Subgroups — C₇×#11269 #13188485

Order 1792.

642 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#423	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#341	C₂	1	2	yes	maximal
3	C₂×C₇×#341	C₂	1	2	yes	maximal
4	C₇×#1728	C₂	1	1	yes	maximal, characteristic
5	C₂×C₇×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
9	C₂³×C₇×C₈	C₂²	1	2	yes	—
10	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₇×#423	—	2	4	no	—
12	C₂×C₇×C₄⋊C₈	—	2	2	no	—
13	C₇×#341	—	2	4	no	—
14	C₇×#341	—	2	4	no	—
15	C₇×#336	—	2	2	no	—
16	C₇×#336	C₄	1	2	yes	—
17	C₇×#336	C₄	1	2	yes	—
18	#11269	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂³×C₄×C₇	Q₈	1	2	yes	—
20	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₂×C₇×M₁₆	—	2	2	no	—
22	C₂×C₇×M₁₆	—	2	1	no	—
23	C₂×C₇×M₁₆	—	2	1	no	—
24	C₂×C₇×M₁₆	—	2	2	no	—
25	C₂×C₇×M₁₆	—	2	2	no	—
26	C₂×C₇×M₁₆	—	2	2	no	—
27	C₂×C₇×M₁₆	—	2	2	no	—
28	C₂×C₇×M₁₆	—	2	2	no	—
29	C₂×C₇×M₁₆	—	2	2	no	—
30	C₂×C₇×M₁₆	—	2	2	no	—
31	C₂×C₇×M₁₆	—	2	2	no	—
32	C₂×C₇×M₁₆	D₄	1	2	yes	—
33	C₂×C₇×M₁₆	D₄	1	4	yes	—
34	C₂×C₇×M₁₆	Q₈	1	2	yes	—
35	C₂²×C₇×C₈	—	2	2	no	—
36	C₂²×C₇×C₈	—	2	2	no	—
37	C₂²×C₇×C₈	—	2	1	no	—
38	C₂²×C₇×C₈	—	2	1	no	—
39	C₂²×C₇×C₈	—	2	2	no	—
40	C₂²×C₇×C₈	—	2	2	no	—
41	C₂²×C₇×C₈	—	2	2	no	—
42	C₂²×C₇×C₈	—	2	2	no	—
43	C₂²×C₇×C₈	—	2	2	no	—
44	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₇	—	2	2	no	—
46	C₂×C₄²×C₇	—	2	2	no	—
47	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₇	D₄	1	2	yes	—
50	C₇×C₄⋊C₈	—	2	8	no	—
51	C₂×#423	C₂×C₇	1	1	yes	characteristic
52	C₂×#341	C₂×C₇	1	2	yes	—
53	C₂×#341	C₂×C₇	1	2	yes	—
54	#1728	C₂×C₇	1	1	yes	characteristic
55	C₂×#336	C₂×C₇	1	1	yes	characteristic
56	C₂⁴×C₇	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₇	—	2	4	no	—
58	C₂²×C₄×C₇	—	2	4	no	—
59	C₂²×C₄×C₇	—	4	2	no	—
60	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₇	#41	1	2	yes	—
68	C₂²×C₄×C₇	#41	1	2	yes	—
69	C₂²×C₄×C₇	#41	1	1	yes	characteristic
70	C₂²×C₄×C₇	#41	1	2	yes	—
71	C₂²×C₄×C₇	#41	1	1	yes	characteristic
72	C₇×M₁₆	—	2	4	no	—
73	C₇×M₁₆	—	2	4	no	—
74	C₇×M₁₆	—	2	4	no	—
75	C₇×M₁₆	—	2	4	no	—
76	C₇×M₁₆	—	4	2	no	—
77	C₇×M₁₆	—	4	2	no	—
78	C₇×M₁₆	—	4	2	no	—
79	C₇×M₁₆	—	4	2	no	—
80	C₂×C₇×C₈	—	2	4	no	—
81	C₂×C₇×C₈	—	2	4	no	—
82	C₂×C₇×C₈	—	2	4	no	—
83	C₂×C₇×C₈	—	2	4	no	—
84	C₂×C₇×C₈	—	2	2	no	—
85	C₂×C₇×C₈	—	2	2	no	—
86	C₂×C₇×C₈	—	4	1	no	—
87	C₂×C₇×C₈	—	4	2	no	—
88	C₂×C₇×C₈	—	4	2	no	—
89	C₂×C₇×C₈	—	4	2	no	—
90	C₂×C₇×C₈	—	4	1	no	—
91	C₂×C₇×C₈	—	4	2	no	—
92	C₂×C₇×C₈	—	4	1	no	—
93	C₂×C₇×C₈	—	4	2	no	—
94	C₂×C₇×C₈	—	4	1	no	—
95	C₂×C₇×C₈	—	4	2	no	—
96	C₄²×C₇	—	2	2	no	—
97	C₄²×C₇	—	2	2	no	—
98	C₄²×C₇	—	2	2	no	—
99	C₄²×C₇	—	2	2	no	—
100	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
101	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
102	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
103	C₂³×C₈	C₂²×C₇	1	2	yes	—
104	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
105	#423	—	2	4	no	—
106	C₂×C₄⋊C₈	—	2	2	no	—
107	#341	—	2	4	no	—
108	#341	—	2	4	no	—
109	#336	—	2	2	no	—
110	#336	C₄×C₇	1	2	yes	—
111	#336	C₄×C₇	1	2	yes	—
112	C₂³×C₇	—	2	2	no	—
113	C₂³×C₇	—	2	2	no	—
114	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂³×C₇	C₂²⋊Q₈	1	2	yes	—
116	C₂³×C₇	#125	1	2	yes	—
117	C₂³×C₇	#125	1	1	yes	characteristic
118	C₂³×C₇	C₄⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₇	—	2	2	no	—
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	—	2	2	no	—
122	C₂×C₄×C₇	—	2	2	no	—
123	C₂×C₄×C₇	—	2	2	no	—
124	C₂×C₄×C₇	—	4	8	no	—
125	C₂×C₄×C₇	#108	1	4	yes	—
126	C₂×C₄×C₇	#117	1	2	yes	—
127	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₇	C₄×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₇	#120	1	1	yes	characteristic
131	C₂×C₄×C₇	#121	1	2	yes	—
132	C₂×C₄×C₇	#121	1	1	yes	characteristic
133	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₇	#123	1	2	yes	—
136	C₂×C₄×C₇	#123	1	2	yes	—
137	C₂×C₄×C₇	#123	1	1	yes	characteristic
138	C₂×C₄×C₇	#124	1	2	yes	—
139	C₂×C₄×C₇	#124	1	1	yes	characteristic
140	C₂×C₄×C₇	#124	1	1	yes	characteristic
141	C₂×C₄×C₇	#125	1	2	yes	—
142	C₂×C₄×C₇	#126	1	2	yes	—
143	C₂×C₄×C₇	#126	1	2	yes	—
144	C₇×C₈	—	4	4	no	—
145	C₇×C₈	—	4	2	no	—
146	C₇×C₈	—	4	4	no	—
147	C₇×C₈	—	4	2	no	—
148	C₂³×C₄	C₇×Q₈	1	2	yes	—
149	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
150	C₂×M₁₆	—	2	2	no	—
151	C₂×M₁₆	—	2	1	no	—
152	C₂×M₁₆	—	2	1	no	—
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	—	2	2	no	—
157	C₂×M₁₆	—	2	2	no	—
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	C₇×D₄	1	2	yes	—
162	C₂×M₁₆	C₇×D₄	1	4	yes	—
163	C₂×M₁₆	C₇×Q₈	1	2	yes	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	1	no	—
167	C₂²×C₈	—	2	1	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
174	C₂×C₄²	—	2	2	no	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
177	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
178	C₂×C₄²	C₇×D₄	1	2	yes	—
179	C₄⋊C₈	—	2	8	no	—
180	C₂²×C₇	—	2	2	no	—
181	C₂²×C₇	—	2	2	no	—
182	C₂²×C₇	—	2	4	no	—
183	C₂²×C₇	—	2	2	no	—
184	C₂²×C₇	—	2	2	no	—
185	C₂²×C₇	#382	1	1	yes	characteristic
186	C₂²×C₇	#385	1	1	yes	characteristic
187	C₂²×C₇	#389	1	1	yes	characteristic
188	C₂²×C₇	#398	1	2	yes	—
189	C₂²×C₇	#400	1	2	yes	—
190	C₂²×C₇	C₂×#108	1	2	yes	—
191	C₂²×C₇	#430	1	2	yes	—
192	C₄×C₇	—	2	4	no	—
193	C₄×C₇	—	4	2	no	—
194	C₄×C₇	#387	1	2	yes	—
195	C₄×C₇	#388	1	2	yes	—
196	C₄×C₇	#393	1	1	yes	characteristic
197	C₄×C₇	#394	1	1	yes	characteristic
198	C₄×C₇	#397	1	1	yes	characteristic
199	C₄×C₇	#400	1	1	yes	characteristic
200	C₂⁴	C₂×C₇×Q₈	1	1	yes	characteristic
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
205	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
206	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
208	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
211	C₂²×C₄	C₇×#41	1	2	yes	—
212	C₂²×C₄	C₇×#41	1	2	yes	—
213	C₂²×C₄	C₇×#41	1	1	yes	characteristic
214	C₂²×C₄	C₇×#41	1	2	yes	—
215	C₂²×C₄	C₇×#41	1	1	yes	characteristic
216	M₁₆	—	2	4	no	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	4	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	—	4	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₂×C₇	—	2	2	no	—
245	C₂×C₇	—	2	2	no	—
246	C₂×C₇	#1442	1	1	yes	characteristic
247	C₂×C₇	#1837	1	1	yes	characteristic
248	C₂×C₇	#1838	1	1	yes	characteristic
249	C₂×C₇	#1855	1	2	yes	—
250	C₂×C₇	#1950	1	2	yes	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
254	C₂³	C₇×C₂²⋊Q₈	1	2	yes	—
255	C₂³	C₇×#125	1	2	yes	—
256	C₂³	C₇×#125	1	1	yes	characteristic
257	C₂³	C₇×C₄⋊Q₈	1	1	yes	characteristic
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	4	8	no	—
264	C₂×C₄	C₇×#108	1	4	yes	—
265	C₂×C₄	C₇×#117	1	2	yes	—
266	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
267	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
268	C₂×C₄	C₄×C₇×Q₈	1	1	yes	characteristic
269	C₂×C₄	C₇×#120	1	1	yes	characteristic
270	C₂×C₄	C₇×#121	1	2	yes	—
271	C₂×C₄	C₇×#121	1	1	yes	characteristic
272	C₂×C₄	C₇×C₂²⋊Q₈	1	2	yes	—
273	C₂×C₄	C₇×C₂²⋊Q₈	1	1	yes	characteristic
274	C₂×C₄	C₇×#123	1	2	yes	—
275	C₂×C₄	C₇×#123	1	2	yes	—
276	C₂×C₄	C₇×#123	1	1	yes	characteristic
277	C₂×C₄	C₇×#124	1	2	yes	—
278	C₂×C₄	C₇×#124	1	1	yes	characteristic
279	C₂×C₄	C₇×#124	1	1	yes	characteristic
280	C₂×C₄	C₇×#125	1	2	yes	—
281	C₂×C₄	C₇×#126	1	2	yes	—
282	C₂×C₄	C₇×#126	1	2	yes	—
283	C₈	—	4	4	no	—
284	C₈	—	4	2	no	—
285	C₈	—	4	4	no	—
286	C₈	—	4	2	no	—
287	C₇	#11269	1	1	yes	characteristic, direct product factor, Sylow
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	C₇×#382	1	1	yes	characteristic
294	C₂²	C₇×#385	1	1	yes	characteristic
295	C₂²	C₇×#389	1	1	yes	characteristic
296	C₂²	C₇×#398	1	2	yes	—
297	C₂²	C₇×#400	1	2	yes	—
298	C₂²	C₂×C₇×#108	1	2	yes	—
299	C₂²	C₇×#430	1	2	yes	—
300	C₄	—	2	4	no	—
301	C₄	—	4	2	no	—
302	C₄	C₇×#387	1	2	yes	—
303	C₄	C₇×#388	1	2	yes	—
304	C₄	C₇×#393	1	1	yes	characteristic
305	C₄	C₇×#394	1	1	yes	characteristic
306	C₄	C₇×#397	1	1	yes	characteristic
307	C₄	C₇×#400	1	1	yes	characteristic
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	C₇×#1442	1	1	yes	characteristic
311	C₂	C₇×#1837	1	1	yes	characteristic
312	C₂	C₇×#1838	1	1	yes	characteristic
313	C₂	C₇×#1855	1	2	yes	—
314	C₂	C₇×#1950	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
socle	C₂³×C₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
18. #11269	C₇	direct (N × H)
287. C₇	#11269	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×#423	2
C₂×C₇×#341	2
C₇×#1728	2
C₂×C₇×#336	2
#11269	7

Subgroups — C7×#11269 #13188485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#11269 #13188485