Subgroups — C₇×#2782 #11742091

Order 896.

448 conjugacy classes of subgroups.

The subgroup lattice has 230 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	2	yes	maximal, complemented
2	C₇×#400	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#397	C₂	1	2	yes	maximal, complemented
4	C₇×#394	C₂	1	1	yes	maximal, characteristic
5	C₇×#388	C₂	1	2	yes	maximal, complemented
6	C₇×#386	C₂	1	2	yes	maximal
7	C₇×#385	C₂	1	2	yes	maximal, complemented
8	C₇×#385	C₂	1	2	yes	maximal, complemented
9	C₇×#382	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₇	C₂²	1	2	yes	—
12	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₇×C₄⋊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	2	yes	—
20	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
23	C₇×#95	C₂²	1	2	yes	—
24	C₇×#95	C₂²	1	2	yes	—
25	C₇×#95	C₂²	1	2	yes	—
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	1	yes	characteristic
31	#2782	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂⁴×C₇	C₂³	1	2	yes	—
33	C₂×C₇×D₄	—	2	4	no	—
34	C₂×C₇×D₄	—	4	1	no	—
35	C₂²×C₄×C₇	—	4	2	no	—
36	C₂²×C₄×C₇	—	4	2	no	—
37	C₂²×C₄×C₇	—	4	2	no	—
38	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
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	1	no	—
48	C₇×C₄⋊C₄	—	4	2	no	—
49	C₇×C₄⋊C₄	—	4	1	no	—
50	C₇×C₄⋊C₄	—	4	2	no	—
51	C₇×C₂²⋊C₄	—	2	4	no	—
52	C₇×C₂²⋊C₄	—	2	4	no	—
53	C₇×C₂²⋊C₄	—	2	4	no	—
54	C₇×C₂²⋊C₄	—	4	2	no	—
55	C₇×C₂²⋊C₄	—	4	2	no	—
56	C₄²×C₇	—	2	2	no	—
57	#400	C₂×C₇	1	2	yes	complemented
58	#400	C₂×C₇	1	1	yes	characteristic, complemented
59	#397	C₂×C₇	1	2	yes	complemented
60	#394	C₂×C₇	1	1	yes	characteristic
61	#388	C₂×C₇	1	2	yes	complemented
62	#386	C₂×C₇	1	2	yes	—
63	#385	C₂×C₇	1	2	yes	complemented
64	#385	C₂×C₇	1	2	yes	complemented
65	#382	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	#41	1	4	yes	—
73	C₂³×C₇	C₂⁴	1	1	yes	characteristic
74	C₇×D₄	—	4	4	no	—
75	C₂×C₄×C₇	—	2	1	no	—
76	C₂×C₄×C₇	—	2	1	no	—
77	C₂×C₄×C₇	—	2	2	no	—
78	C₂×C₄×C₇	—	2	2	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	1	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	1	no	—
95	C₂×C₄×C₇	—	2	2	no	—
96	C₂×C₄×C₇	—	4	8	no	—
97	C₂×C₄×C₇	#41	1	2	yes	—
98	C₂²×D₄	C₂²×C₇	1	1	yes	characteristic
99	C₂³×C₄	C₂²×C₇	1	2	yes	—
100	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
101	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
102	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
103	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
104	C₂×C₄⋊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	#95	C₂²×C₇	1	2	yes	—
112	#95	C₂²×C₇	1	2	yes	—
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	2	yes	—
117	#95	C₂²×C₇	1	1	yes	characteristic
118	#95	C₂²×C₇	1	1	yes	characteristic
119	C₂²×C₇	—	2	4	no	—
120	C₂²×C₇	—	2	4	no	—
121	C₂²×C₇	—	2	4	no	—
122	C₂²×C₇	—	4	2	no	—
123	C₂²×C₇	—	4	2	no	—
124	C₂²×C₇	—	4	2	no	—
125	C₂²×C₇	—	4	2	no	—
126	C₂²×C₇	C₂×#41	1	2	yes	—
127	C₂²×C₇	C₂×#41	1	1	yes	characteristic
128	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂²×C₇	H(5,𝔽₂)	1	2	yes	—
130	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
131	C₄×C₇	—	2	2	no	—
132	C₄×C₇	—	4	2	no	—
133	C₄×C₇	—	4	1	no	—
134	C₄×C₇	—	4	1	no	—
135	C₄×C₇	—	4	2	no	—
136	C₄×C₇	—	4	2	no	—
137	C₄×C₇	—	4	2	no	—
138	C₄×C₇	—	4	2	no	—
139	C₂⁴	C₂³×C₇	1	2	yes	—
140	C₂×D₄	—	2	4	no	—
141	C₂×D₄	—	4	1	no	—
142	C₂²×C₄	—	4	2	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
146	C₂²×C₄	C₂³×C₇	1	2	yes	—
147	C₂²×C₄	C₂³×C₇	1	2	yes	—
148	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₇	1	2	yes	—
150	C₂²×C₄	C₂³×C₇	1	2	yes	—
151	C₂²×C₄	C₂³×C₇	1	2	yes	—
152	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
153	C₄⋊C₄	—	4	1	no	—
154	C₄⋊C₄	—	4	1	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	4	1	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₂²⋊C₄	—	2	4	no	—
159	C₂²⋊C₄	—	2	4	no	—
160	C₂²⋊C₄	—	2	4	no	—
161	C₂²⋊C₄	—	4	2	no	—
162	C₂²⋊C₄	—	4	2	no	—
163	C₄²	—	2	2	no	—
164	C₂×C₇	—	4	4	no	—
165	C₂×C₇	#539	1	1	yes	characteristic
166	C₂×C₇	#539	1	2	yes	—
167	C₂×C₇	#551	1	1	yes	characteristic
168	C₂×C₇	#553	1	2	yes	—
169	C₂×C₇	H(3,𝔽₄)	1	1	yes	characteristic
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	C₇×#41	1	4	yes	—
177	C₂³	C₂⁴×C₇	1	1	yes	characteristic
178	D₄	—	4	4	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	2	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	1	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	8	no	—
201	C₂×C₄	C₇×#41	1	2	yes	—
202	C₇	#2782	1	1	yes	characteristic, direct product factor, Sylow
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	4	no	—
206	C₂²	—	4	2	no	—
207	C₂²	—	4	2	no	—
208	C₂²	—	4	2	no	—
209	C₂²	—	4	2	no	—
210	C₂²	C₂×C₇×#41	1	2	yes	—
211	C₂²	C₂×C₇×#41	1	1	yes	characteristic
212	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
213	C₂²	C₇×H(5,𝔽₂)	1	2	yes	—
214	C₂²	C₇×2₋¹⁺⁴	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	4	2	no	—
217	C₄	—	4	1	no	—
218	C₄	—	4	1	no	—
219	C₄	—	4	2	no	—
220	C₄	—	4	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	2	no	—
223	C₂	—	4	4	no	—
224	C₂	C₇×#539	1	1	yes	characteristic
225	C₂	C₇×#539	1	2	yes	—
226	C₂	C₇×#551	1	1	yes	characteristic
227	C₂	C₇×#553	1	2	yes	—
228	C₂	C₇×H(3,𝔽₄)	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×#400	C₂	semidirect
3. C₇×#397	C₂	semidirect
5. C₇×#388	C₂	semidirect
7. C₇×#385	C₂	semidirect
8. C₇×#385	C₂	semidirect
9. C₇×#382	C₂	semidirect
31. #2782	C₇	direct (N × H)
57. #400	C₂×C₇	semidirect
58. #400	C₂×C₇	semidirect
59. #397	C₂×C₇	semidirect
61. #388	C₂×C₇	semidirect
63. #385	C₂×C₇	semidirect
64. #385	C₂×C₇	semidirect
65. #382	C₂×C₇	semidirect
202. C₇	#2782	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×#397	2
C₇×#394	2
C₇×#388	2
C₇×#386	2
C₇×#385	2
C₇×#382	2
#2782	7

Subgroups — C7×#2782 #11742091

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2782 #11742091