Subgroups — C₇×#7981 #13185197

Order 1792.

714 conjugacy classes of subgroups.

The subgroup lattice has 290 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	1	yes	maximal, characteristic
2	C₇×#2285	C₂	1	2	yes	maximal
3	C₇×#1289	C₂	1	4	yes	maximal
4	C₂×C₇×#117	—	2	2	no	—
5	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
6	C₇×#431	—	2	2	no	—
7	C₇×#431	—	2	2	no	—
8	C₂×C₇×C₄⋊C₈	C₄	1	2	yes	complemented
9	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₇×C₈	C₄	1	2	yes	complemented
11	C₇×#384	—	2	2	no	—
12	C₇×#382	C₄	1	4	yes	complemented
13	C₇×#380	—	2	2	no	—
14	C₇×#380	C₂²	1	4	yes	—
15	C₄×C₇×C₄⋊C₄	—	2	2	no	—
16	C₄×C₇×C₄⋊C₄	C₄	1	4	yes	complemented
17	#7981	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
22	C₇×#117	—	4	4	no	—
23	C₇×#117	—	4	4	no	—
24	C₂×C₇×C₄⋊C₄	—	2	4	no	—
25	C₂×C₇×C₄⋊C₄	—	2	4	no	—
26	C₂×C₇×C₄⋊C₄	—	2	4	no	—
27	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
28	C₂×C₄²×C₇	—	2	4	no	—
29	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₇	D₄	1	2	yes	—
34	C₂×C₄²×C₇	D₄	1	2	yes	—
35	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
37	C₇×C₄⋊C₈	—	2	4	no	—
38	C₇×C₂²⋊C₈	—	2	4	no	—
39	C₄×C₇×C₈	—	2	2	no	—
40	C₄×C₇×C₈	—	2	2	no	—
41	C₇×#95	—	2	4	no	—
42	C₇×#95	C₂×C₄	1	8	yes	—
43	C₂×#431	C₂×C₇	1	1	yes	characteristic
44	#2285	C₂×C₇	1	2	yes	—
45	#1289	C₂×C₇	1	4	yes	—
46	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	—	2	8	no	—
48	C₂²×C₄×C₇	—	4	2	no	—
49	C₂²×C₄×C₇	—	4	2	no	—
50	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
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	1	yes	characteristic
58	C₂²×C₄×C₇	D₈	1	2	yes	—
59	C₂²×C₄×C₇	SD₁₆	1	4	yes	—
60	C₂²×C₄×C₇	Q₁₆	1	2	yes	—
61	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₇×C₈	—	2	4	no	—
66	C₂×C₇×C₈	—	4	4	no	—
67	C₇×C₄⋊C₄	—	4	8	no	—
68	C₇×C₄⋊C₄	—	4	8	no	—
69	C₇×C₄⋊C₄	—	4	8	no	—
70	C₇×C₂²⋊C₄	—	4	4	no	—
71	C₇×C₂²⋊C₄	—	4	4	no	—
72	C₄²×C₇	—	4	8	no	—
73	C₄²×C₇	C₄²	1	1	yes	characteristic
74	C₄²×C₇	C₄²	1	2	yes	—
75	C₄²×C₇	C₄²	1	1	yes	characteristic
76	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₇	C₄⋊C₄	1	2	yes	—
80	C₄²×C₇	C₄⋊C₄	1	2	yes	—
81	C₄²×C₇	C₄⋊C₄	1	2	yes	—
82	C₄²×C₇	C₄⋊C₄	1	2	yes	—
83	C₂×#117	—	2	2	no	—
84	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
85	#431	—	2	2	no	—
86	#431	—	2	2	no	—
87	C₂×C₄⋊C₈	C₄×C₇	1	2	yes	complemented
88	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
89	C₂×C₄×C₈	C₄×C₇	1	2	yes	complemented
90	#384	—	2	2	no	—
91	#382	C₄×C₇	1	4	yes	complemented
92	#380	—	2	2	no	—
93	#380	C₂²×C₇	1	4	yes	—
94	C₄×C₄⋊C₄	—	2	2	no	—
95	C₄×C₄⋊C₄	C₄×C₇	1	4	yes	complemented
96	C₂³×C₇	—	4	1	no	—
97	C₂³×C₇	—	4	1	no	—
98	C₂³×C₇	C₂×C₂²⋊C₄	1	2	yes	—
99	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
101	C₂³×C₇	C₂×SD₁₆	1	2	yes	—
102	C₂³×C₇	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄×C₇	—	2	4	no	—
104	C₂×C₄×C₇	—	2	2	no	—
105	C₂×C₄×C₇	—	2	4	no	—
106	C₂×C₄×C₇	—	2	2	no	—
107	C₂×C₄×C₇	—	4	2	no	—
108	C₂×C₄×C₇	—	4	1	no	—
109	C₂×C₄×C₇	—	4	2	no	—
110	C₂×C₄×C₇	—	4	1	no	—
111	C₂×C₄×C₇	—	4	8	no	—
112	C₂×C₄×C₇	—	4	8	no	—
113	C₂×C₄×C₇	—	4	8	no	—
114	C₂×C₄×C₇	#95	1	2	yes	—
115	C₂×C₄×C₇	#95	1	1	yes	characteristic
116	C₂×C₄×C₇	#95	1	1	yes	characteristic
117	C₂×C₄×C₇	#95	1	1	yes	characteristic
118	C₂×C₄×C₇	#95	1	2	yes	—
119	C₂×C₄×C₇	#95	1	1	yes	characteristic
120	C₂×C₄×C₇	#102	1	4	yes	—
121	C₂×C₄×C₇	#102	1	4	yes	—
122	C₂×C₄×C₇	#103	1	4	yes	—
123	C₂×C₄×C₇	#103	1	4	yes	—
124	C₂×C₄×C₇	#106	1	2	yes	—
125	C₂×C₄×C₇	#106	1	2	yes	—
126	C₂×C₄×C₇	#107	1	2	yes	—
127	C₂×C₄×C₇	#107	1	2	yes	—
128	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
129	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
131	C₇×C₈	—	4	4	no	—
132	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
133	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
134	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
135	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
136	#117	—	4	4	no	—
137	#117	—	4	4	no	—
138	C₂×C₄⋊C₄	—	2	4	no	—
139	C₂×C₄⋊C₄	—	2	4	no	—
140	C₂×C₄⋊C₄	—	2	4	no	—
141	C₂×C₂²⋊C₄	—	2	4	no	—
142	C₂×C₄²	—	2	4	no	—
143	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
144	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
145	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
146	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
147	C₂×C₄²	C₇×D₄	1	2	yes	—
148	C₂×C₄²	C₇×D₄	1	2	yes	—
149	C₂×C₄²	C₇×Q₈	1	1	yes	characteristic
150	C₂×C₄²	C₇×Q₈	1	1	yes	characteristic
151	C₄⋊C₈	—	2	4	no	—
152	C₂²⋊C₈	—	2	4	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	—	2	2	no	—
155	#95	—	2	4	no	—
156	#95	C₂×C₄×C₇	1	8	yes	—
157	C₂²×C₇	—	2	2	no	—
158	C₂²×C₇	—	2	1	no	—
159	C₂²×C₇	—	2	2	no	—
160	C₂²×C₇	—	2	1	no	—
161	C₂²×C₇	—	4	1	no	—
162	C₂²×C₇	—	4	2	no	—
163	C₂²×C₇	—	4	1	no	—
164	C₂²×C₇	C₂×#95	1	1	yes	characteristic
165	C₂²×C₇	C₂×#102	1	2	yes	—
166	C₂²×C₇	C₂×#103	1	2	yes	—
167	C₂²×C₇	C₂×#106	1	1	yes	characteristic
168	C₂²×C₇	C₂×#107	1	1	yes	characteristic
169	C₄×C₇	—	2	2	no	—
170	C₄×C₇	—	2	1	no	—
171	C₄×C₇	—	2	1	no	—
172	C₄×C₇	—	2	2	no	—
173	C₄×C₇	—	8	8	no	—
174	C₄×C₇	H(3,ℤ/4)	1	2	yes	—
175	C₄×C₇	#340	1	4	yes	—
176	C₄×C₇	#340	1	2	yes	—
177	C₄×C₇	#340	1	2	yes	—
178	C₄×C₇	#410	1	2	yes	—
179	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
180	C₂²×C₄	—	2	8	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
184	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
185	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
186	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
187	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
188	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
189	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
190	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	C₇×D₈	1	2	yes	—
192	C₂²×C₄	C₇×SD₁₆	1	4	yes	—
193	C₂²×C₄	C₇×Q₁₆	1	2	yes	—
194	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
195	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
196	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
197	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	4	4	no	—
200	C₄⋊C₄	—	4	8	no	—
201	C₄⋊C₄	—	4	8	no	—
202	C₄⋊C₄	—	4	8	no	—
203	C₂²⋊C₄	—	4	4	no	—
204	C₂²⋊C₄	—	4	4	no	—
205	C₄²	—	4	8	no	—
206	C₄²	C₄²×C₇	1	1	yes	characteristic
207	C₄²	C₄²×C₇	1	2	yes	—
208	C₄²	C₄²×C₇	1	1	yes	characteristic
209	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
210	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
211	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
212	C₄²	C₇×C₄⋊C₄	1	2	yes	—
213	C₄²	C₇×C₄⋊C₄	1	2	yes	—
214	C₄²	C₇×C₄⋊C₄	1	2	yes	—
215	C₄²	C₇×C₄⋊C₄	1	2	yes	—
216	C₂×C₇	—	2	2	no	—
217	C₂×C₇	—	2	2	no	—
218	C₂×C₇	—	2	1	no	—
219	C₂×C₇	—	2	1	no	—
220	C₂×C₇	C₂×H(3,ℤ/4)	1	1	yes	characteristic
221	C₂×C₇	C₂×#340	1	1	yes	characteristic
222	C₂×C₇	#1740	1	1	yes	characteristic
223	C₂³	—	4	1	no	—
224	C₂³	—	4	1	no	—
225	C₂³	C₂×C₇×C₂²⋊C₄	1	2	yes	—
226	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
227	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
228	C₂³	C₂×C₇×SD₁₆	1	2	yes	—
229	C₂³	C₂×C₇×Q₁₆	1	1	yes	characteristic
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	—	4	8	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	C₇×#95	1	1	yes	characteristic
242	C₂×C₄	C₇×#95	1	2	yes	—
243	C₂×C₄	C₇×#95	1	1	yes	characteristic
244	C₂×C₄	C₇×#95	1	1	yes	characteristic
245	C₂×C₄	C₇×#95	1	2	yes	—
246	C₂×C₄	C₇×#95	1	1	yes	characteristic
247	C₂×C₄	C₇×#102	1	4	yes	—
248	C₂×C₄	C₇×#102	1	4	yes	—
249	C₂×C₄	C₇×#103	1	4	yes	—
250	C₂×C₄	C₇×#103	1	4	yes	—
251	C₂×C₄	C₇×#106	1	2	yes	—
252	C₂×C₄	C₇×#106	1	2	yes	—
253	C₂×C₄	C₇×#107	1	2	yes	—
254	C₂×C₄	C₇×#107	1	2	yes	—
255	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
256	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
258	C₈	—	4	4	no	—
259	C₇	#7981	1	1	yes	characteristic, direct product factor, Sylow
260	C₂²	—	2	2	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	C₂×C₇×#95	1	1	yes	characteristic
268	C₂²	C₂×C₇×#102	1	2	yes	—
269	C₂²	C₂×C₇×#103	1	2	yes	—
270	C₂²	C₂×C₇×#106	1	1	yes	characteristic
271	C₂²	C₂×C₇×#107	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	2	no	—
276	C₄	—	8	8	no	—
277	C₄	C₇×H(3,ℤ/4)	1	2	yes	—
278	C₄	C₇×#340	1	4	yes	—
279	C₄	C₇×#340	1	2	yes	—
280	C₄	C₇×#340	1	2	yes	—
281	C₄	C₇×#410	1	2	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	2	2	no	—
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	C₂×C₇×H(3,ℤ/4)	1	1	yes	characteristic
287	C₂	C₂×C₇×#340	1	1	yes	characteristic
288	C₂	C₇×#1740	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	#95
commutator	C₂×C₄	C₂×C₄²×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
socle	C₂²×C₇	C₂×#95
prime radical	C₂⁴×C₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ 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
8. C₂×C₇×C₄⋊C₈	C₄	semidirect
10. C₂×C₄×C₇×C₈	C₄	semidirect
12. C₇×#382	C₄	semidirect
16. C₄×C₇×C₄⋊C₄	C₄	semidirect
17. #7981	C₇	direct (N × H)
87. C₂×C₄⋊C₈	C₄×C₇	semidirect
89. C₂×C₄×C₈	C₄×C₇	semidirect
91. #382	C₄×C₇	semidirect
95. C₄×C₄⋊C₄	C₄×C₇	semidirect
259. C₇	#7981	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
C₇×#2285	2
C₇×#1289	2
#7981	7

Subgroups — C7×#7981 #13185197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#7981 #13185197