Subgroups — C₂³×#6548 #14267682

Order 1792.

65726 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₂²⋊D₇	C₂	1	4	yes	maximal, complemented
3	C₂²×#6548	C₂	1	112	yes	maximal, direct product factor
4	C₂³×D₄×D₇	C₂	1	4	yes	maximal, complemented
5	C₂³×#1159	C₂	1	4	yes	maximal, complemented
6	C₂³×#1157	C₂	1	2	yes	maximal, complemented
7	C₂⁶×C₇	C₂²	1	2	yes	complemented
8	C₂⁵×D₇	C₂²	1	4	yes	complemented
9	C₂³×C₇×D₄	C₂²	1	4	yes	complemented
10	C₂³×C₇×D₄	C₂²	1	56	yes	complemented
11	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic, complemented
12	C₂³×C₂²⋊D₇	C₂²	1	4	yes	complemented
13	C₂³×C₂²⋊D₇	C₂²	1	8	yes	complemented
14	C₂³×C₂²⋊D₇	C₂²	1	224	yes	complemented
15	C₂⁴×Dic₇	C₂²	1	4	yes	complemented
16	C₂×#6548	C₂²	1	1792	yes	direct product factor
17	C₂²×#1159	C₂²	1	224	yes	complemented
18	C₂²×D₄×D₇	C₂²	1	224	yes	complemented
19	C₂²×#1157	C₂²	1	112	yes	complemented
20	C₂³×D₂₈	C₂²	1	2	yes	complemented
21	C₂³×C₄×D₇	C₂²	1	4	yes	complemented
22	C₂³×Dic₁₄	C₂²	1	2	yes	complemented
23	C₂³×H(5,𝔽₂)	—	7	1	no	maximal, Sylow
24	C₂⁵×C₇	—	4	16	no	—
25	C₂⁵×C₇	C₂³	1	4	yes	—
26	C₂⁵×C₇	C₂³	1	1	yes	characteristic
27	C₂⁵×C₇	C₂³	1	56	yes	complemented
28	C₂⁴×D₇	—	4	32	no	—
29	C₂⁴×D₇	C₂³	1	4	yes	—
30	C₂⁴×D₇	C₂³	1	112	yes	complemented
31	C₂²×C₇×D₄	C₂³	1	112	yes	complemented
32	C₂²×C₇×D₄	C₂³	1	448	yes	complemented
33	C₂³×C₄×C₇	C₂³	1	2	yes	—
34	C₂³×C₄×C₇	C₂³	1	28	yes	complemented
35	C₂²×C₂²⋊D₇	C₂³	1	112	yes	complemented
36	C₂²×C₂²⋊D₇	C₂³	1	224	yes	complemented
37	C₂²×C₂²⋊D₇	C₂³	1	1792	yes	complemented
38	C₂³×Dic₇	C₂³	1	4	yes	—
39	C₂³×Dic₇	C₂³	1	112	yes	complemented
40	#6548	C₂³	1	4096	yes	direct product factor
41	C₂×#1159	C₂³	1	1792	yes	complemented
42	C₂×D₄×D₇	C₂³	1	1792	yes	complemented
43	C₂×#1157	C₂³	1	896	yes	complemented
44	C₂²×D₂₈	C₂³	1	56	yes	complemented
45	C₂²×C₄×D₇	C₂³	1	112	yes	complemented
46	C₂²×Dic₁₄	C₂³	1	56	yes	complemented
47	C₂²×H(5,𝔽₂)	—	7	112	no	—
48	C₂³×#41	—	7	4	no	—
49	C₂³×#41	—	7	2	no	—
50	C₂⁴×D₄	—	7	4	no	—
51	C₂⁴×D₄	—	7	4	no	—
52	C₂⁴×D₄	D₇	1	1	yes	characteristic, complemented
53	C₂⁴×C₇	—	2	32	no	—
54	C₂⁴×C₇	—	2	8	no	—
55	C₂⁴×C₇	—	4	224	no	—
56	C₂⁴×C₇	C₂⁴	1	56	yes	—
57	C₂⁴×C₇	C₂⁴	1	14	yes	—
58	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
59	C₂⁴×C₇	C₂⁴	1	224	yes	complemented
60	C₂³×D₇	—	2	32	no	—
61	C₂³×D₇	—	4	448	no	—
62	C₂³×D₇	C₂⁴	1	56	yes	—
63	C₂³×D₇	C₂⁴	1	448	yes	complemented
64	C₂×C₇×D₄	C₂⁴	1	448	yes	complemented
65	C₂×C₇×D₄	C₂⁴	1	512	yes	complemented
66	C₂²×C₄×C₇	C₂⁴	1	28	yes	—
67	C₂²×C₄×C₇	C₂⁴	1	112	yes	complemented
68	C₂×C₂²⋊D₇	C₂⁴	1	448	yes	complemented
69	C₂×C₂²⋊D₇	C₂⁴	1	896	yes	complemented
70	C₂×C₂²⋊D₇	C₂⁴	1	2048	yes	complemented
71	C₂²×Dic₇	C₂⁴	1	56	yes	—
72	C₂²×Dic₇	C₂⁴	1	448	yes	complemented
73	#1159	C₂⁴	1	2048	yes	complemented
74	D₄×D₇	C₂⁴	1	2048	yes	complemented
75	#1157	C₂⁴	1	1024	yes	complemented
76	C₂×D₂₈	C₂⁴	1	224	yes	complemented
77	C₂×C₄×D₇	C₂⁴	1	448	yes	complemented
78	C₂×Dic₁₄	C₂⁴	1	224	yes	complemented
79	C₂⁶	—	7	4	no	—
80	C₂⁶	C₂×D₇	1	2	yes	complemented
81	C₂×H(5,𝔽₂)	—	7	1792	no	—
82	C₂²×#41	—	7	224	no	—
83	C₂²×#41	—	7	112	no	—
84	C₂³×Q₈	—	7	2	no	—
85	C₂³×D₄	—	7	8	no	—
86	C₂³×D₄	—	7	4	no	—
87	C₂³×D₄	—	7	224	no	—
88	C₂³×D₄	—	7	224	no	—
89	C₂³×D₄	—	7	2	no	—
90	C₂³×D₄	C₂×D₇	1	4	yes	complemented
91	C₂³×D₄	C₂×D₇	1	56	yes	complemented
92	C₂⁴×C₄	—	7	4	no	—
93	C₂⁴×C₄	—	7	4	no	—
94	C₂⁴×C₄	C₂×D₇	1	1	yes	characteristic, complemented
95	C₂³×C₇	—	2	224	no	—
96	C₂³×C₇	—	2	56	no	—
97	C₂³×C₇	—	4	448	no	—
98	C₂³×C₇	H(5,𝔽₂)	1	8	yes	complemented
99	C₂³×C₇	C₂⁵	1	112	yes	—
100	C₂³×C₇	C₂⁵	1	28	yes	—
101	C₂³×C₇	C₂⁵	1	7	yes	—
102	C₂³×C₇	C₂⁵	1	128	yes	complemented
103	C₂²×D₇	—	2	224	no	—
104	C₂²×D₇	—	4	896	no	—
105	C₂²×D₇	C₂⁵	1	112	yes	—
106	C₂²×D₇	C₂⁵	1	256	yes	complemented
107	C₇×D₄	C₂⁵	1	256	yes	complemented
108	C₂×C₄×C₇	C₂⁵	1	56	yes	—
109	C₂×C₄×C₇	C₂⁵	1	64	yes	complemented
110	C₂²⋊D₇	C₂⁵	1	256	yes	complemented
111	C₂²⋊D₇	C₂⁵	1	512	yes	complemented
112	C₂×Dic₇	C₂⁵	1	112	yes	—
113	C₂×Dic₇	C₂⁵	1	256	yes	complemented
114	D₂₈	C₂⁵	1	128	yes	complemented
115	C₄×D₇	C₂⁵	1	256	yes	complemented
116	Dic₁₄	C₂⁵	1	128	yes	complemented
117	C₂⁵	—	4	16	no	—
118	C₂⁵	—	7	4	no	—
119	C₂⁵	—	7	112	no	—
120	C₂⁵	—	28	32	no	—
121	C₂⁵	C₂²×D₇	1	4	yes	—
122	C₂⁵	C₂²×D₇	1	1	yes	characteristic
123	C₂⁵	C₂²×D₇	1	56	yes	complemented
124	H(5,𝔽₂)	—	7	4096	no	—
125	C₂×#41	—	7	1792	no	—
126	C₂×#41	—	7	896	no	—
127	C₂²×Q₈	—	7	56	no	—
128	C₂²×D₄	—	7	224	no	—
129	C₂²×D₄	—	7	112	no	—
130	C₂²×D₄	—	7	1792	no	—
131	C₂²×D₄	—	7	1792	no	—
132	C₂²×D₄	—	7	56	no	—
133	C₂²×D₄	C₂²×D₇	1	112	yes	complemented
134	C₂²×D₄	C₂²×D₇	1	448	yes	complemented
135	C₂³×C₄	—	7	112	no	—
136	C₂³×C₄	—	7	4	no	—
137	C₂³×C₄	—	7	112	no	—
138	C₂³×C₄	C₂²×D₇	1	2	yes	—
139	C₂³×C₄	C₂²×D₇	1	28	yes	complemented
140	C₂²×C₇	—	2	224	no	—
141	C₂²×C₇	—	2	56	no	—
142	C₂²×C₇	—	4	128	no	—
143	C₂²×C₇	C₂×H(5,𝔽₂)	1	28	yes	complemented
144	C₂²×C₇	C₂⁶	1	32	yes	—
145	C₂²×C₇	C₂⁶	1	8	yes	—
146	C₂²×C₇	C₂⁶	1	7	yes	—
147	C₂×D₇	—	2	224	no	—
148	C₂×D₇	—	4	256	no	—
149	C₂×D₇	C₂⁶	1	32	yes	—
150	C₄×C₇	C₂⁶	1	16	yes	—
151	Dic₇	C₂⁶	1	32	yes	—
152	C₂⁴	—	2	8	no	—
153	C₂⁴	—	2	32	no	—
154	C₂⁴	—	4	224	no	—
155	C₂⁴	—	7	56	no	—
156	C₂⁴	—	7	448	no	—
157	C₂⁴	—	14	32	no	—
158	C₂⁴	—	28	448	no	—
159	C₂⁴	C₂³×D₇	1	56	yes	—
160	C₂⁴	C₂³×D₇	1	14	yes	—
161	C₂⁴	C₂³×D₇	1	1	yes	characteristic
162	C₂⁴	C₂³×D₇	1	224	yes	complemented
163	#41	—	7	2048	no	—
164	#41	—	7	1024	no	—
165	C₂×Q₈	—	7	224	no	—
166	C₂×D₄	—	7	896	no	—
167	C₂×D₄	—	7	448	no	—
168	C₂×D₄	—	7	2048	no	—
169	C₂×D₄	—	7	2048	no	—
170	C₂×D₄	—	7	224	no	—
171	C₂×D₄	C₂³×D₇	1	448	yes	complemented
172	C₂×D₄	C₂³×D₇	1	512	yes	complemented
173	C₂²×C₄	—	7	56	no	—
174	C₂²×C₄	—	7	448	no	—
175	C₂²×C₄	—	7	448	no	—
176	C₂²×C₄	C₂³×D₇	1	28	yes	—
177	C₂²×C₄	C₂³×D₇	1	112	yes	complemented
178	C₂×C₇	—	2	32	no	—
179	C₂×C₇	—	2	8	no	—
180	C₂×C₇	C₂²×H(5,𝔽₂)	1	14	yes	complemented
181	C₂×C₇	C₂⁷	1	1	yes	characteristic
182	D₇	—	2	32	no	—
183	C₂³	—	2	56	no	—
184	C₂³	—	2	224	no	—
185	C₂³	—	4	448	no	—
186	C₂³	—	7	112	no	—
187	C₂³	—	7	256	no	—
188	C₂³	—	14	224	no	—
189	C₂³	—	28	896	no	—
190	C₂³	#6548	1	8	yes	direct product factor
191	C₂³	C₂⁴×D₇	1	112	yes	—
192	C₂³	C₂⁴×D₇	1	28	yes	—
193	C₂³	C₂⁴×D₇	1	7	yes	—
194	C₂³	C₂⁴×D₇	1	128	yes	complemented
195	Q₈	—	7	128	no	—
196	D₄	—	7	512	no	—
197	D₄	—	7	256	no	—
198	D₄	—	7	128	no	—
199	D₄	C₂⁴×D₇	1	256	yes	complemented
200	C₂×C₄	—	7	112	no	—
201	C₂×C₄	—	7	256	no	—
202	C₂×C₄	—	7	256	no	—
203	C₂×C₄	C₂⁴×D₇	1	56	yes	—
204	C₂×C₄	C₂⁴×D₇	1	64	yes	complemented
205	C₇	C₂³×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
206	C₂²	—	2	56	no	—
207	C₂²	—	2	224	no	—
208	C₂²	—	4	128	no	—
209	C₂²	—	7	32	no	—
210	C₂²	—	14	224	no	—
211	C₂²	—	28	256	no	—
212	C₂²	C₂×#6548	1	28	yes	direct product factor
213	C₂²	C₂⁵×D₇	1	32	yes	—
214	C₂²	C₂⁵×D₇	1	8	yes	—
215	C₂²	C₂⁵×D₇	1	7	yes	—
216	C₄	—	7	32	no	—
217	C₄	C₂⁵×D₇	1	16	yes	—
218	C₂	—	2	8	no	—
219	C₂	—	2	32	no	—
220	C₂	—	14	32	no	—
221	C₂	C₂²×#6548	1	14	yes	direct product factor
222	C₂	C₂⁶×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₇
commutator, cosocle	C₂×C₇	C₂⁷
Frattini	C₂	C₂⁶×D₇
Fitting	C₂⁴×C₇×D₄	C₂
socle	C₂⁴×C₇	C₂⁴
hypercenter	C₂⁴×D₄	D₇
hypocenter	C₇	C₂³×H(5,𝔽₂)
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×D₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₇×D₄	C₂	semidirect
2. C₂⁴×C₂²⋊D₇	C₂	semidirect
3. C₂²×#6548	C₂	direct (N × H)
4. C₂³×D₄×D₇	C₂	semidirect
5. C₂³×#1159	C₂	semidirect
6. C₂³×#1157	C₂	semidirect
7. C₂⁶×C₇	C₂²	semidirect
8. C₂⁵×D₇	C₂²	semidirect
9. C₂³×C₇×D₄	C₂²	semidirect
10. C₂³×C₇×D₄	C₂²	semidirect
11. C₂⁴×C₄×C₇	C₂²	semidirect
12. C₂³×C₂²⋊D₇	C₂²	semidirect
13. C₂³×C₂²⋊D₇	C₂²	semidirect
14. C₂³×C₂²⋊D₇	C₂²	semidirect
15. C₂⁴×Dic₇	C₂²	semidirect
16. C₂×#6548	C₂²	direct (N × H)
17. C₂²×#1159	C₂²	semidirect
18. C₂²×D₄×D₇	C₂²	semidirect
19. C₂²×#1157	C₂²	semidirect
20. C₂³×D₂₈	C₂²	semidirect
21. C₂³×C₄×D₇	C₂²	semidirect
22. C₂³×Dic₁₄	C₂²	semidirect
27. C₂⁵×C₇	C₂³	semidirect
30. C₂⁴×D₇	C₂³	semidirect
31. C₂²×C₇×D₄	C₂³	semidirect
32. C₂²×C₇×D₄	C₂³	semidirect
34. C₂³×C₄×C₇	C₂³	semidirect
35. C₂²×C₂²⋊D₇	C₂³	semidirect
36. C₂²×C₂²⋊D₇	C₂³	semidirect
37. C₂²×C₂²⋊D₇	C₂³	semidirect
39. C₂³×Dic₇	C₂³	semidirect
40. #6548	C₂³	direct (N × H)
41. C₂×#1159	C₂³	semidirect
42. C₂×D₄×D₇	C₂³	semidirect
43. C₂×#1157	C₂³	semidirect
44. C₂²×D₂₈	C₂³	semidirect
45. C₂²×C₄×D₇	C₂³	semidirect
46. C₂²×Dic₁₄	C₂³	semidirect
52. C₂⁴×D₄	D₇	semidirect
59. C₂⁴×C₇	C₂⁴	semidirect
63. C₂³×D₇	C₂⁴	semidirect
64. C₂×C₇×D₄	C₂⁴	semidirect
65. C₂×C₇×D₄	C₂⁴	semidirect
67. C₂²×C₄×C₇	C₂⁴	semidirect
68. C₂×C₂²⋊D₇	C₂⁴	semidirect
69. C₂×C₂²⋊D₇	C₂⁴	semidirect
70. C₂×C₂²⋊D₇	C₂⁴	semidirect
72. C₂²×Dic₇	C₂⁴	semidirect
73. #1159	C₂⁴	semidirect
74. D₄×D₇	C₂⁴	semidirect
75. #1157	C₂⁴	semidirect
76. C₂×D₂₈	C₂⁴	semidirect
77. C₂×C₄×D₇	C₂⁴	semidirect
78. C₂×Dic₁₄	C₂⁴	semidirect
80. C₂⁶	C₂×D₇	semidirect
90. C₂³×D₄	C₂×D₇	semidirect
91. C₂³×D₄	C₂×D₇	semidirect
94. C₂⁴×C₄	C₂×D₇	semidirect
98. C₂³×C₇	H(5,𝔽₂)	semidirect
102. C₂³×C₇	C₂⁵	semidirect
106. C₂²×D₇	C₂⁵	semidirect
107. C₇×D₄	C₂⁵	semidirect
109. C₂×C₄×C₇	C₂⁵	semidirect
110. C₂²⋊D₇	C₂⁵	semidirect
111. C₂²⋊D₇	C₂⁵	semidirect
113. C₂×Dic₇	C₂⁵	semidirect
114. D₂₈	C₂⁵	semidirect
115. C₄×D₇	C₂⁵	semidirect
116. Dic₁₄	C₂⁵	semidirect
123. C₂⁵	C₂²×D₇	semidirect
133. C₂²×D₄	C₂²×D₇	semidirect
134. C₂²×D₄	C₂²×D₇	semidirect
139. C₂³×C₄	C₂²×D₇	semidirect
143. C₂²×C₇	C₂×H(5,𝔽₂)	semidirect
162. C₂⁴	C₂³×D₇	semidirect
171. C₂×D₄	C₂³×D₇	semidirect
172. C₂×D₄	C₂³×D₇	semidirect
177. C₂²×C₄	C₂³×D₇	semidirect
180. C₂×C₇	C₂²×H(5,𝔽₂)	semidirect
190. C₂³	#6548	direct (N × H)
194. C₂³	C₂⁴×D₇	semidirect
199. D₄	C₂⁴×D₇	semidirect
204. C₂×C₄	C₂⁴×D₇	semidirect
205. C₇	C₂³×H(5,𝔽₂)	semidirect
212. C₂²	C₂×#6548	direct (N × H)
221. C₂	C₂²×#6548	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×D₄	2
C₂⁴×C₂²⋊D₇	2
C₂²×#6548	2
C₂³×D₄×D₇	2
C₂³×#1159	2
C₂³×#1157	2
C₂³×H(5,𝔽₂)	7

Subgroups — C23×#6548 #14267682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#6548 #14267682