Subgroups — C₇×#2599 #11741908

Order 896.

492 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#117	C₂	1	1	yes	maximal, characteristic
3	C₇×#398	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#397	C₂	1	1	yes	maximal, characteristic
5	C₇×#395	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#393	C₂	1	1	yes	maximal, characteristic
7	C₇×#388	C₂	1	2	yes	maximal
8	C₇×#387	C₂	1	2	yes	maximal, complemented
9	C₇×#387	C₂	1	2	yes	maximal, complemented
10	C₇×#385	C₂	1	2	yes	maximal
11	C₇×#383	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₄×C₇×Q₈	—	2	4	no	—
15	C₇×#117	—	2	4	no	—
16	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₇	C₂²	1	2	yes	—
28	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
29	C₇×#95	C₂²	1	2	yes	—
30	C₇×#95	C₂²	1	2	yes	—
31	C₇×#95	C₂²	1	1	yes	characteristic
32	C₇×#95	C₂²	1	1	yes	characteristic
33	C₇×#95	C₂²	1	2	yes	—
34	C₇×#95	C₂²	1	2	yes	—
35	C₇×#95	C₂²	1	2	yes	—
36	C₇×#95	C₂²	1	2	yes	—
37	#2599	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂⁴×C₇	C₂³	1	1	yes	characteristic
39	C₂×C₇×Q₈	—	2	2	no	—
40	C₂×C₇×Q₈	—	2	4	no	—
41	C₂²×C₄×C₇	—	2	2	no	—
42	C₂²×C₄×C₇	—	2	2	no	—
43	C₂²×C₄×C₇	—	2	2	no	—
44	C₂²×C₄×C₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂³	1	2	yes	—
49	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₇	C₂³	1	2	yes	—
53	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
54	C₇×C₄⋊C₄	—	2	2	no	—
55	C₇×C₄⋊C₄	—	2	2	no	—
56	C₇×C₄⋊C₄	—	2	4	no	—
57	C₇×C₄⋊C₄	—	2	2	no	—
58	C₇×C₄⋊C₄	—	2	4	no	—
59	C₇×C₄⋊C₄	—	2	4	no	—
60	C₇×C₂²⋊C₄	—	2	4	no	—
61	C₇×C₂²⋊C₄	—	2	2	no	—
62	C₇×C₂²⋊C₄	—	2	2	no	—
63	C₇×C₂²⋊C₄	—	2	4	no	—
64	C₄²×C₇	—	2	2	no	—
65	C₄²×C₇	—	2	4	no	—
66	C₄²×C₇	D₄	1	4	yes	—
67	C₂×C₄×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
68	C₂×#117	C₂×C₇	1	1	yes	characteristic
69	#398	C₂×C₇	1	1	yes	characteristic, complemented
70	#397	C₂×C₇	1	1	yes	characteristic
71	#395	C₂×C₇	1	1	yes	characteristic, complemented
72	#393	C₂×C₇	1	1	yes	characteristic
73	#388	C₂×C₇	1	2	yes	—
74	#387	C₂×C₇	1	2	yes	complemented
75	#387	C₂×C₇	1	2	yes	complemented
76	#385	C₂×C₇	1	2	yes	—
77	#383	C₂×C₇	1	1	yes	characteristic, complemented
78	C₂³×C₇	—	2	1	no	—
79	C₂³×C₇	—	2	2	no	—
80	C₂³×C₇	—	2	1	no	—
81	C₂³×C₇	—	2	2	no	—
82	C₂³×C₇	#41	1	2	yes	—
83	C₂³×C₇	C₂⁴	1	1	yes	characteristic
84	C₇×Q₈	—	2	8	no	—
85	C₂×C₄×C₇	—	2	8	no	—
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	2	1	no	—
88	C₂×C₄×C₇	—	2	1	no	—
89	C₂×C₄×C₇	—	2	1	no	—
90	C₂×C₄×C₇	—	2	1	no	—
91	C₂×C₄×C₇	—	2	2	no	—
92	C₂×C₄×C₇	—	2	2	no	—
93	C₂×C₄×C₇	—	2	2	no	—
94	C₂×C₄×C₇	—	2	1	no	—
95	C₂×C₄×C₇	—	2	1	no	—
96	C₂×C₄×C₇	—	2	2	no	—
97	C₂×C₄×C₇	—	2	2	no	—
98	C₂×C₄×C₇	—	2	2	no	—
99	C₂×C₄×C₇	—	2	2	no	—
100	C₂×C₄×C₇	—	2	2	no	—
101	C₂×C₄×C₇	—	2	2	no	—
102	C₂×C₄×C₇	—	2	2	no	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	2	2	no	—
105	C₂×C₄×C₇	—	2	2	no	—
106	C₂×C₄×C₇	—	2	2	no	—
107	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₇	#41	1	4	yes	—
111	C₂×C₄×C₇	#41	1	2	yes	—
112	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic
113	C₂³×C₄	C₂²×C₇	1	1	yes	characteristic
114	C₄×Q₈	—	2	4	no	—
115	#117	—	2	4	no	—
116	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
117	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
118	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
119	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
120	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
121	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
122	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
123	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
124	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
125	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
126	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
127	C₂×C₄²	C₂²×C₇	1	2	yes	—
128	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
129	#95	C₂²×C₇	1	2	yes	—
130	#95	C₂²×C₇	1	2	yes	—
131	#95	C₂²×C₇	1	1	yes	characteristic
132	#95	C₂²×C₇	1	1	yes	characteristic
133	#95	C₂²×C₇	1	2	yes	—
134	#95	C₂²×C₇	1	2	yes	—
135	#95	C₂²×C₇	1	2	yes	—
136	#95	C₂²×C₇	1	2	yes	—
137	C₂²×C₇	—	2	2	no	—
138	C₂²×C₇	—	2	4	no	—
139	C₂²×C₇	—	4	1	no	—
140	C₂²×C₇	—	4	1	no	—
141	C₂²×C₇	—	4	2	no	—
142	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
143	C₂²×C₇	C₂×#41	1	1	yes	characteristic
144	C₂²×C₇	C₂×#41	1	2	yes	—
145	C₂²×C₇	C₂×#41	1	1	yes	characteristic
146	C₂²×C₇	2₋¹⁺⁴	1	2	yes	—
147	C₄×C₇	—	2	2	no	—
148	C₄×C₇	—	2	2	no	—
149	C₄×C₇	—	2	2	no	—
150	C₄×C₇	—	2	4	no	—
151	C₄×C₇	—	2	2	no	—
152	C₄×C₇	—	4	1	no	—
153	C₄×C₇	—	4	2	no	—
154	C₄×C₇	—	4	1	no	—
155	C₄×C₇	—	4	2	no	—
156	C₄×C₇	—	4	2	no	—
157	C₂⁴	C₂³×C₇	1	1	yes	characteristic
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	—	2	4	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	C₂³×C₇	1	2	yes	—
164	C₂²×C₄	C₂³×C₇	1	2	yes	—
165	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
166	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
167	C₂²×C₄	C₂³×C₇	1	2	yes	—
168	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₇	1	2	yes	—
172	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	4	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	2	4	no	—
179	C₂²⋊C₄	—	2	4	no	—
180	C₂²⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	—	2	4	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	4	no	—
185	C₄²	C₇×D₄	1	4	yes	—
186	C₂×C₇	—	4	2	no	—
187	C₂×C₇	#525	1	1	yes	characteristic
188	C₂×C₇	#525	1	1	yes	characteristic
189	C₂×C₇	#536	1	1	yes	characteristic
190	C₂×C₇	#552	1	2	yes	—
191	C₂×C₇	#556	1	2	yes	—
192	C₂³	—	2	1	no	—
193	C₂³	—	2	2	no	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	2	no	—
196	C₂³	C₇×#41	1	2	yes	—
197	C₂³	C₂⁴×C₇	1	1	yes	characteristic
198	Q₈	—	2	8	no	—
199	C₂×C₄	—	2	8	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
222	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
223	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
224	C₂×C₄	C₇×#41	1	4	yes	—
225	C₂×C₄	C₇×#41	1	2	yes	—
226	C₇	#2599	1	1	yes	characteristic, direct product factor, Sylow
227	C₂²	—	2	2	no	—
228	C₂²	—	2	4	no	—
229	C₂²	—	4	1	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	4	2	no	—
232	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
233	C₂²	C₂×C₇×#41	1	1	yes	characteristic
234	C₂²	C₂×C₇×#41	1	2	yes	—
235	C₂²	C₂×C₇×#41	1	1	yes	characteristic
236	C₂²	C₇×2₋¹⁺⁴	1	2	yes	—
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	2	4	no	—
241	C₄	—	2	2	no	—
242	C₄	—	4	1	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	1	no	—
245	C₄	—	4	2	no	—
246	C₄	—	4	2	no	—
247	C₂	—	4	2	no	—
248	C₂	C₇×#525	1	1	yes	characteristic
249	C₂	C₇×#525	1	1	yes	characteristic
250	C₂	C₇×#536	1	1	yes	characteristic
251	C₂	C₇×#552	1	2	yes	—
252	C₂	C₇×#556	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
prime radical	C₂⁴×C₇	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×C₇×Q₈	C₂	semidirect
3. C₇×#398	C₂	semidirect
5. C₇×#395	C₂	semidirect
8. C₇×#387	C₂	semidirect
9. C₇×#387	C₂	semidirect
11. C₇×#383	C₂	semidirect
37. #2599	C₇	direct (N × H)
67. C₂×C₄×Q₈	C₂×C₇	semidirect
69. #398	C₂×C₇	semidirect
71. #395	C₂×C₇	semidirect
74. #387	C₂×C₇	semidirect
75. #387	C₂×C₇	semidirect
77. #383	C₂×C₇	semidirect
226. C₇	#2599	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇×Q₈	2
C₂×C₇×#117	2
C₇×#398	2
C₇×#397	2
C₇×#395	2
C₇×#393	2
C₇×#388	2
C₇×#387	2
C₇×#385	2
C₇×#383	2
#2599	7

Subgroups — C7×#2599 #11741908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2599 #11741908