Subgroups — C₇×#23265 #13200481

Order 1792.

624 conjugacy classes of subgroups.

The subgroup lattice has 360 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
2	C₇×#1917	C₂	1	2	yes	maximal
3	C₇×#1917	C₂	1	2	yes	maximal
4	C₇×#1917	C₂	1	2	yes	maximal
5	C₇×#1916	C₂	1	2	yes	maximal
6	C₇×#1771	C₂	1	2	yes	maximal
7	C₇×#1770	C₂	1	2	yes	maximal
8	C₇×#1745	C₂	1	1	yes	maximal, characteristic
9	C₇×#1745	C₂	1	1	yes	maximal, characteristic
10	C₂×C₇×#125	C₄	1	1	yes	characteristic
11	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₇×#97	C₂²	1	2	yes	—
14	C₂×C₇×#97	C₂²	1	2	yes	—
15	C₂×C₇×#97	C₂²	1	2	yes	—
16	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
17	C₇×#389	C₄	1	1	yes	characteristic
18	C₇×#389	C₄	1	1	yes	characteristic
19	C₇×#387	C₄	1	2	yes	—
20	C₇×#387	C₄	1	2	yes	—
21	C₇×#382	C₄	1	1	yes	characteristic
22	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
23	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
24	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₄³×C₇	C₂²	1	1	yes	characteristic
27	C₇×#336	C₂²	1	1	yes	characteristic
28	C₇×#336	C₂²	1	1	yes	characteristic
29	C₇×#336	C₂²	1	2	yes	—
30	C₇×#336	C₂²	1	2	yes	—
31	C₇×#336	C₂²	1	1	yes	characteristic
32	C₇×#336	C₂²	1	1	yes	characteristic
33	C₇×#336	C₂²	1	2	yes	—
34	C₇×#336	C₂²	1	2	yes	—
35	C₇×#336	C₂²	1	2	yes	—
36	C₇×#336	C₂²	1	2	yes	—
37	#23265	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂²×C₇×C₈	C₂³	1	2	yes	—
39	C₂²×C₇×C₈	C₂³	1	2	yes	—
40	C₂²×C₇×C₈	C₂³	1	2	yes	—
41	C₂²×C₇×C₈	C₂³	1	2	yes	—
42	C₇×#125	—	2	4	no	—
43	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
46	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
47	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
48	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₇	C₂³	1	2	yes	—
59	C₂×C₄²×C₇	C₂³	1	2	yes	—
60	C₇×C₄⋊C₈	—	2	4	no	—
61	C₇×C₄⋊C₈	—	2	4	no	—
62	C₇×#97	—	2	4	no	—
63	C₇×#97	—	2	4	no	—
64	C₇×#97	—	2	4	no	—
65	C₄×C₇×C₈	—	2	4	no	—
66	C₇×#95	C₂×C₄	1	2	yes	—
67	C₇×#95	C₂×C₄	1	2	yes	—
68	C₇×#95	C₂×C₄	1	2	yes	—
69	C₇×#95	C₂×C₄	1	2	yes	—
70	C₇×#95	C₂×C₄	1	2	yes	—
71	C₇×#95	C₂×C₄	1	2	yes	—
72	C₄×#125	C₂×C₇	1	1	yes	characteristic
73	#1917	C₂×C₇	1	2	yes	—
74	#1917	C₂×C₇	1	2	yes	—
75	#1917	C₂×C₇	1	2	yes	—
76	#1916	C₂×C₇	1	2	yes	—
77	#1771	C₂×C₇	1	2	yes	—
78	#1770	C₂×C₇	1	2	yes	—
79	#1745	C₂×C₇	1	1	yes	characteristic
80	#1745	C₂×C₇	1	1	yes	characteristic
81	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
92	C₂×C₇×C₈	—	2	2	no	—
93	C₂×C₇×C₈	—	2	2	no	—
94	C₂×C₇×C₈	—	2	2	no	—
95	C₂×C₇×C₈	—	2	2	no	—
96	C₂×C₇×C₈	—	2	2	no	—
97	C₂×C₇×C₈	—	2	2	no	—
98	C₂×C₇×C₈	—	2	2	no	—
99	C₂×C₇×C₈	—	2	2	no	—
100	C₂×C₇×C₈	—	2	2	no	—
101	C₂×C₇×C₈	—	2	2	no	—
102	C₂×C₇×C₈	#41	1	4	yes	—
103	C₂×C₇×C₈	#41	1	4	yes	—
104	C₇×C₄⋊C₄	—	2	4	no	—
105	C₇×C₄⋊C₄	—	2	4	no	—
106	C₇×C₄⋊C₄	—	2	2	no	—
107	C₇×C₄⋊C₄	—	4	1	no	—
108	C₇×C₄⋊C₄	—	4	1	no	—
109	C₇×C₄⋊C₄	—	4	2	no	—
110	C₇×C₄⋊C₄	—	4	2	no	—
111	C₇×C₄⋊C₄	M₁₆	1	4	yes	—
112	C₄²×C₇	—	2	2	no	—
113	C₄²×C₇	—	2	2	no	—
114	C₄²×C₇	—	2	2	no	—
115	C₄²×C₇	—	2	2	no	—
116	C₄²×C₇	—	2	2	no	—
117	C₄²×C₇	—	2	2	no	—
118	C₄²×C₇	—	2	2	no	—
119	C₄²×C₇	—	2	1	no	—
120	C₄²×C₇	—	2	1	no	—
121	C₄²×C₇	—	4	1	no	—
122	C₄²×C₇	—	4	1	no	—
123	C₄²×C₇	—	4	1	no	—
124	C₂×#125	C₄×C₇	1	1	yes	characteristic
125	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
126	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
127	C₂×#97	C₂²×C₇	1	2	yes	—
128	C₂×#97	C₂²×C₇	1	2	yes	—
129	C₂×#97	C₂²×C₇	1	2	yes	—
130	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
131	#389	C₄×C₇	1	1	yes	characteristic
132	#389	C₄×C₇	1	1	yes	characteristic
133	#387	C₄×C₇	1	2	yes	—
134	#387	C₄×C₇	1	2	yes	—
135	#382	C₄×C₇	1	1	yes	characteristic
136	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
137	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
138	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
139	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
140	C₄³	C₂²×C₇	1	1	yes	characteristic
141	#336	C₂²×C₇	1	2	yes	—
142	#336	C₂²×C₇	1	2	yes	—
143	#336	C₂²×C₇	1	2	yes	—
144	#336	C₂²×C₇	1	2	yes	—
145	#336	C₂²×C₇	1	2	yes	—
146	#336	C₂²×C₇	1	1	yes	characteristic
147	#336	C₂²×C₇	1	1	yes	characteristic
148	#336	C₂²×C₇	1	2	yes	—
149	#336	C₂²×C₇	1	1	yes	characteristic
150	#336	C₂²×C₇	1	1	yes	characteristic
151	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₄×C₇	—	2	2	no	—
153	C₂×C₄×C₇	—	2	1	no	—
154	C₂×C₄×C₇	—	2	1	no	—
155	C₂×C₄×C₇	—	2	2	no	—
156	C₂×C₄×C₇	—	2	2	no	—
157	C₂×C₄×C₇	—	2	1	no	—
158	C₂×C₄×C₇	—	2	1	no	—
159	C₂×C₄×C₇	—	2	2	no	—
160	C₂×C₄×C₇	—	2	1	no	—
161	C₂×C₄×C₇	—	2	1	no	—
162	C₂×C₄×C₇	—	2	2	no	—
163	C₂×C₄×C₇	—	2	1	no	—
164	C₂×C₄×C₇	—	2	1	no	—
165	C₂×C₄×C₇	—	2	1	no	—
166	C₂×C₄×C₇	—	2	2	no	—
167	C₂×C₄×C₇	—	2	1	no	—
168	C₂×C₄×C₇	—	2	1	no	—
169	C₂×C₄×C₇	—	2	1	no	—
170	C₂×C₄×C₇	—	2	1	no	—
171	C₂×C₄×C₇	—	2	1	no	—
172	C₂×C₄×C₇	—	2	1	no	—
173	C₂×C₄×C₇	—	2	1	no	—
174	C₂×C₄×C₇	—	2	1	no	—
175	C₂×C₄×C₇	—	2	2	no	—
176	C₂×C₄×C₇	—	2	2	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	C₂×M₁₆	1	2	yes	—
181	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
182	C₂×C₄×C₇	C₂×#41	1	2	yes	—
183	C₂×C₄×C₇	C₂×#41	1	2	yes	—
184	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
185	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
186	C₇×C₈	—	2	4	no	—
187	C₇×C₈	—	2	4	no	—
188	C₇×C₈	—	4	2	no	—
189	C₇×C₈	—	4	2	no	—
190	C₂²×C₈	C₂³×C₇	1	2	yes	—
191	C₂²×C₈	C₂³×C₇	1	2	yes	—
192	C₂²×C₈	C₂³×C₇	1	2	yes	—
193	C₂²×C₈	C₂³×C₇	1	2	yes	—
194	#125	—	2	4	no	—
195	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
196	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
197	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
198	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
199	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
200	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
201	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
202	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
203	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
204	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
205	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
206	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
209	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
210	C₂×C₄²	C₂³×C₇	1	2	yes	—
211	C₂×C₄²	C₂³×C₇	1	2	yes	—
212	C₄⋊C₈	—	2	4	no	—
213	C₄⋊C₈	—	2	4	no	—
214	#97	—	2	4	no	—
215	#97	—	2	4	no	—
216	#97	—	2	4	no	—
217	C₄×C₈	—	2	4	no	—
218	#95	C₂×C₄×C₇	1	2	yes	—
219	#95	C₂×C₄×C₇	1	2	yes	—
220	#95	C₂×C₄×C₇	1	2	yes	—
221	#95	C₂×C₄×C₇	1	2	yes	—
222	#95	C₂×C₄×C₇	1	2	yes	—
223	#95	C₂×C₄×C₇	1	2	yes	—
224	C₂²×C₇	C₄×#41	1	1	yes	characteristic
225	C₂²×C₇	C₄×#41	1	1	yes	characteristic
226	C₂²×C₇	#519	1	1	yes	characteristic
227	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
228	C₂²×C₇	#568	1	1	yes	characteristic
229	C₂²×C₇	#568	1	1	yes	characteristic
230	C₂²×C₇	#568	1	1	yes	characteristic
231	C₄×C₇	—	2	4	no	—
232	C₄×C₇	—	2	2	no	—
233	C₄×C₇	—	4	1	no	—
234	C₄×C₇	—	4	1	no	—
235	C₄×C₇	—	4	2	no	—
236	C₄×C₇	—	4	2	no	—
237	C₄×C₇	—	4	1	no	—
238	C₄×C₇	—	4	1	no	—
239	C₄×C₇	—	4	2	no	—
240	C₄×C₇	—	4	1	no	—
241	C₄×C₇	#552	1	2	yes	—
242	C₄×C₇	#552	1	2	yes	—
243	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
244	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
245	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
246	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
247	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
248	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
249	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
251	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
252	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
253	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	C₇×#41	1	4	yes	—
265	C₂×C₈	C₇×#41	1	4	yes	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	C₇×M₁₆	1	4	yes	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	4	1	no	—
284	C₄²	—	4	1	no	—
285	C₄²	—	4	1	no	—
286	C₂×C₇	#2371	1	1	yes	characteristic
287	C₂×C₇	#2967	1	1	yes	characteristic
288	C₂×C₇	#2967	1	1	yes	characteristic
289	C₂×C₇	#2970	1	1	yes	characteristic
290	C₂×C₇	#2971	1	1	yes	characteristic
291	C₂×C₇	#2983	1	1	yes	characteristic
292	C₂×C₇	#2987	1	1	yes	characteristic
293	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
323	C₂×C₄	C₂×C₇×M₁₆	1	4	yes	—
324	C₂×C₄	C₂×C₇×#41	1	2	yes	—
325	C₂×C₄	C₂×C₇×#41	1	2	yes	—
326	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
327	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
328	C₈	—	2	4	no	—
329	C₈	—	2	4	no	—
330	C₈	—	4	2	no	—
331	C₈	—	4	2	no	—
332	C₇	#23265	1	1	yes	characteristic, direct product factor, Sylow
333	C₂²	C₄×C₇×#41	1	1	yes	characteristic
334	C₂²	C₄×C₇×#41	1	1	yes	characteristic
335	C₂²	C₇×#519	1	1	yes	characteristic
336	C₂²	C₂²×C₇×M₁₆	1	1	yes	characteristic
337	C₂²	C₇×#568	1	1	yes	characteristic
338	C₂²	C₇×#568	1	1	yes	characteristic
339	C₂²	C₇×#568	1	1	yes	characteristic
340	C₄	—	2	4	no	—
341	C₄	—	2	2	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	2	no	—
345	C₄	—	4	2	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	2	no	—
349	C₄	—	4	1	no	—
350	C₄	C₇×#552	1	2	yes	—
351	C₄	C₇×#552	1	2	yes	—
352	C₂	C₇×#2371	1	1	yes	characteristic
353	C₂	C₇×#2967	1	1	yes	characteristic
354	C₂	C₇×#2967	1	1	yes	characteristic
355	C₂	C₇×#2970	1	1	yes	characteristic
356	C₂	C₇×#2971	1	1	yes	characteristic
357	C₂	C₇×#2983	1	1	yes	characteristic
358	C₂	C₇×#2987	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
37. #23265	C₇	direct (N × H)
332. C₇	#23265	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
C₇×#1917	2
C₇×#1916	2
C₇×#1771	2
C₇×#1770	2
C₇×#1745	2
#23265	7

Subgroups — C7×#23265 #13200481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23265 #13200481