Subgroups — C₇×#10800 #13188016

Order 1792.

650 conjugacy classes of subgroups.

The subgroup lattice has 312 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	2	no	—
12	C₇×#405	—	2	2	no	—
13	C₂×C₇×#97	C₄	1	2	yes	complemented
14	C₂×C₄×C₇×C₈	C₄	1	2	yes	complemented
15	C₇×#338	—	2	2	no	—
16	C₇×H(3,ℤ/4)	—	2	2	no	—
17	C₇×H(3,ℤ/4)	—	2	4	no	—
18	C₇×#336	C₄	1	8	yes	complemented
19	#10800	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
29	C₂×C₇×M₁₆	D₄	1	2	yes	—
30	C₂×C₇×M₁₆	D₄	1	2	yes	—
31	C₂²×C₇×C₈	—	2	1	no	—
32	C₂²×C₇×C₈	—	2	1	no	—
33	C₂²×C₇×C₈	—	2	2	no	—
34	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
35	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
36	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
39	C₇×#117	—	2	1	no	—
40	C₇×#117	—	2	1	no	—
41	C₇×#117	—	2	2	no	—
42	C₇×#117	—	2	2	no	—
43	C₇×#117	—	2	2	no	—
44	C₇×#117	—	2	2	no	—
45	C₇×#117	D₄	1	2	yes	—
46	C₇×#117	Q₈	1	2	yes	—
47	C₂×C₇×C₄⋊C₄	—	2	1	no	—
48	C₂×C₇×C₄⋊C₄	—	2	2	no	—
49	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
50	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
51	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
53	C₇×#97	—	2	4	no	—
54	C₄×C₇×C₈	—	2	4	no	—
55	C₂×#405	C₂×C₇	1	1	yes	characteristic
56	C₂×#338	C₂×C₇	1	1	yes	characteristic
57	C₂×H(3,ℤ/4)	C₂×C₇	1	1	yes	characteristic
58	C₂×H(3,ℤ/4)	C₂×C₇	1	2	yes	—
59	#1729	C₂×C₇	1	2	yes	—
60	C₂⁴×C₇	#41	1	1	yes	characteristic
61	C₂²×C₄×C₇	—	2	2	no	—
62	C₂²×C₄×C₇	—	2	4	no	—
63	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₇	#41	1	1	yes	characteristic
70	C₂²×C₄×C₇	#41	1	2	yes	—
71	C₂²×C₄×C₇	#41	1	2	yes	—
72	C₂²×C₄×C₇	#41	1	2	yes	—
73	C₇×M₁₆	—	2	4	no	—
74	C₇×M₁₆	—	2	4	no	—
75	C₇×M₁₆	—	4	2	no	—
76	C₇×M₁₆	—	4	2	no	—
77	C₇×M₁₆	—	4	2	no	—
78	C₇×M₁₆	—	4	2	no	—
79	C₂×C₇×C₈	—	2	1	no	—
80	C₂×C₇×C₈	—	2	1	no	—
81	C₂×C₇×C₈	—	2	1	no	—
82	C₂×C₇×C₈	—	2	1	no	—
83	C₂×C₇×C₈	—	2	2	no	—
84	C₂×C₇×C₈	—	2	2	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	—	2	4	no	—
90	C₂×C₇×C₈	C₄²	1	2	yes	—
91	C₂×C₇×C₈	C₄²	1	2	yes	—
92	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
93	C₇×C₄⋊C₄	—	2	2	no	—
94	C₇×C₄⋊C₄	—	2	2	no	—
95	C₇×C₄⋊C₄	—	4	2	no	—
96	C₇×C₄⋊C₄	—	4	2	no	—
97	C₇×C₂²⋊C₄	—	2	2	no	—
98	C₇×C₂²⋊C₄	—	2	2	no	—
99	C₇×C₂²⋊C₄	—	4	2	no	—
100	C₇×C₂²⋊C₄	—	4	2	no	—
101	C₄²×C₇	—	2	2	no	—
102	C₄²×C₇	—	2	4	no	—
103	C₄²×C₇	—	2	4	no	—
104	C₄²×C₇	C₄⋊C₄	1	4	yes	—
105	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
106	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
107	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
108	C₂×#117	C₂²×C₇	1	2	yes	—
109	C₂×#117	C₂²×C₇	1	1	yes	characteristic
110	#405	—	2	2	no	—
111	#405	—	2	2	no	—
112	C₂×#97	C₄×C₇	1	2	yes	complemented
113	C₂×C₄×C₈	C₄×C₇	1	2	yes	complemented
114	#338	—	2	2	no	—
115	H(3,ℤ/4)	—	2	2	no	—
116	H(3,ℤ/4)	—	2	4	no	—
117	#336	C₄×C₇	1	8	yes	complemented
118	C₂³×C₇	—	4	2	no	—
119	C₂³×C₇	#117	1	1	yes	characteristic
120	C₂³×C₇	#117	1	2	yes	—
121	C₂³×C₇	#121	1	1	yes	characteristic
122	C₂³×C₇	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂³×C₇	#126	1	2	yes	—
124	C₂×C₄×C₇	—	2	4	no	—
125	C₂×C₄×C₇	—	4	4	no	—
126	C₂×C₄×C₇	—	4	8	no	—
127	C₂×C₄×C₇	—	4	4	no	—
128	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
129	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₇	#117	1	1	yes	characteristic
132	C₂×C₄×C₇	#117	1	1	yes	characteristic
133	C₂×C₄×C₇	#117	1	1	yes	characteristic
134	C₂×C₄×C₇	#117	1	2	yes	—
135	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
137	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
138	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
139	C₂×C₄×C₇	#121	1	1	yes	characteristic
140	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂×C₄×C₇	#123	1	2	yes	—
142	C₂×C₄×C₇	#123	1	2	yes	—
143	C₂×C₄×C₇	#124	1	2	yes	—
144	C₂×C₄×C₇	#125	1	2	yes	—
145	C₂×C₄×C₇	#126	1	2	yes	—
146	C₇×C₈	—	2	2	no	—
147	C₇×C₈	—	2	2	no	—
148	C₇×C₈	—	2	4	no	—
149	C₇×C₈	—	4	4	no	—
150	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	—	2	2	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	C₇×D₄	1	2	yes	—
160	C₂×M₁₆	C₇×D₄	1	2	yes	—
161	C₂²×C₈	—	2	1	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
165	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
166	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
167	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
168	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
169	#117	—	2	1	no	—
170	#117	—	2	1	no	—
171	#117	—	2	2	no	—
172	#117	—	2	2	no	—
173	#117	—	2	2	no	—
174	#117	—	2	2	no	—
175	#117	C₇×D₄	1	2	yes	—
176	#117	C₇×Q₈	1	2	yes	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₂²⋊C₄	—	2	1	no	—
180	C₂×C₂²⋊C₄	—	2	2	no	—
181	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
182	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
183	#97	—	2	4	no	—
184	C₄×C₈	—	2	4	no	—
185	C₂²×C₇	—	2	2	no	—
186	C₂²×C₇	—	2	4	no	—
187	C₂²×C₇	—	4	1	no	—
188	C₂²×C₇	—	4	1	no	—
189	C₂²×C₇	—	4	2	no	—
190	C₂²×C₇	#376	1	1	yes	characteristic
191	C₂²×C₇	#380	1	1	yes	characteristic
192	C₂²×C₇	#383	1	1	yes	characteristic
193	C₂²×C₇	#387	1	2	yes	—
194	C₂²×C₇	#388	1	2	yes	—
195	C₄×C₇	—	4	4	no	—
196	C₄×C₇	—	8	4	no	—
197	C₄×C₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
198	C₄×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
199	C₄×C₇	#381	1	1	yes	characteristic
200	C₄×C₇	#382	1	1	yes	characteristic
201	C₄×C₇	#387	1	2	yes	—
202	C₄×C₇	#388	1	2	yes	—
203	C₂⁴	C₇×#41	1	1	yes	characteristic
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
207	C₂²×C₄	C₂²×C₄×C₇	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×D₄	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
212	C₂²×C₄	C₇×#41	1	1	yes	characteristic
213	C₂²×C₄	C₇×#41	1	2	yes	—
214	C₂²×C₄	C₇×#41	1	2	yes	—
215	C₂²×C₄	C₇×#41	1	2	yes	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	C₄²×C₇	1	2	yes	—
234	C₂×C₈	C₄²×C₇	1	2	yes	—
235	C₂×C₈	C₇×C₂²⋊C₄	1	4	yes	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₂²⋊C₄	—	4	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	2	4	no	—
247	C₄²	C₇×C₄⋊C₄	1	4	yes	—
248	C₂×C₇	—	2	1	no	—
249	C₂×C₇	—	2	2	no	—
250	C₂×C₇	—	2	1	no	—
251	C₂×C₇	—	2	2	no	—
252	C₂×C₇	#1434	1	1	yes	characteristic
253	C₂×C₇	#1747	1	2	yes	—
254	C₂³	—	4	2	no	—
255	C₂³	C₇×#117	1	1	yes	characteristic
256	C₂³	C₇×#117	1	2	yes	—
257	C₂³	C₇×#121	1	1	yes	characteristic
258	C₂³	C₇×C₂²⋊Q₈	1	1	yes	characteristic
259	C₂³	C₇×#126	1	2	yes	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	8	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
265	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
267	C₂×C₄	C₇×#117	1	1	yes	characteristic
268	C₂×C₄	C₇×#117	1	1	yes	characteristic
269	C₂×C₄	C₇×#117	1	1	yes	characteristic
270	C₂×C₄	C₇×#117	1	2	yes	—
271	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
272	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
273	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
274	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
275	C₂×C₄	C₇×#121	1	1	yes	characteristic
276	C₂×C₄	C₇×C₂²⋊Q₈	1	1	yes	characteristic
277	C₂×C₄	C₇×#123	1	2	yes	—
278	C₂×C₄	C₇×#123	1	2	yes	—
279	C₂×C₄	C₇×#124	1	2	yes	—
280	C₂×C₄	C₇×#125	1	2	yes	—
281	C₂×C₄	C₇×#126	1	2	yes	—
282	C₈	—	2	2	no	—
283	C₈	—	2	2	no	—
284	C₈	—	2	4	no	—
285	C₈	—	4	4	no	—
286	C₇	#10800	1	1	yes	characteristic, direct product factor, Sylow
287	C₂²	—	2	2	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	2	no	—
292	C₂²	C₇×#376	1	1	yes	characteristic
293	C₂²	C₇×#380	1	1	yes	characteristic
294	C₂²	C₇×#383	1	1	yes	characteristic
295	C₂²	C₇×#387	1	2	yes	—
296	C₂²	C₇×#388	1	2	yes	—
297	C₄	—	4	4	no	—
298	C₄	—	8	4	no	—
299	C₄	C₄×C₇×C₂²⋊C₄	1	1	yes	characteristic
300	C₄	C₄×C₇×C₄⋊C₄	1	1	yes	characteristic
301	C₄	C₇×#381	1	1	yes	characteristic
302	C₄	C₇×#382	1	1	yes	characteristic
303	C₄	C₇×#387	1	2	yes	—
304	C₄	C₇×#388	1	2	yes	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	C₇×#1434	1	1	yes	characteristic
310	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
13. C₂×C₇×#97	C₄	semidirect
14. C₂×C₄×C₇×C₈	C₄	semidirect
18. C₇×#336	C₄	semidirect
19. #10800	C₇	direct (N × H)
112. C₂×#97	C₄×C₇	semidirect
113. C₂×C₄×C₈	C₄×C₇	semidirect
117. #336	C₄×C₇	semidirect
286. C₇	#10800	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
#10800	7

Subgroups — C7×#10800 #13188016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#10800 #13188016