Subgroups — C₇×#10798 #13188014

Order 1792.

870 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#336	C₂	1	2	yes	maximal
3	C₂×C₇×#336	C₂	1	2	yes	maximal
4	C₂×C₇×#336	C₂	1	1	yes	maximal, characteristic
5	C₄×C₇×#95	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₇×C₈	C₂²	1	2	yes	—
7	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₇×#97	—	2	4	no	—
12	C₂×C₇×#97	C₄	1	4	yes	complemented
13	C₂×C₇×#95	C₄	1	4	yes	—
14	C₇×#336	—	2	4	no	—
15	C₇×#336	—	2	4	no	—
16	C₇×#336	C₄	1	4	yes	complemented
17	#10798	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₇	C₂×C₄	1	4	yes	—
20	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
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	1	no	—
25	C₂²×C₇×C₈	—	2	1	no	—
26	C₂²×C₇×C₈	—	2	2	no	—
27	C₂²×C₇×C₈	—	2	2	no	—
28	C₂²×C₇×C₈	—	2	2	no	—
29	C₂²×C₇×C₈	—	2	2	no	—
30	C₂²×C₇×C₈	—	2	2	no	—
31	C₂²×C₇×C₈	—	2	2	no	—
32	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
33	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₇×C₈	D₄	1	4	yes	—
36	C₂²×C₇×C₈	D₄	1	2	yes	—
37	C₂²×C₇×C₈	Q₈	1	2	yes	—
38	C₂×C₄²×C₇	—	2	2	no	—
39	C₂×C₄²×C₇	—	2	2	no	—
40	C₂×C₄²×C₇	—	2	2	no	—
41	C₂×C₄²×C₇	—	2	2	no	—
42	C₂×C₄²×C₇	—	2	2	no	—
43	C₂×C₄²×C₇	—	2	2	no	—
44	C₂×C₄²×C₇	—	2	2	no	—
45	C₂×C₄²×C₇	—	2	2	no	—
46	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
48	C₇×#97	—	2	8	no	—
49	C₇×#95	—	2	8	no	—
50	C₂²×#97	C₂×C₇	1	1	yes	characteristic
51	C₂×#336	C₂×C₇	1	2	yes	—
52	C₂×#336	C₂×C₇	1	2	yes	—
53	C₂×#336	C₂×C₇	1	1	yes	characteristic
54	C₄×#95	C₂×C₇	1	1	yes	characteristic
55	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇	—	2	4	no	—
57	C₂²×C₄×C₇	—	2	2	no	—
58	C₂²×C₄×C₇	—	2	2	no	—
59	C₂²×C₄×C₇	—	2	4	no	—
60	C₂²×C₄×C₇	—	2	4	no	—
61	C₂²×C₄×C₇	—	2	4	no	—
62	C₂²×C₄×C₇	—	2	4	no	—
63	C₂²×C₄×C₇	—	2	4	no	—
64	C₂²×C₄×C₇	—	2	4	no	—
65	C₂²×C₄×C₇	—	2	4	no	—
66	C₂²×C₄×C₇	C₄²	1	4	yes	—
67	C₂²×C₄×C₇	M₁₆	1	8	yes	—
68	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₇	#41	1	2	yes	—
74	C₂²×C₄×C₇	#41	1	2	yes	—
75	C₂²×C₄×C₇	#41	1	1	yes	characteristic
76	C₂²×C₄×C₇	#41	1	2	yes	—
77	C₂²×C₄×C₇	#41	1	1	yes	characteristic
78	C₂×C₇×C₈	—	2	2	no	—
79	C₂×C₇×C₈	—	2	2	no	—
80	C₂×C₇×C₈	—	2	2	no	—
81	C₂×C₇×C₈	—	2	2	no	—
82	C₂×C₇×C₈	—	2	4	no	—
83	C₂×C₇×C₈	—	2	4	no	—
84	C₂×C₇×C₈	—	2	4	no	—
85	C₂×C₇×C₈	—	2	4	no	—
86	C₂×C₇×C₈	—	2	4	no	—
87	C₂×C₇×C₈	—	2	4	no	—
88	C₂×C₇×C₈	—	2	4	no	—
89	C₂×C₇×C₈	—	4	2	no	—
90	C₂×C₇×C₈	—	4	2	no	—
91	C₂×C₇×C₈	—	4	2	no	—
92	C₂×C₇×C₈	—	4	2	no	—
93	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
94	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
95	C₄²×C₇	—	2	2	no	—
96	C₄²×C₇	—	2	2	no	—
97	C₄²×C₇	—	2	4	no	—
98	C₄²×C₇	—	4	2	no	—
99	C₄²×C₇	—	4	2	no	—
100	C₄²×C₇	—	4	2	no	—
101	C₄²×C₇	—	4	2	no	—
102	C₂³×C₈	C₂²×C₇	1	2	yes	—
103	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
104	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
105	C₂²×C₄²	C₂²×C₇	1	2	yes	—
106	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
107	C₂×#97	—	2	4	no	—
108	C₂×#97	C₄×C₇	1	4	yes	complemented
109	C₂×#95	C₄×C₇	1	4	yes	—
110	#336	—	2	4	no	—
111	#336	—	2	4	no	—
112	#336	C₄×C₇	1	4	yes	complemented
113	C₂³×C₇	C₂×C₄²	1	1	yes	characteristic
114	C₂³×C₇	#117	1	2	yes	—
115	C₂³×C₇	C₄×D₄	1	2	yes	—
116	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
117	C₂³×C₇	C₄×Q₈	1	1	yes	characteristic
118	C₂³×C₇	C₂×M₁₆	1	4	yes	—
119	C₂³×C₇	#131	1	4	yes	—
120	C₂×C₄×C₇	—	2	4	no	—
121	C₂×C₄×C₇	—	2	4	no	—
122	C₂×C₄×C₇	—	2	4	no	—
123	C₂×C₄×C₇	—	2	8	no	—
124	C₂×C₄×C₇	—	2	8	no	—
125	C₂×C₄×C₇	—	2	8	no	—
126	C₂×C₄×C₇	—	4	4	no	—
127	C₂×C₄×C₇	—	4	4	no	—
128	C₂×C₄×C₇	—	4	4	no	—
129	C₂×C₄×C₇	—	4	4	no	—
130	C₂×C₄×C₇	—	4	4	no	—
131	C₂×C₄×C₇	—	4	4	no	—
132	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₇	#117	1	1	yes	characteristic
135	C₂×C₄×C₇	#117	1	1	yes	characteristic
136	C₂×C₄×C₇	#117	1	1	yes	characteristic
137	C₂×C₄×C₇	#117	1	1	yes	characteristic
138	C₂×C₄×C₇	#117	1	2	yes	—
139	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
140	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₇	C₄×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₇	#121	1	2	yes	—
143	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
144	C₂×C₄×C₇	#123	1	2	yes	—
145	C₂×C₄×C₇	#123	1	2	yes	—
146	C₂×C₄×C₇	#124	1	2	yes	—
147	C₂×C₄×C₇	#125	1	2	yes	—
148	C₂×C₄×C₇	#126	1	2	yes	—
149	C₂×C₄×C₇	#126	1	2	yes	—
150	C₇×C₈	—	2	4	no	—
151	C₇×C₈	—	2	4	no	—
152	C₇×C₈	—	4	4	no	—
153	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
154	C₂³×C₄	C₂×C₄×C₇	1	4	yes	—
155	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	—	2	1	no	—
160	C₂²×C₈	—	2	1	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
168	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
169	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
170	C₂²×C₈	C₇×D₄	1	4	yes	—
171	C₂²×C₈	C₇×D₄	1	2	yes	—
172	C₂²×C₈	C₇×Q₈	1	2	yes	—
173	C₂×C₄²	—	2	2	no	—
174	C₂×C₄²	—	2	2	no	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	—	2	2	no	—
177	C₂×C₄²	—	2	2	no	—
178	C₂×C₄²	—	2	2	no	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	—	2	2	no	—
181	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
182	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
183	#97	—	2	8	no	—
184	#95	—	2	8	no	—
185	C₂²×C₇	#376	1	1	yes	characteristic
186	C₂²×C₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
187	C₂²×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
188	C₂²×C₇	#387	1	2	yes	—
189	C₂²×C₇	#388	1	2	yes	—
190	C₂²×C₇	C₄×M₁₆	1	2	yes	—
191	C₂²×C₇	#405	1	2	yes	—
192	C₂²×C₇	#432	1	4	yes	—
193	C₂²×C₇	#433	1	4	yes	—
194	C₂²×C₇	#435	1	4	yes	—
195	C₂²×C₇	#435	1	4	yes	—
196	C₂²×C₇	#436	1	4	yes	—
197	C₂²×C₇	#446	1	4	yes	—
198	C₄×C₇	—	4	8	no	—
199	C₄×C₇	—	4	4	no	—
200	C₄×C₇	#380	1	1	yes	characteristic
201	C₄×C₇	#381	1	1	yes	characteristic
202	C₄×C₇	#382	1	1	yes	characteristic
203	C₄×C₇	#383	1	1	yes	characteristic
204	C₄×C₇	#387	1	2	yes	—
205	C₄×C₇	#388	1	2	yes	—
206	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	C₄²×C₇	1	4	yes	—
218	C₂²×C₄	C₇×M₁₆	1	8	yes	—
219	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
222	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
224	C₂²×C₄	C₇×#41	1	2	yes	—
225	C₂²×C₄	C₇×#41	1	2	yes	—
226	C₂²×C₄	C₇×#41	1	1	yes	characteristic
227	C₂²×C₄	C₇×#41	1	2	yes	—
228	C₂²×C₄	C₇×#41	1	1	yes	characteristic
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	C₇×C₂²⋊C₄	1	4	yes	—
245	C₂×C₈	C₇×C₄⋊C₄	1	4	yes	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	4	no	—
249	C₄²	—	4	2	no	—
250	C₄²	—	4	2	no	—
251	C₄²	—	4	2	no	—
252	C₄²	—	4	2	no	—
253	C₂×C₇	#1434	1	1	yes	characteristic
254	C₂×C₇	#1745	1	1	yes	characteristic
255	C₂×C₇	#1746	1	1	yes	characteristic
256	C₂×C₇	#1753	1	2	yes	—
257	C₂×C₇	#1771	1	2	yes	—
258	C₂×C₇	#1917	1	4	yes	—
259	C₂×C₇	#1924	1	4	yes	—
260	C₂³	C₂×C₄²×C₇	1	1	yes	characteristic
261	C₂³	C₇×#117	1	2	yes	—
262	C₂³	C₄×C₇×D₄	1	2	yes	—
263	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
264	C₂³	C₄×C₇×Q₈	1	1	yes	characteristic
265	C₂³	C₂×C₇×M₁₆	1	4	yes	—
266	C₂³	C₇×#131	1	4	yes	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
280	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
281	C₂×C₄	C₇×#117	1	1	yes	characteristic
282	C₂×C₄	C₇×#117	1	1	yes	characteristic
283	C₂×C₄	C₇×#117	1	1	yes	characteristic
284	C₂×C₄	C₇×#117	1	1	yes	characteristic
285	C₂×C₄	C₇×#117	1	2	yes	—
286	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
287	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
288	C₂×C₄	C₄×C₇×Q₈	1	1	yes	characteristic
289	C₂×C₄	C₇×#121	1	2	yes	—
290	C₂×C₄	C₇×C₂²⋊Q₈	1	2	yes	—
291	C₂×C₄	C₇×#123	1	2	yes	—
292	C₂×C₄	C₇×#123	1	2	yes	—
293	C₂×C₄	C₇×#124	1	2	yes	—
294	C₂×C₄	C₇×#125	1	2	yes	—
295	C₂×C₄	C₇×#126	1	2	yes	—
296	C₂×C₄	C₇×#126	1	2	yes	—
297	C₈	—	2	4	no	—
298	C₈	—	2	4	no	—
299	C₈	—	4	4	no	—
300	C₇	#10798	1	1	yes	characteristic, direct product factor, Sylow
301	C₂²	C₇×#376	1	1	yes	characteristic
302	C₂²	C₄×C₇×C₂²⋊C₄	1	1	yes	characteristic
303	C₂²	C₄×C₇×C₄⋊C₄	1	1	yes	characteristic
304	C₂²	C₇×#387	1	2	yes	—
305	C₂²	C₇×#388	1	2	yes	—
306	C₂²	C₄×C₇×M₁₆	1	2	yes	—
307	C₂²	C₇×#405	1	2	yes	—
308	C₂²	C₇×#432	1	4	yes	—
309	C₂²	C₇×#433	1	4	yes	—
310	C₂²	C₇×#435	1	4	yes	—
311	C₂²	C₇×#435	1	4	yes	—
312	C₂²	C₇×#436	1	4	yes	—
313	C₂²	C₇×#446	1	4	yes	—
314	C₄	—	4	4	no	—
315	C₄	—	4	8	no	—
316	C₄	C₇×#380	1	1	yes	characteristic
317	C₄	C₇×#381	1	1	yes	characteristic
318	C₄	C₇×#382	1	1	yes	characteristic
319	C₄	C₇×#383	1	1	yes	characteristic
320	C₄	C₇×#387	1	2	yes	—
321	C₄	C₇×#388	1	2	yes	—
322	C₂	C₇×#1434	1	1	yes	characteristic
323	C₂	C₇×#1745	1	1	yes	characteristic
324	C₂	C₇×#1746	1	1	yes	characteristic
325	C₂	C₇×#1753	1	2	yes	—
326	C₂	C₇×#1771	1	2	yes	—
327	C₂	C₇×#1917	1	4	yes	—
328	C₂	C₇×#1924	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₇	C₂³
commutator	C₂³	C₂×C₄²×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
socle, prime radical	C₂⁴×C₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₂×C₇×#97	C₄	semidirect
16. C₇×#336	C₄	semidirect
17. #10798	C₇	direct (N × H)
108. C₂×#97	C₄×C₇	semidirect
112. #336	C₄×C₇	semidirect
300. C₇	#10798	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×#97	2
C₂×C₇×#336	2
C₄×C₇×#95	2
#10798	7

Subgroups — C7×#10798 #13188014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#10798 #13188014