Subgroups — C₇×#23263 #13200479

Order 1792.

692 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×#125	C₂	1	1	yes	maximal, characteristic
2	C₇×#1916	C₂	1	4	yes	maximal
3	C₇×#1916	C₂	1	4	yes	maximal
4	C₇×C₈×C₄⋊C₄	C₂	1	4	yes	maximal
5	C₇×#1744	C₂	1	2	yes	maximal
6	C₂×C₇×#125	C₄	1	1	yes	characteristic
7	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
8	C₂×C₄×C₇×C₈	C₂²	1	4	yes	—
9	C₂×C₄×C₇×C₈	C₂²	1	4	yes	—
10	C₇×#389	C₄	1	2	yes	—
11	C₇×#387	C₄	1	4	yes	—
12	C₇×#382	C₄	1	1	yes	characteristic
13	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
14	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
15	C₄³×C₇	C₂²	1	1	yes	characteristic
16	C₇×#336	C₂²	1	2	yes	—
17	C₇×#336	C₂²	1	4	yes	—
18	C₇×#336	C₂²	1	2	yes	—
19	C₇×#336	C₂²	1	4	yes	—
20	C₇×#336	C₂²	1	4	yes	—
21	#23263	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂²×C₇×C₈	C₂³	1	4	yes	—
23	C₂²×C₇×C₈	C₂³	1	4	yes	—
24	C₇×#125	C₈	1	8	yes	complemented
25	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
26	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
27	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
28	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₇	C₂³	1	2	yes	—
34	C₂×C₄²×C₇	C₂³	1	4	yes	—
35	C₇×C₄⋊C₈	—	2	8	no	—
36	C₄×C₇×C₈	—	2	8	no	—
37	C₄×C₇×C₈	—	2	8	no	—
38	C₇×#95	C₂×C₄	1	4	yes	—
39	C₇×#95	C₂×C₄	1	4	yes	—
40	C₇×#95	C₂×C₄	1	4	yes	—
41	C₄×#125	C₂×C₇	1	1	yes	characteristic
42	#1916	C₂×C₇	1	4	yes	—
43	#1916	C₂×C₇	1	4	yes	—
44	C₈×C₄⋊C₄	C₂×C₇	1	4	yes	—
45	#1744	C₂×C₇	1	2	yes	—
46	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
48	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
53	C₂×C₇×C₈	—	2	4	no	—
54	C₂×C₇×C₈	—	2	4	no	—
55	C₂×C₇×C₈	—	2	4	no	—
56	C₂×C₇×C₈	—	2	4	no	—
57	C₂×C₇×C₈	—	2	4	no	—
58	C₂×C₇×C₈	#41	1	8	yes	—
59	C₇×C₄⋊C₄	—	2	8	no	—
60	C₇×C₄⋊C₄	—	4	2	no	—
61	C₇×C₄⋊C₄	C₂×C₈	1	16	yes	—
62	C₇×C₄⋊C₄	C₂×C₈	1	8	yes	—
63	C₄²×C₇	—	2	4	no	—
64	C₄²×C₇	—	2	4	no	—
65	C₄²×C₇	—	2	4	no	—
66	C₄²×C₇	—	2	4	no	—
67	C₄²×C₇	—	2	2	no	—
68	C₄²×C₇	—	4	1	no	—
69	C₄²×C₇	C₂×C₈	1	4	yes	—
70	C₂×#125	C₄×C₇	1	1	yes	characteristic
71	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
72	C₂×C₄×C₈	C₂²×C₇	1	4	yes	—
73	C₂×C₄×C₈	C₂²×C₇	1	4	yes	—
74	#389	C₄×C₇	1	2	yes	—
75	#387	C₄×C₇	1	4	yes	—
76	#382	C₄×C₇	1	1	yes	characteristic
77	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
78	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
79	C₄³	C₂²×C₇	1	1	yes	characteristic
80	#336	C₂²×C₇	1	4	yes	—
81	#336	C₂²×C₇	1	4	yes	—
82	#336	C₂²×C₇	1	2	yes	—
83	#336	C₂²×C₇	1	4	yes	—
84	#336	C₂²×C₇	1	2	yes	—
85	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	2	4	no	—
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	2	1	no	—
90	C₂×C₄×C₇	—	2	2	no	—
91	C₂×C₄×C₇	—	2	4	no	—
92	C₂×C₄×C₇	—	2	1	no	—
93	C₂×C₄×C₇	—	2	2	no	—
94	C₂×C₄×C₇	—	2	4	no	—
95	C₂×C₄×C₇	—	2	1	no	—
96	C₂×C₄×C₇	—	2	2	no	—
97	C₂×C₄×C₇	—	2	4	no	—
98	C₂×C₄×C₇	—	2	4	no	—
99	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
100	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
101	C₂×C₄×C₇	C₂²×C₈	1	8	yes	—
102	C₂×C₄×C₇	#131	1	4	yes	—
103	C₂×C₄×C₇	C₂×#41	1	4	yes	—
104	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
105	C₇×C₈	—	2	8	no	—
106	C₇×C₈	—	4	4	no	—
107	C₂²×C₈	C₂³×C₇	1	4	yes	—
108	C₂²×C₈	C₂³×C₇	1	4	yes	—
109	#125	C₇×C₈	1	8	yes	complemented
110	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
111	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
112	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
113	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
114	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
115	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
116	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
117	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
118	C₂×C₄²	C₂³×C₇	1	2	yes	—
119	C₂×C₄²	C₂³×C₇	1	4	yes	—
120	C₄⋊C₈	—	2	8	no	—
121	C₄×C₈	—	2	8	no	—
122	C₄×C₈	—	2	8	no	—
123	#95	C₂×C₄×C₇	1	4	yes	—
124	#95	C₂×C₄×C₇	1	4	yes	—
125	#95	C₂×C₄×C₇	1	4	yes	—
126	C₂²×C₇	C₄×#41	1	2	yes	—
127	C₂²×C₇	#519	1	1	yes	characteristic
128	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
129	C₂²×C₇	C₂×#131	1	2	yes	—
130	C₂²×C₇	#568	1	1	yes	characteristic
131	C₄×C₇	—	2	4	no	—
132	C₄×C₇	—	2	8	no	—
133	C₄×C₇	—	2	4	no	—
134	C₄×C₇	—	2	2	no	—
135	C₄×C₇	—	4	4	no	—
136	C₄×C₇	—	4	1	no	—
137	C₄×C₇	#552	1	4	yes	—
138	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
139	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
140	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
141	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
142	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
143	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
144	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	2	4	no	—
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	C₇×#41	1	8	yes	—
151	C₄⋊C₄	—	2	8	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	C₂×C₇×C₈	1	16	yes	—
154	C₄⋊C₄	C₂×C₇×C₈	1	8	yes	—
155	C₄²	—	2	4	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	2	4	no	—
158	C₄²	—	2	4	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	4	1	no	—
161	C₄²	C₂×C₇×C₈	1	4	yes	—
162	C₂×C₇	#2371	1	1	yes	characteristic
163	C₂×C₇	C₈×#41	1	2	yes	—
164	C₂×C₇	#2968	1	2	yes	—
165	C₂×C₇	#2982	1	1	yes	characteristic
166	C₂×C₇	#2986	1	1	yes	characteristic
167	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	C₂²×C₇×C₈	1	4	yes	—
182	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
183	C₂×C₄	C₂²×C₇×C₈	1	8	yes	—
184	C₂×C₄	C₇×#131	1	4	yes	—
185	C₂×C₄	C₂×C₇×#41	1	4	yes	—
186	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
187	C₈	—	2	8	no	—
188	C₈	—	4	4	no	—
189	C₇	#23263	1	1	yes	characteristic, direct product factor, Sylow
190	C₂²	C₄×C₇×#41	1	2	yes	—
191	C₂²	C₇×#519	1	1	yes	characteristic
192	C₂²	C₂³×C₇×C₈	1	1	yes	characteristic
193	C₂²	C₂×C₇×#131	1	2	yes	—
194	C₂²	C₇×#568	1	1	yes	characteristic
195	C₄	—	2	4	no	—
196	C₄	—	2	8	no	—
197	C₄	—	2	4	no	—
198	C₄	—	2	2	no	—
199	C₄	—	4	4	no	—
200	C₄	—	4	1	no	—
201	C₄	C₇×#552	1	4	yes	—
202	C₂	C₇×#2371	1	1	yes	characteristic
203	C₂	C₇×C₈×#41	1	2	yes	—
204	C₂	C₇×#2968	1	2	yes	—
205	C₂	C₇×#2982	1	1	yes	characteristic
206	C₂	C₇×#2986	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator	C₂²	C₂³×C₇×C₈
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle, prime radical	C₂³×C₇	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
21. #23263	C₇	direct (N × H)
24. C₇×#125	C₈	semidirect
109. #125	C₇×C₈	semidirect
189. C₇	#23263	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₇×#125	2
C₇×#1916	2
C₇×C₈×C₄⋊C₄	2
C₇×#1744	2
#23263	7

Subgroups — C7×#23263 #13200479

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23263 #13200479