Subgroups — C₇×#12783 #13189999

Order 1792.

566 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	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂×C₇×C₂²⋊C₁₆	C₂	1	2	yes	maximal
4	C₇×#1374	C₂	1	2	yes	maximal, complemented
5	C₂³×C₇×C₈	C₄	1	1	yes	characteristic
6	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₇	C₄	1	1	yes	characteristic
8	C₂²×C₇×C₁₆	C₂²	1	4	yes	—
9	C₂×C₄×C₇×C₈	—	2	4	no	—
10	C₂×C₄×C₇×C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₇×C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₇×C₄⋊C₁₆	—	2	4	no	—
15	C₇×C₂²⋊C₁₆	—	2	4	no	—
16	#12783	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₄×C₇	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₇	C₈	1	1	yes	characteristic
19	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	—	2	2	no	—
21	C₂²×C₇×C₈	—	2	2	no	—
22	C₂²×C₇×C₈	—	2	2	no	—
23	C₂²×C₇×C₈	—	2	2	no	—
24	C₂²×C₇×C₈	—	2	2	no	—
25	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇×C₈	D₄	1	2	yes	—
29	C₂²×C₇×C₈	Q₈	1	2	yes	—
30	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₇	—	2	2	no	—
32	C₂×C₄²×C₇	—	2	2	no	—
33	C₂×C₄²×C₇	C₈	1	1	yes	characteristic
34	C₂×C₄²×C₇	C₈	1	1	yes	characteristic
35	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₇×C₁₆	—	2	4	no	—
38	C₂×C₇×C₁₆	—	2	4	no	—
39	C₂×C₇×C₁₆	—	2	4	no	—
40	C₄×C₇×C₈	—	2	2	no	—
41	C₄×C₇×C₈	—	2	2	no	—
42	C₄×C₇×C₈	—	2	2	no	—
43	C₄×C₇×C₈	D₄	1	4	yes	—
44	C₂²×C₄×C₈	C₂×C₇	1	1	yes	characteristic
45	C₂×C₄⋊C₁₆	C₂×C₇	1	2	yes	complemented
46	C₂×C₂²⋊C₁₆	C₂×C₇	1	2	yes	—
47	#1374	C₂×C₇	1	2	yes	complemented
48	C₂⁴×C₇	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₇	—	2	2	no	—
50	C₂²×C₄×C₇	—	2	2	no	—
51	C₂²×C₄×C₇	—	2	2	no	—
52	C₂²×C₄×C₇	—	2	2	no	—
53	C₂²×C₄×C₇	—	2	2	no	—
54	C₂²×C₄×C₇	—	2	1	no	—
55	C₂²×C₄×C₇	—	2	1	no	—
56	C₂²×C₄×C₇	—	2	1	no	—
57	C₂²×C₄×C₇	—	2	1	no	—
58	C₂²×C₄×C₇	C₁₆	1	4	yes	complemented
59	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₇	M₁₆	1	2	yes	—
66	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₇×C₈	—	2	8	no	—
68	C₂×C₇×C₈	—	2	1	no	—
69	C₂×C₇×C₈	—	2	1	no	—
70	C₂×C₇×C₈	—	2	1	no	—
71	C₂×C₇×C₈	—	2	1	no	—
72	C₂×C₇×C₈	—	2	1	no	—
73	C₂×C₇×C₈	—	2	1	no	—
74	C₂×C₇×C₈	—	2	4	no	—
75	C₂×C₇×C₈	—	2	4	no	—
76	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
77	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
78	C₂×C₇×C₈	M₁₆	1	2	yes	—
79	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
81	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₇×C₈	#41	1	2	yes	—
83	C₄²×C₇	—	2	2	no	—
84	C₄²×C₇	—	2	2	no	—
85	C₄²×C₇	—	2	1	no	—
86	C₄²×C₇	—	2	1	no	—
87	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₇	M₁₆	1	2	yes	—
89	C₇×C₁₆	—	4	4	no	—
90	C₂³×C₈	C₄×C₇	1	1	yes	characteristic
91	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
92	C₂²×C₄²	C₄×C₇	1	1	yes	characteristic
93	C₂²×C₁₆	C₂²×C₇	1	4	yes	—
94	C₂×C₄×C₈	—	2	4	no	—
95	C₂×C₄×C₈	C₄×C₇	1	1	yes	characteristic
96	C₂×C₄×C₈	C₄×C₇	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
98	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
99	C₄⋊C₁₆	—	2	4	no	—
100	C₂²⋊C₁₆	—	2	4	no	—
101	C₂³×C₇	—	2	1	no	—
102	C₂³×C₇	—	2	1	no	—
103	C₂³×C₇	—	2	1	no	—
104	C₂³×C₇	—	2	1	no	—
105	C₂³×C₇	C₄⋊C₈	1	2	yes	—
106	C₂³×C₇	C₂×C₁₆	1	2	yes	—
107	C₂³×C₇	M₃₂	1	2	yes	—
108	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₇	—	2	8	no	—
110	C₂×C₄×C₇	—	2	1	no	—
111	C₂×C₄×C₇	—	2	1	no	—
112	C₂×C₄×C₇	—	2	1	no	—
113	C₂×C₄×C₇	—	2	1	no	—
114	C₂×C₄×C₇	—	2	1	no	—
115	C₂×C₄×C₇	—	2	1	no	—
116	C₂×C₄×C₇	—	2	1	no	—
117	C₂×C₄×C₇	—	2	1	no	—
118	C₂×C₄×C₇	—	2	8	no	—
119	C₂×C₄×C₇	—	2	2	no	—
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	—	2	2	no	—
122	C₂×C₄×C₇	—	2	2	no	—
123	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
124	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
125	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
126	C₂×C₄×C₇	C₂×C₁₆	1	2	yes	—
127	C₂×C₄×C₇	C₂×C₁₆	1	2	yes	—
128	C₂×C₄×C₇	M₃₂	1	2	yes	—
129	C₂×C₄×C₇	M₃₂	1	2	yes	—
130	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₇	#117	1	1	yes	characteristic
133	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
136	C₇×C₈	—	2	2	no	—
137	C₇×C₈	—	2	2	no	—
138	C₇×C₈	—	2	2	no	—
139	C₇×C₈	#121	1	2	yes	—
140	C₇×C₈	C₂²⋊Q₈	1	2	yes	—
141	C₂³×C₄	C₇×C₈	1	1	yes	characteristic
142	C₂³×C₄	C₇×C₈	1	1	yes	characteristic
143	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
150	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
151	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
152	C₂²×C₈	C₇×D₄	1	2	yes	—
153	C₂²×C₈	C₇×Q₈	1	2	yes	—
154	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
155	C₂×C₄²	—	2	2	no	—
156	C₂×C₄²	—	2	2	no	—
157	C₂×C₄²	C₇×C₈	1	1	yes	characteristic
158	C₂×C₄²	C₇×C₈	1	1	yes	characteristic
159	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
160	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
161	C₂×C₁₆	—	2	4	no	—
162	C₂×C₁₆	—	2	4	no	—
163	C₂×C₁₆	—	2	4	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	C₇×D₄	1	4	yes	—
168	C₂²×C₇	—	2	2	no	—
169	C₂²×C₇	—	2	2	no	—
170	C₂²×C₇	—	2	2	no	—
171	C₂²×C₇	—	2	2	no	—
172	C₂²×C₇	—	2	2	no	—
173	C₂²×C₇	—	2	2	no	—
174	C₂²×C₇	C₄⋊C₁₆	1	4	yes	complemented
175	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
176	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
177	C₂²×C₇	#431	1	1	yes	characteristic
178	C₂²×C₇	C₂²×C₁₆	1	1	yes	characteristic
179	C₂²×C₇	C₂×M₃₂	1	1	yes	characteristic
180	C₂²×C₇	C₂×M₃₂	1	1	yes	characteristic
181	C₂²×C₇	C₂×M₃₂	1	1	yes	characteristic
182	C₄×C₇	—	2	2	no	—
183	C₄×C₇	—	2	2	no	—
184	C₄×C₇	—	2	2	no	—
185	C₄×C₇	—	2	2	no	—
186	C₄×C₇	C₂²⋊C₁₆	1	4	yes	complemented
187	C₄×C₇	#380	1	1	yes	characteristic
188	C₄×C₇	#407	1	1	yes	characteristic
189	C₄×C₇	#423	1	1	yes	characteristic
190	C₄×C₇	#432	1	1	yes	characteristic
191	C₂⁴	C₂×C₇×C₈	1	1	yes	characteristic
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	C₇×C₁₆	1	4	yes	complemented
202	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
203	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
208	C₂²×C₄	C₇×M₁₆	1	2	yes	—
209	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
210	C₂×C₈	—	2	8	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	C₇×C₂²⋊C₄	1	2	yes	—
220	C₂×C₈	C₇×C₄⋊C₄	1	2	yes	—
221	C₂×C₈	C₇×M₁₆	1	2	yes	—
222	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
223	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
224	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
225	C₂×C₈	C₇×#41	1	2	yes	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	1	no	—
229	C₄²	—	2	1	no	—
230	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
231	C₄²	C₇×M₁₆	1	2	yes	—
232	C₁₆	—	4	4	no	—
233	C₂×C₇	—	2	4	no	—
234	C₂×C₇	#1797	1	1	yes	characteristic
235	C₂×C₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
236	C₂×C₇	#2112	1	1	yes	characteristic
237	C₂×C₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
238	C₂×C₇	#2150	1	1	yes	characteristic
239	C₂×C₇	#2162	1	1	yes	characteristic
240	C₂×C₇	#2163	1	1	yes	characteristic
241	C₂³	—	2	1	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	2	1	no	—
245	C₂³	C₇×C₄⋊C₈	1	2	yes	—
246	C₂³	C₂×C₇×C₁₆	1	2	yes	—
247	C₂³	C₇×M₃₂	1	2	yes	—
248	C₂³	C₂²×C₇×C₈	1	1	yes	characteristic
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	C₇×C₂²⋊C₈	1	2	yes	—
264	C₂×C₄	C₇×C₂²⋊C₈	1	2	yes	—
265	C₂×C₄	C₇×C₄⋊C₈	1	2	yes	—
266	C₂×C₄	C₂×C₇×C₁₆	1	2	yes	—
267	C₂×C₄	C₂×C₇×C₁₆	1	2	yes	—
268	C₂×C₄	C₇×M₃₂	1	2	yes	—
269	C₂×C₄	C₇×M₃₂	1	2	yes	—
270	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
271	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
272	C₂×C₄	C₇×#117	1	1	yes	characteristic
273	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
274	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
275	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	2	2	no	—
278	C₈	—	2	2	no	—
279	C₈	C₇×#121	1	2	yes	—
280	C₈	C₇×C₂²⋊Q₈	1	2	yes	—
281	C₇	#12783	1	1	yes	characteristic, direct product factor, Sylow
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	C₇×C₄⋊C₁₆	1	4	yes	complemented
289	C₂²	C₂×C₇×C₂²⋊C₈	1	1	yes	characteristic
290	C₂²	C₂×C₇×C₄⋊C₈	1	1	yes	characteristic
291	C₂²	C₇×#431	1	1	yes	characteristic
292	C₂²	C₂²×C₇×C₁₆	1	1	yes	characteristic
293	C₂²	C₂×C₇×M₃₂	1	1	yes	characteristic
294	C₂²	C₂×C₇×M₃₂	1	1	yes	characteristic
295	C₂²	C₂×C₇×M₃₂	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	C₇×C₂²⋊C₁₆	1	4	yes	complemented
301	C₄	C₇×#380	1	1	yes	characteristic
302	C₄	C₇×#407	1	1	yes	characteristic
303	C₄	C₇×#423	1	1	yes	characteristic
304	C₄	C₇×#432	1	1	yes	characteristic
305	C₂	—	2	4	no	—
306	C₂	C₇×#1797	1	1	yes	characteristic
307	C₂	C₂×C₇×C₂²⋊C₁₆	1	1	yes	characteristic
308	C₂	C₇×#2112	1	1	yes	characteristic
309	C₂	C₂×C₇×C₄⋊C₁₆	1	1	yes	characteristic
310	C₂	C₇×#2150	1	1	yes	characteristic
311	C₂	C₇×#2162	1	1	yes	characteristic
312	C₂	C₇×#2163	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇×C₈	C₂³
commutator	C₂²	C₂²×C₇×C₁₆
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂³×C₇	C₂²×C₈
prime radical	C₂⁴×C₇	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×C₄⋊C₁₆	C₂	semidirect
4. C₇×#1374	C₂	semidirect
16. #12783	C₇	direct (N × H)
45. C₂×C₄⋊C₁₆	C₂×C₇	semidirect
47. #1374	C₂×C₇	semidirect
58. C₂²×C₄×C₇	C₁₆	semidirect
174. C₂²×C₇	C₄⋊C₁₆	semidirect
186. C₄×C₇	C₂²⋊C₁₆	semidirect
201. C₂²×C₄	C₇×C₁₆	semidirect
281. C₇	#12783	direct (N × H)
288. C₂²	C₇×C₄⋊C₁₆	semidirect
300. C₄	C₇×C₂²⋊C₁₆	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄×C₇×C₈	2
C₂×C₇×C₄⋊C₁₆	2
C₂×C₇×C₂²⋊C₁₆	2
C₇×#1374	2
#12783	7

Subgroups — C7×#12783 #13189999

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12783 #13189999