Subgroups — C₂³×#6389 #14217178

Order 1792.

4126 conjugacy classes of subgroups.

The subgroup lattice has 138 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	1	yes	maximal, characteristic
3	C₂⁴×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂²×#6389	C₂	1	28	yes	maximal, direct product factor
5	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₂⁴×Dic₇	C₄	1	2	yes	—
7	C₂³×C₇×C₈	C₄	1	16	yes	complemented
8	C₂³×C₇×C₈	C₂²	1	14	yes	—
9	C₂²×C₄×Dic₇	C₄	1	14	yes	—
10	C₂²×C₄×Dic₇	C₂²	1	14	yes	—
11	C₂³×C₇⋊C₈	C₄	1	16	yes	complemented
12	C₂³×C₇⋊C₈	C₂²	1	14	yes	—
13	C₂×#6389	C₂²	1	112	yes	direct product factor
14	C₂³×#97	—	7	1	no	maximal, Sylow
15	C₂⁵×C₇	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₇	C₂×C₄	1	7	yes	—
17	C₂³×C₄×C₇	C₂×C₄	1	8	yes	—
18	C₂³×C₄×C₇	C₂×C₄	1	8	yes	—
19	C₂³×C₄×C₇	C₂³	1	7	yes	—
20	C₂³×Dic₇	—	2	16	no	—
21	C₂³×Dic₇	C₂×C₄	1	28	yes	—
22	C₂²×C₇×C₈	C₂×C₄	1	112	yes	complemented
23	C₂²×C₇×C₈	C₂³	1	28	yes	—
24	C₂×C₄×Dic₇	C₂×C₄	1	84	yes	—
25	C₂×C₄×Dic₇	C₂³	1	28	yes	—
26	C₂²×C₇⋊C₈	C₂×C₄	1	112	yes	complemented
27	C₂²×C₇⋊C₈	C₂³	1	28	yes	—
28	#6389	C₂³	1	64	yes	direct product factor
29	C₂⁴×C₈	—	7	1	no	—
30	C₂⁴×C₈	D₇	1	1	yes	characteristic
31	C₂³×C₄²	—	7	1	no	—
32	C₂²×#97	—	7	28	no	—
33	C₂⁴×C₇	C₄²	1	8	yes	—
34	C₂⁴×C₇	M₁₆	1	16	yes	—
35	C₂⁴×C₇	C₂²×C₄	1	7	yes	—
36	C₂²×C₄×C₇	C₄²	1	56	yes	—
37	C₂²×C₄×C₇	C₂²×C₄	1	21	yes	—
38	C₂²×C₄×C₇	C₂²×C₄	1	28	yes	—
39	C₂²×C₄×C₇	C₂²×C₄	1	28	yes	—
40	C₂²×C₄×C₇	C₂⁴	1	7	yes	—
41	C₂²×Dic₇	—	2	112	no	—
42	C₂²×Dic₇	C₂²×C₄	1	56	yes	—
43	C₂×C₇×C₈	C₂²×C₄	1	112	yes	complemented
44	C₂×C₇×C₈	C₂⁴	1	8	yes	—
45	C₄×Dic₇	C₂²×C₄	1	56	yes	—
46	C₄×Dic₇	C₂⁴	1	8	yes	—
47	C₂×C₇⋊C₈	C₂²×C₄	1	112	yes	complemented
48	C₂×C₇⋊C₈	C₂⁴	1	8	yes	—
49	C₂⁴×C₄	—	7	2	no	—
50	C₂⁴×C₄	C₂×D₇	1	1	yes	characteristic
51	C₂³×C₈	—	7	16	no	—
52	C₂³×C₈	—	7	14	no	—
53	C₂³×C₈	Dic₇	1	16	yes	complemented
54	C₂³×C₈	C₂×D₇	1	14	yes	—
55	C₂²×C₄²	—	7	14	no	—
56	C₂²×C₄²	—	7	14	no	—
57	C₂×#97	—	7	112	no	—
58	C₂³×C₇	#97	1	64	yes	complemented
59	C₂³×C₇	C₂×C₄²	1	28	yes	—
60	C₂³×C₇	C₂×M₁₆	1	56	yes	—
61	C₂³×C₇	C₂³×C₄	1	7	yes	—
62	C₂×C₄×C₇	C₂×C₄²	1	84	yes	—
63	C₂×C₄×C₇	C₂³×C₄	1	14	yes	—
64	C₂×C₄×C₇	C₂³×C₄	1	7	yes	—
65	C₂×C₄×C₇	C₂³×C₄	1	14	yes	—
66	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
67	C₂×Dic₇	—	2	112	no	—
68	C₂×Dic₇	C₂³×C₄	1	16	yes	—
69	C₇×C₈	C₂³×C₄	1	16	yes	complemented
70	C₇⋊C₈	C₂³×C₄	1	16	yes	complemented
71	C₂⁵	C₄×D₇	1	1	yes	characteristic
72	C₂³×C₄	—	7	28	no	—
73	C₂³×C₄	—	14	16	no	—
74	C₂³×C₄	C₄×D₇	1	7	yes	—
75	C₂³×C₄	C₄×D₇	1	8	yes	—
76	C₂³×C₄	C₂×Dic₇	1	8	yes	—
77	C₂³×C₄	C₂²×D₇	1	7	yes	—
78	C₂²×C₈	—	7	112	no	—
79	C₂²×C₈	—	7	28	no	—
80	C₂²×C₈	C₂×Dic₇	1	112	yes	complemented
81	C₂²×C₈	C₂²×D₇	1	28	yes	—
82	C₂×C₄²	—	7	84	no	—
83	C₂×C₄²	—	7	28	no	—
84	#97	—	7	64	no	—
85	C₂²×C₇	C₂×#97	1	112	yes	complemented
86	C₂²×C₇	C₂²×C₄²	1	14	yes	—
87	C₂²×C₇	C₂²×M₁₆	1	28	yes	—
88	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
89	C₄×C₇	C₂²×C₄²	1	14	yes	—
90	C₄×C₇	C₂⁴×C₄	1	1	yes	characteristic
91	C₄×C₇	C₂⁴×C₄	1	1	yes	characteristic
92	Dic₇	—	2	16	no	—
93	C₂⁴	#1131	1	16	yes	—
94	C₂⁴	C₄×Dic₇	1	8	yes	—
95	C₂⁴	C₂×C₄×D₇	1	7	yes	—
96	C₂²×C₄	—	7	56	no	—
97	C₂²×C₄	—	14	112	no	—
98	C₂²×C₄	C₄×Dic₇	1	56	yes	—
99	C₂²×C₄	C₂×C₄×D₇	1	21	yes	—
100	C₂²×C₄	C₂×C₄×D₇	1	28	yes	—
101	C₂²×C₄	C₂²×Dic₇	1	28	yes	—
102	C₂²×C₄	C₂³×D₇	1	7	yes	—
103	C₂×C₈	—	7	112	no	—
104	C₂×C₈	—	7	8	no	—
105	C₂×C₈	C₂²×Dic₇	1	112	yes	complemented
106	C₂×C₈	C₂³×D₇	1	8	yes	—
107	C₄²	—	7	56	no	—
108	C₄²	—	7	8	no	—
109	C₂×C₇	C₂²×#97	1	28	yes	complemented
110	C₂×C₇	C₂³×C₄²	1	1	yes	characteristic
111	C₂×C₇	C₂³×M₁₆	1	2	yes	—
112	C₂³	#6389	1	64	yes	direct product factor
113	C₂³	C₂×#1131	1	56	yes	—
114	C₂³	C₂×C₄×Dic₇	1	28	yes	—
115	C₂³	C₂²×C₄×D₇	1	7	yes	—
116	C₂×C₄	—	7	16	no	—
117	C₂×C₄	—	14	112	no	—
118	C₂×C₄	C₂×C₄×Dic₇	1	84	yes	—
119	C₂×C₄	C₂²×C₄×D₇	1	7	yes	—
120	C₂×C₄	C₂²×C₄×D₇	1	14	yes	—
121	C₂×C₄	C₂³×Dic₇	1	14	yes	—
122	C₂×C₄	C₂⁴×D₇	1	1	yes	characteristic
123	C₈	—	7	16	no	—
124	C₈	C₂³×Dic₇	1	16	yes	complemented
125	C₇	C₂³×#97	1	1	yes	characteristic, complemented, Sylow
126	C₂²	C₂×#6389	1	112	yes	direct product factor
127	C₂²	C₂²×#1131	1	28	yes	—
128	C₂²	C₂²×C₄×Dic₇	1	14	yes	—
129	C₂²	C₂³×C₄×D₇	1	1	yes	characteristic
130	C₄	—	14	16	no	—
131	C₄	C₂²×C₄×Dic₇	1	14	yes	—
132	C₄	C₂³×C₄×D₇	1	1	yes	characteristic
133	C₄	C₂⁴×Dic₇	1	1	yes	characteristic
134	C₂	C₂²×#6389	1	28	yes	direct product factor
135	C₂	C₂³×#1131	1	2	yes	—
136	C₂	C₂³×C₄×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₇
commutator	C₂×C₇	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₇
Fitting	C₂⁴×C₇×C₈	C₂
socle, prime radical	C₂⁵×C₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂⁵
hypercenter	C₂⁴×C₈	D₇
hypocenter	C₇	C₂³×#97
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
4. C₂²×#6389	C₂	direct (N × H)
7. C₂³×C₇×C₈	C₄	semidirect
11. C₂³×C₇⋊C₈	C₄	semidirect
13. C₂×#6389	C₂²	direct (N × H)
22. C₂²×C₇×C₈	C₂×C₄	semidirect
26. C₂²×C₇⋊C₈	C₂×C₄	semidirect
28. #6389	C₂³	direct (N × H)
43. C₂×C₇×C₈	C₂²×C₄	semidirect
47. C₂×C₇⋊C₈	C₂²×C₄	semidirect
53. C₂³×C₈	Dic₇	semidirect
58. C₂³×C₇	#97	semidirect
69. C₇×C₈	C₂³×C₄	semidirect
70. C₇⋊C₈	C₂³×C₄	semidirect
80. C₂²×C₈	C₂×Dic₇	semidirect
85. C₂²×C₇	C₂×#97	semidirect
105. C₂×C₈	C₂²×Dic₇	semidirect
109. C₂×C₇	C₂²×#97	semidirect
112. C₂³	#6389	direct (N × H)
124. C₈	C₂³×Dic₇	semidirect
125. C₇	C₂³×#97	semidirect
126. C₂²	C₂×#6389	direct (N × H)
134. C₂	C₂²×#6389	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
C₂⁴×C₇⋊C₈	2
C₂²×#6389	2
C₂³×#97	7

Subgroups — C23×#6389 #14217178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#6389 #14217178