Subgroups — C₇×#7867 #13185083

Order 1792.

922 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2920	C₂	1	2	yes	maximal
2	C₂×C₄³×C₇	C₂	1	1	yes	maximal, characteristic
3	C₇×#1276	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄²×C₇	—	2	2	no	—
5	C₂²×C₄²×C₇	C₄	1	2	yes	—
6	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
7	C₇×#432	—	2	2	no	—
8	C₇×#432	C₄	1	4	yes	complemented
9	C₇×#431	—	2	2	no	—
10	C₇×#431	C₂²	1	4	yes	—
11	C₇×#423	C₄	1	4	yes	complemented
12	C₇×#407	—	2	2	no	—
13	C₄×C₇×M₁₆	—	2	2	no	—
14	C₄³×C₇	—	2	2	no	—
15	C₄³×C₇	C₄	1	2	yes	—
16	C₄³×C₇	C₂²	1	2	yes	—
17	#7867	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₇	—	2	2	no	—
19	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₇	D₄	1	2	yes	—
21	C₂×C₇×M₁₆	—	2	4	no	—
22	C₂×C₄²×C₇	—	2	16	no	—
23	C₂×C₄²×C₇	—	2	2	no	—
24	C₂×C₄²×C₇	—	2	4	no	—
25	C₂×C₄²×C₇	—	2	4	no	—
26	C₂×C₄²×C₇	—	2	8	no	—
27	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₇	D₄	1	2	yes	—
32	C₂×C₄²×C₇	D₄	1	2	yes	—
33	C₂×C₄²×C₇	Q₈	1	2	yes	—
34	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
35	C₇×C₄⋊C₈	—	2	4	no	—
36	C₇×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
37	C₇×C₂²⋊C₈	—	2	4	no	—
38	C₇×C₂²⋊C₈	—	2	4	no	—
39	C₇×#97	—	2	4	no	—
40	C₄×C₇×C₈	—	2	4	no	—
41	#2920	C₂×C₇	1	2	yes	—
42	C₂×C₄³	C₂×C₇	1	1	yes	characteristic
43	#1276	C₂×C₇	1	4	yes	complemented
44	C₂⁴×C₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇	—	2	16	no	—
46	C₂²×C₄×C₇	—	2	8	no	—
47	C₂²×C₄×C₇	—	2	4	no	—
48	C₂²×C₄×C₇	—	2	8	no	—
49	C₂²×C₄×C₇	—	2	4	no	—
50	C₂²×C₄×C₇	C₄²	1	2	yes	—
51	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
60	C₇×M₁₆	—	4	8	no	—
61	C₂×C₇×C₈	—	2	8	no	—
62	C₄²×C₇	—	2	16	no	—
63	C₄²×C₇	—	2	16	no	—
64	C₄²×C₇	—	2	8	no	—
65	C₄²×C₇	—	2	4	no	—
66	C₄²×C₇	C₄²	1	1	yes	characteristic
67	C₄²×C₇	C₄²	1	1	yes	characteristic
68	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₇	C₄⋊C₄	1	2	yes	—
71	C₄²×C₇	C₄⋊C₄	1	2	yes	—
72	C₄²×C₇	C₄⋊C₄	1	2	yes	—
73	C₄²×C₇	M₁₆	1	8	yes	complemented
74	C₄²×C₇	C₂²×C₄	1	2	yes	—
75	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄²	—	2	2	no	—
78	C₂²×C₄²	C₄×C₇	1	2	yes	—
79	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
80	#432	—	2	2	no	—
81	#432	C₄×C₇	1	4	yes	complemented
82	#431	—	2	2	no	—
83	#431	C₂²×C₇	1	4	yes	—
84	#423	C₄×C₇	1	4	yes	complemented
85	#407	—	2	2	no	—
86	C₄×M₁₆	—	2	2	no	—
87	C₄³	—	2	2	no	—
88	C₄³	C₄×C₇	1	2	yes	—
89	C₄³	C₂²×C₇	1	2	yes	—
90	C₂³×C₇	—	2	4	no	—
91	C₂³×C₇	—	2	2	no	—
92	C₂³×C₇	#95	1	2	yes	—
93	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₇	—	2	16	no	—
95	C₂×C₄×C₇	—	2	32	no	—
96	C₂×C₄×C₇	—	2	4	no	—
97	C₂×C₄×C₇	—	2	2	no	—
98	C₂×C₄×C₇	—	2	8	no	—
99	C₂×C₄×C₇	—	2	4	no	—
100	C₂×C₄×C₇	—	2	4	no	—
101	C₂×C₄×C₇	—	2	4	no	—
102	C₂×C₄×C₇	—	2	4	no	—
103	C₂×C₄×C₇	—	2	4	no	—
104	C₂×C₄×C₇	—	2	4	no	—
105	C₂×C₄×C₇	#95	1	2	yes	—
106	C₂×C₄×C₇	#95	1	2	yes	—
107	C₂×C₄×C₇	#95	1	2	yes	—
108	C₂×C₄×C₇	#104	1	8	yes	—
109	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
111	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
114	C₇×C₈	—	4	8	no	—
115	C₂³×C₄	—	2	2	no	—
116	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
117	C₂³×C₄	C₇×D₄	1	2	yes	—
118	C₂×M₁₆	—	2	4	no	—
119	C₂×C₄²	—	2	16	no	—
120	C₂×C₄²	—	2	2	no	—
121	C₂×C₄²	—	2	4	no	—
122	C₂×C₄²	—	2	4	no	—
123	C₂×C₄²	—	2	8	no	—
124	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
125	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
126	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
127	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
128	C₂×C₄²	C₇×D₄	1	2	yes	—
129	C₂×C₄²	C₇×D₄	1	2	yes	—
130	C₂×C₄²	C₇×Q₈	1	2	yes	—
131	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
132	C₄⋊C₈	—	2	4	no	—
133	C₄⋊C₈	C₂×C₄×C₇	1	8	yes	complemented
134	C₂²⋊C₈	—	2	4	no	—
135	C₂²⋊C₈	—	2	4	no	—
136	#97	—	2	4	no	—
137	C₄×C₈	—	2	4	no	—
138	C₂²×C₇	—	2	8	no	—
139	C₂²×C₇	—	2	2	no	—
140	C₂²×C₇	—	2	2	no	—
141	C₂²×C₇	—	2	2	no	—
142	C₂²×C₇	#339	1	4	yes	—
143	C₂²×C₇	C₂×#95	1	1	yes	characteristic
144	C₂²×C₇	#407	1	2	yes	—
145	C₄×C₇	—	2	16	no	—
146	C₄×C₇	—	2	2	no	—
147	C₄×C₇	—	2	2	no	—
148	C₄×C₇	—	2	2	no	—
149	C₄×C₇	#339	1	4	yes	—
150	C₄×C₇	C₄×M₁₆	1	2	yes	—
151	C₄×C₇	C₂×#104	1	4	yes	—
152	C₄×C₇	#423	1	2	yes	—
153	C₂⁴	C₇×C₂²⋊C₄	1	1	yes	characteristic
154	C₂²×C₄	—	2	16	no	—
155	C₂²×C₄	—	2	8	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	2	8	no	—
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	C₄²×C₇	1	2	yes	—
160	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
161	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
162	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
163	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
164	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
165	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
166	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
167	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
169	M₁₆	—	4	8	no	—
170	C₂×C₈	—	2	8	no	—
171	C₄²	—	2	16	no	—
172	C₄²	—	2	16	no	—
173	C₄²	—	2	8	no	—
174	C₄²	—	2	4	no	—
175	C₄²	C₄²×C₇	1	1	yes	characteristic
176	C₄²	C₄²×C₇	1	1	yes	characteristic
177	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
178	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
179	C₄²	C₇×C₄⋊C₄	1	2	yes	—
180	C₄²	C₇×C₄⋊C₄	1	2	yes	—
181	C₄²	C₇×C₄⋊C₄	1	2	yes	—
182	C₄²	C₇×M₁₆	1	8	yes	complemented
183	C₄²	C₂²×C₄×C₇	1	2	yes	—
184	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
185	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
186	C₂×C₇	—	2	2	no	—
187	C₂×C₇	—	2	4	no	—
188	C₂×C₇	#1728	1	1	yes	characteristic
189	C₂×C₇	C₂×#339	1	2	yes	—
190	C₂³	—	2	4	no	—
191	C₂³	—	2	2	no	—
192	C₂³	C₇×#95	1	2	yes	—
193	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
194	C₂×C₄	—	2	16	no	—
195	C₂×C₄	—	2	32	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	C₇×#95	1	2	yes	—
206	C₂×C₄	C₇×#95	1	2	yes	—
207	C₂×C₄	C₇×#95	1	2	yes	—
208	C₂×C₄	C₇×#104	1	8	yes	—
209	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
210	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
211	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
212	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
213	C₂×C₄	C₂×C₇×M₁₆	1	4	yes	—
214	C₈	—	4	8	no	—
215	C₇	#7867	1	1	yes	characteristic, direct product factor, Sylow
216	C₂²	—	2	8	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	C₇×#339	1	4	yes	—
221	C₂²	C₂×C₇×#95	1	1	yes	characteristic
222	C₂²	C₇×#407	1	2	yes	—
223	C₄	—	2	16	no	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	C₇×#339	1	4	yes	—
228	C₄	C₄×C₇×M₁₆	1	2	yes	—
229	C₄	C₂×C₇×#104	1	4	yes	—
230	C₄	C₇×#423	1	2	yes	—
231	C₂	—	2	2	no	—
232	C₂	—	2	4	no	—
233	C₂	C₇×#1728	1	1	yes	characteristic
234	C₂	C₂×C₇×#339	1	2	yes	—

	subgroup	G/H
center	C₄²×C₇	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	C₂×#95
prime radical	C₂⁴×C₇	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1276	C₂	semidirect
8. C₇×#432	C₄	semidirect
11. C₇×#423	C₄	semidirect
17. #7867	C₇	direct (N × H)
36. C₇×C₄⋊C₈	C₂×C₄	semidirect
43. #1276	C₂×C₇	semidirect
73. C₄²×C₇	M₁₆	semidirect
81. #432	C₄×C₇	semidirect
84. #423	C₄×C₇	semidirect
133. C₄⋊C₈	C₂×C₄×C₇	semidirect
182. C₄²	C₇×M₁₆	semidirect
215. C₇	#7867	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×#2920	2
C₂×C₄³×C₇	2
C₇×#1276	2
#7867	7

Subgroups — C7×#7867 #13185083

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#7867 #13185083