Subgroups — C₇×#23367 #13200583

Order 1792.

600 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2399	C₂	1	1	yes	maximal, characteristic
2	C₇×#1985	C₂	1	1	yes	maximal, characteristic
3	C₇×#1984	C₂	1	1	yes	maximal, characteristic
4	C₇×#1984	C₂	1	2	yes	maximal
5	C₇×#1917	C₂	1	2	yes	maximal
6	C₇×#1916	C₂	1	2	yes	maximal
7	C₇×#1916	C₂	1	2	yes	maximal
8	C₇×#1916	C₂	1	2	yes	maximal
9	C₇×#1840	C₂	1	2	yes	maximal
10	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₇×#97	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
18	C₇×#398	C₄	1	2	yes	—
19	C₇×#398	C₄	1	1	yes	characteristic
20	C₇×#398	C₄	1	1	yes	characteristic
21	C₇×#395	C₄	1	1	yes	characteristic
22	C₇×#395	C₄	1	1	yes	characteristic
23	C₇×#395	C₄	1	2	yes	—
24	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
25	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
26	C₇×#376	C₂²	1	1	yes	characteristic
27	C₇×#336	C₂²	1	2	yes	—
28	C₇×#336	C₂²	1	2	yes	—
29	C₇×#336	C₂²	1	2	yes	—
30	C₇×#336	C₂²	1	2	yes	—
31	C₇×#336	C₂²	1	2	yes	—
32	C₇×#336	C₂²	1	2	yes	—
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	#23367	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₇×C₈	C₂³	1	2	yes	—
41	C₂²×C₇×C₈	C₂³	1	2	yes	—
42	C₂²×C₇×C₈	C₂³	1	2	yes	—
43	C₂²×C₇×C₈	C₂³	1	2	yes	—
44	C₄×C₇×Q₈	—	4	1	no	—
45	C₄×C₇×Q₈	—	4	1	no	—
46	C₄×C₇×Q₈	—	4	1	no	—
47	C₄×C₇×Q₈	—	4	1	no	—
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	2	yes	—
51	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
52	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
53	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
54	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
55	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
56	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₇	C₂³	1	2	yes	—
58	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₇	C₂³	1	2	yes	—
60	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
62	C₇×C₄⋊C₈	—	4	1	no	—
63	C₇×C₄⋊C₈	—	4	1	no	—
64	C₇×C₄⋊C₈	—	4	2	no	—
65	C₇×#97	—	4	1	no	—
66	C₄×C₇×C₈	—	4	1	no	—
67	C₄×C₇×C₈	—	4	2	no	—
68	C₇×#95	C₂×C₄	1	1	yes	characteristic
69	C₇×#95	C₂×C₄	1	2	yes	—
70	C₇×#95	C₂×C₄	1	1	yes	characteristic
71	C₇×#95	C₂×C₄	1	2	yes	—
72	C₇×#95	C₂×C₄	1	2	yes	—
73	C₇×#95	C₂×C₄	1	2	yes	—
74	C₇×#95	C₂×C₄	1	2	yes	—
75	#2399	C₂×C₇	1	1	yes	characteristic
76	#1985	C₂×C₇	1	1	yes	characteristic
77	#1984	C₂×C₇	1	2	yes	—
78	#1984	C₂×C₇	1	1	yes	characteristic
79	#1917	C₂×C₇	1	2	yes	—
80	#1916	C₂×C₇	1	2	yes	—
81	#1916	C₂×C₇	1	2	yes	—
82	#1916	C₂×C₇	1	2	yes	—
83	#1840	C₂×C₇	1	2	yes	—
84	C₂×C₇×Q₈	—	2	4	no	—
85	C₂×C₇×Q₈	—	2	2	no	—
86	C₂×C₇×Q₈	—	4	1	no	—
87	C₂×C₇×Q₈	—	4	1	no	—
88	C₂×C₇×Q₈	—	4	1	no	—
89	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
100	C₂×C₇×C₈	—	2	2	no	—
101	C₂×C₇×C₈	—	2	2	no	—
102	C₂×C₇×C₈	—	2	2	no	—
103	C₂×C₇×C₈	—	2	2	no	—
104	C₂×C₇×C₈	—	2	2	no	—
105	C₂×C₇×C₈	—	2	2	no	—
106	C₂×C₇×C₈	—	2	4	no	—
107	C₂×C₇×C₈	—	2	4	no	—
108	C₂×C₇×C₈	—	2	2	no	—
109	C₂×C₇×C₈	—	2	2	no	—
110	C₇×C₄⋊C₄	—	2	4	no	—
111	C₇×C₄⋊C₄	—	2	4	no	—
112	C₇×C₄⋊C₄	—	2	4	no	—
113	C₇×C₄⋊C₄	—	4	2	no	—
114	C₇×C₄⋊C₄	—	4	1	no	—
115	C₇×C₄⋊C₄	—	4	2	no	—
116	C₇×C₄⋊C₄	—	4	1	no	—
117	C₇×C₄⋊C₄	—	4	1	no	—
118	C₇×C₄⋊C₄	—	4	1	no	—
119	C₄²×C₇	—	2	1	no	—
120	C₄²×C₇	—	2	2	no	—
121	C₄²×C₇	—	2	1	no	—
122	C₄²×C₇	—	2	1	no	—
123	C₄²×C₇	—	2	2	no	—
124	C₄²×C₇	—	2	1	no	—
125	C₄²×C₇	—	2	1	no	—
126	C₄²×C₇	—	2	1	no	—
127	C₄²×C₇	—	2	2	no	—
128	C₄²×C₇	—	2	2	no	—
129	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
130	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
131	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
132	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
133	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
134	C₂×#97	C₂²×C₇	1	1	yes	characteristic
135	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
136	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
137	#398	C₄×C₇	1	2	yes	—
138	#398	C₄×C₇	1	1	yes	characteristic
139	#398	C₄×C₇	1	1	yes	characteristic
140	#395	C₄×C₇	1	1	yes	characteristic
141	#395	C₄×C₇	1	1	yes	characteristic
142	#395	C₄×C₇	1	2	yes	—
143	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
144	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
145	#376	C₂²×C₇	1	1	yes	characteristic
146	#336	C₂²×C₇	1	2	yes	—
147	#336	C₂²×C₇	1	2	yes	—
148	#336	C₂²×C₇	1	2	yes	—
149	#336	C₂²×C₇	1	2	yes	—
150	#336	C₂²×C₇	1	2	yes	—
151	#336	C₂²×C₇	1	2	yes	—
152	#336	C₂²×C₇	1	2	yes	—
153	#336	C₂²×C₇	1	2	yes	—
154	#336	C₂²×C₇	1	2	yes	—
155	#336	C₂²×C₇	1	2	yes	—
156	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
157	C₇×Q₈	—	4	4	no	—
158	C₇×Q₈	—	4	4	no	—
159	C₂×C₄×C₇	—	2	1	no	—
160	C₂×C₄×C₇	—	2	2	no	—
161	C₂×C₄×C₇	—	2	2	no	—
162	C₂×C₄×C₇	—	2	1	no	—
163	C₂×C₄×C₇	—	2	1	no	—
164	C₂×C₄×C₇	—	2	1	no	—
165	C₂×C₄×C₇	—	2	2	no	—
166	C₂×C₄×C₇	—	2	1	no	—
167	C₂×C₄×C₇	—	2	1	no	—
168	C₂×C₄×C₇	—	2	1	no	—
169	C₂×C₄×C₇	—	2	2	no	—
170	C₂×C₄×C₇	—	2	2	no	—
171	C₂×C₄×C₇	—	2	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	2	no	—
177	C₂×C₄×C₇	—	2	1	no	—
178	C₂×C₄×C₇	—	2	1	no	—
179	C₂×C₄×C₇	—	2	2	no	—
180	C₂×C₄×C₇	—	2	1	no	—
181	C₂×C₄×C₇	—	2	2	no	—
182	C₂×C₄×C₇	—	2	2	no	—
183	C₂×C₄×C₇	—	2	1	no	—
184	C₂×C₄×C₇	#131	1	2	yes	—
185	C₂×C₄×C₇	#131	1	4	yes	—
186	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
187	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
188	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
189	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
190	C₇×C₈	—	4	2	no	—
191	C₇×C₈	—	4	2	no	—
192	C₇×C₈	—	4	2	no	—
193	C₇×C₈	—	4	2	no	—
194	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
195	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₇	1	2	yes	—
197	C₂²×C₈	C₂³×C₇	1	2	yes	—
198	C₂²×C₈	C₂³×C₇	1	2	yes	—
199	C₂²×C₈	C₂³×C₇	1	2	yes	—
200	C₄×Q₈	—	4	1	no	—
201	C₄×Q₈	—	4	1	no	—
202	C₄×Q₈	—	4	1	no	—
203	C₄×Q₈	—	4	1	no	—
204	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
205	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
206	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
207	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
208	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
209	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
210	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
211	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
212	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
213	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
214	C₂×C₄²	C₂³×C₇	1	2	yes	—
215	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
217	C₂×C₄²	C₂³×C₇	1	2	yes	—
218	C₄⋊C₈	—	4	1	no	—
219	C₄⋊C₈	—	4	1	no	—
220	C₄⋊C₈	—	4	2	no	—
221	#97	—	4	1	no	—
222	C₄×C₈	—	4	1	no	—
223	C₄×C₈	—	4	2	no	—
224	#95	C₂×C₄×C₇	1	1	yes	characteristic
225	#95	C₂×C₄×C₇	1	2	yes	—
226	#95	C₂×C₄×C₇	1	1	yes	characteristic
227	#95	C₂×C₄×C₇	1	2	yes	—
228	#95	C₂×C₄×C₇	1	2	yes	—
229	#95	C₂×C₄×C₇	1	2	yes	—
230	#95	C₂×C₄×C₇	1	2	yes	—
231	C₂²×C₇	#518	1	1	yes	characteristic
232	C₂²×C₇	#519	1	2	yes	—
233	C₂²×C₇	C₂×#131	1	1	yes	characteristic
234	C₂²×C₇	C₂×#131	1	2	yes	—
235	C₂²×C₇	#568	1	1	yes	characteristic
236	C₄×C₇	—	2	2	no	—
237	C₄×C₇	—	2	4	no	—
238	C₄×C₇	—	4	2	no	—
239	C₄×C₇	—	4	1	no	—
240	C₄×C₇	—	4	1	no	—
241	C₄×C₇	—	4	1	no	—
242	C₄×C₇	—	4	1	no	—
243	C₄×C₇	—	4	1	no	—
244	C₄×C₇	—	4	2	no	—
245	C₄×C₇	—	4	2	no	—
246	C₄×C₇	#563	1	1	yes	characteristic
247	C₄×C₇	#563	1	1	yes	characteristic
248	C₄×C₇	#563	1	2	yes	—
249	C₂×Q₈	—	2	4	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	—	4	1	no	—
253	C₂×Q₈	—	4	1	no	—
254	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
255	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
256	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
257	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
258	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
259	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
260	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
261	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
262	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
263	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
264	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	4	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄²	—	2	1	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	2	1	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	2	1	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	2	no	—
294	C₂×C₇	#2381	1	1	yes	characteristic
295	C₂×C₇	#2976	1	1	yes	characteristic
296	C₂×C₇	#2977	1	1	yes	characteristic
297	C₂×C₇	#2985	1	2	yes	—
298	C₂×C₇	#2986	1	2	yes	—
299	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
300	Q₈	—	4	4	no	—
301	Q₈	—	4	4	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	2	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	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	C₇×#131	1	2	yes	—
328	C₂×C₄	C₇×#131	1	4	yes	—
329	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
330	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
331	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
332	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
333	C₈	—	4	2	no	—
334	C₈	—	4	2	no	—
335	C₈	—	4	2	no	—
336	C₈	—	4	2	no	—
337	C₇	#23367	1	1	yes	characteristic, direct product factor, Sylow
338	C₂²	C₇×#518	1	1	yes	characteristic
339	C₂²	C₇×#519	1	2	yes	—
340	C₂²	C₂×C₇×#131	1	1	yes	characteristic
341	C₂²	C₂×C₇×#131	1	2	yes	—
342	C₂²	C₇×#568	1	1	yes	characteristic
343	C₄	—	2	2	no	—
344	C₄	—	2	4	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	2	no	—
348	C₄	—	4	1	no	—
349	C₄	—	4	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	C₇×#563	1	1	yes	characteristic
354	C₄	C₇×#563	1	1	yes	characteristic
355	C₄	C₇×#563	1	2	yes	—
356	C₂	C₇×#2381	1	1	yes	characteristic
357	C₂	C₇×#2976	1	1	yes	characteristic
358	C₂	C₇×#2977	1	1	yes	characteristic
359	C₂	C₇×#2985	1	2	yes	—
360	C₂	C₇×#2986	1	2	yes	—

	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. #23367	C₇	direct (N × H)
337. C₇	#23367	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#2399	2
C₇×#1985	2
C₇×#1984	2
C₇×#1917	2
C₇×#1916	2
C₇×#1840	2
#23367	7

Subgroups — C7×#23367 #13200583

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23367 #13200583