Subgroups — C₇×#21268 #13198484

Order 1792.

838 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#433	C₂	1	2	yes	maximal, complemented
2	C₇×#2293	C₂	1	1	yes	maximal, characteristic
3	C₇×#1954	C₂	1	2	yes	maximal
4	C₇×#1939	C₂	1	2	yes	maximal, complemented
5	C₇×#1924	C₂	1	4	yes	maximal, complemented
6	C₄×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1750	C₂	1	2	yes	maximal, complemented
8	C₄×C₇×#97	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
10	C₂×C₇×#117	C₂²	1	2	yes	—
11	C₇×#433	—	2	8	no	—
12	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
13	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₇×C₂²⋊C₈	C₂²	1	4	yes	—
15	C₂×C₇×C₂²⋊C₈	C₂²	1	4	yes	—
16	C₂×C₇×#97	C₂²	1	1	yes	characteristic
17	C₂×C₇×#97	C₂²	1	2	yes	—
18	C₂×C₇×#97	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
20	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
21	C₇×#390	C₄	1	2	yes	—
22	C₇×#389	C₄	1	2	yes	—
23	C₇×#388	C₄	1	4	yes	—
24	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₄³×C₇	C₂²	1	1	yes	characteristic
27	C₇×#336	C₂²	1	4	yes	—
28	C₇×#336	C₂²	1	2	yes	—
29	#21268	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₇×D₄	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₇	C₂³	1	2	yes	—
32	C₂²×C₇×C₈	C₂³	1	4	yes	—
33	C₂²×C₇×C₈	C₂³	1	2	yes	—
34	C₂²×C₇×C₈	C₂³	1	2	yes	—
35	C₄×C₇×D₄	—	2	4	no	—
36	C₇×#117	—	2	4	no	—
37	C₇×#117	—	2	4	no	—
38	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
39	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
41	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	4	yes	—
42	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
43	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₇	C₂³	1	2	yes	—
48	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
50	C₇×C₄⋊C₈	—	2	8	no	—
51	C₇×C₄⋊C₈	—	2	4	no	—
52	C₇×C₂²⋊C₈	—	2	8	no	—
53	C₇×C₂²⋊C₈	—	2	8	no	—
54	C₇×#97	—	2	4	no	—
55	C₇×#97	—	2	2	no	—
56	C₇×#97	D₄	1	4	yes	complemented
57	C₄×C₇×C₈	—	2	4	no	—
58	C₄×C₇×C₈	—	2	4	no	—
59	C₇×#95	C₂×C₄	1	4	yes	—
60	C₇×#95	C₂×C₄	1	2	yes	—
61	C₂×#433	C₂×C₇	1	2	yes	complemented
62	#2293	C₂×C₇	1	1	yes	characteristic
63	#1954	C₂×C₇	1	2	yes	—
64	#1939	C₂×C₇	1	2	yes	complemented
65	#1924	C₂×C₇	1	4	yes	complemented
66	C₄×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
67	#1750	C₂×C₇	1	2	yes	complemented
68	C₄×#97	C₂×C₇	1	1	yes	characteristic, complemented
69	C₂⁴×C₇	C₂²×C₄	1	2	yes	—
70	C₂×C₇×D₄	—	2	2	no	—
71	C₂×C₇×D₄	—	2	4	no	—
72	C₂²×C₄×C₇	—	2	2	no	—
73	C₂²×C₄×C₇	—	2	4	no	—
74	C₂²×C₄×C₇	—	2	2	no	—
75	C₂²×C₄×C₇	—	2	4	no	—
76	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
83	C₂×C₇×C₈	—	2	2	no	—
84	C₂×C₇×C₈	—	2	2	no	—
85	C₂×C₇×C₈	—	2	2	no	—
86	C₂×C₇×C₈	—	2	2	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	C₂×D₄	1	4	yes	—
91	C₂×C₇×C₈	#41	1	4	yes	—
92	C₇×C₄⋊C₄	—	2	2	no	—
93	C₇×C₄⋊C₄	—	2	8	no	—
94	C₇×C₄⋊C₄	—	2	4	no	—
95	C₇×C₂²⋊C₄	—	2	4	no	—
96	C₇×C₂²⋊C₄	—	2	8	no	—
97	C₇×C₂²⋊C₄	—	4	4	no	—
98	C₄²×C₇	—	2	4	no	—
99	C₄²×C₇	—	2	2	no	—
100	C₄²×C₇	—	2	2	no	—
101	C₄²×C₇	—	2	2	no	—
102	C₄²×C₇	—	4	2	no	—
103	C₄²×C₇	C₂×D₄	1	2	yes	—
104	C₄²×C₇	#41	1	4	yes	—
105	C₄²×C₇	#41	1	2	yes	—
106	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
107	C₂×#117	C₂²×C₇	1	2	yes	—
108	#433	—	2	8	no	—
109	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
110	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
111	C₂×C₂²⋊C₈	C₂²×C₇	1	4	yes	—
112	C₂×C₂²⋊C₈	C₂²×C₇	1	4	yes	—
113	C₂×#97	C₂²×C₇	1	2	yes	—
114	C₂×#97	C₂²×C₇	1	1	yes	characteristic
115	C₂×#97	C₂²×C₇	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
117	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
118	#390	C₄×C₇	1	2	yes	—
119	#389	C₄×C₇	1	2	yes	—
120	#388	C₄×C₇	1	4	yes	—
121	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
122	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
123	C₄³	C₂²×C₇	1	1	yes	characteristic
124	#336	C₂²×C₇	1	2	yes	—
125	#336	C₂²×C₇	1	4	yes	—
126	C₂³×C₇	—	2	2	no	—
127	C₂³×C₇	—	2	2	no	—
128	C₂³×C₇	—	2	2	no	—
129	C₂³×C₇	—	2	4	no	—
130	C₂³×C₇	—	2	4	no	—
131	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
132	C₇×D₄	—	4	4	no	—
133	C₂×C₄×C₇	—	2	2	no	—
134	C₂×C₄×C₇	—	2	2	no	—
135	C₂×C₄×C₇	—	2	2	no	—
136	C₂×C₄×C₇	—	2	2	no	—
137	C₂×C₄×C₇	—	2	4	no	—
138	C₂×C₄×C₇	—	2	2	no	—
139	C₂×C₄×C₇	—	2	2	no	—
140	C₂×C₄×C₇	—	2	2	no	—
141	C₂×C₄×C₇	—	2	2	no	—
142	C₂×C₄×C₇	—	2	4	no	—
143	C₂×C₄×C₇	—	2	2	no	—
144	C₂×C₄×C₇	—	2	4	no	—
145	C₂×C₄×C₇	—	4	4	no	—
146	C₂×C₄×C₇	—	4	4	no	—
147	C₂×C₄×C₇	#117	1	2	yes	—
148	C₂×C₄×C₇	#117	1	2	yes	—
149	C₂×C₄×C₇	C₄×D₄	1	4	yes	—
150	C₂×C₄×C₇	#131	1	4	yes	—
151	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₇	C₂×#41	1	2	yes	—
154	C₂×C₄×C₇	C₂×#41	1	2	yes	—
155	C₇×C₈	—	2	4	no	—
156	C₇×C₈	—	2	4	no	—
157	C₇×C₈	—	4	4	no	—
158	C₂²×D₄	C₂×C₄×C₇	1	1	yes	characteristic
159	C₂³×C₄	C₂³×C₇	1	2	yes	—
160	C₂²×C₈	C₂³×C₇	1	4	yes	—
161	C₂²×C₈	C₂³×C₇	1	2	yes	—
162	C₂²×C₈	C₂³×C₇	1	2	yes	—
163	C₄×D₄	—	2	4	no	—
164	#117	—	2	4	no	—
165	#117	—	2	4	no	—
166	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
167	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
168	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
169	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	4	yes	—
170	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
171	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	4	yes	—
172	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
173	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
174	C₂×C₄²	C₂³×C₇	1	2	yes	—
175	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
176	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
177	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
178	C₄⋊C₈	—	2	8	no	—
179	C₄⋊C₈	—	2	4	no	—
180	C₂²⋊C₈	—	2	8	no	—
181	C₂²⋊C₈	—	2	8	no	—
182	#97	—	2	4	no	—
183	#97	—	2	2	no	—
184	#97	C₇×D₄	1	4	yes	complemented
185	C₄×C₈	—	2	4	no	—
186	C₄×C₈	—	2	4	no	—
187	#95	C₂×C₄×C₇	1	4	yes	—
188	#95	C₂×C₄×C₇	1	2	yes	—
189	C₂²×C₇	—	4	2	no	—
190	C₂²×C₇	—	4	2	no	—
191	C₂²×C₇	—	4	4	no	—
192	C₂²×C₇	—	4	4	no	—
193	C₂²×C₇	—	4	2	no	—
194	C₂²×C₇	C₂×#117	1	1	yes	characteristic
195	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
196	C₂²×C₇	C₄×#41	1	1	yes	characteristic
197	C₂²×C₇	C₂×#131	1	2	yes	—
198	C₂²×C₇	#568	1	2	yes	—
199	C₄×C₇	—	2	2	no	—
200	C₄×C₇	—	2	4	no	—
201	C₄×C₇	—	4	4	no	—
202	C₄×C₇	—	4	2	no	—
203	C₄×C₇	—	4	2	no	—
204	C₄×C₇	#433	1	4	yes	complemented
205	C₄×C₇	#529	1	2	yes	—
206	C₄×C₇	#532	1	2	yes	—
207	C₂⁴	C₂²×C₄×C₇	1	2	yes	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
215	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
216	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
217	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
218	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
219	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
220	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
229	C₂×C₈	C₇×#41	1	4	yes	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	8	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₂²⋊C₄	—	2	8	no	—
235	C₂²⋊C₄	—	4	4	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	4	2	no	—
241	C₄²	C₂×C₇×D₄	1	2	yes	—
242	C₄²	C₇×#41	1	4	yes	—
243	C₄²	C₇×#41	1	2	yes	—
244	C₂×C₇	—	8	2	no	—
245	C₂×C₇	#2293	1	1	yes	characteristic
246	C₂×C₇	C₂×#433	1	1	yes	characteristic
247	C₂×C₇	#2923	1	1	yes	characteristic
248	C₂×C₇	#2935	1	2	yes	—
249	C₂×C₇	#2969	1	2	yes	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	4	no	—
254	C₂³	—	2	4	no	—
255	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
256	D₄	—	4	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	C₇×#117	1	2	yes	—
272	C₂×C₄	C₇×#117	1	2	yes	—
273	C₂×C₄	C₄×C₇×D₄	1	4	yes	—
274	C₂×C₄	C₇×#131	1	4	yes	—
275	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
276	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
277	C₂×C₄	C₂×C₇×#41	1	2	yes	—
278	C₂×C₄	C₂×C₇×#41	1	2	yes	—
279	C₈	—	2	4	no	—
280	C₈	—	2	4	no	—
281	C₈	—	4	4	no	—
282	C₇	#21268	1	1	yes	characteristic, direct product factor, Sylow
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	4	4	no	—
287	C₂²	—	4	2	no	—
288	C₂²	C₂×C₇×#117	1	1	yes	characteristic
289	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
290	C₂²	C₄×C₇×#41	1	1	yes	characteristic
291	C₂²	C₂×C₇×#131	1	2	yes	—
292	C₂²	C₇×#568	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	2	4	no	—
295	C₄	—	4	4	no	—
296	C₄	—	4	2	no	—
297	C₄	—	4	2	no	—
298	C₄	C₇×#433	1	4	yes	complemented
299	C₄	C₇×#529	1	2	yes	—
300	C₄	C₇×#532	1	2	yes	—
301	C₂	—	8	2	no	—
302	C₂	C₇×#2293	1	1	yes	characteristic
303	C₂	C₂×C₇×#433	1	1	yes	characteristic
304	C₂	C₇×#2923	1	1	yes	characteristic
305	C₂	C₇×#2935	1	2	yes	—
306	C₂	C₇×#2969	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#433	C₂	semidirect
4. C₇×#1939	C₂	semidirect
5. C₇×#1924	C₂	semidirect
6. C₄×C₇×C₄⋊C₈	C₂	semidirect
7. C₇×#1750	C₂	semidirect
8. C₄×C₇×#97	C₂	semidirect
29. #21268	C₇	direct (N × H)
56. C₇×#97	D₄	semidirect
61. C₂×#433	C₂×C₇	semidirect
64. #1939	C₂×C₇	semidirect
65. #1924	C₂×C₇	semidirect
66. C₄×C₄⋊C₈	C₂×C₇	semidirect
67. #1750	C₂×C₇	semidirect
68. C₄×#97	C₂×C₇	semidirect
184. #97	C₇×D₄	semidirect
204. C₄×C₇	#433	semidirect
282. C₇	#21268	direct (N × H)
298. C₄	C₇×#433	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#433	2
C₇×#2293	2
C₇×#1954	2
C₇×#1939	2
C₇×#1924	2
C₄×C₇×C₄⋊C₈	2
C₇×#1750	2
C₄×C₇×#97	2
#21268	7

Subgroups — C7×#21268 #13198484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21268 #13198484