Subgroups — C₇×#2795 #11742104

Order 896.

420 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#400	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#396	C₂	1	2	yes	maximal, complemented
3	C₇×#394	C₂	1	1	yes	maximal, characteristic
4	C₇×#394	C₂	1	2	yes	maximal
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×#387	C₂	1	2	yes	maximal, complemented
9	C₇×#384	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic
11	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
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	2	yes	—
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×C₄²×C₇	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₇	C₂²	1	2	yes	—
25	C₇×#95	C₂²	1	1	yes	characteristic
26	C₇×#95	C₂²	1	2	yes	—
27	C₇×#95	C₂²	1	2	yes	—
28	C₇×#95	C₂²	1	2	yes	—
29	C₇×#95	C₂²	1	1	yes	characteristic
30	C₇×#95	C₂²	1	2	yes	—
31	#2795	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂⁴×C₇	C₂³	1	2	yes	—
33	C₂×C₇×D₄	—	4	1	no	—
34	C₂×C₇×D₄	—	4	2	no	—
35	C₂²×C₄×C₇	C₂³	1	2	yes	—
36	C₂²×C₄×C₇	C₂³	1	2	yes	—
37	C₂²×C₄×C₇	C₂³	1	2	yes	—
38	C₂²×C₄×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	1	yes	characteristic
42	C₂²×C₄×C₇	C₂³	1	2	yes	—
43	C₇×C₄⋊C₄	—	2	4	no	—
44	C₇×C₄⋊C₄	—	4	2	no	—
45	C₇×C₄⋊C₄	—	4	1	no	—
46	C₇×C₄⋊C₄	—	4	2	no	—
47	C₇×C₄⋊C₄	—	4	1	no	—
48	C₇×C₄⋊C₄	—	4	1	no	—
49	C₇×C₂²⋊C₄	—	2	4	no	—
50	C₇×C₂²⋊C₄	—	2	4	no	—
51	C₇×C₂²⋊C₄	—	2	4	no	—
52	C₇×C₂²⋊C₄	—	4	2	no	—
53	C₇×C₂²⋊C₄	—	4	2	no	—
54	C₇×C₂²⋊C₄	—	4	2	no	—
55	C₄²×C₇	—	2	2	no	—
56	C₄²×C₇	—	4	2	no	—
57	#400	C₂×C₇	1	1	yes	characteristic, complemented
58	#396	C₂×C₇	1	2	yes	complemented
59	#394	C₂×C₇	1	1	yes	characteristic
60	#394	C₂×C₇	1	2	yes	—
61	#388	C₂×C₇	1	2	yes	complemented
62	#388	C₂×C₇	1	2	yes	complemented
63	#388	C₂×C₇	1	2	yes	complemented
64	#387	C₂×C₇	1	2	yes	complemented
65	#384	C₂×C₇	1	1	yes	characteristic, complemented
66	C₂³×C₇	—	2	2	no	—
67	C₂³×C₇	—	2	2	no	—
68	C₂³×C₇	—	2	2	no	—
69	C₂³×C₇	—	2	2	no	—
70	C₂³×C₇	—	2	2	no	—
71	C₂³×C₇	—	2	2	no	—
72	C₂³×C₇	—	2	2	no	—
73	C₂³×C₇	C₂⁴	1	1	yes	characteristic
74	C₇×D₄	—	4	4	no	—
75	C₂×C₄×C₇	—	2	2	no	—
76	C₂×C₄×C₇	—	2	1	no	—
77	C₂×C₄×C₇	—	2	2	no	—
78	C₂×C₄×C₇	—	2	1	no	—
79	C₂×C₄×C₇	—	2	2	no	—
80	C₂×C₄×C₇	—	2	2	no	—
81	C₂×C₄×C₇	—	2	2	no	—
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	2	2	no	—
84	C₂×C₄×C₇	—	2	2	no	—
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	2	2	no	—
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	1	no	—
92	C₂×C₄×C₇	—	2	1	no	—
93	C₂×C₄×C₇	—	2	2	no	—
94	C₂×C₄×C₇	—	2	2	no	—
95	C₂×C₄×C₇	#41	1	2	yes	—
96	C₂×C₄×C₇	#41	1	4	yes	—
97	C₂²×D₄	C₂²×C₇	1	1	yes	characteristic
98	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
99	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
100	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
101	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
102	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
103	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
104	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
105	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
106	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
107	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
108	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
109	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
110	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
111	C₂×C₄²	C₂²×C₇	1	2	yes	—
112	#95	C₂²×C₇	1	1	yes	characteristic
113	#95	C₂²×C₇	1	2	yes	—
114	#95	C₂²×C₇	1	2	yes	—
115	#95	C₂²×C₇	1	2	yes	—
116	#95	C₂²×C₇	1	1	yes	characteristic
117	#95	C₂²×C₇	1	2	yes	—
118	C₂²×C₇	—	4	2	no	—
119	C₂²×C₇	—	4	2	no	—
120	C₂²×C₇	—	4	2	no	—
121	C₂²×C₇	—	4	2	no	—
122	C₂²×C₇	—	4	2	no	—
123	C₂²×C₇	—	4	2	no	—
124	C₂²×C₇	—	4	2	no	—
125	C₂²×C₇	C₂×#41	1	1	yes	characteristic
126	C₂²×C₇	C₂×#41	1	2	yes	—
127	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂²×C₇	H(5,𝔽₂)	1	2	yes	—
129	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
130	C₄×C₇	—	2	2	no	—
131	C₄×C₇	—	2	4	no	—
132	C₄×C₇	—	4	2	no	—
133	C₄×C₇	—	4	2	no	—
134	C₄×C₇	—	4	2	no	—
135	C₄×C₇	—	4	2	no	—
136	C₄×C₇	—	4	1	no	—
137	C₄×C₇	—	4	1	no	—
138	C₂⁴	C₂³×C₇	1	2	yes	—
139	C₂×D₄	—	4	1	no	—
140	C₂×D₄	—	4	2	no	—
141	C₂²×C₄	C₂³×C₇	1	2	yes	—
142	C₂²×C₄	C₂³×C₇	1	2	yes	—
143	C₂²×C₄	C₂³×C₇	1	2	yes	—
144	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
145	C₂²×C₄	C₂³×C₇	1	2	yes	—
146	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
147	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
148	C₂²×C₄	C₂³×C₇	1	2	yes	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	4	1	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	1	no	—
154	C₄⋊C₄	—	4	1	no	—
155	C₂²⋊C₄	—	2	4	no	—
156	C₂²⋊C₄	—	2	4	no	—
157	C₂²⋊C₄	—	2	4	no	—
158	C₂²⋊C₄	—	4	2	no	—
159	C₂²⋊C₄	—	4	2	no	—
160	C₂²⋊C₄	—	4	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	4	2	no	—
163	C₂×C₇	—	8	2	no	—
164	C₂×C₇	#539	1	1	yes	characteristic
165	C₂×C₇	#542	1	2	yes	—
166	C₂×C₇	#553	1	2	yes	—
167	C₂×C₇	#555	1	1	yes	characteristic
168	C₂×C₇	#560	1	1	yes	characteristic
169	C₂³	—	2	2	no	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	C₂³	C₂⁴×C₇	1	1	yes	characteristic
177	D₄	—	4	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	C₇×#41	1	2	yes	—
199	C₂×C₄	C₇×#41	1	4	yes	—
200	C₇	#2795	1	1	yes	characteristic, direct product factor, Sylow
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	2	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	—	4	2	no	—
207	C₂²	—	4	2	no	—
208	C₂²	C₂×C₇×#41	1	1	yes	characteristic
209	C₂²	C₂×C₇×#41	1	2	yes	—
210	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
211	C₂²	C₇×H(5,𝔽₂)	1	2	yes	—
212	C₂²	C₇×2₋¹⁺⁴	1	1	yes	characteristic
213	C₄	—	2	2	no	—
214	C₄	—	2	4	no	—
215	C₄	—	4	2	no	—
216	C₄	—	4	2	no	—
217	C₄	—	4	2	no	—
218	C₄	—	4	1	no	—
219	C₄	—	4	2	no	—
220	C₄	—	4	1	no	—
221	C₂	—	8	2	no	—
222	C₂	C₇×#539	1	1	yes	characteristic
223	C₂	C₇×#542	1	2	yes	—
224	C₂	C₇×#553	1	2	yes	—
225	C₂	C₇×#555	1	1	yes	characteristic
226	C₂	C₇×#560	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	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×#400	C₂	semidirect
2. C₇×#396	C₂	semidirect
5. C₇×#388	C₂	semidirect
6. C₇×#388	C₂	semidirect
7. C₇×#388	C₂	semidirect
8. C₇×#387	C₂	semidirect
9. C₇×#384	C₂	semidirect
31. #2795	C₇	direct (N × H)
57. #400	C₂×C₇	semidirect
58. #396	C₂×C₇	semidirect
61. #388	C₂×C₇	semidirect
62. #388	C₂×C₇	semidirect
63. #388	C₂×C₇	semidirect
64. #387	C₂×C₇	semidirect
65. #384	C₂×C₇	semidirect
200. C₇	#2795	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×#400	2
C₇×#396	2
C₇×#394	2
C₇×#388	2
C₇×#387	2
C₇×#384	2
#2795	7

Subgroups — C7×#2795 #11742104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2795 #11742104