Subgroups — C₇×#8039 #13185255

Order 1792.

706 conjugacy classes of subgroups.

The subgroup lattice has 328 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	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	—	2	1	no	—
7	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
8	C₇×#431	—	2	2	no	—
9	C₇×#431	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×C₄×C₇×C₈	—	2	1	no	—
17	C₇×#384	C₂²	1	2	yes	—
18	C₇×#382	—	2	1	no	—
19	C₇×#380	—	2	1	no	—
20	C₇×#380	C₄	1	2	yes	—
21	C₄×C₇×C₄⋊C₄	—	2	1	no	—
22	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
23	#8039	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	2	no	—
31	C₂²×C₇×C₈	—	4	2	no	—
32	C₇×#117	—	4	2	no	—
33	C₇×#117	—	4	2	no	—
34	C₂×C₇×C₄⋊C₄	—	2	2	no	—
35	C₂×C₇×C₄⋊C₄	—	2	2	no	—
36	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
38	C₂×C₄²×C₇	—	2	2	no	—
39	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₇	D₄	1	2	yes	—
45	C₂×C₄²×C₇	Q₈	1	2	yes	—
46	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
47	C₇×C₄⋊C₈	—	2	4	no	—
48	C₇×C₄⋊C₈	—	2	2	no	—
49	C₇×C₄⋊C₈	—	2	2	no	—
50	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
51	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
52	C₇×C₂²⋊C₈	—	2	4	no	—
53	C₄×C₇×C₈	—	2	4	no	—
54	C₇×#95	—	2	2	no	—
55	C₇×#95	—	2	2	no	—
56	C₂×#431	C₂×C₇	1	1	yes	characteristic
57	C₂²×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
58	#2285	C₂×C₇	1	1	yes	characteristic
59	#1296	C₂×C₇	1	4	yes	complemented
60	C₂⁴×C₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇	—	2	4	no	—
62	C₂²×C₄×C₇	—	2	4	no	—
63	C₂²×C₄×C₇	—	4	2	no	—
64	C₂²×C₄×C₇	C₄²	1	2	yes	—
65	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇	C₄⋊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	D₈	1	2	yes	—
74	C₂²×C₄×C₇	SD₁₆	1	2	yes	—
75	C₂²×C₄×C₇	SD₁₆	1	2	yes	—
76	C₂²×C₄×C₇	Q₁₆	1	2	yes	—
77	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₇×C₈	—	2	4	no	—
81	C₂×C₇×C₈	—	2	4	no	—
82	C₂×C₇×C₈	—	4	8	no	—
83	C₂×C₇×C₈	—	4	2	no	—
84	C₂×C₇×C₈	—	4	2	no	—
85	C₂×C₇×C₈	—	4	4	no	—
86	C₇×C₄⋊C₄	—	4	4	no	—
87	C₇×C₄⋊C₄	—	4	2	no	—
88	C₇×C₄⋊C₄	—	4	2	no	—
89	C₇×C₄⋊C₄	—	4	4	no	—
90	C₇×C₂²⋊C₄	—	4	4	no	—
91	C₄²×C₇	—	4	4	no	—
92	C₄²×C₇	C₄²	1	2	yes	—
93	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
94	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
95	C₄²×C₇	C₄⋊C₄	1	2	yes	—
96	C₄²×C₇	C₄⋊C₄	1	2	yes	—
97	C₄²×C₇	C₄⋊C₄	1	2	yes	—
98	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
102	C₂³×C₈	—	2	1	no	—
103	C₂×#117	—	2	1	no	—
104	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
105	#431	—	2	2	no	—
106	#431	C₄×C₇	1	4	yes	complemented
107	C₂×C₄⋊C₈	—	2	1	no	—
108	C₂×C₄⋊C₈	—	2	2	no	—
109	C₂×C₄⋊C₈	C₄×C₇	1	4	yes	complemented
110	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
111	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
112	C₂×C₂²⋊C₈	—	2	1	no	—
113	C₂×C₄×C₈	—	2	1	no	—
114	#384	C₂²×C₇	1	2	yes	—
115	#382	—	2	1	no	—
116	#380	—	2	1	no	—
117	#380	C₄×C₇	1	2	yes	—
118	C₄×C₄⋊C₄	—	2	1	no	—
119	C₄×C₄⋊C₄	C₄×C₇	1	2	yes	—
120	C₂³×C₇	—	2	2	no	—
121	C₂³×C₇	—	2	2	no	—
122	C₂³×C₇	#95	1	2	yes	—
123	C₂³×C₇	#102	1	2	yes	—
124	C₂³×C₇	#103	1	2	yes	—
125	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₇	—	2	2	no	—
127	C₂×C₄×C₇	—	2	2	no	—
128	C₂×C₄×C₇	—	2	2	no	—
129	C₂×C₄×C₇	—	2	4	no	—
130	C₂×C₄×C₇	—	2	4	no	—
131	C₂×C₄×C₇	—	4	2	no	—
132	C₂×C₄×C₇	—	4	2	no	—
133	C₂×C₄×C₇	—	4	4	no	—
134	C₂×C₄×C₇	—	4	4	no	—
135	C₂×C₄×C₇	—	4	4	no	—
136	C₂×C₄×C₇	#95	1	2	yes	—
137	C₂×C₄×C₇	#95	1	2	yes	—
138	C₂×C₄×C₇	#95	1	2	yes	—
139	C₂×C₄×C₇	#102	1	2	yes	—
140	C₂×C₄×C₇	#102	1	4	yes	—
141	C₂×C₄×C₇	#103	1	2	yes	—
142	C₂×C₄×C₇	#103	1	4	yes	—
143	C₂×C₄×C₇	#106	1	4	yes	—
144	C₂×C₄×C₇	#107	1	4	yes	—
145	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
146	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
153	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
154	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
155	C₂×C₄×C₇	#135	1	4	yes	—
156	C₇×C₈	—	4	4	no	—
157	C₇×C₈	—	4	4	no	—
158	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
159	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
160	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
161	C₂²×C₈	—	2	4	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	4	2	no	—
165	C₂²×C₈	—	4	2	no	—
166	#117	—	4	2	no	—
167	#117	—	4	2	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
171	C₂×C₂²⋊C₄	—	2	2	no	—
172	C₂×C₄²	—	2	2	no	—
173	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
174	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
175	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
176	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
177	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
178	C₂×C₄²	C₇×D₄	1	2	yes	—
179	C₂×C₄²	C₇×Q₈	1	2	yes	—
180	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
181	C₄⋊C₈	—	2	4	no	—
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	—	2	2	no	—
184	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	—
185	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	—
186	C₂²⋊C₈	—	2	4	no	—
187	C₄×C₈	—	2	4	no	—
188	#95	—	2	2	no	—
189	#95	—	2	2	no	—
190	C₂²×C₇	—	2	4	no	—
191	C₂²×C₇	—	2	4	no	—
192	C₂²×C₇	—	2	2	no	—
193	C₂²×C₇	—	2	2	no	—
194	C₂²×C₇	#340	1	4	yes	—
195	C₂²×C₇	C₂×#95	1	1	yes	characteristic
196	C₂²×C₇	#413	1	2	yes	—
197	C₂²×C₇	C₂×#102	1	1	yes	characteristic
198	C₂²×C₇	C₂×#103	1	1	yes	characteristic
199	C₂²×C₇	#416	1	2	yes	—
200	C₄×C₇	—	2	2	no	—
201	C₄×C₇	—	2	2	no	—
202	C₄×C₇	—	2	1	no	—
203	C₄×C₇	—	2	1	no	—
204	C₄×C₇	—	8	4	no	—
205	C₄×C₇	#340	1	4	yes	—
206	C₄×C₇	C₂×#102	1	1	yes	characteristic
207	C₄×C₇	C₂×#103	1	1	yes	characteristic
208	C₄×C₇	#416	1	2	yes	—
209	C₄×C₇	C₂×#106	1	1	yes	characteristic
210	C₄×C₇	C₂×#107	1	1	yes	characteristic
211	C₄×C₇	#427	1	2	yes	—
212	C₂⁴	C₇×C₂²⋊C₄	1	1	yes	characteristic
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	C₄²×C₇	1	2	yes	—
217	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
218	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
219	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
220	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
221	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
222	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
223	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
224	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
225	C₂²×C₄	C₇×D₈	1	2	yes	—
226	C₂²×C₄	C₇×SD₁₆	1	2	yes	—
227	C₂²×C₄	C₇×SD₁₆	1	2	yes	—
228	C₂²×C₄	C₇×Q₁₆	1	2	yes	—
229	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	4	8	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	4	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	4	no	—
242	C₂²⋊C₄	—	4	4	no	—
243	C₄²	—	4	4	no	—
244	C₄²	C₄²×C₇	1	2	yes	—
245	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
246	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
247	C₄²	C₇×C₄⋊C₄	1	2	yes	—
248	C₄²	C₇×C₄⋊C₄	1	2	yes	—
249	C₄²	C₇×C₄⋊C₄	1	2	yes	—
250	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
251	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
252	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
253	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
254	C₂×C₇	—	2	4	no	—
255	C₂×C₇	#1540	1	2	yes	—
256	C₂×C₇	#1542	1	2	yes	—
257	C₂×C₇	C₂×#340	1	1	yes	characteristic
258	C₂×C₇	#1735	1	1	yes	characteristic
259	C₂×C₇	#1742	1	1	yes	characteristic
260	C₂³	—	2	2	no	—
261	C₂³	—	2	2	no	—
262	C₂³	C₇×#95	1	2	yes	—
263	C₂³	C₇×#102	1	2	yes	—
264	C₂³	C₇×#103	1	2	yes	—
265	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	C₇×#95	1	2	yes	—
277	C₂×C₄	C₇×#95	1	2	yes	—
278	C₂×C₄	C₇×#95	1	2	yes	—
279	C₂×C₄	C₇×#102	1	2	yes	—
280	C₂×C₄	C₇×#102	1	4	yes	—
281	C₂×C₄	C₇×#103	1	2	yes	—
282	C₂×C₄	C₇×#103	1	4	yes	—
283	C₂×C₄	C₇×#106	1	4	yes	—
284	C₂×C₄	C₇×#107	1	4	yes	—
285	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
286	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
287	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
288	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
289	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
290	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
291	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
292	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
293	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
294	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
295	C₂×C₄	C₇×#135	1	4	yes	—
296	C₈	—	4	4	no	—
297	C₈	—	4	4	no	—
298	C₇	#8039	1	1	yes	characteristic, direct product factor, Sylow
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	4	no	—
303	C₂²	C₇×#340	1	4	yes	—
304	C₂²	C₂×C₇×#95	1	1	yes	characteristic
305	C₂²	C₇×#413	1	2	yes	—
306	C₂²	C₂×C₇×#102	1	1	yes	characteristic
307	C₂²	C₂×C₇×#103	1	1	yes	characteristic
308	C₂²	C₇×#416	1	2	yes	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	8	4	no	—
314	C₄	C₇×#340	1	4	yes	—
315	C₄	C₂×C₇×#102	1	1	yes	characteristic
316	C₄	C₂×C₇×#103	1	1	yes	characteristic
317	C₄	C₇×#416	1	2	yes	—
318	C₄	C₂×C₇×#106	1	1	yes	characteristic
319	C₄	C₂×C₇×#107	1	1	yes	characteristic
320	C₄	C₇×#427	1	2	yes	—
321	C₂	—	2	4	no	—
322	C₂	C₇×#1540	1	2	yes	—
323	C₂	C₇×#1542	1	2	yes	—
324	C₂	C₂×C₇×#340	1	1	yes	characteristic
325	C₂	C₇×#1735	1	1	yes	characteristic
326	C₂	C₇×#1742	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂³×C₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×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×#431	C₄	semidirect
12. C₂×C₇×C₄⋊C₈	C₄	semidirect
23. #8039	C₇	direct (N × H)
59. #1296	C₂×C₇	semidirect
106. #431	C₄×C₇	semidirect
109. C₂×C₄⋊C₈	C₄×C₇	semidirect
298. C₇	#8039	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
C₂²×C₇×C₄⋊C₈	2
C₇×#2285	2
C₇×#1296	2
#8039	7

Subgroups — C7×#8039 #13185255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#8039 #13185255