Subgroups — C₂³×#6501 #14260430

Order 1792.

19998 conjugacy classes of subgroups.

The subgroup lattice has 240 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	2	yes	maximal, complemented
3	C₂⁴×C₄×D₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₂²⋊Dic₇	C₂	1	2	yes	maximal, complemented
5	C₂²×#6501	C₂	1	56	yes	maximal, direct product factor
6	C₂³×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁶×C₇	C₂²	1	2	yes	—
8	C₂⁵×D₇	C₂²	1	1	yes	characteristic
9	C₂³×C₇×D₄	—	2	16	no	—
10	C₂³×C₇×D₄	C₂²	1	28	yes	complemented
11	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₂³×C₂²⋊D₇	—	2	32	no	—
13	C₂³×C₂²⋊D₇	C₂²	1	56	yes	complemented
14	C₂⁴×Dic₇	C₂²	1	2	yes	—
15	C₂⁴×Dic₇	C₂²	1	1	yes	characteristic
16	C₂³×C₄×D₇	—	2	16	no	—
17	C₂³×C₄×D₇	C₂²	1	28	yes	complemented
18	C₂²×C₂²⋊Dic₇	C₂²	1	56	yes	complemented
19	C₂×#6501	C₂²	1	448	yes	direct product factor
20	C₂²×C₄⋊Dic₇	C₂²	1	28	yes	complemented
21	C₂³×#121	—	7	1	no	maximal, Sylow
22	C₂⁵×C₇	—	2	16	no	—
23	C₂⁵×C₇	—	2	16	no	—
24	C₂⁵×C₇	—	2	16	no	—
25	C₂⁵×C₇	C₂³	1	28	yes	—
26	C₂⁵×C₇	C₂³	1	1	yes	characteristic
27	C₂⁴×D₇	—	2	8	no	—
28	C₂⁴×D₇	—	2	8	no	—
29	C₂⁴×D₇	D₄	1	16	yes	complemented
30	C₂⁴×D₇	C₂³	1	14	yes	—
31	C₂²×C₇×D₄	—	2	224	no	—
32	C₂²×C₇×D₄	C₂³	1	112	yes	complemented
33	C₂³×C₄×C₇	D₄	1	16	yes	complemented
34	C₂³×C₄×C₇	C₂³	1	14	yes	—
35	C₂²×C₂²⋊D₇	—	2	448	no	—
36	C₂²×C₂²⋊D₇	C₂³	1	224	yes	complemented
37	C₂³×Dic₇	—	2	8	no	—
38	C₂³×Dic₇	—	2	16	no	—
39	C₂³×Dic₇	C₂³	1	28	yes	—
40	C₂³×Dic₇	C₂³	1	14	yes	—
41	C₂²×C₄×D₇	—	2	224	no	—
42	C₂²×C₄×D₇	C₂³	1	112	yes	complemented
43	C₂×C₂²⋊Dic₇	C₂³	1	224	yes	complemented
44	#6501	C₂³	1	512	yes	direct product factor
45	C₂×C₄⋊Dic₇	C₂³	1	112	yes	complemented
46	C₂⁴×D₄	—	7	2	no	—
47	C₂⁴×D₄	D₇	1	1	yes	characteristic, complemented
48	C₂⁵×C₄	—	7	1	no	—
49	C₂²×#121	—	7	56	no	—
50	C₂³×C₄⋊C₄	—	7	1	no	—
51	C₂³×C₂²⋊C₄	—	7	2	no	—
52	C₂⁴×C₇	—	2	112	no	—
53	C₂⁴×C₇	—	2	112	no	—
54	C₂⁴×C₇	—	2	112	no	—
55	C₂⁴×C₇	—	4	128	no	—
56	C₂⁴×C₇	C₂×D₄	1	8	yes	—
57	C₂⁴×C₇	C₂×D₄	1	8	yes	—
58	C₂⁴×C₇	#41	1	8	yes	—
59	C₂⁴×C₇	C₂⁴	1	56	yes	—
60	C₂⁴×C₇	C₂⁴	1	7	yes	—
61	C₂³×D₇	—	2	128	no	—
62	C₂³×D₇	—	2	56	no	—
63	C₂³×D₇	—	2	56	no	—
64	C₂³×D₇	C₂×D₄	1	112	yes	complemented
65	C₂³×D₇	C₂⁴	1	28	yes	—
66	C₂×C₇×D₄	—	2	448	no	—
67	C₂×C₇×D₄	C₂⁴	1	64	yes	complemented
68	C₂²×C₄×C₇	C₂×D₄	1	112	yes	complemented
69	C₂²×C₄×C₇	C₂⁴	1	28	yes	—
70	C₂×C₂²⋊D₇	—	2	896	no	—
71	C₂×C₂²⋊D₇	C₂⁴	1	128	yes	complemented
72	C₂²×Dic₇	—	2	56	no	—
73	C₂²×Dic₇	—	2	112	no	—
74	C₂²×Dic₇	C₂⁴	1	56	yes	—
75	C₂²×Dic₇	C₂⁴	1	28	yes	—
76	C₂×C₄×D₇	—	2	448	no	—
77	C₂×C₄×D₇	C₂⁴	1	64	yes	complemented
78	C₂²⋊Dic₇	C₂⁴	1	128	yes	complemented
79	C₄⋊Dic₇	C₂⁴	1	64	yes	complemented
80	C₂⁶	—	7	1	no	—
81	C₂⁶	C₂×D₇	1	2	yes	—
82	C₂³×D₄	—	2	16	no	—
83	C₂³×D₄	—	7	56	no	—
84	C₂³×D₄	—	14	32	no	—
85	C₂³×D₄	C₂×D₇	1	28	yes	complemented
86	C₂⁴×C₄	—	7	1	no	—
87	C₂⁴×C₄	—	7	2	no	—
88	C₂⁴×C₄	—	7	28	no	—
89	C₂⁴×C₄	—	14	16	no	—
90	C₂⁴×C₄	C₂×D₇	1	1	yes	characteristic
91	C₂×#121	—	7	448	no	—
92	C₂²×C₄⋊C₄	—	7	28	no	—
93	C₂²×C₂²⋊C₄	—	7	56	no	—
94	C₂³×C₇	—	2	112	no	—
95	C₂³×C₇	—	2	112	no	—
96	C₂³×C₇	—	2	112	no	—
97	C₂³×C₇	—	4	448	no	—
98	C₂³×C₇	#121	1	64	yes	complemented
99	C₂³×C₇	C₂²×D₄	1	28	yes	—
100	C₂³×C₇	C₂²×D₄	1	28	yes	—
101	C₂³×C₇	C₂×#41	1	28	yes	—
102	C₂³×C₇	C₂⁵	1	16	yes	—
103	C₂³×C₇	C₂⁵	1	7	yes	—
104	C₂²×D₇	—	2	448	no	—
105	C₂²×D₇	—	2	56	no	—
106	C₂²×D₇	—	2	56	no	—
107	C₂²×D₇	C₂²×D₄	1	112	yes	complemented
108	C₂²×D₇	C₂⁵	1	8	yes	—
109	C₇×D₄	—	2	128	no	—
110	C₂×C₄×C₇	C₂²×D₄	1	112	yes	complemented
111	C₂×C₄×C₇	C₂⁵	1	8	yes	—
112	C₂²⋊D₇	—	2	256	no	—
113	C₂×Dic₇	—	2	56	no	—
114	C₂×Dic₇	—	2	112	no	—
115	C₂×Dic₇	C₂⁵	1	16	yes	—
116	C₂×Dic₇	C₂⁵	1	8	yes	—
117	C₄×D₇	—	2	128	no	—
118	C₂⁵	—	2	16	no	—
119	C₂⁵	—	2	16	no	—
120	C₂⁵	—	2	16	no	—
121	C₂⁵	—	7	16	no	—
122	C₂⁵	—	7	14	no	—
123	C₂⁵	—	14	8	no	—
124	C₂⁵	—	14	8	no	—
125	C₂⁵	C₂²×D₇	1	28	yes	—
126	C₂⁵	C₂²×D₇	1	1	yes	characteristic
127	C₂²×D₄	—	2	224	no	—
128	C₂²×D₄	—	7	224	no	—
129	C₂²×D₄	—	14	448	no	—
130	C₂²×D₄	C₂²×D₇	1	112	yes	complemented
131	C₂³×C₄	—	7	14	no	—
132	C₂³×C₄	—	7	28	no	—
133	C₂³×C₄	—	7	112	no	—
134	C₂³×C₄	—	14	16	no	—
135	C₂³×C₄	—	14	8	no	—
136	C₂³×C₄	—	14	224	no	—
137	C₂³×C₄	C₂²⋊D₇	1	16	yes	complemented
138	C₂³×C₄	C₂²×D₇	1	14	yes	—
139	#121	—	7	512	no	—
140	C₂×C₄⋊C₄	—	7	112	no	—
141	C₂×C₂²⋊C₄	—	7	224	no	—
142	C₂²×C₇	—	2	16	no	—
143	C₂²×C₇	—	2	16	no	—
144	C₂²×C₇	—	2	16	no	—
145	C₂²×C₇	—	4	224	no	—
146	C₂²×C₇	C₂×#121	1	112	yes	complemented
147	C₂²×C₇	C₂³×D₄	1	14	yes	—
148	C₂²×C₇	C₂³×D₄	1	14	yes	—
149	C₂²×C₇	C₂²×#41	1	14	yes	—
150	C₂²×C₇	C₂⁶	1	1	yes	characteristic
151	C₂×D₇	—	2	8	no	—
152	C₂×D₇	—	2	8	no	—
153	C₂×D₇	—	2	224	no	—
154	C₂×D₇	C₂³×D₄	1	16	yes	complemented
155	C₄×C₇	C₂³×D₄	1	16	yes	complemented
156	Dic₇	—	2	8	no	—
157	Dic₇	—	2	16	no	—
158	C₂⁴	—	2	112	no	—
159	C₂⁴	—	2	112	no	—
160	C₂⁴	—	2	112	no	—
161	C₂⁴	—	4	128	no	—
162	C₂⁴	—	7	112	no	—
163	C₂⁴	—	7	28	no	—
164	C₂⁴	—	14	128	no	—
165	C₂⁴	—	14	56	no	—
166	C₂⁴	—	14	56	no	—
167	C₂⁴	D₄×D₇	1	8	yes	—
168	C₂⁴	#1159	1	8	yes	—
169	C₂⁴	C₂×C₂²⋊D₇	1	8	yes	—
170	C₂⁴	C₂³×D₇	1	56	yes	—
171	C₂⁴	C₂³×D₇	1	7	yes	—
172	C₂×D₄	—	2	448	no	—
173	C₂×D₄	—	7	128	no	—
174	C₂×D₄	—	14	896	no	—
175	C₂×D₄	C₂³×D₇	1	64	yes	complemented
176	C₂²×C₄	—	7	28	no	—
177	C₂²×C₄	—	7	56	no	—
178	C₂²×C₄	—	7	64	no	—
179	C₂²×C₄	—	14	112	no	—
180	C₂²×C₄	—	14	56	no	—
181	C₂²×C₄	—	14	448	no	—
182	C₂²×C₄	C₂×C₂²⋊D₇	1	112	yes	complemented
183	C₂²×C₄	C₂³×D₇	1	28	yes	—
184	C₄⋊C₄	—	7	64	no	—
185	C₂²⋊C₄	—	7	128	no	—
186	C₂×C₇	—	4	16	no	—
187	C₂×C₇	C₂²×#121	1	28	yes	complemented
188	C₂×C₇	C₂⁴×D₄	1	1	yes	characteristic
189	C₂×C₇	C₂⁴×D₄	1	1	yes	characteristic
190	C₂×C₇	C₂³×#41	1	1	yes	characteristic
191	D₇	—	2	16	no	—
192	C₂³	—	2	112	no	—
193	C₂³	—	2	112	no	—
194	C₂³	—	2	112	no	—
195	C₂³	—	4	448	no	—
196	C₂³	—	7	8	no	—
197	C₂³	—	7	112	no	—
198	C₂³	—	14	448	no	—
199	C₂³	—	14	56	no	—
200	C₂³	—	14	56	no	—
201	C₂³	#6501	1	64	yes	direct product factor
202	C₂³	C₂×D₄×D₇	1	28	yes	—
203	C₂³	C₂×#1159	1	28	yes	—
204	C₂³	C₂²×C₂²⋊D₇	1	28	yes	—
205	C₂³	C₂⁴×D₇	1	16	yes	—
206	C₂³	C₂⁴×D₇	1	7	yes	—
207	D₄	—	2	128	no	—
208	D₄	—	14	256	no	—
209	C₂×C₄	—	7	8	no	—
210	C₂×C₄	—	7	16	no	—
211	C₂×C₄	—	14	112	no	—
212	C₂×C₄	—	14	56	no	—
213	C₂×C₄	—	14	128	no	—
214	C₂×C₄	C₂²×C₂²⋊D₇	1	112	yes	complemented
215	C₂×C₄	C₂⁴×D₇	1	8	yes	—
216	C₇	C₂³×#121	1	1	yes	characteristic, complemented, Sylow
217	C₂²	—	2	16	no	—
218	C₂²	—	2	16	no	—
219	C₂²	—	2	16	no	—
220	C₂²	—	4	224	no	—
221	C₂²	—	7	16	no	—
222	C₂²	—	14	8	no	—
223	C₂²	—	14	224	no	—
224	C₂²	—	14	8	no	—
225	C₂²	C₂×#6501	1	112	yes	direct product factor
226	C₂²	C₂²×D₄×D₇	1	14	yes	—
227	C₂²	C₂²×#1159	1	14	yes	—
228	C₂²	C₂³×C₂²⋊D₇	1	14	yes	—
229	C₂²	C₂⁵×D₇	1	1	yes	characteristic
230	C₄	—	14	16	no	—
231	C₄	—	14	8	no	—
232	C₄	C₂³×C₂²⋊D₇	1	16	yes	complemented
233	C₂	—	4	16	no	—
234	C₂	—	14	16	no	—
235	C₂	C₂²×#6501	1	28	yes	direct product factor
236	C₂	C₂³×#1159	1	1	yes	characteristic
237	C₂	C₂³×D₄×D₇	1	1	yes	characteristic
238	C₂	C₂⁴×C₂²⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₇
commutator, cosocle	C₂²×C₇	C₂⁶
Frattini	C₂²	C₂⁵×D₇
Fitting	C₂⁴×C₇×D₄	C₂
socle	C₂⁵×C₇	C₂³
hypercenter	C₂⁴×D₄	D₇
hypocenter	C₇	C₂³×#121
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂⁵ ◁ 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. C₂⁴×C₄×D₇	C₂	semidirect
4. C₂³×C₂²⋊Dic₇	C₂	semidirect
5. C₂²×#6501	C₂	direct (N × H)
6. C₂³×C₄⋊Dic₇	C₂	semidirect
10. C₂³×C₇×D₄	C₂²	semidirect
13. C₂³×C₂²⋊D₇	C₂²	semidirect
17. C₂³×C₄×D₇	C₂²	semidirect
18. C₂²×C₂²⋊Dic₇	C₂²	semidirect
19. C₂×#6501	C₂²	direct (N × H)
20. C₂²×C₄⋊Dic₇	C₂²	semidirect
29. C₂⁴×D₇	D₄	semidirect
32. C₂²×C₇×D₄	C₂³	semidirect
33. C₂³×C₄×C₇	D₄	semidirect
36. C₂²×C₂²⋊D₇	C₂³	semidirect
42. C₂²×C₄×D₇	C₂³	semidirect
43. C₂×C₂²⋊Dic₇	C₂³	semidirect
44. #6501	C₂³	direct (N × H)
45. C₂×C₄⋊Dic₇	C₂³	semidirect
47. C₂⁴×D₄	D₇	semidirect
64. C₂³×D₇	C₂×D₄	semidirect
67. C₂×C₇×D₄	C₂⁴	semidirect
68. C₂²×C₄×C₇	C₂×D₄	semidirect
71. C₂×C₂²⋊D₇	C₂⁴	semidirect
77. C₂×C₄×D₇	C₂⁴	semidirect
78. C₂²⋊Dic₇	C₂⁴	semidirect
79. C₄⋊Dic₇	C₂⁴	semidirect
85. C₂³×D₄	C₂×D₇	semidirect
98. C₂³×C₇	#121	semidirect
107. C₂²×D₇	C₂²×D₄	semidirect
110. C₂×C₄×C₇	C₂²×D₄	semidirect
130. C₂²×D₄	C₂²×D₇	semidirect
137. C₂³×C₄	C₂²⋊D₇	semidirect
146. C₂²×C₇	C₂×#121	semidirect
154. C₂×D₇	C₂³×D₄	semidirect
155. C₄×C₇	C₂³×D₄	semidirect
175. C₂×D₄	C₂³×D₇	semidirect
182. C₂²×C₄	C₂×C₂²⋊D₇	semidirect
187. C₂×C₇	C₂²×#121	semidirect
201. C₂³	#6501	direct (N × H)
214. C₂×C₄	C₂²×C₂²⋊D₇	semidirect
216. C₇	C₂³×#121	semidirect
225. C₂²	C₂×#6501	direct (N × H)
232. C₄	C₂³×C₂²⋊D₇	semidirect
235. C₂	C₂²×#6501	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
C₂⁴×C₄×D₇	2
C₂³×C₂²⋊Dic₇	2
C₂²×#6501	2
C₂³×C₄⋊Dic₇	2
C₂³×#121	7

Subgroups — C23×#6501 #14260430

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#6501 #14260430