Subgroups — C₂³×#6489 #14260356

Order 1792.

14334 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₂²⋊Dic₇	C₂	1	2	yes	maximal
3	C₂²×#6489	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₄⋊Dic₇	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄×Dic₇	C₂	1	2	yes	maximal, complemented
6	C₂⁶×C₇	C₂²	1	1	yes	characteristic
7	C₂⁴×C₄×C₇	C₄	1	32	yes	complemented
8	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂⁴×C₄×C₇	C₂²	1	28	yes	—
10	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂⁴×Dic₇	C₂²	1	4	yes	—
12	C₂²×C₂²⋊Dic₇	C₂²	1	56	yes	—
13	C₂×#6489	C₂²	1	448	yes	direct product factor
14	C₂²×C₄⋊Dic₇	C₂²	1	56	yes	complemented
15	C₂²×C₄×Dic₇	C₂²	1	56	yes	complemented
16	C₂³×#117	—	7	1	no	maximal, Sylow
17	C₂⁵×C₇	—	2	16	no	—
18	C₂⁵×C₇	C₂×C₄	1	16	yes	—
19	C₂⁵×C₇	C₂³	1	14	yes	—
20	C₂⁵×C₇	C₂³	1	1	yes	characteristic
21	C₂³×C₄×C₇	C₂×C₄	1	16	yes	—
22	C₂³×C₄×C₇	C₂×C₄	1	16	yes	—
23	C₂³×C₄×C₇	C₂×C₄	1	448	yes	complemented
24	C₂³×C₄×C₇	C₂³	1	14	yes	—
25	C₂³×C₄×C₇	C₂³	1	14	yes	—
26	C₂³×C₄×C₇	C₂³	1	112	yes	—
27	C₂³×Dic₇	—	2	32	no	—
28	C₂³×Dic₇	C₂³	1	56	yes	—
29	C₂×C₂²⋊Dic₇	C₂³	1	224	yes	—
30	#6489	C₂³	1	512	yes	direct product factor
31	C₂×C₄⋊Dic₇	C₂³	1	224	yes	complemented
32	C₂×C₄×Dic₇	C₂³	1	224	yes	complemented
33	C₂⁵×C₄	D₇	1	1	yes	characteristic
34	C₂²×#117	—	7	56	no	—
35	C₂³×C₄⋊C₄	—	7	2	no	—
36	C₂³×C₂²⋊C₄	—	7	2	no	—
37	C₂³×C₄²	—	7	2	no	—
38	C₂⁴×C₇	—	2	128	no	—
39	C₂⁴×C₇	—	2	112	no	—
40	C₂⁴×C₇	C₂²×C₄	1	112	yes	—
41	C₂⁴×C₇	C₂²×C₄	1	8	yes	—
42	C₂⁴×C₇	#41	1	16	yes	—
43	C₂⁴×C₇	C₂⁴	1	28	yes	—
44	C₂⁴×C₇	C₂⁴	1	7	yes	—
45	C₂²×C₄×C₇	C₂²×C₄	1	112	yes	—
46	C₂²×C₄×C₇	C₂²×C₄	1	112	yes	—
47	C₂²×C₄×C₇	C₂²×C₄	1	896	yes	complemented
48	C₂²×C₄×C₇	C₂⁴	1	28	yes	—
49	C₂²×C₄×C₇	C₂⁴	1	28	yes	—
50	C₂²×C₄×C₇	C₂⁴	1	64	yes	—
51	C₂²×Dic₇	—	2	224	no	—
52	C₂²×Dic₇	C₂⁴	1	112	yes	—
53	C₂²⋊Dic₇	C₂⁴	1	128	yes	—
54	C₄⋊Dic₇	C₂⁴	1	128	yes	complemented
55	C₄×Dic₇	C₂⁴	1	128	yes	complemented
56	C₂⁶	C₂×D₇	1	1	yes	characteristic
57	C₂⁴×C₄	—	7	4	no	—
58	C₂⁴×C₄	Dic₇	1	32	yes	complemented
59	C₂⁴×C₄	C₂×D₇	1	1	yes	characteristic
60	C₂⁴×C₄	C₂×D₇	1	28	yes	—
61	C₂⁴×C₄	C₂×D₇	1	1	yes	characteristic
62	C₂×#117	—	7	448	no	—
63	C₂²×C₄⋊C₄	—	7	56	no	—
64	C₂²×C₂²⋊C₄	—	7	56	no	—
65	C₂²×C₄²	—	7	56	no	—
66	C₂³×C₇	—	2	448	no	—
67	C₂³×C₇	—	2	112	no	—
68	C₂³×C₇	#117	1	64	yes	complemented
69	C₂³×C₇	C₂³×C₄	1	112	yes	—
70	C₂³×C₇	C₂³×C₄	1	28	yes	—
71	C₂³×C₇	C₂×#41	1	56	yes	—
72	C₂³×C₇	C₂⁵	1	8	yes	—
73	C₂³×C₇	C₂⁵	1	7	yes	—
74	C₂×C₄×C₇	C₂³×C₄	1	112	yes	—
75	C₂×C₄×C₇	C₂³×C₄	1	112	yes	—
76	C₂×C₄×C₇	C₂³×C₄	1	256	yes	complemented
77	C₂×C₄×C₇	C₂⁵	1	8	yes	—
78	C₂×C₄×C₇	C₂⁵	1	8	yes	—
79	C₂×Dic₇	—	2	224	no	—
80	C₂×Dic₇	C₂⁵	1	32	yes	—
81	C₂⁵	—	2	16	no	—
82	C₂⁵	C₂×Dic₇	1	16	yes	—
83	C₂⁵	C₂²×D₇	1	14	yes	—
84	C₂⁵	C₂²×D₇	1	1	yes	characteristic
85	C₂³×C₄	—	7	56	no	—
86	C₂³×C₄	—	14	32	no	—
87	C₂³×C₄	C₂×Dic₇	1	16	yes	—
88	C₂³×C₄	C₂×Dic₇	1	16	yes	—
89	C₂³×C₄	C₂×Dic₇	1	448	yes	complemented
90	C₂³×C₄	C₂²×D₇	1	14	yes	—
91	C₂³×C₄	C₂²×D₇	1	14	yes	—
92	C₂³×C₄	C₂²×D₇	1	112	yes	—
93	#117	—	7	512	no	—
94	C₂×C₄⋊C₄	—	7	224	no	—
95	C₂×C₂²⋊C₄	—	7	224	no	—
96	C₂×C₄²	—	7	224	no	—
97	C₂²×C₇	—	2	16	no	—
98	C₂²×C₇	—	2	224	no	—
99	C₂²×C₇	C₂×#117	1	112	yes	complemented
100	C₂²×C₇	C₂⁴×C₄	1	16	yes	—
101	C₂²×C₇	C₂⁴×C₄	1	14	yes	—
102	C₂²×C₇	C₂²×#41	1	28	yes	—
103	C₂²×C₇	C₂⁶	1	1	yes	characteristic
104	C₄×C₇	C₂⁴×C₄	1	16	yes	—
105	C₄×C₇	C₂⁴×C₄	1	16	yes	—
106	Dic₇	—	2	32	no	—
107	C₂⁴	—	2	128	no	—
108	C₂⁴	—	2	112	no	—
109	C₂⁴	#1157	1	16	yes	—
110	C₂⁴	C₂²×Dic₇	1	112	yes	—
111	C₂⁴	C₂²×Dic₇	1	8	yes	—
112	C₂⁴	C₂³×D₇	1	28	yes	—
113	C₂⁴	C₂³×D₇	1	7	yes	—
114	C₂²×C₄	—	7	112	no	—
115	C₂²×C₄	—	14	224	no	—
116	C₂²×C₄	C₂²×Dic₇	1	112	yes	—
117	C₂²×C₄	C₂²×Dic₇	1	112	yes	—
118	C₂²×C₄	C₂²×Dic₇	1	896	yes	complemented
119	C₂²×C₄	C₂³×D₇	1	28	yes	—
120	C₂²×C₄	C₂³×D₇	1	28	yes	—
121	C₂²×C₄	C₂³×D₇	1	64	yes	—
122	C₄⋊C₄	—	7	128	no	—
123	C₂²⋊C₄	—	7	128	no	—
124	C₄²	—	7	128	no	—
125	C₂×C₇	—	2	16	no	—
126	C₂×C₇	C₂²×#117	1	28	yes	complemented
127	C₂×C₇	C₂⁵×C₄	1	1	yes	characteristic
128	C₂×C₇	C₂³×#41	1	2	yes	—
129	C₂³	—	2	448	no	—
130	C₂³	—	2	112	no	—
131	C₂³	#6489	1	64	yes	direct product factor
132	C₂³	C₂×#1157	1	56	yes	—
133	C₂³	C₂³×Dic₇	1	112	yes	—
134	C₂³	C₂³×Dic₇	1	28	yes	—
135	C₂³	C₂⁴×D₇	1	8	yes	—
136	C₂³	C₂⁴×D₇	1	7	yes	—
137	C₂×C₄	—	7	32	no	—
138	C₂×C₄	—	14	224	no	—
139	C₂×C₄	C₂³×Dic₇	1	112	yes	—
140	C₂×C₄	C₂³×Dic₇	1	112	yes	—
141	C₂×C₄	C₂³×Dic₇	1	256	yes	complemented
142	C₂×C₄	C₂⁴×D₇	1	8	yes	—
143	C₂×C₄	C₂⁴×D₇	1	8	yes	—
144	C₇	C₂³×#117	1	1	yes	characteristic, complemented, Sylow
145	C₂²	—	2	224	no	—
146	C₂²	—	2	16	no	—
147	C₂²	C₂×#6489	1	112	yes	direct product factor
148	C₂²	C₂²×#1157	1	28	yes	—
149	C₂²	C₂⁴×Dic₇	1	16	yes	—
150	C₂²	C₂⁴×Dic₇	1	14	yes	—
151	C₂²	C₂⁵×D₇	1	1	yes	characteristic
152	C₄	—	14	32	no	—
153	C₄	C₂⁴×Dic₇	1	16	yes	—
154	C₄	C₂⁴×Dic₇	1	16	yes	—
155	C₂	—	2	16	no	—
156	C₂	C₂²×#6489	1	28	yes	direct product factor
157	C₂	C₂³×#1157	1	2	yes	—
158	C₂	C₂⁵×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₇
commutator	C₂×C₇	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₇
Fitting	C₂⁵×C₄×C₇	C₂
socle	C₂⁵×C₇	C₂³
prime radical	C₂⁶×C₇	C₂²
cosocle	C₂²×C₇	C₂⁶
hypercenter	C₂⁵×C₄	D₇
hypocenter	C₇	C₂³×#117
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
3. C₂²×#6489	C₂	direct (N × H)
4. C₂³×C₄⋊Dic₇	C₂	semidirect
5. C₂³×C₄×Dic₇	C₂	semidirect
7. C₂⁴×C₄×C₇	C₄	semidirect
13. C₂×#6489	C₂²	direct (N × H)
14. C₂²×C₄⋊Dic₇	C₂²	semidirect
15. C₂²×C₄×Dic₇	C₂²	semidirect
23. C₂³×C₄×C₇	C₂×C₄	semidirect
30. #6489	C₂³	direct (N × H)
31. C₂×C₄⋊Dic₇	C₂³	semidirect
32. C₂×C₄×Dic₇	C₂³	semidirect
47. C₂²×C₄×C₇	C₂²×C₄	semidirect
54. C₄⋊Dic₇	C₂⁴	semidirect
55. C₄×Dic₇	C₂⁴	semidirect
58. C₂⁴×C₄	Dic₇	semidirect
68. C₂³×C₇	#117	semidirect
76. C₂×C₄×C₇	C₂³×C₄	semidirect
89. C₂³×C₄	C₂×Dic₇	semidirect
99. C₂²×C₇	C₂×#117	semidirect
118. C₂²×C₄	C₂²×Dic₇	semidirect
126. C₂×C₇	C₂²×#117	semidirect
131. C₂³	#6489	direct (N × H)
141. C₂×C₄	C₂³×Dic₇	semidirect
144. C₇	C₂³×#117	semidirect
147. C₂²	C₂×#6489	direct (N × H)
156. C₂	C₂²×#6489	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₄×C₇	2
C₂³×C₂²⋊Dic₇	2
C₂²×#6489	2
C₂³×C₄⋊Dic₇	2
C₂³×C₄×Dic₇	2
C₂³×#117	7

Subgroups — C23×#6489 #14260356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#6489 #14260356