Subgroups — C₇×#19203 #13196419

Order 1792.

774 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3208	C₂	1	2	yes	maximal, complemented
2	C₂×C₇×#493	C₂	1	1	yes	maximal, characteristic
3	C₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1688	C₂	1	4	yes	maximal, complemented
5	C₇×#1668	C₂	1	4	yes	maximal, complemented
6	C₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1545	C₂	1	2	yes	maximal, complemented
8	C₂²×C₇×D₈	—	2	1	no	—
9	C₂²×C₇×D₈	—	2	1	no	—
10	C₇×#533	C₂²	1	1	yes	characteristic
11	C₂×C₇×#127	C₂²	1	2	yes	—
12	C₇×#493	—	2	1	no	—
13	C₇×#493	—	2	1	no	—
14	C₇×#493	C₂²	1	2	yes	—
15	C₇×#493	C₂²	1	2	yes	—
16	C₇×#492	—	2	2	no	—
17	C₇×#486	—	2	2	no	—
18	C₇×#460	C₂²	1	8	yes	complemented
19	C₇×#450	—	2	4	no	—
20	C₇×#446	—	2	1	no	—
21	C₇×#446	C₂²	1	2	yes	complemented
22	C₇×C₈×Q₈	—	2	1	no	—
23	C₇×C₈×Q₈	C₂²	1	2	yes	complemented
24	C₇×#431	C₂²	1	2	yes	—
25	C₇×#424	—	2	1	no	—
26	C₇×#423	C₂²	1	1	yes	characteristic
27	C₇×#405	—	2	1	no	—
28	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
29	C₇×#335	C₂²	1	2	yes	complemented
30	C₇×#334	C₂²	1	2	yes	complemented
31	C₇×#331	C₂²	1	8	yes	complemented
32	#19203	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂²×C₇×D₄	—	2	2	no	—
34	C₂²×C₇×D₄	—	2	2	no	—
35	C₂²×C₇×D₄	—	2	2	no	—
36	C₂×C₇×SD₁₆	—	2	8	no	—
37	C₂×C₇×D₈	—	2	2	no	—
38	C₂×C₇×D₈	—	2	2	no	—
39	C₂×C₇×D₈	—	2	2	no	—
40	C₂×C₇×D₈	—	2	2	no	—
41	C₂×C₇×D₈	—	4	4	no	—
42	C₂×C₇×D₈	—	4	4	no	—
43	C₂×C₇×M₁₆	—	2	1	no	—
44	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
45	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
46	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
47	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
48	C₇×#127	—	2	2	no	—
49	C₇×#127	—	2	2	no	—
50	C₇×#127	—	2	2	no	—
51	C₇×#127	C₂³	1	4	yes	—
52	C₇×#125	—	2	1	no	—
53	C₇×C₂²⋊Q₈	—	2	1	no	—
54	C₇×C₂²⋊Q₈	D₄	1	2	yes	—
55	C₄×C₇×Q₈	—	2	1	no	—
56	C₄×C₇×Q₈	C₂³	1	2	yes	—
57	C₇×#117	—	2	1	no	—
58	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
59	C₇×C₄⋊C₈	—	2	2	no	—
60	C₇×C₄⋊C₈	—	2	2	no	—
61	C₇×C₄⋊C₈	—	2	1	no	—
62	C₇×C₄⋊C₈	C₂³	1	2	yes	—
63	C₇×C₄⋊C₈	C₂³	1	4	yes	—
64	C₇×#102	—	2	8	no	—
65	C₇×C₂²⋊C₈	—	2	2	no	—
66	C₇×#97	—	2	2	no	—
67	C₄×C₇×C₈	—	2	2	no	—
68	C₄×C₇×C₈	—	2	2	no	—
69	C₄×C₇×C₈	D₄	1	1	yes	characteristic
70	C₄×C₇×C₈	D₄	1	1	yes	characteristic
71	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
72	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
73	#3208	C₂×C₇	1	2	yes	complemented
74	C₂×#493	C₂×C₇	1	1	yes	characteristic
75	#2960	C₂×C₇	1	1	yes	characteristic, complemented
76	#1688	C₂×C₇	1	4	yes	complemented
77	#1668	C₂×C₇	1	4	yes	complemented
78	#1564	C₂×C₇	1	1	yes	characteristic, complemented
79	#1545	C₂×C₇	1	2	yes	complemented
80	C₂⁴×C₇	—	4	2	no	—
81	C₂×C₇×Q₈	—	2	1	no	—
82	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
83	C₂×C₇×D₄	—	2	4	no	—
84	C₂×C₇×D₄	—	2	4	no	—
85	C₂×C₇×D₄	—	2	4	no	—
86	C₂×C₇×D₄	—	4	2	no	—
87	C₂×C₇×D₄	—	4	2	no	—
88	C₂×C₇×D₄	—	4	2	no	—
89	C₂×C₇×D₄	—	4	4	no	—
90	C₂×C₇×D₄	—	4	4	no	—
91	C₂×C₇×D₄	—	8	4	no	—
92	C₂×C₇×D₄	—	8	4	no	—
93	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₇	#41	1	1	yes	characteristic
96	C₇×SD₁₆	—	8	8	no	—
97	C₇×D₈	—	4	4	no	—
98	C₇×D₈	—	4	4	no	—
99	C₇×D₈	—	4	4	no	—
100	C₇×D₈	—	4	4	no	—
101	C₇×M₁₆	—	4	2	no	—
102	C₂×C₇×C₈	—	2	2	no	—
103	C₂×C₇×C₈	—	2	2	no	—
104	C₂×C₇×C₈	—	2	4	no	—
105	C₂×C₇×C₈	SD₁₆	1	4	yes	complemented
106	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
110	C₇×C₄⋊C₄	—	2	1	no	—
111	C₇×C₄⋊C₄	—	2	1	no	—
112	C₇×C₄⋊C₄	—	2	1	no	—
113	C₇×C₄⋊C₄	—	2	2	no	—
114	C₇×C₄⋊C₄	—	2	2	no	—
115	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
116	C₇×C₂²⋊C₄	—	2	2	no	—
117	C₄²×C₇	—	2	2	no	—
118	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₇	#41	1	1	yes	characteristic
121	C₄²×C₇	C₂⁴	1	1	yes	characteristic
122	C₂²×D₈	—	2	1	no	—
123	C₂²×D₈	—	2	1	no	—
124	#533	C₂²×C₇	1	1	yes	characteristic
125	C₂×#127	C₂²×C₇	1	2	yes	—
126	#493	—	2	1	no	—
127	#493	—	2	1	no	—
128	#493	C₂²×C₇	1	2	yes	—
129	#493	C₂²×C₇	1	2	yes	—
130	#492	—	2	2	no	—
131	#486	—	2	2	no	—
132	#460	C₂²×C₇	1	8	yes	complemented
133	#450	—	2	4	no	—
134	#446	—	2	1	no	—
135	#446	C₂²×C₇	1	2	yes	complemented
136	C₈×Q₈	—	2	1	no	—
137	C₈×Q₈	C₂²×C₇	1	2	yes	complemented
138	#431	C₂²×C₇	1	2	yes	—
139	#424	—	2	1	no	—
140	#423	C₂²×C₇	1	1	yes	characteristic
141	#405	—	2	1	no	—
142	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
143	#335	C₂²×C₇	1	2	yes	complemented
144	#334	C₂²×C₇	1	2	yes	complemented
145	#331	C₂²×C₇	1	8	yes	complemented
146	C₂³×C₇	—	4	4	no	—
147	C₂³×C₇	—	8	4	no	—
148	C₂³×C₇	—	8	4	no	—
149	C₂³×C₇	—	8	4	no	—
150	C₂³×C₇	#121	1	1	yes	characteristic
151	C₇×Q₈	—	2	4	no	—
152	C₇×Q₈	—	4	2	no	—
153	C₇×D₄	—	4	4	no	—
154	C₇×D₄	—	4	4	no	—
155	C₇×D₄	—	4	8	no	—
156	C₇×D₄	—	8	4	no	—
157	C₇×D₄	—	8	4	no	—
158	C₇×D₄	—	8	4	no	—
159	C₇×D₄	—	8	4	no	—
160	C₂×C₄×C₇	—	2	2	no	—
161	C₂×C₄×C₇	—	2	2	no	—
162	C₂×C₄×C₇	—	2	2	no	—
163	C₂×C₄×C₇	—	2	2	no	—
164	C₂×C₄×C₇	#121	1	1	yes	characteristic
165	C₂×C₄×C₇	#121	1	1	yes	characteristic
166	C₂×C₄×C₇	#121	1	1	yes	characteristic
167	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
168	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
169	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
172	C₇×C₈	—	2	1	no	—
173	C₇×C₈	—	2	1	no	—
174	C₇×C₈	—	4	2	no	—
175	C₇×C₈	—	4	4	no	—
176	C₇×C₈	C₂×SD₁₆	1	2	yes	—
177	C₇×C₈	C₂×SD₁₆	1	2	yes	—
178	C₂²×D₄	—	2	2	no	—
179	C₂²×D₄	—	2	2	no	—
180	C₂²×D₄	—	2	2	no	—
181	C₂×SD₁₆	—	2	8	no	—
182	C₂×D₈	—	2	2	no	—
183	C₂×D₈	—	2	2	no	—
184	C₂×D₈	—	2	2	no	—
185	C₂×D₈	—	2	2	no	—
186	C₂×D₈	—	4	4	no	—
187	C₂×D₈	—	4	4	no	—
188	C₂×M₁₆	—	2	1	no	—
189	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
190	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
191	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
192	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
193	#127	—	2	2	no	—
194	#127	—	2	2	no	—
195	#127	—	2	2	no	—
196	#127	C₂³×C₇	1	4	yes	—
197	#125	—	2	1	no	—
198	C₂²⋊Q₈	—	2	1	no	—
199	C₂²⋊Q₈	C₇×D₄	1	2	yes	—
200	C₄×Q₈	—	2	1	no	—
201	C₄×Q₈	C₂³×C₇	1	2	yes	—
202	#117	—	2	1	no	—
203	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	—	2	2	no	—
206	C₄⋊C₈	—	2	1	no	—
207	C₄⋊C₈	C₂³×C₇	1	2	yes	—
208	C₄⋊C₈	C₂³×C₇	1	4	yes	—
209	#102	—	2	8	no	—
210	C₂²⋊C₈	—	2	2	no	—
211	#97	—	2	2	no	—
212	C₄×C₈	—	2	2	no	—
213	C₄×C₈	—	2	2	no	—
214	C₄×C₈	C₇×D₄	1	1	yes	characteristic
215	C₄×C₈	C₇×D₄	1	1	yes	characteristic
216	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
217	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
218	C₂²×C₇	—	2	1	no	—
219	C₂²×C₇	—	2	1	no	—
220	C₂²×C₇	—	8	4	no	—
221	C₂²×C₇	—	8	4	no	—
222	C₂²×C₇	—	8	4	no	—
223	C₂²×C₇	—	16	8	no	—
224	C₂²×C₇	#460	1	2	yes	—
225	C₂²×C₇	C₂×#121	1	1	yes	characteristic
226	C₄×C₇	—	2	1	no	—
227	C₄×C₇	—	2	1	no	—
228	C₄×C₇	—	4	2	no	—
229	C₄×C₇	—	4	2	no	—
230	C₄×C₇	#460	1	2	yes	—
231	C₄×C₇	#472	1	2	yes	—
232	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
233	C₄×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
234	C₄×C₇	#576	1	2	yes	—
235	C₂⁴	—	4	2	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
238	C₂×D₄	—	2	4	no	—
239	C₂×D₄	—	2	4	no	—
240	C₂×D₄	—	2	4	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂×D₄	—	4	4	no	—
245	C₂×D₄	—	4	4	no	—
246	C₂×D₄	—	8	4	no	—
247	C₂×D₄	—	8	4	no	—
248	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
250	C₂²×C₄	C₇×#41	1	1	yes	characteristic
251	SD₁₆	—	8	8	no	—
252	D₈	—	4	4	no	—
253	D₈	—	4	4	no	—
254	D₈	—	4	4	no	—
255	D₈	—	4	4	no	—
256	M₁₆	—	4	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	C₇×SD₁₆	1	4	yes	complemented
261	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
262	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
263	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
264	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
274	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
275	C₄²	C₇×#41	1	1	yes	characteristic
276	C₄²	C₂⁴×C₇	1	1	yes	characteristic
277	C₂×C₇	—	2	2	no	—
278	C₂×C₇	—	16	4	no	—
279	C₂×C₇	C₂×#460	1	1	yes	characteristic
280	C₂×C₇	#3058	1	1	yes	characteristic
281	C₂×C₇	C₂×#472	1	1	yes	characteristic
282	C₂³	—	4	4	no	—
283	C₂³	—	8	4	no	—
284	C₂³	—	8	4	no	—
285	C₂³	—	8	4	no	—
286	C₂³	C₇×#121	1	1	yes	characteristic
287	Q₈	—	2	4	no	—
288	Q₈	—	4	2	no	—
289	D₄	—	4	4	no	—
290	D₄	—	4	4	no	—
291	D₄	—	4	8	no	—
292	D₄	—	8	4	no	—
293	D₄	—	8	4	no	—
294	D₄	—	8	4	no	—
295	D₄	—	8	4	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	C₇×#121	1	1	yes	characteristic
301	C₂×C₄	C₇×#121	1	1	yes	characteristic
302	C₂×C₄	C₇×#121	1	1	yes	characteristic
303	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
304	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
305	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
307	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	4	2	no	—
311	C₈	—	4	4	no	—
312	C₈	C₂×C₇×SD₁₆	1	2	yes	—
313	C₈	C₂×C₇×SD₁₆	1	2	yes	—
314	C₇	#19203	1	1	yes	characteristic, direct product factor, Sylow
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	8	4	no	—
318	C₂²	—	8	4	no	—
319	C₂²	—	8	4	no	—
320	C₂²	—	16	8	no	—
321	C₂²	C₇×#460	1	2	yes	—
322	C₂²	C₂×C₇×#121	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	C₇×#460	1	2	yes	—
328	C₄	C₇×#472	1	2	yes	—
329	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
330	C₄	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
331	C₄	C₇×#576	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	16	4	no	—
334	C₂	C₂×C₇×#460	1	1	yes	characteristic
335	C₂	C₇×#3058	1	1	yes	characteristic
336	C₂	C₂×C₇×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂×C₇×#493	C₂
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×#3208	C₂	semidirect
3. C₇×#2960	C₂	semidirect
4. C₇×#1688	C₂	semidirect
5. C₇×#1668	C₂	semidirect
6. C₇×#1564	C₂	semidirect
7. C₇×#1545	C₂	semidirect
18. C₇×#460	C₂²	semidirect
21. C₇×#446	C₂²	semidirect
23. C₇×C₈×Q₈	C₂²	semidirect
29. C₇×#335	C₂²	semidirect
30. C₇×#334	C₂²	semidirect
31. C₇×#331	C₂²	semidirect
32. #19203	C₇	direct (N × H)
73. #3208	C₂×C₇	semidirect
75. #2960	C₂×C₇	semidirect
76. #1688	C₂×C₇	semidirect
77. #1668	C₂×C₇	semidirect
78. #1564	C₂×C₇	semidirect
79. #1545	C₂×C₇	semidirect
105. C₂×C₇×C₈	SD₁₆	semidirect
132. #460	C₂²×C₇	semidirect
135. #446	C₂²×C₇	semidirect
137. C₈×Q₈	C₂²×C₇	semidirect
143. #335	C₂²×C₇	semidirect
144. #334	C₂²×C₇	semidirect
145. #331	C₂²×C₇	semidirect
260. C₂×C₈	C₇×SD₁₆	semidirect
314. C₇	#19203	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×#3208	2
C₂×C₇×#493	2
C₇×#2960	2
C₇×#1688	2
C₇×#1668	2
C₇×#1564	2
C₇×#1545	2
#19203	7

Subgroups — C7×#19203 #13196419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19203 #13196419