Subgroups — C₇×#2735 #11742044

Order 896.

552 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#399	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#397	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#397	C₂	1	2	yes	maximal, complemented
4	C₇×#396	C₂	1	2	yes	maximal, complemented
5	C₇×#388	C₂	1	2	yes	maximal, complemented
6	C₇×#388	C₂	1	2	yes	maximal, complemented
7	C₇×#388	C₂	1	2	yes	maximal, complemented
8	C₇×#379	C₂	1	1	yes	maximal, characteristic
9	C₇×#379	C₂	1	2	yes	maximal
10	C₂⁵×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	complemented
13	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	complemented
14	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
16	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
17	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₇	C₂²	1	2	yes	complemented
26	C₇×#95	C₂²	1	2	yes	complemented
27	C₇×#95	C₂²	1	1	yes	characteristic, complemented
28	C₇×#95	C₂²	1	2	yes	complemented
29	C₇×#95	C₂²	1	2	yes	complemented
30	C₇×#95	C₂²	1	1	yes	characteristic, complemented
31	#2735	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂⁴×C₇	—	2	2	no	—
33	C₂⁴×C₇	—	2	4	no	—
34	C₂⁴×C₇	—	4	2	no	—
35	C₂⁴×C₇	—	4	1	no	—
36	C₂⁴×C₇	—	4	1	no	—
37	C₂⁴×C₇	C₂³	1	2	yes	—
38	C₂⁴×C₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₇	C₂³	1	2	yes	—
44	C₂²×C₄×C₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
46	C₇×C₄⋊C₄	—	4	1	no	—
47	C₇×C₄⋊C₄	—	4	2	no	—
48	C₇×C₄⋊C₄	—	4	2	no	—
49	C₇×C₄⋊C₄	—	4	1	no	—
50	C₇×C₂²⋊C₄	—	2	4	no	—
51	C₇×C₂²⋊C₄	—	2	4	no	—
52	C₇×C₂²⋊C₄	—	2	4	no	—
53	C₇×C₂²⋊C₄	—	4	2	no	—
54	C₇×C₂²⋊C₄	—	4	2	no	—
55	C₇×C₂²⋊C₄	—	4	2	no	—
56	C₇×C₂²⋊C₄	—	4	1	no	—
57	C₇×C₂²⋊C₄	—	4	2	no	—
58	C₇×C₂²⋊C₄	—	4	2	no	—
59	C₇×C₂²⋊C₄	—	4	1	no	—
60	C₄²×C₇	—	4	2	no	—
61	#399	C₂×C₇	1	1	yes	characteristic, complemented
62	#397	C₂×C₇	1	1	yes	characteristic, complemented
63	#397	C₂×C₇	1	2	yes	complemented
64	#396	C₂×C₇	1	2	yes	complemented
65	#388	C₂×C₇	1	2	yes	complemented
66	#388	C₂×C₇	1	2	yes	complemented
67	#388	C₂×C₇	1	2	yes	complemented
68	#379	C₂×C₇	1	1	yes	characteristic
69	#379	C₂×C₇	1	2	yes	—
70	C₂³×C₇	—	2	2	no	—
71	C₂³×C₇	—	2	2	no	—
72	C₂³×C₇	—	2	2	no	—
73	C₂³×C₇	—	2	1	no	—
74	C₂³×C₇	—	2	2	no	—
75	C₂³×C₇	—	2	2	no	—
76	C₂³×C₇	—	2	1	no	—
77	C₂³×C₇	—	2	2	no	—
78	C₂³×C₇	—	2	2	no	—
79	C₂³×C₇	—	2	2	no	—
80	C₂³×C₇	—	4	8	no	—
81	C₂³×C₇	—	4	4	no	—
82	C₂³×C₇	—	4	8	no	—
83	C₂³×C₇	—	4	4	no	—
84	C₂³×C₇	—	4	4	no	—
85	C₂³×C₇	#41	1	4	yes	—
86	C₂³×C₇	#41	1	2	yes	—
87	C₂³×C₇	C₂⁴	1	1	yes	characteristic
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	2	2	no	—
90	C₂×C₄×C₇	—	2	2	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	2	no	—
95	C₂×C₄×C₇	—	2	2	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	1	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	1	no	—
107	C₂⁵	C₂²×C₇	1	1	yes	characteristic
108	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic, complemented
109	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	complemented
110	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	complemented
111	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic, complemented
112	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
113	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
114	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
115	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
116	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
117	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
118	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
119	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
120	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
121	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
122	C₂×C₄²	C₂²×C₇	1	2	yes	complemented
123	#95	C₂²×C₇	1	2	yes	complemented
124	#95	C₂²×C₇	1	1	yes	characteristic, complemented
125	#95	C₂²×C₇	1	2	yes	complemented
126	#95	C₂²×C₇	1	2	yes	complemented
127	#95	C₂²×C₇	1	1	yes	characteristic, complemented
128	C₂²×C₇	—	2	4	no	—
129	C₂²×C₇	—	2	2	no	—
130	C₂²×C₇	—	2	4	no	—
131	C₂²×C₇	—	2	4	no	—
132	C₂²×C₇	—	2	4	no	—
133	C₂²×C₇	—	4	16	no	—
134	C₂²×C₇	—	4	2	no	—
135	C₂²×C₇	—	4	2	no	—
136	C₂²×C₇	—	4	2	no	—
137	C₂²×C₇	—	4	2	no	—
138	C₂²×C₇	—	4	1	no	—
139	C₂²×C₇	—	4	2	no	—
140	C₂²×C₇	—	4	1	no	—
141	C₂²×C₇	C₂×#41	1	2	yes	—
142	C₂²×C₇	C₂×#41	1	1	yes	characteristic
143	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂²×C₇	H(5,𝔽₂)	1	2	yes	—
145	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
146	C₄×C₇	—	4	2	no	—
147	C₄×C₇	—	4	2	no	—
148	C₄×C₇	—	4	2	no	—
149	C₄×C₇	—	4	2	no	—
150	C₄×C₇	—	4	1	no	—
151	C₄×C₇	—	4	2	no	—
152	C₄×C₇	—	4	1	no	—
153	C₂⁴	—	2	2	no	—
154	C₂⁴	—	2	4	no	—
155	C₂⁴	—	4	2	no	—
156	C₂⁴	—	4	1	no	—
157	C₂⁴	—	4	1	no	—
158	C₂⁴	C₂³×C₇	1	2	yes	—
159	C₂⁴	C₂³×C₇	1	1	yes	characteristic
160	C₂²×C₄	C₂³×C₇	1	2	yes	—
161	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
162	C₂²×C₄	C₂³×C₇	1	2	yes	—
163	C₂²×C₄	C₂³×C₇	1	2	yes	—
164	C₂²×C₄	C₂³×C₇	1	2	yes	—
165	C₂²×C₄	C₂³×C₇	1	2	yes	—
166	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
167	C₄⋊C₄	—	4	1	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	4	1	no	—
171	C₂²⋊C₄	—	2	4	no	—
172	C₂²⋊C₄	—	2	4	no	—
173	C₂²⋊C₄	—	2	4	no	—
174	C₂²⋊C₄	—	4	2	no	—
175	C₂²⋊C₄	—	4	2	no	—
176	C₂²⋊C₄	—	4	2	no	—
177	C₂²⋊C₄	—	4	1	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₂²⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	—	4	1	no	—
181	C₄²	—	4	2	no	—
182	C₂×C₇	—	4	4	no	—
183	C₂×C₇	—	4	2	no	—
184	C₂×C₇	#538	1	2	yes	—
185	C₂×C₇	#538	1	1	yes	characteristic
186	C₂×C₇	#551	1	2	yes	—
187	C₂×C₇	#551	1	1	yes	characteristic
188	C₂×C₇	#560	1	1	yes	characteristic
189	C₂³	—	2	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	1	no	—
193	C₂³	—	2	2	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	1	no	—
196	C₂³	—	2	2	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	4	8	no	—
200	C₂³	—	4	4	no	—
201	C₂³	—	4	8	no	—
202	C₂³	—	4	4	no	—
203	C₂³	—	4	4	no	—
204	C₂³	C₇×#41	1	4	yes	—
205	C₂³	C₇×#41	1	2	yes	—
206	C₂³	C₂⁴×C₇	1	1	yes	characteristic
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	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	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₇	#2735	1	1	yes	characteristic, direct product factor, Sylow
227	C₂²	—	2	4	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	2	4	no	—
231	C₂²	—	2	4	no	—
232	C₂²	—	4	16	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	1	no	—
240	C₂²	C₂×C₇×#41	1	2	yes	—
241	C₂²	C₂×C₇×#41	1	1	yes	characteristic
242	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
243	C₂²	C₇×H(5,𝔽₂)	1	2	yes	—
244	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
245	C₄	—	4	2	no	—
246	C₄	—	4	2	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	2	no	—
251	C₄	—	4	1	no	—
252	C₂	—	4	4	no	—
253	C₂	—	4	2	no	—
254	C₂	C₇×#538	1	2	yes	—
255	C₂	C₇×#538	1	1	yes	characteristic
256	C₂	C₇×#551	1	2	yes	—
257	C₂	C₇×#551	1	1	yes	characteristic
258	C₂	C₇×#560	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
prime radical	C₂⁵×C₇	C₂²
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. C₇×#399	C₂	semidirect
2. C₇×#397	C₂	semidirect
3. C₇×#397	C₂	semidirect
4. C₇×#396	C₂	semidirect
5. C₇×#388	C₂	semidirect
6. C₇×#388	C₂	semidirect
7. C₇×#388	C₂	semidirect
11. C₂×C₇×C₄⋊C₄	C₂²	semidirect
12. C₂×C₇×C₄⋊C₄	C₂²	semidirect
13. C₂×C₇×C₄⋊C₄	C₂²	semidirect
14. C₂×C₇×C₄⋊C₄	C₂²	semidirect
25. C₂×C₄²×C₇	C₂²	semidirect
26. C₇×#95	C₂²	semidirect
27. C₇×#95	C₂²	semidirect
28. C₇×#95	C₂²	semidirect
29. C₇×#95	C₂²	semidirect
30. C₇×#95	C₂²	semidirect
31. #2735	C₇	direct (N × H)
61. #399	C₂×C₇	semidirect
62. #397	C₂×C₇	semidirect
63. #397	C₂×C₇	semidirect
64. #396	C₂×C₇	semidirect
65. #388	C₂×C₇	semidirect
66. #388	C₂×C₇	semidirect
67. #388	C₂×C₇	semidirect
108. C₂×C₄⋊C₄	C₂²×C₇	semidirect
109. C₂×C₄⋊C₄	C₂²×C₇	semidirect
110. C₂×C₄⋊C₄	C₂²×C₇	semidirect
111. C₂×C₄⋊C₄	C₂²×C₇	semidirect
122. C₂×C₄²	C₂²×C₇	semidirect
123. #95	C₂²×C₇	semidirect
124. #95	C₂²×C₇	semidirect
125. #95	C₂²×C₇	semidirect
126. #95	C₂²×C₇	semidirect
127. #95	C₂²×C₇	semidirect
226. C₇	#2735	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#399	2
C₇×#397	2
C₇×#396	2
C₇×#388	2
C₇×#379	2
#2735	7

Subgroups — C7×#2735 #11742044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2735 #11742044