Subgroups — C₇×#8044 #13185260

Order 1792.

686 conjugacy classes of subgroups.

The subgroup lattice has 314 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	1	yes	maximal, characteristic
2	C₂²×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#2285	C₂	1	1	yes	maximal, characteristic
4	C₇×#1296	C₂	1	4	yes	maximal, complemented
5	C₂³×C₇×C₈	—	2	1	no	—
6	C₂×C₇×#117	C₄	1	2	yes	—
7	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
8	C₇×#432	—	2	2	no	—
9	C₇×#432	C₄	1	4	yes	complemented
10	C₂×C₇×C₄⋊C₈	—	2	1	no	—
11	C₂×C₇×C₄⋊C₈	—	2	2	no	—
12	C₂×C₇×C₄⋊C₈	C₄	1	4	yes	complemented
13	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₇×C₂²⋊C₈	—	2	1	no	—
16	C₂×C₇×#97	—	2	1	no	—
17	C₇×#384	C₂²	1	2	yes	—
18	C₇×#382	C₄	1	2	yes	—
19	C₇×#380	—	2	1	no	—
20	C₇×#380	—	2	1	no	—
21	C₄×C₇×C₄⋊C₄	—	2	1	no	—
22	C₄×C₇×C₄⋊C₄	—	2	1	no	—
23	#8044	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	—	2	4	no	—
28	C₂²×C₇×C₈	—	2	2	no	—
29	C₂²×C₇×C₈	—	2	2	no	—
30	C₂²×C₇×C₈	—	4	4	no	—
31	C₇×#117	—	4	4	no	—
32	C₂×C₇×C₄⋊C₄	—	2	2	no	—
33	C₂×C₇×C₄⋊C₄	—	2	2	no	—
34	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
35	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
36	C₂×C₄²×C₇	—	2	2	no	—
37	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₇	D₄	1	2	yes	—
43	C₂×C₄²×C₇	Q₈	1	2	yes	—
44	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
45	C₇×C₄⋊C₈	—	2	4	no	—
46	C₇×C₄⋊C₈	—	2	2	no	—
47	C₇×C₄⋊C₈	—	2	2	no	—
48	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
49	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
50	C₇×C₂²⋊C₈	—	2	4	no	—
51	C₇×#97	—	2	4	no	—
52	C₇×#95	—	2	2	no	—
53	C₇×#95	—	2	2	no	—
54	C₂×#432	C₂×C₇	1	1	yes	characteristic
55	C₂²×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
56	#2285	C₂×C₇	1	1	yes	characteristic
57	#1296	C₂×C₇	1	4	yes	complemented
58	C₂⁴×C₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇	—	2	2	no	—
60	C₂²×C₄×C₇	—	2	2	no	—
61	C₂²×C₄×C₇	—	2	4	no	—
62	C₂²×C₄×C₇	—	2	2	no	—
63	C₂²×C₄×C₇	—	2	2	no	—
64	C₂²×C₄×C₇	—	2	4	no	—
65	C₂²×C₄×C₇	C₄²	1	2	yes	—
66	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₇×C₈	—	2	4	no	—
78	C₂×C₇×C₈	—	2	4	no	—
79	C₂×C₇×C₈	—	4	8	no	—
80	C₂×C₇×C₈	—	4	4	no	—
81	C₂×C₇×C₈	—	4	4	no	—
82	C₇×C₄⋊C₄	—	4	4	no	—
83	C₇×C₄⋊C₄	—	4	4	no	—
84	C₇×C₄⋊C₄	—	4	2	no	—
85	C₇×C₄⋊C₄	—	4	2	no	—
86	C₇×C₂²⋊C₄	—	4	4	no	—
87	C₄²×C₇	—	4	4	no	—
88	C₄²×C₇	C₄²	1	2	yes	—
89	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
90	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
91	C₄²×C₇	C₄⋊C₄	1	2	yes	—
92	C₄²×C₇	C₄⋊C₄	1	2	yes	—
93	C₄²×C₇	C₄⋊C₄	1	2	yes	—
94	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
98	C₂³×C₈	—	2	1	no	—
99	C₂×#117	C₄×C₇	1	2	yes	—
100	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
101	#432	—	2	2	no	—
102	#432	C₄×C₇	1	4	yes	complemented
103	C₂×C₄⋊C₈	—	2	1	no	—
104	C₂×C₄⋊C₈	—	2	2	no	—
105	C₂×C₄⋊C₈	C₄×C₇	1	4	yes	complemented
106	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
107	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
108	C₂×C₂²⋊C₈	—	2	1	no	—
109	C₂×#97	—	2	1	no	—
110	#384	C₂²×C₇	1	2	yes	—
111	#382	C₄×C₇	1	2	yes	—
112	#380	—	2	1	no	—
113	#380	—	2	1	no	—
114	C₄×C₄⋊C₄	—	2	1	no	—
115	C₄×C₄⋊C₄	—	2	1	no	—
116	C₂³×C₇	—	2	2	no	—
117	C₂³×C₇	—	2	2	no	—
118	C₂³×C₇	#95	1	2	yes	—
119	C₂³×C₇	#100	1	2	yes	—
120	C₂³×C₇	#101	1	2	yes	—
121	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₇	—	2	4	no	—
123	C₂×C₄×C₇	—	2	4	no	—
124	C₂×C₄×C₇	—	2	1	no	—
125	C₂×C₄×C₇	—	2	1	no	—
126	C₂×C₄×C₇	—	2	1	no	—
127	C₂×C₄×C₇	—	2	1	no	—
128	C₂×C₄×C₇	—	2	2	no	—
129	C₂×C₄×C₇	—	2	2	no	—
130	C₂×C₄×C₇	—	2	4	no	—
131	C₂×C₄×C₇	—	2	2	no	—
132	C₂×C₄×C₇	—	2	2	no	—
133	C₂×C₄×C₇	—	2	4	no	—
134	C₂×C₄×C₇	—	2	4	no	—
135	C₂×C₄×C₇	—	4	4	no	—
136	C₂×C₄×C₇	—	4	4	no	—
137	C₂×C₄×C₇	—	4	4	no	—
138	C₂×C₄×C₇	#95	1	2	yes	—
139	C₂×C₄×C₇	#95	1	2	yes	—
140	C₂×C₄×C₇	#95	1	2	yes	—
141	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
142	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₇	#135	1	4	yes	—
148	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
149	C₂×C₄×C₇	#137	1	2	yes	—
150	C₇×C₈	—	4	4	no	—
151	C₇×C₈	—	4	4	no	—
152	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
153	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
154	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
155	C₂²×C₈	—	2	4	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	4	4	no	—
159	#117	—	4	4	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
163	C₂×C₂²⋊C₄	—	2	2	no	—
164	C₂×C₄²	—	2	2	no	—
165	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
166	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
167	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
168	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
169	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
170	C₂×C₄²	C₇×D₄	1	2	yes	—
171	C₂×C₄²	C₇×Q₈	1	2	yes	—
172	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
173	C₄⋊C₈	—	2	4	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	—
177	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	—
178	C₂²⋊C₈	—	2	4	no	—
179	#97	—	2	4	no	—
180	#95	—	2	2	no	—
181	#95	—	2	2	no	—
182	C₂²×C₇	—	2	4	no	—
183	C₂²×C₇	—	2	4	no	—
184	C₂²×C₇	—	2	2	no	—
185	C₂²×C₇	—	2	2	no	—
186	C₂²×C₇	#343	1	4	yes	—
187	C₂²×C₇	C₂×#95	1	1	yes	characteristic
188	C₂²×C₇	C₂×#100	1	1	yes	characteristic
189	C₂²×C₇	C₂×#101	1	1	yes	characteristic
190	C₂²×C₇	#416	1	2	yes	—
191	C₂²×C₇	#417	1	2	yes	—
192	C₄×C₇	—	2	2	no	—
193	C₄×C₇	—	2	2	no	—
194	C₄×C₇	—	2	2	no	—
195	C₄×C₇	—	2	1	no	—
196	C₄×C₇	—	2	1	no	—
197	C₄×C₇	—	8	4	no	—
198	C₄×C₇	#416	1	2	yes	—
199	C₄×C₇	#418	1	1	yes	characteristic
200	C₄×C₇	#419	1	1	yes	characteristic
201	C₄×C₇	#427	1	2	yes	—
202	C₄×C₇	#428	1	2	yes	—
203	C₂⁴	C₇×C₂²⋊C₄	1	1	yes	characteristic
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	C₄²×C₇	1	2	yes	—
211	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
213	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
214	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
215	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
216	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
217	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
218	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
219	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	4	8	no	—
225	C₂×C₈	—	4	4	no	—
226	C₂×C₈	—	4	4	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	4	4	no	—
232	C₄²	—	4	4	no	—
233	C₄²	C₄²×C₇	1	2	yes	—
234	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
235	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
236	C₄²	C₇×C₄⋊C₄	1	2	yes	—
237	C₄²	C₇×C₄⋊C₄	1	2	yes	—
238	C₄²	C₇×C₄⋊C₄	1	2	yes	—
239	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
240	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
241	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
242	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
243	C₂×C₇	—	2	4	no	—
244	C₂×C₇	#1551	1	2	yes	—
245	C₂×C₇	#1552	1	2	yes	—
246	C₂×C₇	#1735	1	1	yes	characteristic
247	C₂×C₇	#1736	1	1	yes	characteristic
248	C₂×C₇	C₂×#343	1	1	yes	characteristic
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	C₇×#95	1	2	yes	—
252	C₂³	C₇×#100	1	2	yes	—
253	C₂³	C₇×#101	1	2	yes	—
254	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	C₇×#95	1	2	yes	—
272	C₂×C₄	C₇×#95	1	2	yes	—
273	C₂×C₄	C₇×#95	1	2	yes	—
274	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
275	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
277	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
278	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
280	C₂×C₄	C₇×#135	1	4	yes	—
281	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
282	C₂×C₄	C₇×#137	1	2	yes	—
283	C₈	—	4	4	no	—
284	C₈	—	4	4	no	—
285	C₇	#8044	1	1	yes	characteristic, direct product factor, Sylow
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	4	no	—
290	C₂²	C₇×#343	1	4	yes	—
291	C₂²	C₂×C₇×#95	1	1	yes	characteristic
292	C₂²	C₂×C₇×#100	1	1	yes	characteristic
293	C₂²	C₂×C₇×#101	1	1	yes	characteristic
294	C₂²	C₇×#416	1	2	yes	—
295	C₂²	C₇×#417	1	2	yes	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	8	4	no	—
302	C₄	C₇×#416	1	2	yes	—
303	C₄	C₇×#418	1	1	yes	characteristic
304	C₄	C₇×#419	1	1	yes	characteristic
305	C₄	C₇×#427	1	2	yes	—
306	C₄	C₇×#428	1	2	yes	—
307	C₂	—	2	4	no	—
308	C₂	C₇×#1551	1	2	yes	—
309	C₂	C₇×#1552	1	2	yes	—
310	C₂	C₇×#1735	1	1	yes	characteristic
311	C₂	C₇×#1736	1	1	yes	characteristic
312	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
4. C₇×#1296	C₂	semidirect
9. C₇×#432	C₄	semidirect
12. C₂×C₇×C₄⋊C₈	C₄	semidirect
23. #8044	C₇	direct (N × H)
57. #1296	C₂×C₇	semidirect
102. #432	C₄×C₇	semidirect
105. C₂×C₄⋊C₈	C₄×C₇	semidirect
285. C₇	#8044	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
C₂²×C₇×C₄⋊C₈	2
C₇×#2285	2
C₇×#1296	2
#8044	7

Subgroups — C7×#8044 #13185260

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#8044 #13185260