Subgroups — C₇×#11271 #13188487

Order 1792.

634 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#106	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#343	C₂	1	2	yes	maximal
3	C₂×C₇×#341	C₂	1	1	yes	maximal, characteristic
4	C₇×#1736	C₂	1	2	yes	maximal
5	C₇×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
8	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₇×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₇×#106	—	2	2	no	—
12	C₂×C₇×#106	—	2	4	no	—
13	C₇×#343	—	2	4	no	—
14	C₇×#341	C₄	1	4	yes	complemented
15	C₇×#340	—	2	4	no	—
16	C₇×#336	—	2	2	no	—
17	#11271	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₇	—	2	2	no	—
19	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₂×C₇×M₁₆	—	2	2	no	—
21	C₂×C₇×M₁₆	—	2	1	no	—
22	C₂×C₇×M₁₆	—	2	2	no	—
23	C₂×C₇×M₁₆	—	2	2	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	C₂×C₄	1	4	yes	—
29	C₂×C₇×M₁₆	Q₈	1	2	yes	—
30	C₂²×C₇×C₈	—	2	2	no	—
31	C₂²×C₇×C₈	—	2	2	no	—
32	C₂²×C₇×C₈	—	2	1	no	—
33	C₂²×C₇×C₈	—	2	2	no	—
34	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₇×C₈	D₄	1	2	yes	—
36	C₂²×C₇×C₈	Q₈	1	2	yes	—
37	C₇×#117	—	2	2	no	—
38	C₇×#117	—	2	2	no	—
39	C₂×C₇×C₄⋊C₄	—	2	2	no	—
40	C₂×C₇×C₄⋊C₄	—	2	2	no	—
41	C₂×C₇×C₄⋊C₄	—	2	2	no	—
42	C₂×C₇×C₄⋊C₄	—	2	2	no	—
43	C₂×C₇×C₄⋊C₄	—	2	2	no	—
44	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
45	C₂×C₇×C₄⋊C₄	D₄	1	4	yes	—
46	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
47	C₂×C₄²×C₇	—	2	1	no	—
48	C₇×#106	—	2	8	no	—
49	C₂²×#106	C₂×C₇	1	1	yes	characteristic
50	C₂×#343	C₂×C₇	1	2	yes	—
51	C₂×#341	C₂×C₇	1	1	yes	characteristic
52	#1736	C₂×C₇	1	2	yes	—
53	#1729	C₂×C₇	1	1	yes	characteristic
54	C₂⁴×C₇	#41	1	1	yes	characteristic
55	C₂²×C₄×C₇	—	2	2	no	—
56	C₂²×C₄×C₇	—	2	4	no	—
57	C₂²×C₄×C₇	—	4	2	no	—
58	C₂²×C₄×C₇	—	4	2	no	—
59	C₂²×C₄×C₇	—	8	4	no	—
60	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₇	#41	1	2	yes	—
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	C₇×M₁₆	—	2	4	no	—
71	C₇×M₁₆	—	2	4	no	—
72	C₇×M₁₆	—	2	4	no	—
73	C₇×M₁₆	—	4	2	no	—
74	C₇×M₁₆	—	4	2	no	—
75	C₇×M₁₆	—	4	2	no	—
76	C₂×C₇×C₈	—	2	4	no	—
77	C₂×C₇×C₈	—	2	2	no	—
78	C₂×C₇×C₈	—	2	4	no	—
79	C₂×C₇×C₈	—	2	2	no	—
80	C₂×C₇×C₈	—	2	4	no	—
81	C₂×C₇×C₈	—	2	2	no	—
82	C₂×C₇×C₈	—	4	1	no	—
83	C₂×C₇×C₈	—	4	2	no	—
84	C₂×C₇×C₈	—	4	2	no	—
85	C₂×C₇×C₈	—	4	1	no	—
86	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
87	C₇×C₄⋊C₄	—	2	4	no	—
88	C₇×C₄⋊C₄	—	2	4	no	—
89	C₇×C₄⋊C₄	—	2	4	no	—
90	C₇×C₄⋊C₄	—	4	2	no	—
91	C₇×C₄⋊C₄	—	4	2	no	—
92	C₇×C₂²⋊C₄	—	4	2	no	—
93	C₄²×C₇	—	4	1	no	—
94	C₄²×C₇	—	4	1	no	—
95	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
96	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
97	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
98	C₂×#117	C₂²×C₇	1	1	yes	characteristic
99	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
100	C₂×#106	—	2	2	no	—
101	C₂×#106	—	2	4	no	—
102	#343	—	2	4	no	—
103	#341	C₄×C₇	1	4	yes	complemented
104	#340	—	2	4	no	—
105	#336	—	2	2	no	—
106	C₂³×C₇	—	2	4	no	—
107	C₂³×C₇	C₄×Q₈	1	1	yes	characteristic
108	C₂³×C₇	#121	1	1	yes	characteristic
109	C₂³×C₇	#123	1	2	yes	—
110	C₂³×C₇	#125	1	1	yes	characteristic
111	C₂³×C₇	#126	1	2	yes	—
112	C₂×C₄×C₇	—	4	4	no	—
113	C₂×C₄×C₇	—	4	2	no	—
114	C₂×C₄×C₇	—	4	4	no	—
115	C₂×C₄×C₇	—	8	4	no	—
116	C₂×C₄×C₇	—	8	4	no	—
117	C₂×C₄×C₇	—	8	4	no	—
118	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₇	#117	1	2	yes	—
120	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇	#120	1	1	yes	characteristic
123	C₂×C₄×C₇	#121	1	2	yes	—
124	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₇	#123	1	1	yes	characteristic
128	C₂×C₄×C₇	#123	1	2	yes	—
129	C₂×C₄×C₇	#124	1	1	yes	characteristic
130	C₂×C₄×C₇	#124	1	1	yes	characteristic
131	C₂×C₄×C₇	#124	1	2	yes	—
132	C₂×C₄×C₇	#125	1	2	yes	—
133	C₂×C₄×C₇	#125	1	2	yes	—
134	C₂×C₄×C₇	#126	1	2	yes	—
135	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
136	C₇×C₈	—	2	4	no	—
137	C₇×C₈	—	4	4	no	—
138	C₇×C₈	—	4	2	no	—
139	C₂³×C₄	—	2	2	no	—
140	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	2	1	no	—
143	C₂×M₁₆	—	2	2	no	—
144	C₂×M₁₆	—	2	2	no	—
145	C₂×M₁₆	—	2	2	no	—
146	C₂×M₁₆	—	2	2	no	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	C₂×C₄×C₇	1	4	yes	—
150	C₂×M₁₆	C₇×Q₈	1	2	yes	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
156	C₂²×C₈	C₇×D₄	1	2	yes	—
157	C₂²×C₈	C₇×Q₈	1	2	yes	—
158	#117	—	2	2	no	—
159	#117	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	C₇×D₄	1	4	yes	—
166	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
167	C₂×C₂²⋊C₄	—	2	1	no	—
168	C₂×C₄²	—	2	1	no	—
169	#106	—	2	8	no	—
170	C₂²×C₇	—	2	4	no	—
171	C₂²×C₇	—	2	2	no	—
172	C₂²×C₇	—	2	2	no	—
173	C₂²×C₇	—	2	2	no	—
174	C₂²×C₇	—	2	2	no	—
175	C₂²×C₇	#387	1	2	yes	—
176	C₂²×C₇	#389	1	1	yes	characteristic
177	C₂²×C₇	#394	1	1	yes	characteristic
178	C₂²×C₇	#397	1	1	yes	characteristic
179	C₂²×C₇	#400	1	2	yes	—
180	C₂²×C₇	#471	1	4	yes	—
181	C₄×C₇	—	8	2	no	—
182	C₄×C₇	—	8	4	no	—
183	C₄×C₇	#382	1	1	yes	characteristic
184	C₄×C₇	#385	1	1	yes	characteristic
185	C₄×C₇	#388	1	2	yes	—
186	C₄×C₇	#393	1	1	yes	characteristic
187	C₄×C₇	#398	1	2	yes	—
188	C₄×C₇	#400	1	1	yes	characteristic
189	C₂⁴	C₇×#41	1	1	yes	characteristic
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	8	4	no	—
195	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
198	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
201	C₂²×C₄	C₇×#41	1	2	yes	—
202	C₂²×C₄	C₇×#41	1	2	yes	—
203	C₂²×C₄	C₇×#41	1	2	yes	—
204	C₂²×C₄	C₇×#41	1	1	yes	characteristic
205	M₁₆	—	2	4	no	—
206	M₁₆	—	2	4	no	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	4	2	no	—
209	M₁₆	—	4	2	no	—
210	M₁₆	—	4	2	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	C₇×C₄⋊C₄	1	4	yes	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₂²⋊C₄	—	4	2	no	—
228	C₄²	—	4	1	no	—
229	C₄²	—	4	1	no	—
230	C₂×C₇	—	2	2	no	—
231	C₂×C₇	—	2	2	no	—
232	C₂×C₇	#1442	1	1	yes	characteristic
233	C₂×C₇	#1951	1	2	yes	—
234	C₂×C₇	#2046	1	2	yes	—
235	C₂×C₇	#2079	1	2	yes	—
236	C₂³	—	2	4	no	—
237	C₂³	C₄×C₇×Q₈	1	1	yes	characteristic
238	C₂³	C₇×#121	1	1	yes	characteristic
239	C₂³	C₇×#123	1	2	yes	—
240	C₂³	C₇×#125	1	1	yes	characteristic
241	C₂³	C₇×#126	1	2	yes	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	—	8	4	no	—
247	C₂×C₄	—	8	4	no	—
248	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
249	C₂×C₄	C₇×#117	1	2	yes	—
250	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
251	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
252	C₂×C₄	C₇×#120	1	1	yes	characteristic
253	C₂×C₄	C₇×#121	1	2	yes	—
254	C₂×C₄	C₇×C₂²⋊Q₈	1	2	yes	—
255	C₂×C₄	C₇×C₂²⋊Q₈	1	2	yes	—
256	C₂×C₄	C₇×C₂²⋊Q₈	1	1	yes	characteristic
257	C₂×C₄	C₇×#123	1	1	yes	characteristic
258	C₂×C₄	C₇×#123	1	2	yes	—
259	C₂×C₄	C₇×#124	1	1	yes	characteristic
260	C₂×C₄	C₇×#124	1	1	yes	characteristic
261	C₂×C₄	C₇×#124	1	2	yes	—
262	C₂×C₄	C₇×#125	1	2	yes	—
263	C₂×C₄	C₇×#125	1	2	yes	—
264	C₂×C₄	C₇×#126	1	2	yes	—
265	C₂×C₄	C₇×C₄⋊Q₈	1	1	yes	characteristic
266	C₈	—	2	4	no	—
267	C₈	—	4	4	no	—
268	C₈	—	4	2	no	—
269	C₇	#11271	1	1	yes	characteristic, direct product factor, Sylow
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	C₇×#387	1	2	yes	—
276	C₂²	C₇×#389	1	1	yes	characteristic
277	C₂²	C₇×#394	1	1	yes	characteristic
278	C₂²	C₇×#397	1	1	yes	characteristic
279	C₂²	C₇×#400	1	2	yes	—
280	C₂²	C₇×#471	1	4	yes	—
281	C₄	—	8	4	no	—
282	C₄	—	8	2	no	—
283	C₄	C₇×#382	1	1	yes	characteristic
284	C₄	C₇×#385	1	1	yes	characteristic
285	C₄	C₇×#388	1	2	yes	—
286	C₄	C₇×#393	1	1	yes	characteristic
287	C₄	C₇×#398	1	2	yes	—
288	C₄	C₇×#400	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	—	2	2	no	—
291	C₂	C₇×#1442	1	1	yes	characteristic
292	C₂	C₇×#1951	1	2	yes	—
293	C₂	C₇×#2046	1	2	yes	—
294	C₂	C₇×#2079	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	#121
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
prime radical	C₂⁴×C₇	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂³×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₇×#341	C₄	semidirect
17. #11271	C₇	direct (N × H)
103. #341	C₄×C₇	semidirect
269. C₇	#11271	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×#106	2
C₂×C₇×#343	2
C₂×C₇×#341	2
C₇×#1736	2
C₇×#1729	2
#11271	7

Subgroups — C7×#11271 #13188487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#11271 #13188487