Subgroups — C₇×#2562 #11741871

Order 896.

532 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
3	C₇×#400	C₂	1	2	yes	maximal, complemented
4	C₇×#394	C₂	1	2	yes	maximal
5	C₇×#388	C₂	1	4	yes	maximal, complemented
6	C₇×#380	C₂	1	2	yes	maximal, complemented
7	C₇×#380	C₂	1	2	yes	maximal, complemented
8	C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₇	C₂²	1	2	yes	—
11	C₇×#125	—	2	4	no	—
12	C₄×C₇×D₄	—	2	4	no	—
13	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
21	C₂×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
22	C₂×C₄²×C₇	C₂²	1	2	yes	—
23	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
24	C₇×#95	C₂²	1	2	yes	—
25	C₇×#95	C₂²	1	4	yes	—
26	C₇×#95	C₂²	1	4	yes	—
27	#2562	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂⁴×C₇	C₂³	1	2	yes	—
29	C₂×C₇×D₄	—	2	4	no	—
30	C₂×C₇×D₄	—	2	2	no	—
31	C₂²×C₄×C₇	—	2	4	no	—
32	C₂²×C₄×C₇	—	2	4	no	—
33	C₂²×C₄×C₇	—	2	4	no	—
34	C₂²×C₄×C₇	C₂³	1	2	yes	—
35	C₂²×C₄×C₇	C₂³	1	4	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	1	yes	characteristic
40	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₇×C₄⋊C₄	—	2	2	no	—
42	C₇×C₄⋊C₄	—	2	4	no	—
43	C₇×C₄⋊C₄	—	2	4	no	—
44	C₇×C₄⋊C₄	—	2	2	no	—
45	C₇×C₄⋊C₄	—	2	2	no	—
46	C₇×C₄⋊C₄	—	2	4	no	—
47	C₇×C₂²⋊C₄	—	2	8	no	—
48	C₇×C₂²⋊C₄	—	2	4	no	—
49	C₇×C₂²⋊C₄	—	4	4	no	—
50	C₄²×C₇	—	4	2	no	—
51	C₄²×C₇	D₄	1	4	yes	—
52	C₂×#125	C₂×C₇	1	1	yes	characteristic, complemented
53	C₂×C₄×D₄	C₂×C₇	1	1	yes	characteristic
54	#400	C₂×C₇	1	2	yes	complemented
55	#394	C₂×C₇	1	2	yes	—
56	#388	C₂×C₇	1	4	yes	complemented
57	#380	C₂×C₇	1	2	yes	complemented
58	#380	C₂×C₇	1	2	yes	complemented
59	C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
60	C₂³×C₇	—	2	2	no	—
61	C₂³×C₇	—	2	2	no	—
62	C₂³×C₇	—	2	2	no	—
63	C₂³×C₇	—	2	2	no	—
64	C₂³×C₇	—	2	2	no	—
65	C₂³×C₇	—	2	2	no	—
66	C₂³×C₇	#41	1	4	yes	—
67	C₂³×C₇	C₂⁴	1	1	yes	characteristic
68	C₇×D₄	—	2	8	no	—
69	C₂×C₄×C₇	—	2	2	no	—
70	C₂×C₄×C₇	—	2	1	no	—
71	C₂×C₄×C₇	—	2	1	no	—
72	C₂×C₄×C₇	—	2	2	no	—
73	C₂×C₄×C₇	—	2	2	no	—
74	C₂×C₄×C₇	—	2	2	no	—
75	C₂×C₄×C₇	—	2	4	no	—
76	C₂×C₄×C₇	—	2	4	no	—
77	C₂×C₄×C₇	—	2	4	no	—
78	C₂×C₄×C₇	—	2	2	no	—
79	C₂×C₄×C₇	—	2	2	no	—
80	C₂×C₄×C₇	—	2	4	no	—
81	C₂×C₄×C₇	—	2	2	no	—
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	4	8	no	—
84	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₇	#41	1	4	yes	—
88	C₂²×D₄	C₂²×C₇	1	1	yes	characteristic
89	C₂³×C₄	C₂²×C₇	1	2	yes	—
90	#125	—	2	4	no	—
91	C₄×D₄	—	2	4	no	—
92	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
93	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
94	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
95	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
96	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
97	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
98	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
99	C₂×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
100	C₂×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
101	C₂×C₄²	C₂²×C₇	1	2	yes	—
102	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
103	#95	C₂²×C₇	1	2	yes	—
104	#95	C₂²×C₇	1	4	yes	—
105	#95	C₂²×C₇	1	4	yes	—
106	C₂²×C₇	—	2	4	no	—
107	C₂²×C₇	—	2	4	no	—
108	C₂²×C₇	—	2	4	no	—
109	C₂²×C₇	—	4	2	no	—
110	C₂²×C₇	—	4	2	no	—
111	C₂²×C₇	—	4	2	no	—
112	C₂²×C₇	—	4	2	no	—
113	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
114	C₂²×C₇	C₂×#41	1	2	yes	—
115	C₂²×C₇	C₂×#41	1	2	yes	—
116	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
118	C₄×C₇	—	2	2	no	—
119	C₄×C₇	—	2	2	no	—
120	C₄×C₇	—	4	2	no	—
121	C₄×C₇	—	4	2	no	—
122	C₄×C₇	—	4	2	no	—
123	C₄×C₇	—	4	4	no	—
124	C₄×C₇	#124	1	4	yes	—
125	C₂⁴	C₂³×C₇	1	2	yes	—
126	C₂×D₄	—	2	4	no	—
127	C₂×D₄	—	2	2	no	—
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	—	2	4	no	—
130	C₂²×C₄	—	2	4	no	—
131	C₂²×C₄	C₂³×C₇	1	2	yes	—
132	C₂²×C₄	C₂³×C₇	1	4	yes	—
133	C₂²×C₄	C₂³×C₇	1	2	yes	—
134	C₂²×C₄	C₂³×C₇	1	2	yes	—
135	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
136	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
137	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	—	2	4	no	—
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	—	2	2	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₂²⋊C₄	—	2	8	no	—
145	C₂²⋊C₄	—	2	4	no	—
146	C₂²⋊C₄	—	4	4	no	—
147	C₄²	—	4	2	no	—
148	C₄²	C₇×D₄	1	4	yes	—
149	C₂×C₇	—	4	4	no	—
150	C₂×C₇	#525	1	1	yes	characteristic
151	C₂×C₇	C₂×#124	1	1	yes	characteristic
152	C₂×C₇	#537	1	1	yes	characteristic
153	C₂×C₇	#552	1	2	yes	—
154	C₂×C₇	#553	1	2	yes	—
155	C₂³	—	2	2	no	—
156	C₂³	—	2	2	no	—
157	C₂³	—	2	2	no	—
158	C₂³	—	2	2	no	—
159	C₂³	—	2	2	no	—
160	C₂³	—	2	2	no	—
161	C₂³	C₇×#41	1	4	yes	—
162	C₂³	C₂⁴×C₇	1	1	yes	characteristic
163	D₄	—	2	8	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
180	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
181	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
182	C₂×C₄	C₇×#41	1	4	yes	—
183	C₇	#2562	1	1	yes	characteristic, direct product factor, Sylow
184	C₂²	—	2	4	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	2	4	no	—
187	C₂²	—	4	2	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	4	2	no	—
191	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
192	C₂²	C₂×C₇×#41	1	2	yes	—
193	C₂²	C₂×C₇×#41	1	2	yes	—
194	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
195	C₂²	C₇×2₋¹⁺⁴	1	1	yes	characteristic
196	C₄	—	2	2	no	—
197	C₄	—	2	2	no	—
198	C₄	—	4	2	no	—
199	C₄	—	4	2	no	—
200	C₄	—	4	2	no	—
201	C₄	—	4	4	no	—
202	C₄	C₇×#124	1	4	yes	—
203	C₂	—	4	4	no	—
204	C₂	C₇×#525	1	1	yes	characteristic
205	C₂	C₂×C₇×#124	1	1	yes	characteristic
206	C₂	C₇×#537	1	1	yes	characteristic
207	C₂	C₇×#552	1	2	yes	—
208	C₂	C₇×#553	1	2	yes	—

	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. C₂×C₇×#125	C₂	semidirect
3. C₇×#400	C₂	semidirect
5. C₇×#388	C₂	semidirect
6. C₇×#380	C₂	semidirect
7. C₇×#380	C₂	semidirect
8. C₄×C₇×C₄⋊C₄	C₂	semidirect
27. #2562	C₇	direct (N × H)
52. C₂×#125	C₂×C₇	semidirect
54. #400	C₂×C₇	semidirect
56. #388	C₂×C₇	semidirect
57. #380	C₂×C₇	semidirect
58. #380	C₂×C₇	semidirect
59. C₄×C₄⋊C₄	C₂×C₇	semidirect
183. C₇	#2562	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×C₇×#125	2
C₂×C₄×C₇×D₄	2
C₇×#400	2
C₇×#394	2
C₇×#388	2
C₇×#380	2
C₄×C₇×C₄⋊C₄	2
#2562	7

Subgroups — C7×#2562 #11741871

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2562 #11741871