Subgroups — C₇×#23165 #13200381

Order 1792.

706 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#499	C₂	1	1	yes	maximal, characteristic
2	C₇×#2370	C₂	1	2	yes	maximal
3	C₇×#1989	C₂	1	2	yes	maximal
4	C₇×#1944	C₂	1	2	yes	maximal
5	C₇×#1921	C₂	1	2	yes	maximal
6	C₇×#1918	C₂	1	2	yes	maximal
7	C₇×#1776	C₂	1	2	yes	maximal
8	C₄×C₇×#107	C₂	1	1	yes	maximal, characteristic
9	C₇×#1770	C₂	1	1	yes	maximal, characteristic
10	C₂×C₇×#125	C₂²	1	2	yes	—
11	C₇×#499	C₄	1	8	yes	complemented
12	C₂×C₇×#107	C₂²	1	1	yes	characteristic
13	C₂×C₇×#107	C₂²	1	1	yes	characteristic
14	C₂×C₇×#107	C₂²	1	1	yes	characteristic
15	C₂×C₇×#107	C₂²	1	1	yes	characteristic
16	C₂×C₇×#106	C₂²	1	2	yes	—
17	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₇×#97	C₂²	1	2	yes	—
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₇×#389	—	2	2	no	—
21	C₇×#389	C₂²	1	2	yes	—
22	C₇×#389	C₂²	1	2	yes	—
23	C₇×#387	—	2	2	no	—
24	C₇×#387	—	2	2	no	—
25	C₄×C₇×C₄⋊C₄	—	2	2	no	—
26	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
27	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
28	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
29	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
30	C₇×#340	C₂²	1	2	yes	—
31	C₇×#340	C₂²	1	2	yes	—
32	C₇×#340	C₂²	1	2	yes	—
33	C₇×#340	C₂²	1	2	yes	—
34	C₇×#340	C₂²	1	2	yes	—
35	C₇×#336	C₂²	1	2	yes	—
36	#23165	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂²×C₇×C₈	C₂³	1	2	yes	—
38	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
40	C₇×#125	—	2	4	no	—
41	C₇×#125	C₂×C₄	1	8	yes	—
42	C₂×C₇×C₄⋊C₄	—	2	2	no	—
43	C₂×C₇×C₄⋊C₄	—	2	2	no	—
44	C₂×C₇×C₄⋊C₄	—	2	2	no	—
45	C₂×C₇×C₄⋊C₄	—	2	2	no	—
46	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
47	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
48	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
51	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
52	C₂×C₄²×C₇	—	2	2	no	—
53	C₂×C₄²×C₇	—	2	2	no	—
54	C₂×C₄²×C₇	—	2	2	no	—
55	C₂×C₄²×C₇	C₂³	1	2	yes	—
56	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
57	C₇×#107	—	2	2	no	—
58	C₇×#107	C₂×C₄	1	4	yes	—
59	C₇×#107	C₂×C₄	1	4	yes	—
60	C₇×#107	Q₈	1	4	yes	—
61	C₇×#106	C₂×C₄	1	8	yes	—
62	C₇×C₄⋊C₈	—	2	2	no	—
63	C₇×#97	—	2	4	no	—
64	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
65	C₇×#95	—	2	2	no	—
66	C₇×#95	—	2	2	no	—
67	C₇×#95	—	2	2	no	—
68	C₇×#95	—	2	2	no	—
69	C₇×#95	—	2	2	no	—
70	C₇×#95	D₄	1	2	yes	—
71	C₂×#499	C₂×C₇	1	1	yes	characteristic
72	#2370	C₂×C₇	1	2	yes	—
73	#1989	C₂×C₇	1	2	yes	—
74	#1944	C₂×C₇	1	2	yes	—
75	#1921	C₂×C₇	1	2	yes	—
76	#1918	C₂×C₇	1	2	yes	—
77	#1776	C₂×C₇	1	2	yes	—
78	C₄×#107	C₂×C₇	1	1	yes	characteristic
79	#1770	C₂×C₇	1	1	yes	characteristic
80	C₂²×C₄×C₇	—	2	2	no	—
81	C₂²×C₄×C₇	—	2	2	no	—
82	C₂²×C₄×C₇	—	2	2	no	—
83	C₂²×C₄×C₇	—	2	2	no	—
84	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
89	C₂×C₇×C₈	—	2	1	no	—
90	C₂×C₇×C₈	—	2	1	no	—
91	C₂×C₇×C₈	—	2	2	no	—
92	C₂×C₇×C₈	—	2	2	no	—
93	C₂×C₇×C₈	—	2	2	no	—
94	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
97	C₂×C₇×C₈	#41	1	2	yes	—
98	C₇×C₄⋊C₄	—	2	8	no	—
99	C₇×C₄⋊C₄	—	2	8	no	—
100	C₇×C₄⋊C₄	—	2	2	no	—
101	C₇×C₄⋊C₄	—	2	2	no	—
102	C₇×C₄⋊C₄	—	2	2	no	—
103	C₇×C₄⋊C₄	—	2	2	no	—
104	C₇×C₄⋊C₄	—	2	2	no	—
105	C₇×C₄⋊C₄	—	4	1	no	—
106	C₇×C₄⋊C₄	—	4	1	no	—
107	C₇×C₄⋊C₄	—	4	4	no	—
108	C₇×C₄⋊C₄	—	8	2	no	—
109	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
110	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
111	C₇×C₄⋊C₄	C₂×Q₈	1	4	yes	—
112	C₄²×C₇	—	2	2	no	—
113	C₄²×C₇	—	2	2	no	—
114	C₄²×C₇	—	4	4	no	—
115	C₄²×C₇	—	8	2	no	—
116	C₄²×C₇	—	8	2	no	—
117	C₄²×C₇	C₂²×C₄	1	2	yes	—
118	C₄²×C₇	#41	1	2	yes	—
119	C₂×#125	C₂²×C₇	1	2	yes	—
120	#499	C₄×C₇	1	8	yes	complemented
121	C₂×#107	C₂²×C₇	1	1	yes	characteristic
122	C₂×#107	C₂²×C₇	1	1	yes	characteristic
123	C₂×#107	C₂²×C₇	1	1	yes	characteristic
124	C₂×#107	C₂²×C₇	1	1	yes	characteristic
125	C₂×#106	C₂²×C₇	1	2	yes	—
126	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
127	C₂×#97	C₂²×C₇	1	2	yes	—
128	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
129	#389	—	2	2	no	—
130	#389	C₂²×C₇	1	2	yes	—
131	#389	C₂²×C₇	1	2	yes	—
132	#387	—	2	2	no	—
133	#387	—	2	2	no	—
134	C₄×C₄⋊C₄	—	2	2	no	—
135	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
136	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
137	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
138	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
139	#340	C₂²×C₇	1	2	yes	—
140	#340	C₂²×C₇	1	2	yes	—
141	#340	C₂²×C₇	1	2	yes	—
142	#340	C₂²×C₇	1	2	yes	—
143	#340	C₂²×C₇	1	2	yes	—
144	#336	C₂²×C₇	1	2	yes	—
145	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₇	—	2	1	no	—
147	C₂×C₄×C₇	—	2	1	no	—
148	C₂×C₄×C₇	—	2	1	no	—
149	C₂×C₄×C₇	—	2	1	no	—
150	C₂×C₄×C₇	—	2	4	no	—
151	C₂×C₄×C₇	—	2	4	no	—
152	C₂×C₄×C₇	—	2	2	no	—
153	C₂×C₄×C₇	—	2	2	no	—
154	C₂×C₄×C₇	—	2	4	no	—
155	C₂×C₄×C₇	—	2	2	no	—
156	C₂×C₄×C₇	—	2	2	no	—
157	C₂×C₄×C₇	—	2	2	no	—
158	C₂×C₄×C₇	—	2	2	no	—
159	C₂×C₄×C₇	—	4	2	no	—
160	C₂×C₄×C₇	—	4	2	no	—
161	C₂×C₄×C₇	—	4	2	no	—
162	C₂×C₄×C₇	—	4	2	no	—
163	C₂×C₄×C₇	—	4	2	no	—
164	C₂×C₄×C₇	—	4	2	no	—
165	C₂×C₄×C₇	—	4	2	no	—
166	C₂×C₄×C₇	—	4	2	no	—
167	C₂×C₄×C₇	—	4	2	no	—
168	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
169	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
170	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
171	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
172	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
173	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
174	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
175	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
176	C₇×C₈	—	2	2	no	—
177	C₇×C₈	—	4	2	no	—
178	C₇×C₈	C₄×Q₈	1	4	yes	—
179	C₂²×C₈	C₂³×C₇	1	2	yes	—
180	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
182	#125	—	2	4	no	—
183	#125	C₂×C₄×C₇	1	8	yes	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
189	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
190	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
191	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
192	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
193	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	C₂³×C₇	1	2	yes	—
198	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
199	#107	—	2	2	no	—
200	#107	C₂×C₄×C₇	1	4	yes	—
201	#107	C₂×C₄×C₇	1	4	yes	—
202	#107	C₇×Q₈	1	4	yes	—
203	#106	C₂×C₄×C₇	1	8	yes	—
204	C₄⋊C₈	—	2	2	no	—
205	#97	—	2	4	no	—
206	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
207	#95	—	2	2	no	—
208	#95	—	2	2	no	—
209	#95	—	2	2	no	—
210	#95	—	2	2	no	—
211	#95	—	2	2	no	—
212	#95	C₇×D₄	1	2	yes	—
213	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
214	C₂²×C₇	#547	1	1	yes	characteristic
215	C₂²×C₇	D₄×Q₈	1	1	yes	characteristic
216	C₂²×C₇	#575	1	1	yes	characteristic
217	C₂²×C₇	#575	1	1	yes	characteristic
218	C₂²×C₇	#576	1	1	yes	characteristic
219	C₂²×C₇	#578	1	1	yes	characteristic
220	C₄×C₇	—	2	2	no	—
221	C₄×C₇	—	2	2	no	—
222	C₄×C₇	—	4	2	no	—
223	C₄×C₇	—	4	2	no	—
224	C₄×C₇	—	4	4	no	—
225	C₄×C₇	—	4	4	no	—
226	C₄×C₇	—	4	4	no	—
227	C₄×C₇	—	8	2	no	—
228	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
229	C₄×C₇	#519	1	1	yes	characteristic
230	C₄×C₇	#556	1	1	yes	characteristic
231	C₄×C₇	#557	1	1	yes	characteristic
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
237	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
238	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
240	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
247	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
248	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
249	C₂×C₈	C₇×#41	1	2	yes	—
250	C₄⋊C₄	—	2	8	no	—
251	C₄⋊C₄	—	2	8	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	4	no	—
260	C₄⋊C₄	—	8	2	no	—
261	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
262	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
263	C₄⋊C₄	C₂×C₇×Q₈	1	4	yes	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	4	4	no	—
267	C₄²	—	8	2	no	—
268	C₄²	—	8	2	no	—
269	C₄²	C₂²×C₄×C₇	1	2	yes	—
270	C₄²	C₇×#41	1	2	yes	—
271	C₂×C₇	#2365	1	1	yes	characteristic
272	C₂×C₇	#2943	1	1	yes	characteristic
273	C₂×C₇	#2951	1	1	yes	characteristic
274	C₂×C₇	#3352	1	1	yes	characteristic
275	C₂×C₇	#3355	1	1	yes	characteristic
276	C₂×C₇	#3389	1	1	yes	characteristic
277	C₂×C₇	#3390	1	1	yes	characteristic
278	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
302	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
303	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
305	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
306	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
307	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
308	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
309	C₈	—	2	2	no	—
310	C₈	—	4	2	no	—
311	C₈	C₄×C₇×Q₈	1	4	yes	—
312	C₇	#23165	1	1	yes	characteristic, direct product factor, Sylow
313	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
314	C₂²	C₇×#547	1	1	yes	characteristic
315	C₂²	C₇×D₄×Q₈	1	1	yes	characteristic
316	C₂²	C₇×#575	1	1	yes	characteristic
317	C₂²	C₇×#575	1	1	yes	characteristic
318	C₂²	C₇×#576	1	1	yes	characteristic
319	C₂²	C₇×#578	1	1	yes	characteristic
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	—	4	4	no	—
323	C₄	—	4	4	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	4	no	—
327	C₄	—	8	2	no	—
328	C₄	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
329	C₄	C₇×#519	1	1	yes	characteristic
330	C₄	C₇×#556	1	1	yes	characteristic
331	C₄	C₇×#557	1	1	yes	characteristic
332	C₂	C₇×#2365	1	1	yes	characteristic
333	C₂	C₇×#2943	1	1	yes	characteristic
334	C₂	C₇×#2951	1	1	yes	characteristic
335	C₂	C₇×#3352	1	1	yes	characteristic
336	C₂	C₇×#3355	1	1	yes	characteristic
337	C₂	C₇×#3389	1	1	yes	characteristic
338	C₂	C₇×#3390	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
11. C₇×#499	C₄	semidirect
36. #23165	C₇	direct (N × H)
120. #499	C₄×C₇	semidirect
312. C₇	#23165	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×#499	2
C₇×#2370	2
C₇×#1989	2
C₇×#1944	2
C₇×#1921	2
C₇×#1918	2
C₇×#1776	2
C₄×C₇×#107	2
C₇×#1770	2
#23165	7

Subgroups — C7×#23165 #13200381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23165 #13200381