×C₈	C₂²×C₄	1	112	yes	complemented
39	C₂×C₇×C₈	C₂⁴	1	8	yes	—
40	C₄⋊Dic₇	—	2	56	no	—
41	C₄⋊Dic₇	C₂⁴	1	16	yes	—
42	C₂⁴×C₄	—	14	2	no	—
43	C₂⁴×C₄	C₂×D₇	1	1	yes	characteristic
44	C₂³×C₈	Dic₇	1	16	yes	complemented
45	C₂³×C₈	C₂×D₇	1	14	yes	—
46	C₂²×C₄⋊C₄	—	7	28	no	—
47	C₂²×C₄⋊C₄	—	14	14	no	—
48	C₂×#107	—	7	112	no	—
49	C₂³×C₇	#107	1	64	yes	complemented
50	C₂³×C₇	C₂×C₄⋊C₄	1	28	yes	—
51	C₂³×C₇	C₂×D₈	1	28	yes	—
52	C₂³×C₇	C₂×Q₁₆	1	28	yes	—
53	C₂³×C₇	C₂²×D₄	1	7	yes	—
54	C₂×C₄×C₇	C₂×C₄⋊C₄	1	84	yes	—
55	C₂×C₄×C₇	C₂³×C₄	1	14	yes	—
56	C₂×C₄×C₇	C₂²×D₄	1	7	yes	—
57	C₂×C₄×C₇	C₂²×Q₈	1	14	yes	—
58	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
59	C₂×Dic₇	—	2	16	no	—
60	C₂×Dic₇	—	4	112	no	—
61	C₇×C₈	C₂³×C₄	1	16	yes	complemented
62	C₂⁵	D₂₈	1	1	yes	characteristic
63	C₂³×C₄	—	14	28	no	—
64	C₂³×C₄	—	28	16	no	—
65	C₂³×C₄	Dic₁₄	1	8	yes	—
66	C₂³×C₄	D₂₈	1	7	yes	—
67	C₂³×C₄	C₂×Dic₇	1	8	yes	—
68	C₂³×C₄	C₂²×D₇	1	7	yes	—
69	C₂²×C₈	C₂×Dic₇	1	112	yes	complemented
70	C₂²×C₈	C₂²×D₇	1	28	yes	—
71	C₂×C₄⋊C₄	—	7	56	no	—
72	C₂×C₄⋊C₄	—	14	84	no	—
73	#107	—	7	64	no	—
74	C₂²×C₇	C₂×#107	1	112	yes	complemented
75	C₂²×C₇	C₂²×C₄⋊C₄	1	14	yes	—
76	C₂²×C₇	C₂²×D₈	1	14	yes	—
77	C₂²×C₇	C₂²×Q₁₆	1	14	yes	—
78	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
79	C₄×C₇	C₂²×C₄⋊C₄	1	14	yes	—
80	C₄×C₇	C₂⁴×C₄	1	1	yes	characteristic
81	C₄×C₇	C₂³×Q₈	1	1	yes	characteristic
82	Dic₇	—	4	16	no	—
83	C₂⁴	D₅₆	1	8	yes	—
84	C₂⁴	Dic₂₈	1	8	yes	—
85	C₂⁴	C₄⋊Dic₇	1	8	yes	—
86	C₂⁴	C₂×D₂₈	1	7	yes	—
87	C₂²×C₄	—	14	56	no	—
88	C₂²×C₄	—	28	112	no	—
89	C₂²×C₄	C₄⋊Dic₇	1	56	yes	—
90	C₂²×C₄	C₂×Dic₁₄	1	28	yes	—
91	C₂²×C₄	C₂×D₂₈	1	21	yes	—
92	C₂²×C₄	C₂²×Dic₇	1	28	yes	—
93	C₂²×C₄	C₂³×D₇	1	7	yes	—
94	C₂×C₈	C₂²×Dic₇	1	112	yes	complemented
95	C₂×C₈	C₂³×D₇	1	8	yes	—
96	C₄⋊C₄	—	7	16	no	—
97	C₄⋊C₄	—	14	56	no	—
98	C₂×C₇	C₂²×#107	1	28	yes	complemented
99	C₂×C₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₇	C₂³×D₈	1	1	yes	characteristic
101	C₂×C₇	C₂³×Q₁₆	1	1	yes	characteristic
102	C₂³	#6392	1	64	yes	direct product factor
103	C₂³	C₂×D₅₆	1	28	yes	—
104	C₂³	C₂×Dic₂₈	1	28	yes	—
105	C₂³	C₂×C₄⋊Dic₇	1	28	yes	—
106	C₂³	C₂²×D₂₈	1	7	yes	—
107	C₂×C₄	—	14	16	no	—
108	C₂×C₄	—	28	112	no	—
109	C₂×C₄	C₂×C₄⋊Dic₇	1	84	yes	—
110	C₂×C₄	C₂²×Dic₁₄	1	14	yes	—
111	C₂×C₄	C₂²×D₂₈	1	7	yes	—
112	C₂×C₄	C₂³×Dic₇	1	14	yes	—
113	C₂×C₄	C₂⁴×D₇	1	1	yes	characteristic
114	C₈	C₂³×Dic₇	1	16	yes	complemented
115	C₇	C₂³×#107	1	1	yes	characteristic, complemented, Sylow
116	C₂²	C₂×#6392	1	112	yes	direct product factor
117	C₂²	C₂²×D₅₆	1	14	yes	—
118	C₂²	C₂²×Dic₂₈	1	14	yes	—
119	C₂²	C₂²×C₄⋊Dic₇	1	14	yes	—
120	C₂²	C₂³×D₂₈	1	1	yes	characteristic
121	C₄	—	28	16	no	—
122	C₄	C₂²×C₄⋊Dic₇	1	14	yes	—
123	C₄	C₂³×Dic₁₄	1	1	yes	characteristic
124	C₄	C₂⁴×Dic₇	1	1	yes	characteristic
125	C₂	C₂²×#6392	1	28	yes	direct product factor
126	C₂	C₂³×D₅₆	1	1	yes	characteristic
127	C₂	C₂³×Dic₂₈	1	1	yes	characteristic
128	C₂	C₂³×C₄⋊Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	D₂₈
commutator	C₄×C₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₇
Fitting	C₂⁴×C₇×C₈	C₂
socle, prime radical	C₂⁵×C₇	D₄
cosocle	C₂×C₄×C₇	C₂⁵
hypercenter	C₂⁴×C₈	D₇
hypocenter	C₇	C₂³×#107
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
3. C₂²×#6392	C₂	direct (N × H)
6. C₂³×C₇×C₈	C₄	semidirect
10. C₂×#6392	C₂²	direct (N × H)
19. C₂²×C₇×C₈	C₂×C₄	semidirect
23. #6392	C₂³	direct (N × H)
38. C₂×C₇×C₈	C₂²×C₄	semidirect
44. C₂³×C₈	Dic₇	semidirect
49. C₂³×C₇	#107	semidirect
61. C₇×C₈	C₂³×C₄	semidirect
69. C₂²×C₈	C₂×Dic₇	semidirect
74. C₂²×C₇	C₂×#107	semidirect
94. C₂×C₈	C₂²×Dic₇	semidirect
98. C₂×C₇	C₂²×#107	semidirect
102. C₂³	#6392	direct (N × H)
114. C₈	C₂³×Dic₇	semidirect
115. C₇	C₂³×#107	semidirect
116. C₂²	C₂×#6392	direct (N × H)
125. C₂	C₂²×#6392	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×C₈	2
C₂³×C₄⋊Dic₇	2
C₂²×#6392	2
C₂³×#107	7

Subgroups — C23×#6392 #14217441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#6392 #14217441