Subgroups — C₇×#7920 #13185136

Order 1792.

614 conjugacy classes of subgroups.

The subgroup lattice has 244 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×C₄×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₇×#1281	C₂	1	4	yes	maximal, complemented
4	C₂²×C₇×M₁₆	C₄	1	2	yes	—
5	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
6	C₇×#432	—	2	2	no	—
7	C₇×#432	C₄	1	4	yes	—
8	C₇×#431	—	2	2	no	—
9	C₇×#431	C₂²	1	4	yes	—
10	C₇×#423	—	2	2	no	—
11	C₇×#407	C₄	1	4	yes	—
12	C₄×C₇×M₁₆	—	2	2	no	—
13	C₄×C₇×M₁₆	—	2	2	no	—
14	C₄×C₇×M₁₆	—	2	2	no	—
15	C₂×C₇×#97	C₂²	1	2	yes	—
16	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
17	#7920	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
20	C₂×C₇×M₁₆	—	2	2	no	—
21	C₂×C₇×M₁₆	—	2	2	no	—
22	C₂×C₇×M₁₆	—	2	4	no	—
23	C₂×C₇×M₁₆	—	2	8	no	—
24	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₇	D₄	1	2	yes	—
28	C₂×C₄²×C₇	D₄	1	2	yes	—
29	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
34	C₇×C₄⋊C₈	—	2	4	no	—
35	C₇×C₄⋊C₈	—	2	4	no	—
36	C₇×C₂²⋊C₈	—	2	4	no	—
37	C₇×C₂²⋊C₈	C₂×C₄	1	8	yes	—
38	C₇×#97	—	2	4	no	—
39	C₇×#97	—	2	4	no	—
40	C₄×C₇×C₈	—	2	4	no	—
41	C₄×C₇×C₈	—	2	4	no	—
42	#2920	C₂×C₇	1	2	yes	—
43	C₂×C₄×M₁₆	C₂×C₇	1	1	yes	characteristic
44	#1281	C₂×C₇	1	4	yes	complemented
45	C₂⁴×C₇	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₇	—	2	4	no	—
47	C₂²×C₄×C₇	—	4	4	no	—
48	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₄²	1	2	yes	—
50	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
60	C₇×M₁₆	—	2	8	no	—
61	C₇×M₁₆	—	2	8	no	—
62	C₇×M₁₆	—	4	8	no	—
63	C₂×C₇×C₈	—	2	4	no	—
64	C₂×C₇×C₈	—	2	4	no	—
65	C₂×C₇×C₈	—	2	8	no	—
66	C₂×C₇×C₈	M₁₆	1	8	yes	complemented
67	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
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	1	yes	characteristic
72	C₄²×C₇	C₄⋊C₄	1	2	yes	—
73	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
74	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₇	C₂×D₄	1	2	yes	—
76	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
77	C₂²×M₁₆	C₄×C₇	1	2	yes	—
78	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
79	#432	—	2	2	no	—
80	#432	C₄×C₇	1	4	yes	—
81	#431	—	2	2	no	—
82	#431	C₂²×C₇	1	4	yes	—
83	#423	—	2	2	no	—
84	#407	C₄×C₇	1	4	yes	—
85	C₄×M₁₆	—	2	2	no	—
86	C₄×M₁₆	—	2	2	no	—
87	C₄×M₁₆	—	2	2	no	—
88	C₂×#97	C₂²×C₇	1	2	yes	—
89	C₂×C₄×C₈	C₄×C₇	1	2	yes	—
90	C₂³×C₇	—	2	2	no	—
91	C₂³×C₇	—	4	2	no	—
92	C₂³×C₇	#95	1	1	yes	characteristic
93	C₂³×C₇	#95	1	1	yes	characteristic
94	C₂³×C₇	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₇	—	2	2	no	—
96	C₂×C₄×C₇	—	2	4	no	—
97	C₂×C₄×C₇	—	2	4	no	—
98	C₂×C₄×C₇	—	2	4	no	—
99	C₂×C₄×C₇	—	2	2	no	—
100	C₂×C₄×C₇	—	2	2	no	—
101	C₂×C₄×C₇	—	2	2	no	—
102	C₂×C₄×C₇	—	2	2	no	—
103	C₂×C₄×C₇	—	4	2	no	—
104	C₂×C₄×C₇	—	4	2	no	—
105	C₂×C₄×C₇	—	4	2	no	—
106	C₂×C₄×C₇	#95	1	1	yes	characteristic
107	C₂×C₄×C₇	#95	1	1	yes	characteristic
108	C₂×C₄×C₇	#95	1	2	yes	—
109	C₂×C₄×C₇	#95	1	2	yes	—
110	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
111	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
115	C₇×C₈	—	2	8	no	—
116	C₇×C₈	—	4	8	no	—
117	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
118	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
119	C₂×M₁₆	—	2	2	no	—
120	C₂×M₁₆	—	2	2	no	—
121	C₂×M₁₆	—	2	4	no	—
122	C₂×M₁₆	—	2	8	no	—
123	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
124	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
125	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
126	C₂×C₄²	C₇×D₄	1	2	yes	—
127	C₂×C₄²	C₇×D₄	1	2	yes	—
128	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
129	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
130	C₂×C₄²	C₇×Q₈	1	1	yes	characteristic
131	C₂×C₄²	C₇×Q₈	1	1	yes	characteristic
132	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
133	C₄⋊C₈	—	2	4	no	—
134	C₄⋊C₈	—	2	4	no	—
135	C₂²⋊C₈	—	2	4	no	—
136	C₂²⋊C₈	C₂×C₄×C₇	1	8	yes	—
137	#97	—	2	4	no	—
138	#97	—	2	4	no	—
139	C₄×C₈	—	2	4	no	—
140	C₄×C₈	—	2	4	no	—
141	C₂²×C₇	—	2	2	no	—
142	C₂²×C₇	—	2	1	no	—
143	C₂²×C₇	—	2	1	no	—
144	C₂²×C₇	—	2	2	no	—
145	C₂²×C₇	—	2	2	no	—
146	C₂²×C₇	—	4	4	no	—
147	C₂²×C₇	C₂×#95	1	1	yes	characteristic
148	C₂²×C₇	C₄×M₁₆	1	2	yes	—
149	C₄×C₇	—	2	1	no	—
150	C₄×C₇	—	2	1	no	—
151	C₄×C₇	—	2	2	no	—
152	C₄×C₇	—	2	2	no	—
153	C₄×C₇	—	2	2	no	—
154	C₄×C₇	#344	1	4	yes	—
155	C₄×C₇	#407	1	2	yes	—
156	C₄×C₇	#423	1	2	yes	—
157	C₂⁴	C₄²×C₇	1	1	yes	characteristic
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	4	4	no	—
160	C₂²×C₄	C₄²×C₇	1	1	yes	characteristic
161	C₂²×C₄	C₄²×C₇	1	2	yes	—
162	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
163	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
164	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
165	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
166	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
167	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
168	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
169	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
170	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
171	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
172	M₁₆	—	2	8	no	—
173	M₁₆	—	2	8	no	—
174	M₁₆	—	4	8	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	8	no	—
178	C₂×C₈	C₇×M₁₆	1	8	yes	complemented
179	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
180	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
181	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
182	C₄²	C₇×C₄⋊C₄	1	2	yes	—
183	C₄²	C₇×C₄⋊C₄	1	1	yes	characteristic
184	C₄²	C₇×C₄⋊C₄	1	2	yes	—
185	C₄²	C₇×C₄⋊C₄	1	1	yes	characteristic
186	C₄²	C₂×C₇×D₄	1	2	yes	—
187	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
188	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
189	C₂×C₇	—	2	2	no	—
190	C₂×C₇	—	4	1	no	—
191	C₂×C₇	—	4	1	no	—
192	C₂×C₇	#1728	1	1	yes	characteristic
193	C₂×C₇	C₂×#344	1	2	yes	—
194	C₂³	—	2	2	no	—
195	C₂³	—	4	2	no	—
196	C₂³	C₇×#95	1	1	yes	characteristic
197	C₂³	C₇×#95	1	1	yes	characteristic
198	C₂³	C₂×C₄²×C₇	1	1	yes	characteristic
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	C₇×#95	1	1	yes	characteristic
211	C₂×C₄	C₇×#95	1	1	yes	characteristic
212	C₂×C₄	C₇×#95	1	2	yes	—
213	C₂×C₄	C₇×#95	1	2	yes	—
214	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
215	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
216	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
217	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
218	C₂×C₄	C₂×C₇×M₁₆	1	4	yes	—
219	C₈	—	2	8	no	—
220	C₈	—	4	8	no	—
221	C₇	#7920	1	1	yes	characteristic, direct product factor, Sylow
222	C₂²	—	2	2	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	4	4	no	—
228	C₂²	C₂×C₇×#95	1	1	yes	characteristic
229	C₂²	C₄×C₇×M₁₆	1	2	yes	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	C₇×#344	1	4	yes	—
236	C₄	C₇×#407	1	2	yes	—
237	C₄	C₇×#423	1	2	yes	—
238	C₂	—	2	2	no	—
239	C₂	—	4	1	no	—
240	C₂	—	4	1	no	—
241	C₂	C₇×#1728	1	1	yes	characteristic
242	C₂	C₂×C₇×#344	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₇×#1281	C₂	semidirect
17. #7920	C₇	direct (N × H)
44. #1281	C₂×C₇	semidirect
66. C₂×C₇×C₈	M₁₆	semidirect
178. C₂×C₈	C₇×M₁₆	semidirect
221. C₇	#7920	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×C₄×C₇×M₁₆	2
C₇×#1281	2
#7920	7

Subgroups — C7×#7920 #13185136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#7920 #13185136