Subgroups — C₇×#19188 #13196404

Order 1792.

714 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×#3226	C₂	1	2	yes	maximal
2	C₂×C₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2931	C₂	1	1	yes	maximal, characteristic
4	C₇×#1689	C₂	1	4	yes	maximal, complemented
5	C₇×#1669	C₂	1	4	yes	maximal, complemented
6	C₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1546	C₂	1	2	yes	maximal, complemented
8	C₂²×C₇×Q₁₆	—	2	1	no	—
9	C₂²×C₇×Q₁₆	—	2	1	no	—
10	C₂×C₇×#131	—	2	1	no	—
11	C₇×#532	C₂²	1	1	yes	characteristic
12	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₇×#500	—	2	2	no	—
14	C₇×#494	—	2	1	no	—
15	C₇×#494	—	2	1	no	—
16	C₇×#494	C₂²	1	2	yes	—
17	C₇×#494	C₂²	1	2	yes	—
18	C₇×#486	—	2	2	no	—
19	C₇×#480	—	2	4	no	—
20	C₇×#474	C₂²	1	8	yes	—
21	C₇×#436	—	2	1	no	—
22	C₇×#436	C₂²	1	2	yes	—
23	C₇×C₈×D₄	—	2	1	no	—
24	C₇×C₈×D₄	C₂²	1	2	yes	—
25	C₇×#431	C₂²	1	2	yes	—
26	C₇×#423	C₂²	1	1	yes	characteristic
27	C₇×#408	—	2	1	no	—
28	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
29	C₇×#335	C₂²	1	2	yes	—
30	C₇×#334	C₂²	1	2	yes	—
31	C₇×#332	C₂²	1	8	yes	—
32	#19188	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂×C₇×#41	—	2	1	no	—
34	C₂²×C₇×Q₈	—	2	2	no	—
35	C₂×C₇×Q₁₆	—	2	2	no	—
36	C₂×C₇×Q₁₆	—	2	2	no	—
37	C₂×C₇×Q₁₆	—	2	2	no	—
38	C₂×C₇×Q₁₆	—	2	2	no	—
39	C₂×C₇×Q₁₆	—	4	4	no	—
40	C₂×C₇×Q₁₆	—	4	4	no	—
41	C₇×#131	—	4	4	no	—
42	C₂×C₇×M₁₆	—	2	1	no	—
43	C₂×C₇×M₁₆	—	2	2	no	—
44	C₂²×C₇×C₈	—	2	2	no	—
45	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
46	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
47	C₇×C₄⋊Q₈	—	2	2	no	—
48	C₇×C₄⋊Q₈	—	2	2	no	—
49	C₇×C₄⋊Q₈	—	2	2	no	—
50	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
51	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
52	C₇×#127	D₄	1	1	yes	characteristic
53	C₇×#124	—	2	1	no	—
54	C₇×#121	—	2	1	no	—
55	C₇×#121	D₄	1	2	yes	—
56	C₄×C₇×D₄	—	2	1	no	—
57	C₄×C₇×D₄	C₂³	1	2	yes	—
58	C₂×C₇×C₄⋊C₄	—	2	2	no	—
59	C₂×C₇×C₄⋊C₄	—	2	2	no	—
60	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
61	C₇×#107	—	2	8	no	—
62	C₇×C₄⋊C₈	—	2	1	no	—
63	C₇×C₄⋊C₈	C₂³	1	4	yes	—
64	C₇×C₄⋊C₈	C₂³	1	2	yes	—
65	C₇×#102	—	2	8	no	—
66	C₇×C₂²⋊C₈	—	2	2	no	—
67	C₇×C₂²⋊C₈	—	2	2	no	—
68	C₇×C₂²⋊C₈	—	2	2	no	—
69	C₄×C₇×C₈	—	2	2	no	—
70	C₄×C₇×C₈	D₄	1	1	yes	characteristic
71	C₄×C₇×C₈	D₄	1	1	yes	characteristic
72	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
73	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
74	#3226	C₂×C₇	1	2	yes	—
75	C₂×#494	C₂×C₇	1	1	yes	characteristic, complemented
76	#2931	C₂×C₇	1	1	yes	characteristic
77	#1689	C₂×C₇	1	4	yes	complemented
78	#1669	C₂×C₇	1	4	yes	complemented
79	#1564	C₂×C₇	1	1	yes	characteristic, complemented
80	#1546	C₂×C₇	1	2	yes	complemented
81	C₇×#41	—	4	2	no	—
82	C₇×#41	—	4	2	no	—
83	C₂×C₇×Q₈	—	2	1	no	—
84	C₂×C₇×Q₈	—	2	4	no	—
85	C₂×C₇×Q₈	—	4	2	no	—
86	C₂×C₇×Q₈	—	4	4	no	—
87	C₂×C₇×Q₈	—	4	4	no	—
88	C₂×C₇×D₄	—	2	2	no	—
89	C₂×C₇×D₄	—	2	1	no	—
90	C₂×C₇×D₄	—	2	1	no	—
91	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₇	—	2	2	no	—
93	C₂²×C₄×C₇	—	4	2	no	—
94	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₇	#41	1	1	yes	characteristic
97	C₇×Q₁₆	—	4	4	no	—
98	C₇×Q₁₆	—	4	4	no	—
99	C₇×Q₁₆	—	4	4	no	—
100	C₇×Q₁₆	—	4	4	no	—
101	C₇×M₁₆	—	4	2	no	—
102	C₇×M₁₆	—	4	4	no	—
103	C₂×C₇×C₈	—	2	2	no	—
104	C₂×C₇×C₈	—	2	4	no	—
105	C₂×C₇×C₈	—	2	2	no	—
106	C₂×C₇×C₈	—	4	4	no	—
107	C₂×C₇×C₈	D₈	1	4	yes	—
108	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
112	C₇×C₄⋊C₄	—	2	1	no	—
113	C₇×C₄⋊C₄	—	2	4	no	—
114	C₇×C₄⋊C₄	—	2	4	no	—
115	C₇×C₄⋊C₄	—	4	2	no	—
116	C₇×C₄⋊C₄	—	4	2	no	—
117	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
118	C₇×C₂²⋊C₄	—	2	2	no	—
119	C₇×C₂²⋊C₄	—	2	2	no	—
120	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₇	#41	1	1	yes	characteristic
123	C₄²×C₇	C₂⁴	1	1	yes	characteristic
124	C₂²×Q₁₆	—	2	1	no	—
125	C₂²×Q₁₆	—	2	1	no	—
126	C₂×#131	—	2	1	no	—
127	#532	C₂²×C₇	1	1	yes	characteristic
128	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
129	#500	—	2	2	no	—
130	#494	—	2	1	no	—
131	#494	—	2	1	no	—
132	#494	C₂²×C₇	1	2	yes	—
133	#494	C₂²×C₇	1	2	yes	—
134	#486	—	2	2	no	—
135	#480	—	2	4	no	—
136	#474	C₂²×C₇	1	8	yes	—
137	#436	—	2	1	no	—
138	#436	C₂²×C₇	1	2	yes	—
139	C₈×D₄	—	2	1	no	—
140	C₈×D₄	C₂²×C₇	1	2	yes	—
141	#431	C₂²×C₇	1	2	yes	—
142	#423	C₂²×C₇	1	1	yes	characteristic
143	#408	—	2	1	no	—
144	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
145	#335	C₂²×C₇	1	2	yes	—
146	#334	C₂²×C₇	1	2	yes	—
147	#332	C₂²×C₇	1	8	yes	—
148	C₂³×C₇	—	2	2	no	—
149	C₂³×C₇	#121	1	1	yes	characteristic
150	C₇×Q₈	—	4	2	no	—
151	C₇×Q₈	—	4	4	no	—
152	C₇×Q₈	—	4	4	no	—
153	C₇×Q₈	—	8	4	no	—
154	C₇×D₄	—	2	4	no	—
155	C₇×D₄	—	4	4	no	—
156	C₇×D₄	—	4	2	no	—
157	C₇×D₄	—	4	2	no	—
158	C₂×C₄×C₇	—	2	2	no	—
159	C₂×C₄×C₇	—	2	2	no	—
160	C₂×C₄×C₇	—	2	2	no	—
161	C₂×C₄×C₇	—	4	2	no	—
162	C₂×C₄×C₇	—	4	2	no	—
163	C₂×C₄×C₇	—	4	4	no	—
164	C₂×C₄×C₇	—	8	4	no	—
165	C₂×C₄×C₇	#121	1	1	yes	characteristic
166	C₂×C₄×C₇	#121	1	1	yes	characteristic
167	C₂×C₄×C₇	#121	1	1	yes	characteristic
168	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
169	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
170	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
173	C₇×C₈	—	2	1	no	—
174	C₇×C₈	—	2	1	no	—
175	C₇×C₈	—	4	2	no	—
176	C₇×C₈	—	4	4	no	—
177	C₇×C₈	C₂×D₈	1	2	yes	—
178	C₇×C₈	C₂×D₈	1	2	yes	—
179	C₂×#41	—	2	1	no	—
180	C₂²×Q₈	—	2	2	no	—
181	C₂×Q₁₆	—	2	2	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	—	2	2	no	—
184	C₂×Q₁₆	—	2	2	no	—
185	C₂×Q₁₆	—	4	4	no	—
186	C₂×Q₁₆	—	4	4	no	—
187	#131	—	4	4	no	—
188	C₂×M₁₆	—	2	1	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
192	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
193	C₄⋊Q₈	—	2	2	no	—
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
197	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
198	#127	C₇×D₄	1	1	yes	characteristic
199	#124	—	2	1	no	—
200	#121	—	2	1	no	—
201	#121	C₇×D₄	1	2	yes	—
202	C₄×D₄	—	2	1	no	—
203	C₄×D₄	C₂³×C₇	1	2	yes	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
207	#107	—	2	8	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	C₂³×C₇	1	2	yes	—
210	C₄⋊C₈	C₂³×C₇	1	4	yes	—
211	#102	—	2	8	no	—
212	C₂²⋊C₈	—	2	2	no	—
213	C₂²⋊C₈	—	2	2	no	—
214	C₂²⋊C₈	—	2	2	no	—
215	C₄×C₈	—	2	2	no	—
216	C₄×C₈	C₇×D₄	1	1	yes	characteristic
217	C₄×C₈	C₇×D₄	1	1	yes	characteristic
218	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
219	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
220	C₂²×C₇	—	2	1	no	—
221	C₂²×C₇	—	2	1	no	—
222	C₂²×C₇	—	4	2	no	—
223	C₂²×C₇	—	4	2	no	—
224	C₂²×C₇	—	4	2	no	—
225	C₂²×C₇	#459	1	2	yes	—
226	C₂²×C₇	C₂×#121	1	1	yes	characteristic
227	C₄×C₇	—	2	1	no	—
228	C₄×C₇	—	2	1	no	—
229	C₄×C₇	—	4	2	no	—
230	C₄×C₇	—	8	4	no	—
231	C₄×C₇	#459	1	2	yes	—
232	C₄×C₇	#473	1	2	yes	—
233	C₄×C₇	C₂²×D₈	1	1	yes	characteristic
234	C₄×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
235	C₄×C₇	#578	1	2	yes	—
236	#41	—	4	2	no	—
237	#41	—	4	2	no	—
238	C₂×Q₈	—	2	1	no	—
239	C₂×Q₈	—	2	4	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×Q₈	—	4	4	no	—
242	C₂×Q₈	—	4	4	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	4	2	no	—
249	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
251	C₂²×C₄	C₇×#41	1	1	yes	characteristic
252	Q₁₆	—	4	4	no	—
253	Q₁₆	—	4	4	no	—
254	Q₁₆	—	4	4	no	—
255	Q₁₆	—	4	4	no	—
256	M₁₆	—	4	2	no	—
257	M₁₆	—	4	4	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	4	4	no	—
262	C₂×C₈	C₇×D₈	1	4	yes	—
263	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
264	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
265	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
266	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	2	2	no	—
275	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
276	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
277	C₄²	C₇×#41	1	1	yes	characteristic
278	C₄²	C₂⁴×C₇	1	1	yes	characteristic
279	C₂×C₇	—	2	2	no	—
280	C₂×C₇	—	8	2	no	—
281	C₂×C₇	C₂×#459	1	1	yes	characteristic
282	C₂×C₇	#3059	1	1	yes	characteristic
283	C₂×C₇	C₂×#473	1	1	yes	characteristic
284	C₂³	—	2	2	no	—
285	C₂³	C₇×#121	1	1	yes	characteristic
286	Q₈	—	4	2	no	—
287	Q₈	—	4	4	no	—
288	Q₈	—	4	4	no	—
289	Q₈	—	8	4	no	—
290	D₄	—	2	4	no	—
291	D₄	—	4	4	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	8	4	no	—
301	C₂×C₄	C₇×#121	1	1	yes	characteristic
302	C₂×C₄	C₇×#121	1	1	yes	characteristic
303	C₂×C₄	C₇×#121	1	1	yes	characteristic
304	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
305	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
306	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
307	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
308	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	4	2	no	—
312	C₈	—	4	4	no	—
313	C₈	C₂×C₇×D₈	1	2	yes	—
314	C₈	C₂×C₇×D₈	1	2	yes	—
315	C₇	#19188	1	1	yes	characteristic, direct product factor, Sylow
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	C₇×#459	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	—	8	4	no	—
327	C₄	C₇×#459	1	2	yes	—
328	C₄	C₇×#473	1	2	yes	—
329	C₄	C₂²×C₇×D₈	1	1	yes	characteristic
330	C₄	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
331	C₄	C₇×#578	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	8	2	no	—
334	C₂	C₂×C₇×#459	1	1	yes	characteristic
335	C₂	C₇×#3059	1	1	yes	characteristic
336	C₂	C₂×C₇×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#532	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
2. C₂×C₇×#494	C₂	semidirect
4. C₇×#1689	C₂	semidirect
5. C₇×#1669	C₂	semidirect
6. C₇×#1564	C₂	semidirect
7. C₇×#1546	C₂	semidirect
32. #19188	C₇	direct (N × H)
75. C₂×#494	C₂×C₇	semidirect
77. #1689	C₂×C₇	semidirect
78. #1669	C₂×C₇	semidirect
79. #1564	C₂×C₇	semidirect
80. #1546	C₂×C₇	semidirect
315. C₇	#19188	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×#3226	2
C₂×C₇×#494	2
C₇×#2931	2
C₇×#1689	2
C₇×#1669	2
C₇×#1564	2
C₇×#1546	2
#19188	7

Subgroups — C7×#19188 #13196404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19188 #13196404