Subgroups — C₇×#23818 #13201034

Order 1792.

758 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₇×#124	C₂	1	1	yes	maximal, characteristic
3	C₇×#2101	C₂	1	2	yes	maximal, complemented
4	C₇×#2074	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#2073	C₂	1	1	yes	maximal, characteristic
6	C₇×#2042	C₂	1	2	yes	maximal
7	C₇×#2026	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#2025	C₂	1	1	yes	maximal, characteristic
9	C₄×C₇×#103	C₂	1	2	yes	maximal, complemented
10	C₄×C₇×#102	C₂	1	2	yes	maximal
11	C₇×#1744	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#125	C₂²	1	1	yes	characteristic
13	C₂×C₇×#124	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
16	C₂×C₇×#103	C₂²	1	2	yes	—
17	C₂×C₇×#103	C₂²	1	2	yes	—
18	C₂×C₇×#102	C₂²	1	2	yes	—
19	C₂×C₇×#102	C₂²	1	2	yes	—
20	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
21	C₇×#391	C₂²	1	1	yes	characteristic
22	C₇×#390	C₂²	1	1	yes	characteristic
23	C₇×#389	C₂²	1	2	yes	—
24	C₇×#388	—	2	2	no	—
25	C₇×#387	—	2	2	no	—
26	C₇×#382	C₂²	1	1	yes	characteristic
27	C₇×#382	C₂²	1	1	yes	characteristic
28	C₄×C₇×C₄⋊C₄	—	2	2	no	—
29	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
30	C₄×C₇×C₂²⋊C₄	—	2	2	no	—
31	C₄³×C₇	C₂²	1	1	yes	characteristic
32	C₇×#340	C₂²	1	2	yes	—
33	C₇×#340	C₂²	1	2	yes	—
34	C₇×#340	C₂²	1	2	yes	—
35	C₇×#340	C₂²	1	2	yes	—
36	C₇×#336	C₂²	1	1	yes	characteristic
37	C₇×#336	C₂²	1	1	yes	characteristic
38	C₇×#336	C₂²	1	2	yes	—
39	#23818	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
41	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₇	—	2	1	no	—
43	C₂²×C₇×C₈	C₂³	1	2	yes	—
44	C₂²×C₇×C₈	C₂³	1	2	yes	—
45	C₇×#125	—	2	4	no	—
46	C₇×#124	—	2	2	no	—
47	C₇×#124	—	2	2	no	—
48	C₄×C₇×Q₈	—	2	4	no	—
49	C₄×C₇×D₄	—	2	4	no	—
50	C₂×C₇×C₄⋊C₄	—	2	2	no	—
51	C₂×C₇×C₄⋊C₄	—	2	2	no	—
52	C₂×C₇×C₄⋊C₄	—	2	1	no	—
53	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
54	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
55	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
56	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
57	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
58	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
59	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
60	C₂×C₄²×C₇	—	2	1	no	—
61	C₂×C₄²×C₇	—	2	2	no	—
62	C₂×C₄²×C₇	D₄	1	2	yes	—
63	C₂×C₄²×C₇	D₄	1	2	yes	—
64	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
67	C₇×#103	—	2	4	no	—
68	C₇×#103	—	2	4	no	—
69	C₇×#102	—	2	4	no	—
70	C₇×#102	—	2	4	no	—
71	C₄×C₇×C₈	—	2	4	no	—
72	C₇×#95	—	2	2	no	—
73	C₇×#95	—	2	2	no	—
74	C₇×#95	—	2	2	no	—
75	C₇×#95	—	2	2	no	—
76	C₇×#95	—	2	2	no	—
77	C₄×#125	C₂×C₇	1	1	yes	characteristic, complemented
78	C₄×#124	C₂×C₇	1	1	yes	characteristic
79	#2101	C₂×C₇	1	2	yes	complemented
80	#2074	C₂×C₇	1	1	yes	characteristic, complemented
81	#2073	C₂×C₇	1	1	yes	characteristic
82	#2042	C₂×C₇	1	2	yes	—
83	#2026	C₂×C₇	1	1	yes	characteristic, complemented
84	#2025	C₂×C₇	1	1	yes	characteristic
85	C₄×#103	C₂×C₇	1	2	yes	complemented
86	C₄×#102	C₂×C₇	1	2	yes	—
87	#1744	C₂×C₇	1	1	yes	characteristic, complemented
88	C₂⁴×C₇	—	2	1	no	—
89	C₂×C₇×Q₈	—	2	1	no	—
90	C₂×C₇×Q₈	—	2	2	no	—
91	C₂×C₇×Q₈	—	2	2	no	—
92	C₂×C₇×Q₈	#41	1	2	yes	—
93	C₂×C₇×D₄	—	2	2	no	—
94	C₂×C₇×D₄	—	2	2	no	—
95	C₂×C₇×D₄	—	2	1	no	—
96	C₂×C₇×D₄	#41	1	2	yes	—
97	C₂²×C₄×C₇	—	2	1	no	—
98	C₂²×C₄×C₇	—	2	2	no	—
99	C₂²×C₄×C₇	—	2	2	no	—
100	C₂²×C₄×C₇	—	2	1	no	—
101	C₂²×C₄×C₇	—	2	1	no	—
102	C₂²×C₄×C₇	—	4	2	no	—
103	C₂²×C₄×C₇	—	4	4	no	—
104	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
109	C₂×C₇×C₈	—	2	2	no	—
110	C₂×C₇×C₈	—	2	2	no	—
111	C₂×C₇×C₈	—	2	2	no	—
112	C₂×C₇×C₈	—	2	2	no	—
113	C₂×C₇×C₈	—	2	2	no	—
114	C₂×C₇×C₈	—	2	2	no	—
115	C₇×C₄⋊C₄	—	2	8	no	—
116	C₇×C₄⋊C₄	—	2	2	no	—
117	C₇×C₄⋊C₄	—	2	2	no	—
118	C₇×C₄⋊C₄	—	2	2	no	—
119	C₇×C₄⋊C₄	—	2	2	no	—
120	C₇×C₄⋊C₄	—	2	2	no	—
121	C₇×C₄⋊C₄	—	4	4	no	—
122	C₇×C₄⋊C₄	—	4	2	no	—
123	C₇×C₄⋊C₄	#41	1	4	yes	—
124	C₇×C₂²⋊C₄	—	2	8	no	—
125	C₇×C₂²⋊C₄	—	4	4	no	—
126	C₇×C₂²⋊C₄	—	4	2	no	—
127	C₄²×C₇	—	2	4	no	—
128	C₄²×C₇	—	2	1	no	—
129	C₄²×C₇	—	2	1	no	—
130	C₄²×C₇	—	2	2	no	—
131	C₄²×C₇	—	4	2	no	—
132	C₄²×C₇	—	4	2	no	—
133	C₄²×C₇	—	4	4	no	—
134	C₄²×C₇	#41	1	4	yes	—
135	C₂×#125	C₂²×C₇	1	1	yes	characteristic
136	C₂×#124	C₂²×C₇	1	1	yes	characteristic
137	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
138	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
139	C₂×#103	C₂²×C₇	1	2	yes	—
140	C₂×#103	C₂²×C₇	1	2	yes	—
141	C₂×#102	C₂²×C₇	1	2	yes	—
142	C₂×#102	C₂²×C₇	1	2	yes	—
143	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
144	#391	C₂²×C₇	1	1	yes	characteristic
145	#390	C₂²×C₇	1	1	yes	characteristic
146	#389	C₂²×C₇	1	2	yes	—
147	#388	—	2	2	no	—
148	#387	—	2	2	no	—
149	#382	C₂²×C₇	1	1	yes	characteristic
150	#382	C₂²×C₇	1	1	yes	characteristic
151	C₄×C₄⋊C₄	—	2	2	no	—
152	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
153	C₄×C₂²⋊C₄	—	2	2	no	—
154	C₄³	C₂²×C₇	1	1	yes	characteristic
155	#340	C₂²×C₇	1	2	yes	—
156	#340	C₂²×C₇	1	2	yes	—
157	#340	C₂²×C₇	1	2	yes	—
158	#340	C₂²×C₇	1	2	yes	—
159	#336	C₂²×C₇	1	1	yes	characteristic
160	#336	C₂²×C₇	1	1	yes	characteristic
161	#336	C₂²×C₇	1	2	yes	—
162	C₂³×C₇	—	2	2	no	—
163	C₂³×C₇	—	4	1	no	—
164	C₂³×C₇	—	4	1	no	—
165	C₂³×C₇	—	4	2	no	—
166	C₂³×C₇	—	4	2	no	—
167	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
168	C₇×Q₈	—	2	4	no	—
169	C₇×Q₈	—	4	2	no	—
170	C₇×D₄	—	2	4	no	—
171	C₇×D₄	—	4	2	no	—
172	C₂×C₄×C₇	—	2	2	no	—
173	C₂×C₄×C₇	—	2	2	no	—
174	C₂×C₄×C₇	—	2	2	no	—
175	C₂×C₄×C₇	—	2	2	no	—
176	C₂×C₄×C₇	—	2	4	no	—
177	C₂×C₄×C₇	—	2	2	no	—
178	C₂×C₄×C₇	—	2	2	no	—
179	C₂×C₄×C₇	—	2	2	no	—
180	C₂×C₄×C₇	—	2	2	no	—
181	C₂×C₄×C₇	—	4	1	no	—
182	C₂×C₄×C₇	—	4	1	no	—
183	C₂×C₄×C₇	—	4	2	no	—
184	C₂×C₄×C₇	—	4	2	no	—
185	C₂×C₄×C₇	—	4	2	no	—
186	C₂×C₄×C₇	—	4	1	no	—
187	C₂×C₄×C₇	—	4	1	no	—
188	C₂×C₄×C₇	—	4	2	no	—
189	C₂×C₄×C₇	—	4	2	no	—
190	C₂×C₄×C₇	—	4	2	no	—
191	C₂×C₄×C₇	—	4	2	no	—
192	C₂×C₄×C₇	—	8	4	no	—
193	C₂×C₄×C₇	#124	1	2	yes	—
194	C₂×C₄×C₇	#124	1	2	yes	—
195	C₂×C₄×C₇	#135	1	4	yes	—
196	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₇	C₂×#41	1	2	yes	—
199	C₂×C₄×C₇	C₂×#41	1	2	yes	—
200	C₇×C₈	—	4	2	no	—
201	C₇×C₈	—	4	2	no	—
202	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
203	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
204	C₂³×C₄	—	2	1	no	—
205	C₂²×C₈	C₂³×C₇	1	2	yes	—
206	C₂²×C₈	C₂³×C₇	1	2	yes	—
207	#125	—	2	4	no	—
208	#124	—	2	2	no	—
209	#124	—	2	2	no	—
210	C₄×Q₈	—	2	4	no	—
211	C₄×D₄	—	2	4	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
216	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
217	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
218	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
219	C₂×C₂²⋊C₄	—	2	2	no	—
220	C₂×C₂²⋊C₄	—	2	1	no	—
221	C₂×C₂²⋊C₄	—	2	2	no	—
222	C₂×C₄²	—	2	1	no	—
223	C₂×C₄²	—	2	2	no	—
224	C₂×C₄²	C₇×D₄	1	2	yes	—
225	C₂×C₄²	C₇×D₄	1	2	yes	—
226	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
227	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
228	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
229	#103	—	2	4	no	—
230	#103	—	2	4	no	—
231	#102	—	2	4	no	—
232	#102	—	2	4	no	—
233	C₄×C₈	—	2	4	no	—
234	#95	—	2	2	no	—
235	#95	—	2	2	no	—
236	#95	—	2	2	no	—
237	#95	—	2	2	no	—
238	#95	—	2	2	no	—
239	C₂²×C₇	—	4	4	no	—
240	C₂²×C₇	—	4	2	no	—
241	C₂²×C₇	—	8	2	no	—
242	C₂²×C₇	—	8	1	no	—
243	C₂²×C₇	—	8	1	no	—
244	C₂²×C₇	#525	1	1	yes	characteristic
245	C₂²×C₇	#525	1	1	yes	characteristic
246	C₂²×C₇	C₂×#124	1	1	yes	characteristic
247	C₂²×C₇	C₂×#135	1	2	yes	—
248	C₂²×C₇	#575	1	2	yes	—
249	C₄×C₇	—	2	2	no	—
250	C₄×C₇	—	4	2	no	—
251	C₄×C₇	—	4	2	no	—
252	C₄×C₇	—	4	4	no	—
253	C₄×C₇	—	4	2	no	—
254	C₄×C₇	—	8	2	no	—
255	C₄×C₇	—	8	1	no	—
256	C₄×C₇	—	8	1	no	—
257	C₄×C₇	#488	1	4	yes	—
258	C₄×C₇	#529	1	2	yes	—
259	C₄×C₇	#529	1	2	yes	—
260	C₂⁴	—	2	1	no	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	C₇×#41	1	2	yes	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	C₇×#41	1	2	yes	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	2	1	no	—
273	C₂²×C₄	—	2	1	no	—
274	C₂²×C₄	—	4	2	no	—
275	C₂²×C₄	—	4	4	no	—
276	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
277	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
278	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
280	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₄⋊C₄	—	2	8	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	4	4	no	—
294	C₄⋊C₄	—	4	2	no	—
295	C₄⋊C₄	C₇×#41	1	4	yes	—
296	C₂²⋊C₄	—	2	8	no	—
297	C₂²⋊C₄	—	4	4	no	—
298	C₂²⋊C₄	—	4	2	no	—
299	C₄²	—	2	4	no	—
300	C₄²	—	2	1	no	—
301	C₄²	—	2	1	no	—
302	C₄²	—	2	2	no	—
303	C₄²	—	4	2	no	—
304	C₄²	—	4	2	no	—
305	C₄²	—	4	4	no	—
306	C₄²	C₇×#41	1	4	yes	—
307	C₂×C₇	—	8	2	no	—
308	C₂×C₇	#2398	1	1	yes	characteristic
309	C₂×C₇	#3102	1	2	yes	—
310	C₂×C₇	#3102	1	2	yes	—
311	C₂×C₇	C₂×#488	1	1	yes	characteristic
312	C₂×C₇	#3135	1	1	yes	characteristic
313	C₂³	—	2	2	no	—
314	C₂³	—	4	1	no	—
315	C₂³	—	4	1	no	—
316	C₂³	—	4	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
319	Q₈	—	2	4	no	—
320	Q₈	—	4	2	no	—
321	D₄	—	2	4	no	—
322	D₄	—	4	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	4	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	8	4	no	—
344	C₂×C₄	C₇×#124	1	2	yes	—
345	C₂×C₄	C₇×#124	1	2	yes	—
346	C₂×C₄	C₇×#135	1	4	yes	—
347	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
348	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
349	C₂×C₄	C₂×C₇×#41	1	2	yes	—
350	C₂×C₄	C₂×C₇×#41	1	2	yes	—
351	C₈	—	4	2	no	—
352	C₈	—	4	2	no	—
353	C₇	#23818	1	1	yes	characteristic, direct product factor, Sylow
354	C₂²	—	4	2	no	—
355	C₂²	—	4	4	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	8	1	no	—
358	C₂²	—	8	1	no	—
359	C₂²	C₇×#525	1	1	yes	characteristic
360	C₂²	C₇×#525	1	1	yes	characteristic
361	C₂²	C₂×C₇×#124	1	1	yes	characteristic
362	C₂²	C₂×C₇×#135	1	2	yes	—
363	C₂²	C₇×#575	1	2	yes	—
364	C₄	—	2	2	no	—
365	C₄	—	4	2	no	—
366	C₄	—	4	2	no	—
367	C₄	—	4	2	no	—
368	C₄	—	4	4	no	—
369	C₄	—	8	2	no	—
370	C₄	—	8	1	no	—
371	C₄	—	8	1	no	—
372	C₄	C₇×#488	1	4	yes	—
373	C₄	C₇×#529	1	2	yes	—
374	C₄	C₇×#529	1	2	yes	—
375	C₂	—	8	2	no	—
376	C₂	C₇×#2398	1	1	yes	characteristic
377	C₂	C₇×#3102	1	2	yes	—
378	C₂	C₇×#3102	1	2	yes	—
379	C₂	C₂×C₇×#488	1	1	yes	characteristic
380	C₂	C₇×#3135	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂³×C₇	C₂²×D₄
prime radical	C₂²×C₇×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₄³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄×C₇×#125	C₂	semidirect
3. C₇×#2101	C₂	semidirect
4. C₇×#2074	C₂	semidirect
7. C₇×#2026	C₂	semidirect
9. C₄×C₇×#103	C₂	semidirect
11. C₇×#1744	C₂	semidirect
39. #23818	C₇	direct (N × H)
77. C₄×#125	C₂×C₇	semidirect
79. #2101	C₂×C₇	semidirect
80. #2074	C₂×C₇	semidirect
83. #2026	C₂×C₇	semidirect
85. C₄×#103	C₂×C₇	semidirect
87. #1744	C₂×C₇	semidirect
353. C₇	#23818	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₇×#125	2
C₄×C₇×#124	2
C₇×#2101	2
C₇×#2074	2
C₇×#2073	2
C₇×#2042	2
C₇×#2026	2
C₇×#2025	2
C₄×C₇×#103	2
C₄×C₇×#102	2
C₇×#1744	2
#23818	7

Subgroups — C7×#23818 #13201034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23818 #13201034