Subgroups — C₇×#8040 #13185256

Order 1792.

726 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#432	C₂	1	2	yes	maximal
2	C₂×C₇×#384	C₂	1	1	yes	maximal, characteristic
3	C₇×#1296	C₂	1	4	yes	maximal, complemented
4	C₂²×C₇×C₄⋊C₄	—	2	2	no	—
5	C₂²×C₇×C₄⋊C₄	C₄	1	2	yes	—
6	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
7	C₇×#432	—	2	4	no	—
8	C₇×#432	C₄	1	8	yes	complemented
9	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
10	C₂×C₇×C₂²⋊C₈	—	2	2	no	—
11	C₂×C₇×#97	—	2	2	no	—
12	C₇×#384	—	2	2	no	—
13	C₇×#384	C₄	1	2	yes	—
14	C₇×#384	C₂²	1	2	yes	—
15	#8040	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₄×C₇	—	2	2	no	—
17	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₇	D₄	1	2	yes	—
19	C₂²×C₇×C₈	—	2	4	no	—
20	C₂×C₇×C₄⋊C₄	—	2	4	no	—
21	C₂×C₇×C₄⋊C₄	—	2	2	no	—
22	C₂×C₇×C₄⋊C₄	—	2	4	no	—
23	C₂×C₇×C₄⋊C₄	—	4	4	no	—
24	C₂×C₇×C₄⋊C₄	—	4	8	no	—
25	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₇	D₄	1	2	yes	—
29	C₂×C₄²×C₇	D₄	1	2	yes	—
30	C₂×C₄²×C₇	Q₈	1	2	yes	—
31	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
32	C₇×C₄⋊C₈	—	2	4	no	—
33	C₇×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
34	C₇×C₂²⋊C₈	—	2	8	no	—
35	C₇×#97	—	2	8	no	—
36	C₂×#432	C₂×C₇	1	2	yes	—
37	C₂×#384	C₂×C₇	1	1	yes	characteristic
38	#1296	C₂×C₇	1	4	yes	complemented
39	C₂⁴×C₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇	—	2	2	no	—
41	C₂²×C₄×C₇	—	2	2	no	—
42	C₂²×C₄×C₇	—	2	4	no	—
43	C₂²×C₄×C₇	—	2	4	no	—
44	C₂²×C₄×C₇	—	2	4	no	—
45	C₂²×C₄×C₇	—	4	8	no	—
46	C₂²×C₄×C₇	—	4	4	no	—
47	C₂²×C₄×C₇	C₄²	1	2	yes	—
48	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂²⋊C₄	1	4	yes	—
51	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
55	C₂×C₇×C₈	—	2	8	no	—
56	C₂×C₇×C₈	—	4	4	no	—
57	C₂×C₇×C₈	—	4	4	no	—
58	C₇×C₄⋊C₄	—	4	4	no	—
59	C₇×C₄⋊C₄	—	4	4	no	—
60	C₇×C₄⋊C₄	—	4	8	no	—
61	C₇×C₄⋊C₄	—	4	8	no	—
62	C₄²×C₇	C₄²	1	2	yes	—
63	C₄²×C₇	C₂²⋊C₄	1	4	yes	—
64	C₄²×C₇	C₄⋊C₄	1	2	yes	—
65	C₄²×C₇	C₄⋊C₄	1	4	yes	—
66	C₄²×C₇	C₂²×C₄	1	2	yes	—
67	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄⋊C₄	—	2	2	no	—
70	C₂²×C₄⋊C₄	C₄×C₇	1	2	yes	—
71	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
72	#432	—	2	4	no	—
73	#432	C₄×C₇	1	8	yes	complemented
74	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
75	C₂×C₂²⋊C₈	—	2	2	no	—
76	C₂×#97	—	2	2	no	—
77	#384	—	2	2	no	—
78	#384	C₄×C₇	1	2	yes	—
79	#384	C₂²×C₇	1	2	yes	—
80	C₂³×C₇	—	2	2	no	—
81	C₂³×C₇	—	2	2	no	—
82	C₂³×C₇	#95	1	2	yes	—
83	C₂³×C₇	#100	1	2	yes	—
84	C₂³×C₇	#101	1	2	yes	—
85	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₇	—	2	4	no	—
87	C₂×C₄×C₇	—	2	4	no	—
88	C₂×C₄×C₇	—	2	1	no	—
89	C₂×C₄×C₇	—	2	1	no	—
90	C₂×C₄×C₇	—	2	1	no	—
91	C₂×C₄×C₇	—	2	1	no	—
92	C₂×C₄×C₇	—	2	2	no	—
93	C₂×C₄×C₇	—	2	2	no	—
94	C₂×C₄×C₇	—	2	8	no	—
95	C₂×C₄×C₇	—	2	8	no	—
96	C₂×C₄×C₇	—	4	4	no	—
97	C₂×C₄×C₇	—	4	8	no	—
98	C₂×C₄×C₇	—	8	8	no	—
99	C₂×C₄×C₇	#95	1	4	yes	—
100	C₂×C₄×C₇	#95	1	2	yes	—
101	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₇	AGL(1,ℤ/8)	1	4	yes	—
106	C₂×C₄×C₇	#137	1	4	yes	—
107	C₇×C₈	—	4	8	no	—
108	C₂³×C₄	—	2	2	no	—
109	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
110	C₂³×C₄	C₇×D₄	1	2	yes	—
111	C₂²×C₈	—	2	4	no	—
112	C₂×C₄⋊C₄	—	2	4	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	—	4	4	no	—
116	C₂×C₄⋊C₄	—	4	8	no	—
117	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
118	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
119	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
120	C₂×C₄²	C₇×D₄	1	2	yes	—
121	C₂×C₄²	C₇×D₄	1	2	yes	—
122	C₂×C₄²	C₇×Q₈	1	2	yes	—
123	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
124	C₄⋊C₈	—	2	4	no	—
125	C₄⋊C₈	C₂×C₄×C₇	1	8	yes	complemented
126	C₂²⋊C₈	—	2	8	no	—
127	#97	—	2	8	no	—
128	C₂²×C₇	—	2	2	no	—
129	C₂²×C₇	—	2	4	no	—
130	C₂²×C₇	—	2	2	no	—
131	C₂²×C₇	—	2	4	no	—
132	C₂²×C₇	#343	1	4	yes	—
133	C₂²×C₇	C₂×#95	1	1	yes	characteristic
134	C₂²×C₇	C₂×#100	1	1	yes	characteristic
135	C₂²×C₇	C₂×#101	1	1	yes	characteristic
136	C₂²×C₇	#418	1	2	yes	—
137	C₂²×C₇	#419	1	2	yes	—
138	C₄×C₇	—	2	4	no	—
139	C₄×C₇	—	2	2	no	—
140	C₄×C₇	—	2	1	no	—
141	C₄×C₇	—	2	1	no	—
142	C₄×C₇	—	8	4	no	—
143	C₄×C₇	#417	1	4	yes	—
144	C₄×C₇	#428	1	4	yes	—
145	C₂⁴	C₇×C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	4	8	no	—
152	C₂²×C₄	—	4	4	no	—
153	C₂²×C₄	C₄²×C₇	1	2	yes	—
154	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
155	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄	C₇×C₂²⋊C₄	1	4	yes	—
157	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
158	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
159	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
160	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
161	C₂×C₈	—	2	8	no	—
162	C₂×C₈	—	4	4	no	—
163	C₂×C₈	—	4	4	no	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄⋊C₄	—	4	8	no	—
167	C₄⋊C₄	—	4	8	no	—
168	C₄²	C₄²×C₇	1	2	yes	—
169	C₄²	C₇×C₂²⋊C₄	1	4	yes	—
170	C₄²	C₇×C₄⋊C₄	1	2	yes	—
171	C₄²	C₇×C₄⋊C₄	1	4	yes	—
172	C₄²	C₂²×C₄×C₇	1	2	yes	—
173	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
174	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
175	C₂×C₇	—	2	4	no	—
176	C₂×C₇	#1555	1	1	yes	characteristic
177	C₂×C₇	#1556	1	2	yes	—
178	C₂×C₇	#1558	1	1	yes	characteristic
179	C₂×C₇	#1736	1	2	yes	—
180	C₂×C₇	C₂×#343	1	1	yes	characteristic
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	C₇×#95	1	2	yes	—
184	C₂³	C₇×#100	1	2	yes	—
185	C₂³	C₇×#101	1	2	yes	—
186	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	2	8	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	8	no	—
199	C₂×C₄	—	8	8	no	—
200	C₂×C₄	C₇×#95	1	4	yes	—
201	C₂×C₄	C₇×#95	1	2	yes	—
202	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
203	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
204	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
205	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
206	C₂×C₄	C₇×AGL(1,ℤ/8)	1	4	yes	—
207	C₂×C₄	C₇×#137	1	4	yes	—
208	C₈	—	4	8	no	—
209	C₇	#8040	1	1	yes	characteristic, direct product factor, Sylow
210	C₂²	—	2	4	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	4	no	—
213	C₂²	—	2	2	no	—
214	C₂²	C₇×#343	1	4	yes	—
215	C₂²	C₂×C₇×#95	1	1	yes	characteristic
216	C₂²	C₂×C₇×#100	1	1	yes	characteristic
217	C₂²	C₂×C₇×#101	1	1	yes	characteristic
218	C₂²	C₇×#418	1	2	yes	—
219	C₂²	C₇×#419	1	2	yes	—
220	C₄	—	2	4	no	—
221	C₄	—	2	2	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	8	4	no	—
225	C₄	C₇×#417	1	4	yes	—
226	C₄	C₇×#428	1	4	yes	—
227	C₂	—	2	4	no	—
228	C₂	C₇×#1555	1	1	yes	characteristic
229	C₂	C₇×#1556	1	2	yes	—
230	C₂	C₇×#1558	1	1	yes	characteristic
231	C₂	C₇×#1736	1	2	yes	—
232	C₂	C₂×C₇×#343	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₇×#1296	C₂	semidirect
8. C₇×#432	C₄	semidirect
15. #8040	C₇	direct (N × H)
33. C₇×C₄⋊C₈	C₂×C₄	semidirect
38. #1296	C₂×C₇	semidirect
73. #432	C₄×C₇	semidirect
125. C₄⋊C₈	C₂×C₄×C₇	semidirect
209. C₇	#8040	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×#432	2
C₂×C₇×#384	2
C₇×#1296	2
#8040	7

Subgroups — C7×#8040 #13185256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#8040 #13185256