Subgroups — C₇×#7918 #13185134

Order 1792.

666 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	C₂×C₇×#431	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×#431	—	2	2	no	—
7	C₇×#431	—	2	2	no	—
8	C₇×#431	C₄	1	4	yes	—
9	C₇×#431	C₂²	1	4	yes	—
10	C₂×C₇×C₄⋊C₈	—	2	2	no	—
11	C₂×C₇×C₂²⋊C₈	C₄	1	4	yes	—
12	C₄×C₇×M₁₆	—	2	2	no	—
13	C₄×C₇×M₁₆	—	2	2	no	—
14	C₂×C₇×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₇×C₈	—	2	2	no	—
16	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
17	#7918	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₇×M₁₆	—	2	4	no	—
21	C₂×C₇×M₁₆	—	2	4	no	—
22	C₂×C₇×M₁₆	C₈	1	8	yes	complemented
23	C₂²×C₇×C₈	—	2	4	no	—
24	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₇	D₄	1	2	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	—	2	4	no	—
34	C₇×C₂²⋊C₈	—	2	4	no	—
35	C₇×C₂²⋊C₈	C₂×C₄	1	8	yes	—
36	C₇×#97	—	2	4	no	—
37	C₄×C₇×C₈	—	2	4	no	—
38	C₄×C₇×C₈	—	2	2	no	—
39	C₄×C₇×C₈	—	2	2	no	—
40	C₄×C₇×C₈	—	2	4	no	—
41	C₂×#431	C₂×C₇	1	2	yes	—
42	C₂×C₄×M₁₆	C₂×C₇	1	1	yes	characteristic
43	#1281	C₂×C₇	1	4	yes	complemented
44	C₂⁴×C₇	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₇	—	4	4	no	—
46	C₂²×C₄×C₇	C₄²	1	2	yes	—
47	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₇	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₇	M₁₆	1	4	yes	—
56	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
58	C₇×M₁₆	—	2	16	no	—
59	C₂×C₇×C₈	—	2	4	no	—
60	C₂×C₇×C₈	—	2	4	no	—
61	C₂×C₇×C₈	—	2	8	no	—
62	C₂×C₇×C₈	—	4	8	no	—
63	C₂×C₇×C₈	C₂×C₈	1	8	yes	complemented
64	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₇	C₄⋊C₄	1	2	yes	—
69	C₄²×C₇	C₄⋊C₄	1	2	yes	—
70	C₄²×C₇	C₄⋊C₄	1	2	yes	—
71	C₄²×C₇	C₂×D₄	1	2	yes	—
72	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
74	C₂²×M₁₆	C₄×C₇	1	2	yes	—
75	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
76	#431	—	2	2	no	—
77	#431	—	2	2	no	—
78	#431	C₄×C₇	1	4	yes	—
79	#431	C₂²×C₇	1	4	yes	—
80	C₂×C₄⋊C₈	—	2	2	no	—
81	C₂×C₂²⋊C₈	C₄×C₇	1	4	yes	—
82	C₄×M₁₆	—	2	2	no	—
83	C₄×M₁₆	—	2	2	no	—
84	C₂×#97	C₂²×C₇	1	2	yes	—
85	C₂×C₄×C₈	—	2	2	no	—
86	C₂×C₄×C₈	C₄×C₇	1	2	yes	—
87	C₂³×C₇	—	4	2	no	—
88	C₂³×C₇	#95	1	2	yes	—
89	C₂³×C₇	C₄×C₈	1	2	yes	—
90	C₂³×C₇	#97	1	2	yes	—
91	C₂³×C₇	C₂×C₄²	1	1	yes	characteristic
92	C₂×C₄×C₇	—	2	4	no	—
93	C₂×C₄×C₇	—	2	4	no	—
94	C₂×C₄×C₇	—	2	4	no	—
95	C₂×C₄×C₇	—	4	2	no	—
96	C₂×C₄×C₇	—	4	2	no	—
97	C₂×C₄×C₇	—	4	2	no	—
98	C₂×C₄×C₇	#95	1	2	yes	—
99	C₂×C₄×C₇	#95	1	2	yes	—
100	C₂×C₄×C₇	#95	1	2	yes	—
101	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
102	C₂×C₄×C₇	#97	1	2	yes	—
103	C₂×C₄×C₇	C₂²⋊C₈	1	8	yes	—
104	C₂×C₄×C₇	C₄⋊C₈	1	8	yes	—
105	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
106	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
111	C₇×C₈	—	2	8	no	—
112	C₇×C₈	—	4	8	no	—
113	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
114	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
115	C₂×M₁₆	—	2	4	no	—
116	C₂×M₁₆	—	2	4	no	—
117	C₂×M₁₆	C₇×C₈	1	8	yes	complemented
118	C₂²×C₈	—	2	4	no	—
119	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
120	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
121	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
122	C₂×C₄²	C₇×D₄	1	2	yes	—
123	C₂×C₄²	C₇×D₄	1	2	yes	—
124	C₂×C₄²	C₇×D₄	1	2	yes	—
125	C₂×C₄²	C₇×Q₈	1	2	yes	—
126	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
127	C₄⋊C₈	—	2	4	no	—
128	C₄⋊C₈	—	2	4	no	—
129	C₂²⋊C₈	—	2	4	no	—
130	C₂²⋊C₈	C₂×C₄×C₇	1	8	yes	—
131	#97	—	2	4	no	—
132	C₄×C₈	—	2	4	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	—	2	4	no	—
136	C₂²×C₇	—	2	2	no	—
137	C₂²×C₇	—	2	2	no	—
138	C₂²×C₇	—	2	2	no	—
139	C₂²×C₇	—	4	2	no	—
140	C₂²×C₇	—	4	2	no	—
141	C₂²×C₇	#336	1	4	yes	—
142	C₂²×C₇	C₂×#95	1	1	yes	characteristic
143	C₂²×C₇	C₂×C₄×C₈	1	1	yes	characteristic
144	C₂²×C₇	C₂×#97	1	1	yes	characteristic
145	C₄×C₇	—	2	2	no	—
146	C₄×C₇	—	2	2	no	—
147	C₄×C₇	—	2	2	no	—
148	C₄×C₇	#336	1	4	yes	—
149	C₄×C₇	#344	1	4	yes	—
150	C₄×C₇	C₂×C₂²⋊C₈	1	2	yes	—
151	C₄×C₇	C₂×C₄⋊C₈	1	2	yes	—
152	C₂⁴	C₄²×C₇	1	1	yes	characteristic
153	C₂²×C₄	—	4	4	no	—
154	C₂²×C₄	C₄²×C₇	1	2	yes	—
155	C₂²×C₄	C₄²×C₇	1	1	yes	characteristic
156	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
157	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
158	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
159	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
160	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
161	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
162	C₂²×C₄	C₂×C₇×C₈	1	4	yes	—
163	C₂²×C₄	C₇×M₁₆	1	4	yes	—
164	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
165	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
166	M₁₆	—	2	16	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	2	8	no	—
170	C₂×C₈	—	4	8	no	—
171	C₂×C₈	C₂×C₇×C₈	1	8	yes	complemented
172	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
173	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
174	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
175	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
176	C₄²	C₇×C₄⋊C₄	1	2	yes	—
177	C₄²	C₇×C₄⋊C₄	1	2	yes	—
178	C₄²	C₇×C₄⋊C₄	1	2	yes	—
179	C₄²	C₂×C₇×D₄	1	2	yes	—
180	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
181	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
182	C₂×C₇	—	2	4	no	—
183	C₂×C₇	—	2	1	no	—
184	C₂×C₇	—	2	1	no	—
185	C₂×C₇	C₂×#336	1	1	yes	characteristic
186	C₂×C₇	C₂×#344	1	2	yes	—
187	C₂³	—	4	2	no	—
188	C₂³	C₇×#95	1	2	yes	—
189	C₂³	C₄×C₇×C₈	1	2	yes	—
190	C₂³	C₇×#97	1	2	yes	—
191	C₂³	C₂×C₄²×C₇	1	1	yes	characteristic
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	C₇×#95	1	2	yes	—
199	C₂×C₄	C₇×#95	1	2	yes	—
200	C₂×C₄	C₇×#95	1	2	yes	—
201	C₂×C₄	C₄×C₇×C₈	1	2	yes	—
202	C₂×C₄	C₇×#97	1	2	yes	—
203	C₂×C₄	C₇×C₂²⋊C₈	1	8	yes	—
204	C₂×C₄	C₇×C₄⋊C₈	1	8	yes	—
205	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
206	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
207	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
208	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
209	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
210	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
211	C₈	—	2	8	no	—
212	C₈	—	4	8	no	—
213	C₇	#7918	1	1	yes	characteristic, direct product factor, Sylow
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	C₇×#336	1	4	yes	—
220	C₂²	C₂×C₇×#95	1	1	yes	characteristic
221	C₂²	C₂×C₄×C₇×C₈	1	1	yes	characteristic
222	C₂²	C₂×C₇×#97	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	C₇×#336	1	4	yes	—
227	C₄	C₇×#344	1	4	yes	—
228	C₄	C₂×C₇×C₂²⋊C₈	1	2	yes	—
229	C₄	C₂×C₇×C₄⋊C₈	1	2	yes	—
230	C₂	—	2	4	no	—
231	C₂	—	2	1	no	—
232	C₂	—	2	1	no	—
233	C₂	C₂×C₇×#336	1	1	yes	characteristic
234	C₂	C₂×C₇×#344	1	2	yes	—

	subgroup	G/H
center	C₄²×C₇	C₂²⋊C₄
commutator	C₂²	C₂×C₄×C₇×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	C₂×#95
prime radical	C₂⁴×C₇	C₄²
LCS	G ▷ C₂² ▷ 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×#1281	C₂	semidirect
17. #7918	C₇	direct (N × H)
22. C₂×C₇×M₁₆	C₈	semidirect
43. #1281	C₂×C₇	semidirect
63. C₂×C₇×C₈	C₂×C₈	semidirect
117. C₂×M₁₆	C₇×C₈	semidirect
171. C₂×C₈	C₂×C₇×C₈	semidirect
213. C₇	#7918	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×#431	2
C₂×C₄×C₇×M₁₆	2
C₇×#1281	2
#7918	7

Subgroups — C7×#7918 #13185134

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#7918 #13185134