Subgroups — C₇×#13191 #13190407

Order 1792.

450 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	C₇×#2931	C₂	1	1	yes	maximal, characteristic
2	C₇×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#350	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×#131	C₂²	1	2	yes	—
6	C₇×#532	C₄	1	1	yes	characteristic
7	C₂×C₇×M₃₂	C₂²	1	2	yes	—
8	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
9	C₇×#436	—	2	2	no	—
10	C₇×C₈×D₄	—	2	2	no	—
11	C₇×#423	C₄	1	1	yes	characteristic
12	C₇×#408	C₄	1	2	yes	—
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₇×C₄⋊C₁₆	—	2	2	no	—
15	C₇×#350	—	2	8	no	—
16	C₄×C₇×C₁₆	—	2	2	no	—
17	#13191	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂×C₇×#41	C₂×C₄	1	2	yes	—
19	C₇×#131	—	2	4	no	—
20	C₇×#131	—	2	4	no	—
21	C₂×C₇×M₁₆	—	2	2	no	—
22	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
23	C₂²×C₇×C₈	—	2	2	no	—
24	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
26	C₇×C₄⋊Q₈	C₈	1	1	yes	characteristic
27	C₇×#127	C₈	1	1	yes	characteristic
28	C₇×#124	C₈	1	2	yes	—
29	C₇×#121	—	2	2	no	—
30	C₄×C₇×D₄	—	2	2	no	—
31	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
32	C₇×M₃₂	—	2	4	no	—
33	C₂×C₇×C₁₆	—	2	2	no	—
34	C₂×C₇×C₁₆	—	2	2	no	—
35	C₂×C₇×C₁₆	—	2	2	no	—
36	C₂×C₇×C₁₆	D₄	1	4	yes	complemented
37	C₇×C₄⋊C₈	—	2	2	no	—
38	C₇×C₂²⋊C₈	—	2	4	no	—
39	C₄×C₇×C₈	D₄	1	2	yes	—
40	C₄×C₇×C₈	D₄	1	2	yes	—
41	#2931	C₂×C₇	1	1	yes	characteristic
42	#2150	C₂×C₇	1	1	yes	characteristic, complemented
43	C₂×#350	C₂×C₇	1	4	yes	complemented
44	C₂×C₄×C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
45	C₇×#41	—	2	4	no	—
46	C₇×#41	—	2	4	no	—
47	C₂×C₇×Q₈	C₂×C₈	1	2	yes	—
48	C₂×C₇×D₄	—	2	2	no	—
49	C₂×C₇×D₄	—	2	2	no	—
50	C₂×C₇×D₄	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₇	—	2	2	no	—
52	C₂²×C₄×C₇	M₁₆	1	2	yes	—
53	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
54	C₇×M₁₆	—	2	4	no	—
55	C₇×M₁₆	—	4	4	no	—
56	C₂×C₇×C₈	—	2	2	no	—
57	C₂×C₇×C₈	—	2	2	no	—
58	C₂×C₇×C₈	—	4	2	no	—
59	C₂×C₇×C₈	—	4	2	no	—
60	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
61	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
62	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
63	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂×C₇×C₈	#41	1	2	yes	—
65	C₇×C₄⋊C₄	—	2	2	no	—
66	C₇×C₂²⋊C₄	—	2	4	no	—
67	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
69	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
70	C₇×C₁₆	—	2	4	no	—
71	C₇×C₁₆	—	4	2	no	—
72	C₂×#131	C₂²×C₇	1	2	yes	—
73	#532	C₄×C₇	1	1	yes	characteristic
74	C₂×M₃₂	C₂²×C₇	1	2	yes	—
75	C₂²×C₁₆	C₂²×C₇	1	2	yes	—
76	#436	—	2	2	no	—
77	C₈×D₄	—	2	2	no	—
78	#423	C₄×C₇	1	1	yes	characteristic
79	#408	C₄×C₇	1	2	yes	—
80	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
81	C₄⋊C₁₆	—	2	2	no	—
82	#350	—	2	8	no	—
83	C₄×C₁₆	—	2	2	no	—
84	C₂³×C₇	—	2	2	no	—
85	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
86	C₇×Q₈	—	2	4	no	—
87	C₇×D₄	—	2	4	no	—
88	C₇×D₄	—	4	4	no	—
89	C₇×D₄	—	4	4	no	—
90	C₂×C₄×C₇	—	2	4	no	—
91	C₂×C₄×C₇	—	2	2	no	—
92	C₂×C₄×C₇	—	4	2	no	—
93	C₂×C₄×C₇	—	4	2	no	—
94	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
95	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
96	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₇	#131	1	2	yes	—
100	C₇×C₈	—	2	2	no	—
101	C₇×C₈	—	4	2	no	—
102	C₇×C₈	#121	1	2	yes	—
103	C₇×C₈	#124	1	1	yes	characteristic
104	C₇×C₈	#127	1	1	yes	characteristic
105	C₂×#41	C₂×C₄×C₇	1	2	yes	—
106	#131	—	2	4	no	—
107	#131	—	2	4	no	—
108	C₂×M₁₆	—	2	2	no	—
109	C₂×M₁₆	C₂×C₄×C₇	1	2	yes	—
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
112	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
113	C₄⋊Q₈	C₇×C₈	1	1	yes	characteristic
114	#127	C₇×C₈	1	1	yes	characteristic
115	#124	C₇×C₈	1	2	yes	—
116	#121	—	2	2	no	—
117	C₄×D₄	—	2	2	no	—
118	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
119	M₃₂	—	2	4	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	C₇×D₄	1	4	yes	complemented
124	C₄⋊C₈	—	2	2	no	—
125	C₂²⋊C₈	—	2	4	no	—
126	C₄×C₈	C₇×D₄	1	2	yes	—
127	C₄×C₈	C₇×D₄	1	2	yes	—
128	C₂²×C₇	—	2	2	no	—
129	C₂²×C₇	—	4	4	no	—
130	C₂²×C₇	—	4	2	no	—
131	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
132	C₂²×C₇	#436	1	2	yes	—
133	C₄×C₇	—	2	2	no	—
134	C₄×C₇	—	4	2	no	—
135	C₄×C₇	#350	1	4	yes	complemented
136	C₄×C₇	#390	1	1	yes	characteristic
137	C₄×C₇	#408	1	1	yes	characteristic
138	C₄×C₇	C₈×D₄	1	2	yes	—
139	#41	—	2	4	no	—
140	#41	—	2	4	no	—
141	C₂×Q₈	C₂×C₇×C₈	1	2	yes	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	C₂×C₇×C₈	1	2	yes	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	C₇×M₁₆	1	2	yes	—
147	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
148	M₁₆	—	2	4	no	—
149	M₁₆	—	4	4	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	C₇×C₂²⋊C₄	1	2	yes	—
155	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
156	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
157	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
158	C₂×C₈	C₇×#41	1	2	yes	—
159	C₄⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	4	no	—
161	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
162	C₄²	C₂×C₇×C₈	1	1	yes	characteristic
163	C₄²	C₂×C₇×C₈	1	1	yes	characteristic
164	C₁₆	—	2	4	no	—
165	C₁₆	—	4	2	no	—
166	C₂×C₇	—	2	2	no	—
167	C₂×C₇	—	8	2	no	—
168	C₂×C₇	#1954	1	1	yes	characteristic
169	C₂×C₇	C₂×#350	1	2	yes	—
170	C₂³	—	2	2	no	—
171	C₂³	C₂×C₇×M₁₆	1	1	yes	characteristic
172	Q₈	—	2	4	no	—
173	D₄	—	2	4	no	—
174	D₄	—	4	4	no	—
175	D₄	—	4	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	C₇×C₂²⋊C₈	1	2	yes	—
181	C₂×C₄	C₇×C₂²⋊C₈	1	2	yes	—
182	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
183	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
184	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
185	C₂×C₄	C₇×#131	1	2	yes	—
186	C₈	—	2	2	no	—
187	C₈	—	4	2	no	—
188	C₈	C₇×#121	1	2	yes	—
189	C₈	C₇×#124	1	1	yes	characteristic
190	C₈	C₇×#127	1	1	yes	characteristic
191	C₇	#13191	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²	—	2	2	no	—
193	C₂²	—	4	4	no	—
194	C₂²	—	4	2	no	—
195	C₂²	C₂×C₇×C₂²⋊C₈	1	1	yes	characteristic
196	C₂²	C₇×#436	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	4	2	no	—
199	C₄	C₇×#350	1	4	yes	complemented
200	C₄	C₇×#390	1	1	yes	characteristic
201	C₄	C₇×#408	1	1	yes	characteristic
202	C₄	C₇×C₈×D₄	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	8	2	no	—
205	C₂	C₇×#1954	1	1	yes	characteristic
206	C₂	C₂×C₇×#350	1	2	yes	—

	subgroup	G/H
center	C₂×C₇×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₇×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂²×C₇	C₂×C₂²⋊C₈
prime radical	C₇×#532	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2150	C₂	semidirect
3. C₂×C₇×#350	C₂	semidirect
4. C₂×C₄×C₇×C₁₆	C₂	semidirect
17. #13191	C₇	direct (N × H)
36. C₂×C₇×C₁₆	D₄	semidirect
42. #2150	C₂×C₇	semidirect
43. C₂×#350	C₂×C₇	semidirect
44. C₂×C₄×C₁₆	C₂×C₇	semidirect
123. C₂×C₁₆	C₇×D₄	semidirect
135. C₄×C₇	#350	semidirect
191. C₇	#13191	direct (N × H)
199. C₄	C₇×#350	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2931	2
C₇×#2150	2
C₂×C₇×#350	2
C₂×C₄×C₇×C₁₆	2
#13191	7

Subgroups — C7×#13191 #13190407

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#13191 #13190407