Subgroups — C₇×#19717 #13196933

Order 1792.

720 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3231	C₂	1	2	yes	maximal, complemented
2	C₇×#3208	C₂	1	2	yes	maximal
3	C₇×#3185	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3138	C₂	1	1	yes	maximal, characteristic
5	C₇×#1720	C₂	1	2	yes	maximal, complemented
6	C₇×#1719	C₂	1	2	yes	maximal, complemented
7	C₇×#1714	C₂	1	2	yes	maximal, complemented
8	C₇×#1708	C₂	1	2	yes	maximal, complemented
9	C₇×#1453	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#533	C₂²	1	2	yes	—
11	C₇×#533	C₂²	1	1	yes	characteristic
12	C₂×C₇×#127	C₂²	1	1	yes	characteristic
13	C₇×#500	C₂²	1	2	yes	complemented
14	C₇×#498	C₂²	1	2	yes	complemented
15	C₇×#498	C₂²	1	1	yes	characteristic, complemented
16	C₇×#494	C₂²	1	1	yes	characteristic, complemented
17	C₇×#492	C₂²	1	2	yes	—
18	C₇×#492	C₂²	1	2	yes	—
19	C₇×#490	—	2	2	no	—
20	C₇×#487	C₂²	1	2	yes	complemented
21	C₇×#487	C₂²	1	2	yes	complemented
22	C₇×#487	C₂²	1	2	yes	complemented
23	C₇×#486	C₂²	1	2	yes	—
24	C₇×#486	C₂²	1	2	yes	—
25	C₇×#486	C₂²	1	2	yes	—
26	C₇×#486	C₂²	1	2	yes	—
27	C₇×#485	—	2	4	no	—
28	C₇×#479	—	2	4	no	—
29	C₇×#470	—	2	2	no	—
30	C₇×#460	—	2	4	no	—
31	C₇×#450	—	2	4	no	—
32	C₇×#441	—	2	2	no	—
33	C₇×#431	C₂²	1	2	yes	—
34	C₇×#418	—	2	2	no	—
35	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
36	C₇×#322	C₂²	1	2	yes	complemented
37	C₇×#322	C₂²	1	1	yes	characteristic, complemented
38	C₇×C₈²	C₂²	1	1	yes	characteristic, complemented
39	#19717	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂²×C₇×D₄	—	2	1	no	—
41	C₂²×C₇×D₄	—	2	2	no	—
42	C₂×C₇×Q₁₆	—	2	2	no	—
43	C₂×C₇×SD₁₆	—	2	4	no	—
44	C₂×C₇×SD₁₆	—	2	4	no	—
45	C₂×C₇×M₁₆	D₄	1	2	yes	—
46	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₇×#127	—	2	1	no	—
50	C₇×#127	—	2	2	no	—
51	C₇×#127	C₂³	1	2	yes	—
52	C₇×#125	—	2	2	no	—
53	C₇×#125	—	2	1	no	—
54	C₇×C₂²⋊Q₈	—	2	2	no	—
55	C₇×C₂²⋊Q₈	—	2	2	no	—
56	C₇×C₂²⋊Q₈	—	2	2	no	—
57	C₄×C₇×Q₈	—	2	2	no	—
58	C₄×C₇×Q₈	—	2	2	no	—
59	C₄×C₇×Q₈	—	2	2	no	—
60	C₇×#117	—	2	2	no	—
61	C₇×#117	—	2	1	no	—
62	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
63	C₇×#107	—	2	4	no	—
64	C₇×#106	—	2	2	no	—
65	C₇×#106	—	2	4	no	—
66	C₇×C₄⋊C₈	D₄	1	4	yes	—
67	C₇×#103	—	2	4	no	—
68	C₇×#103	—	2	4	no	—
69	C₇×#103	—	2	4	no	—
70	C₇×#102	—	2	4	no	—
71	C₇×#102	—	2	4	no	—
72	C₇×#102	—	2	4	no	—
73	C₇×#102	—	2	4	no	—
74	C₇×C₂²⋊C₈	D₄	1	4	yes	—
75	C₇×#97	D₄	1	2	yes	—
76	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
77	C₄×C₇×C₈	C₂³	1	2	yes	—
78	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
79	C₄×C₇×C₈	C₂³	1	2	yes	—
80	#3231	C₂×C₇	1	2	yes	complemented
81	#3208	C₂×C₇	1	2	yes	—
82	#3185	C₂×C₇	1	1	yes	characteristic, complemented
83	#3138	C₂×C₇	1	1	yes	characteristic
84	#1720	C₂×C₇	1	2	yes	complemented
85	#1719	C₂×C₇	1	2	yes	complemented
86	#1714	C₂×C₇	1	2	yes	complemented
87	#1708	C₂×C₇	1	2	yes	complemented
88	#1453	C₂×C₇	1	1	yes	characteristic, complemented
89	C₂⁴×C₇	—	4	1	no	—
90	C₂×C₇×Q₈	—	2	2	no	—
91	C₂×C₇×Q₈	—	2	2	no	—
92	C₂×C₇×Q₈	—	2	2	no	—
93	C₂×C₇×D₄	—	2	2	no	—
94	C₂×C₇×D₄	—	2	2	no	—
95	C₂×C₇×D₄	—	2	2	no	—
96	C₂×C₇×D₄	—	4	4	no	—
97	C₂×C₇×D₄	—	4	1	no	—
98	C₂×C₇×D₄	—	4	2	no	—
99	C₂×C₇×D₄	—	8	4	no	—
100	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
102	C₇×Q₁₆	—	4	4	no	—
103	C₇×SD₁₆	—	8	4	no	—
104	C₇×SD₁₆	—	8	4	no	—
105	C₇×M₁₆	—	4	2	no	—
106	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
107	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
108	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
109	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
110	C₇×C₄⋊C₄	—	2	2	no	—
111	C₇×C₄⋊C₄	—	2	2	no	—
112	C₇×C₄⋊C₄	—	2	2	no	—
113	C₇×C₄⋊C₄	—	2	2	no	—
114	C₇×C₄⋊C₄	—	2	2	no	—
115	C₇×C₄⋊C₄	—	2	2	no	—
116	C₇×C₄⋊C₄	—	4	2	no	—
117	C₇×C₄⋊C₄	—	4	2	no	—
118	C₇×C₄⋊C₄	—	4	2	no	—
119	C₇×C₂²⋊C₄	—	4	2	no	—
120	C₇×C₂²⋊C₄	—	4	1	no	—
121	C₄²×C₇	—	4	2	no	—
122	C₄²×C₇	—	4	1	no	—
123	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₇	C₂×D₄	1	2	yes	—
125	C₄²×C₇	C₂⁴	1	1	yes	characteristic
126	#533	C₂²×C₇	1	1	yes	characteristic
127	#533	C₂²×C₇	1	2	yes	—
128	C₂×#127	C₂²×C₇	1	1	yes	characteristic
129	#500	C₂²×C₇	1	2	yes	complemented
130	#498	C₂²×C₇	1	1	yes	characteristic, complemented
131	#498	C₂²×C₇	1	2	yes	complemented
132	#494	C₂²×C₇	1	1	yes	characteristic, complemented
133	#492	C₂²×C₇	1	2	yes	—
134	#492	C₂²×C₇	1	2	yes	—
135	#490	—	2	2	no	—
136	#487	C₂²×C₇	1	2	yes	complemented
137	#487	C₂²×C₇	1	2	yes	complemented
138	#487	C₂²×C₇	1	2	yes	complemented
139	#486	C₂²×C₇	1	2	yes	—
140	#486	C₂²×C₇	1	2	yes	—
141	#486	C₂²×C₇	1	2	yes	—
142	#486	C₂²×C₇	1	2	yes	—
143	#485	—	2	4	no	—
144	#479	—	2	4	no	—
145	#470	—	2	2	no	—
146	#460	—	2	4	no	—
147	#450	—	2	4	no	—
148	#441	—	2	2	no	—
149	#431	C₂²×C₇	1	2	yes	—
150	#418	—	2	2	no	—
151	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
152	#322	C₂²×C₇	1	2	yes	complemented
153	#322	C₂²×C₇	1	1	yes	characteristic, complemented
154	C₈²	C₂²×C₇	1	1	yes	characteristic, complemented
155	C₂³×C₇	—	4	2	no	—
156	C₂³×C₇	—	8	2	no	—
157	C₂³×C₇	—	8	4	no	—
158	C₂³×C₇	#127	1	1	yes	characteristic
159	C₇×Q₈	—	4	4	no	—
160	C₇×Q₈	—	4	4	no	—
161	C₇×Q₈	—	4	4	no	—
162	C₇×D₄	—	4	4	no	—
163	C₇×D₄	—	4	4	no	—
164	C₇×D₄	—	4	4	no	—
165	C₇×D₄	—	8	2	no	—
166	C₇×D₄	—	8	4	no	—
167	C₂×C₄×C₇	—	2	2	no	—
168	C₂×C₄×C₇	—	2	4	no	—
169	C₂×C₄×C₇	—	4	2	no	—
170	C₂×C₄×C₇	—	4	2	no	—
171	C₂×C₄×C₇	—	4	2	no	—
172	C₂×C₄×C₇	#127	1	1	yes	characteristic
173	C₂×C₄×C₇	#127	1	2	yes	—
174	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
176	C₇×C₈	—	2	2	no	—
177	C₇×C₈	—	2	2	no	—
178	C₇×C₈	—	2	4	no	—
179	C₇×C₈	—	2	4	no	—
180	C₂²×D₄	—	2	1	no	—
181	C₂²×D₄	—	2	2	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×SD₁₆	—	2	4	no	—
184	C₂×SD₁₆	—	2	4	no	—
185	C₂×M₁₆	C₇×D₄	1	2	yes	—
186	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
187	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
188	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
189	#127	—	2	2	no	—
190	#127	—	2	1	no	—
191	#127	C₂³×C₇	1	2	yes	—
192	#125	—	2	2	no	—
193	#125	—	2	1	no	—
194	C₂²⋊Q₈	—	2	2	no	—
195	C₂²⋊Q₈	—	2	2	no	—
196	C₂²⋊Q₈	—	2	2	no	—
197	C₄×Q₈	—	2	2	no	—
198	C₄×Q₈	—	2	2	no	—
199	C₄×Q₈	—	2	2	no	—
200	#117	—	2	1	no	—
201	#117	—	2	2	no	—
202	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
203	#107	—	2	4	no	—
204	#106	—	2	2	no	—
205	#106	—	2	4	no	—
206	C₄⋊C₈	C₇×D₄	1	4	yes	—
207	#103	—	2	4	no	—
208	#103	—	2	4	no	—
209	#103	—	2	4	no	—
210	#102	—	2	4	no	—
211	#102	—	2	4	no	—
212	#102	—	2	4	no	—
213	#102	—	2	4	no	—
214	C₂²⋊C₈	C₇×D₄	1	4	yes	—
215	#97	C₇×D₄	1	2	yes	—
216	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
217	C₄×C₈	C₂³×C₇	1	2	yes	—
218	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
219	C₄×C₈	C₂³×C₇	1	2	yes	—
220	C₂²×C₇	—	2	1	no	—
221	C₂²×C₇	—	2	2	no	—
222	C₂²×C₇	—	8	2	no	—
223	C₂²×C₇	—	8	2	no	—
224	C₂²×C₇	—	8	2	no	—
225	C₂²×C₇	—	16	4	no	—
226	C₂²×C₇	C₂×#127	1	1	yes	characteristic
227	C₄×C₇	—	2	1	no	—
228	C₄×C₇	—	2	2	no	—
229	C₄×C₇	—	8	2	no	—
230	C₄×C₇	—	8	2	no	—
231	C₄×C₇	—	8	2	no	—
232	C₄×C₇	#575	1	2	yes	—
233	C₄×C₇	#577	1	4	yes	—
234	C₂⁴	—	4	1	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	4	4	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂×D₄	—	8	4	no	—
245	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
246	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
247	Q₁₆	—	4	4	no	—
248	SD₁₆	—	8	4	no	—
249	SD₁₆	—	8	4	no	—
250	M₁₆	—	4	2	no	—
251	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
252	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
253	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
254	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	1	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₄²	—	4	1	no	—
267	C₄²	—	4	2	no	—
268	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
269	C₄²	C₂×C₇×D₄	1	2	yes	—
270	C₄²	C₂⁴×C₇	1	1	yes	characteristic
271	C₂×C₇	—	4	1	no	—
272	C₂×C₇	—	16	2	no	—
273	C₂×C₇	#3153	1	1	yes	characteristic
274	C₂×C₇	#3154	1	2	yes	—
275	C₂³	—	4	2	no	—
276	C₂³	—	8	2	no	—
277	C₂³	—	8	4	no	—
278	C₂³	C₇×#127	1	1	yes	characteristic
279	Q₈	—	4	4	no	—
280	Q₈	—	4	4	no	—
281	Q₈	—	4	4	no	—
282	D₄	—	4	4	no	—
283	D₄	—	4	4	no	—
284	D₄	—	4	4	no	—
285	D₄	—	8	2	no	—
286	D₄	—	8	4	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	C₇×#127	1	1	yes	characteristic
293	C₂×C₄	C₇×#127	1	2	yes	—
294	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
295	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
296	C₈	—	2	2	no	—
297	C₈	—	2	2	no	—
298	C₈	—	2	4	no	—
299	C₈	—	2	4	no	—
300	C₇	#19717	1	1	yes	characteristic, direct product factor, Sylow
301	C₂²	—	2	1	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	16	4	no	—
307	C₂²	C₂×C₇×#127	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	2	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	C₇×#575	1	2	yes	—
314	C₄	C₇×#577	1	4	yes	—
315	C₂	—	4	1	no	—
316	C₂	—	16	2	no	—
317	C₂	C₇×#3153	1	1	yes	characteristic
318	C₂	C₇×#3154	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂×C₇×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3231	C₂	semidirect
3. C₇×#3185	C₂	semidirect
5. C₇×#1720	C₂	semidirect
6. C₇×#1719	C₂	semidirect
7. C₇×#1714	C₂	semidirect
8. C₇×#1708	C₂	semidirect
9. C₇×#1453	C₂	semidirect
13. C₇×#500	C₂²	semidirect
14. C₇×#498	C₂²	semidirect
15. C₇×#498	C₂²	semidirect
16. C₇×#494	C₂²	semidirect
20. C₇×#487	C₂²	semidirect
21. C₇×#487	C₂²	semidirect
22. C₇×#487	C₂²	semidirect
36. C₇×#322	C₂²	semidirect
37. C₇×#322	C₂²	semidirect
38. C₇×C₈²	C₂²	semidirect
39. #19717	C₇	direct (N × H)
80. #3231	C₂×C₇	semidirect
82. #3185	C₂×C₇	semidirect
84. #1720	C₂×C₇	semidirect
85. #1719	C₂×C₇	semidirect
86. #1714	C₂×C₇	semidirect
87. #1708	C₂×C₇	semidirect
88. #1453	C₂×C₇	semidirect
129. #500	C₂²×C₇	semidirect
130. #498	C₂²×C₇	semidirect
131. #498	C₂²×C₇	semidirect
132. #494	C₂²×C₇	semidirect
136. #487	C₂²×C₇	semidirect
137. #487	C₂²×C₇	semidirect
138. #487	C₂²×C₇	semidirect
152. #322	C₂²×C₇	semidirect
153. #322	C₂²×C₇	semidirect
154. C₈²	C₂²×C₇	semidirect
300. C₇	#19717	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#3231	2
C₇×#3208	2
C₇×#3185	2
C₇×#3138	2
C₇×#1720	2
C₇×#1719	2
C₇×#1714	2
C₇×#1708	2
C₇×#1453	2
#19717	7

Subgroups — C7×#19717 #13196933

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19717 #13196933