Subgroups — C₇×#10799 #13188015

Order 1792.

650 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#338	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×H(3,ℤ/4)	C₂	1	2	yes	maximal
5	C₇×#1729	C₂	1	2	yes	maximal
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	2	yes	—
10	C₂×C₇×#117	C₂²	1	1	yes	characteristic
11	C₇×#405	—	2	4	no	—
12	C₂×C₇×#97	C₄	1	2	yes	complemented
13	C₂×C₄×C₇×C₈	C₄	1	2	yes	complemented
14	C₇×#338	—	2	2	no	—
15	C₇×H(3,ℤ/4)	—	2	2	no	—
16	C₇×H(3,ℤ/4)	—	2	4	no	—
17	C₇×#336	C₄	1	8	yes	complemented
18	#10799	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₂×C₇×M₁₆	—	2	2	no	—
21	C₂×C₇×M₁₆	—	2	2	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	D₄	1	2	yes	—
29	C₂×C₇×M₁₆	Q₈	1	2	yes	—
30	C₂²×C₇×C₈	—	2	2	no	—
31	C₂²×C₇×C₈	—	2	1	no	—
32	C₂²×C₇×C₈	—	2	1	no	—
33	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
34	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
35	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
38	C₇×#117	—	2	2	no	—
39	C₇×#117	—	2	2	no	—
40	C₇×#117	—	2	2	no	—
41	C₇×#117	—	2	2	no	—
42	C₇×#117	—	2	2	no	—
43	C₇×#117	D₄	1	4	yes	—
44	C₂×C₇×C₄⋊C₄	—	2	1	no	—
45	C₂×C₇×C₄⋊C₄	—	2	2	no	—
46	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
47	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
48	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
50	C₇×#97	—	2	4	no	—
51	C₄×C₇×C₈	—	2	4	no	—
52	C₂×#405	C₂×C₇	1	1	yes	characteristic
53	C₂×#338	C₂×C₇	1	1	yes	characteristic
54	C₂×H(3,ℤ/4)	C₂×C₇	1	1	yes	characteristic
55	C₂×H(3,ℤ/4)	C₂×C₇	1	2	yes	—
56	#1729	C₂×C₇	1	2	yes	—
57	C₂⁴×C₇	#41	1	1	yes	characteristic
58	C₂²×C₄×C₇	—	2	2	no	—
59	C₂²×C₄×C₇	—	2	4	no	—
60	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
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	#41	1	2	yes	—
66	C₂²×C₄×C₇	#41	1	1	yes	characteristic
67	C₂²×C₄×C₇	#41	1	1	yes	characteristic
68	C₂²×C₄×C₇	#41	1	1	yes	characteristic
69	C₂²×C₄×C₇	#41	1	2	yes	—
70	C₇×M₁₆	—	2	4	no	—
71	C₇×M₁₆	—	2	4	no	—
72	C₇×M₁₆	—	4	2	no	—
73	C₇×M₁₆	—	4	2	no	—
74	C₇×M₁₆	—	4	2	no	—
75	C₇×M₁₆	—	4	2	no	—
76	C₂×C₇×C₈	—	2	1	no	—
77	C₂×C₇×C₈	—	2	2	no	—
78	C₂×C₇×C₈	—	2	1	no	—
79	C₂×C₇×C₈	—	2	1	no	—
80	C₂×C₇×C₈	—	2	1	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	C₄²	1	2	yes	—
88	C₂×C₇×C₈	C₄²	1	2	yes	—
89	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
90	C₇×C₄⋊C₄	—	2	4	no	—
91	C₇×C₄⋊C₄	—	4	2	no	—
92	C₇×C₄⋊C₄	—	4	2	no	—
93	C₇×C₂²⋊C₄	—	2	4	no	—
94	C₇×C₂²⋊C₄	—	4	2	no	—
95	C₇×C₂²⋊C₄	—	4	2	no	—
96	C₄²×C₇	—	2	2	no	—
97	C₄²×C₇	—	2	4	no	—
98	C₄²×C₇	—	2	4	no	—
99	C₄²×C₇	C₂²⋊C₄	1	4	yes	—
100	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
101	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
102	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
103	C₂×#117	C₂²×C₇	1	2	yes	—
104	C₂×#117	C₂²×C₇	1	1	yes	characteristic
105	#405	—	2	4	no	—
106	C₂×#97	C₄×C₇	1	2	yes	complemented
107	C₂×C₄×C₈	C₄×C₇	1	2	yes	complemented
108	#338	—	2	2	no	—
109	H(3,ℤ/4)	—	2	2	no	—
110	H(3,ℤ/4)	—	2	4	no	—
111	#336	C₄×C₇	1	8	yes	complemented
112	C₂³×C₇	—	4	2	no	—
113	C₂³×C₇	#117	1	1	yes	characteristic
114	C₂³×C₇	#117	1	2	yes	—
115	C₂³×C₇	#123	1	2	yes	—
116	C₂³×C₇	#124	1	1	yes	characteristic
117	C₂³×C₇	#125	1	1	yes	characteristic
118	C₂×C₄×C₇	—	2	4	no	—
119	C₂×C₄×C₇	—	4	4	no	—
120	C₂×C₄×C₇	—	4	8	no	—
121	C₂×C₄×C₇	—	4	4	no	—
122	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₇	#117	1	1	yes	characteristic
126	C₂×C₄×C₇	#117	1	1	yes	characteristic
127	C₂×C₄×C₇	#117	1	1	yes	characteristic
128	C₂×C₄×C₇	#117	1	2	yes	—
129	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
130	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
133	C₂×C₄×C₇	#121	1	2	yes	—
134	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₇	#123	1	2	yes	—
136	C₂×C₄×C₇	#124	1	1	yes	characteristic
137	C₂×C₄×C₇	#125	1	1	yes	characteristic
138	C₂×C₄×C₇	#126	1	2	yes	—
139	C₂×C₄×C₇	#126	1	2	yes	—
140	C₇×C₈	—	2	2	no	—
141	C₇×C₈	—	2	2	no	—
142	C₇×C₈	—	2	4	no	—
143	C₇×C₈	—	4	4	no	—
144	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
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	—	2	2	no	—
150	C₂×M₁₆	—	2	2	no	—
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	C₇×D₄	1	2	yes	—
154	C₂×M₁₆	C₇×Q₈	1	2	yes	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
159	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
160	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
161	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
162	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
163	#117	—	2	2	no	—
164	#117	—	2	2	no	—
165	#117	—	2	2	no	—
166	#117	—	2	2	no	—
167	#117	—	2	2	no	—
168	#117	C₇×D₄	1	4	yes	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₂²⋊C₄	—	2	1	no	—
172	C₂×C₂²⋊C₄	—	2	2	no	—
173	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
174	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
175	#97	—	2	4	no	—
176	C₄×C₈	—	2	4	no	—
177	C₂²×C₇	—	2	2	no	—
178	C₂²×C₇	—	2	4	no	—
179	C₂²×C₇	—	4	2	no	—
180	C₂²×C₇	—	4	1	no	—
181	C₂²×C₇	—	4	1	no	—
182	C₂²×C₇	#376	1	1	yes	characteristic
183	C₂²×C₇	#381	1	1	yes	characteristic
184	C₂²×C₇	#382	1	1	yes	characteristic
185	C₂²×C₇	#387	1	2	yes	—
186	C₂²×C₇	#388	1	2	yes	—
187	C₄×C₇	—	4	4	no	—
188	C₄×C₇	—	8	4	no	—
189	C₄×C₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
190	C₄×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
191	C₄×C₇	#380	1	1	yes	characteristic
192	C₄×C₇	#383	1	1	yes	characteristic
193	C₄×C₇	#387	1	2	yes	—
194	C₄×C₇	#388	1	2	yes	—
195	C₂⁴	C₇×#41	1	1	yes	characteristic
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
199	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
200	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
201	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
203	C₂²×C₄	C₇×#41	1	2	yes	—
204	C₂²×C₄	C₇×#41	1	1	yes	characteristic
205	C₂²×C₄	C₇×#41	1	1	yes	characteristic
206	C₂²×C₄	C₇×#41	1	1	yes	characteristic
207	C₂²×C₄	C₇×#41	1	2	yes	—
208	M₁₆	—	2	4	no	—
209	M₁₆	—	2	4	no	—
210	M₁₆	—	4	2	no	—
211	M₁₆	—	4	2	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	4	2	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	C₄²×C₇	1	2	yes	—
226	C₂×C₈	C₄²×C₇	1	2	yes	—
227	C₂×C₈	C₇×C₄⋊C₄	1	4	yes	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	—	4	2	no	—
233	C₂²⋊C₄	—	4	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	2	4	no	—
237	C₄²	C₇×C₂²⋊C₄	1	4	yes	—
238	C₂×C₇	—	2	1	no	—
239	C₂×C₇	—	2	2	no	—
240	C₂×C₇	—	2	1	no	—
241	C₂×C₇	—	2	2	no	—
242	C₂×C₇	#1434	1	1	yes	characteristic
243	C₂×C₇	#1747	1	2	yes	—
244	C₂³	—	4	2	no	—
245	C₂³	C₇×#117	1	1	yes	characteristic
246	C₂³	C₇×#117	1	2	yes	—
247	C₂³	C₇×#123	1	2	yes	—
248	C₂³	C₇×#124	1	1	yes	characteristic
249	C₂³	C₇×#125	1	1	yes	characteristic
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	8	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
255	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
257	C₂×C₄	C₇×#117	1	1	yes	characteristic
258	C₂×C₄	C₇×#117	1	1	yes	characteristic
259	C₂×C₄	C₇×#117	1	1	yes	characteristic
260	C₂×C₄	C₇×#117	1	2	yes	—
261	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
262	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
263	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
264	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
265	C₂×C₄	C₇×#121	1	2	yes	—
266	C₂×C₄	C₇×C₂²⋊Q₈	1	2	yes	—
267	C₂×C₄	C₇×#123	1	2	yes	—
268	C₂×C₄	C₇×#124	1	1	yes	characteristic
269	C₂×C₄	C₇×#125	1	1	yes	characteristic
270	C₂×C₄	C₇×#126	1	2	yes	—
271	C₂×C₄	C₇×#126	1	2	yes	—
272	C₈	—	2	2	no	—
273	C₈	—	2	2	no	—
274	C₈	—	2	4	no	—
275	C₈	—	4	4	no	—
276	C₇	#10799	1	1	yes	characteristic, direct product factor, Sylow
277	C₂²	—	2	2	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	C₇×#376	1	1	yes	characteristic
283	C₂²	C₇×#381	1	1	yes	characteristic
284	C₂²	C₇×#382	1	1	yes	characteristic
285	C₂²	C₇×#387	1	2	yes	—
286	C₂²	C₇×#388	1	2	yes	—
287	C₄	—	4	4	no	—
288	C₄	—	8	4	no	—
289	C₄	C₄×C₇×C₂²⋊C₄	1	1	yes	characteristic
290	C₄	C₄×C₇×C₄⋊C₄	1	1	yes	characteristic
291	C₄	C₇×#380	1	1	yes	characteristic
292	C₄	C₇×#383	1	1	yes	characteristic
293	C₄	C₇×#387	1	2	yes	—
294	C₄	C₇×#388	1	2	yes	—
295	C₂	—	2	1	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	C₇×#1434	1	1	yes	characteristic
300	C₂	C₇×#1747	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
socle	C₂²×C₇	#376
prime radical	C₂⁴×C₇	#41
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×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
13. C₂×C₄×C₇×C₈	C₄	semidirect
17. C₇×#336	C₄	semidirect
18. #10799	C₇	direct (N × H)
106. C₂×#97	C₄×C₇	semidirect
107. C₂×C₄×C₈	C₄×C₇	semidirect
111. #336	C₄×C₇	semidirect
276. C₇	#10799	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×#405	2
C₂×C₇×#338	2
C₂×C₇×H(3,ℤ/4)	2
C₇×#1729	2
#10799	7

Subgroups — C7×#10799 #13188015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#10799 #13188015