Subgroups — C₇×#19187 #13196403

Order 1792.

842 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3205	C₂	1	2	yes	maximal, complemented
2	C₂×C₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1687	C₂	1	4	yes	maximal, complemented
5	C₇×#1667	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	C₂×C₇×#131	—	2	1	no	—
11	C₇×#532	C₂²	1	1	yes	characteristic
12	C₂×C₇×#127	C₂²	1	2	yes	—
13	C₇×#493	—	2	2	no	—
14	C₇×#493	—	2	1	no	—
15	C₇×#493	—	2	1	no	—
16	C₇×#493	C₂²	1	2	yes	—
17	C₇×#493	C₂²	1	2	yes	—
18	C₇×#486	—	2	2	no	—
19	C₇×#459	C₂²	1	8	yes	complemented
20	C₇×#447	—	2	4	no	—
21	C₇×#436	—	2	1	no	—
22	C₇×#436	C₂²	1	2	yes	complemented
23	C₇×C₈×D₄	—	2	1	no	—
24	C₇×C₈×D₄	C₂²	1	2	yes	complemented
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	complemented
30	C₇×#334	C₂²	1	2	yes	complemented
31	C₇×#331	C₂²	1	8	yes	complemented
32	#19187	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂×C₇×#41	—	2	1	no	—
34	C₂²×C₇×D₄	—	2	2	no	—
35	C₂²×C₇×D₄	—	2	2	no	—
36	C₂²×C₇×D₄	—	2	2	no	—
37	C₂×C₇×D₈	—	2	8	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	—	2	2	no	—
42	C₂×C₇×D₈	—	4	4	no	—
43	C₂×C₇×D₈	—	4	4	no	—
44	C₇×#131	—	4	4	no	—
45	C₂×C₇×M₁₆	—	2	1	no	—
46	C₂×C₇×M₁₆	—	2	2	no	—
47	C₂²×C₇×C₈	—	2	2	no	—
48	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
49	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
50	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
51	C₇×#127	—	2	2	no	—
52	C₇×#127	—	2	2	no	—
53	C₇×#127	—	2	2	no	—
54	C₇×#127	D₄	1	1	yes	characteristic
55	C₇×#127	C₂³	1	4	yes	—
56	C₇×#124	—	2	1	no	—
57	C₇×#121	—	2	1	no	—
58	C₇×#121	D₄	1	2	yes	—
59	C₄×C₇×D₄	—	2	1	no	—
60	C₄×C₇×D₄	C₂³	1	2	yes	—
61	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
62	C₇×C₄⋊C₈	—	2	1	no	—
63	C₇×C₄⋊C₈	C₂³	1	2	yes	—
64	C₇×C₄⋊C₈	C₂³	1	4	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	#3205	C₂×C₇	1	2	yes	complemented
75	C₂×#493	C₂×C₇	1	1	yes	characteristic, complemented
76	#2931	C₂×C₇	1	1	yes	characteristic, complemented
77	#1687	C₂×C₇	1	4	yes	complemented
78	#1667	C₂×C₇	1	4	yes	complemented
79	#1564	C₂×C₇	1	1	yes	characteristic, complemented
80	#1545	C₂×C₇	1	2	yes	complemented
81	C₂⁴×C₇	—	4	2	no	—
82	C₇×#41	—	4	2	no	—
83	C₇×#41	—	4	2	no	—
84	C₂×C₇×Q₈	—	2	1	no	—
85	C₂×C₇×D₄	—	2	2	no	—
86	C₂×C₇×D₄	—	2	1	no	—
87	C₂×C₇×D₄	—	2	1	no	—
88	C₂×C₇×D₄	—	2	4	no	—
89	C₂×C₇×D₄	—	2	4	no	—
90	C₂×C₇×D₄	—	2	4	no	—
91	C₂×C₇×D₄	—	4	2	no	—
92	C₂×C₇×D₄	—	4	2	no	—
93	C₂×C₇×D₄	—	4	2	no	—
94	C₂×C₇×D₄	—	4	4	no	—
95	C₂×C₇×D₄	—	4	4	no	—
96	C₂×C₇×D₄	—	8	4	no	—
97	C₂×C₇×D₄	—	8	4	no	—
98	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₇	—	2	2	no	—
100	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₇	#41	1	1	yes	characteristic
103	C₇×D₈	—	4	4	no	—
104	C₇×D₈	—	4	4	no	—
105	C₇×D₈	—	4	4	no	—
106	C₇×D₈	—	4	4	no	—
107	C₇×D₈	—	8	8	no	—
108	C₇×M₁₆	—	4	2	no	—
109	C₇×M₁₆	—	4	4	no	—
110	C₂×C₇×C₈	—	2	2	no	—
111	C₂×C₇×C₈	—	2	4	no	—
112	C₂×C₇×C₈	—	2	2	no	—
113	C₂×C₇×C₈	—	4	4	no	—
114	C₂×C₇×C₈	D₈	1	4	yes	complemented
115	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
116	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
117	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
118	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
119	C₇×C₄⋊C₄	—	2	1	no	—
120	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
121	C₇×C₂²⋊C₄	—	2	2	no	—
122	C₇×C₂²⋊C₄	—	2	2	no	—
123	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₇	#41	1	1	yes	characteristic
126	C₄²×C₇	C₂⁴	1	1	yes	characteristic
127	C₂²×D₈	—	2	1	no	—
128	C₂²×D₈	—	2	1	no	—
129	C₂×#131	—	2	1	no	—
130	#532	C₂²×C₇	1	1	yes	characteristic
131	C₂×#127	C₂²×C₇	1	2	yes	—
132	#493	—	2	2	no	—
133	#493	—	2	1	no	—
134	#493	—	2	1	no	—
135	#493	C₂²×C₇	1	2	yes	—
136	#493	C₂²×C₇	1	2	yes	—
137	#486	—	2	2	no	—
138	#459	C₂²×C₇	1	8	yes	complemented
139	#447	—	2	4	no	—
140	#436	—	2	1	no	—
141	#436	C₂²×C₇	1	2	yes	complemented
142	C₈×D₄	—	2	1	no	—
143	C₈×D₄	C₂²×C₇	1	2	yes	complemented
144	#431	C₂²×C₇	1	2	yes	—
145	#423	C₂²×C₇	1	1	yes	characteristic
146	#408	—	2	1	no	—
147	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
148	#335	C₂²×C₇	1	2	yes	complemented
149	#334	C₂²×C₇	1	2	yes	complemented
150	#331	C₂²×C₇	1	8	yes	complemented
151	C₂³×C₇	—	2	2	no	—
152	C₂³×C₇	—	4	4	no	—
153	C₂³×C₇	—	8	4	no	—
154	C₂³×C₇	—	8	4	no	—
155	C₂³×C₇	—	8	4	no	—
156	C₂³×C₇	#121	1	1	yes	characteristic
157	C₇×Q₈	—	4	2	no	—
158	C₇×D₄	—	2	4	no	—
159	C₇×D₄	—	4	4	no	—
160	C₇×D₄	—	4	4	no	—
161	C₇×D₄	—	4	4	no	—
162	C₇×D₄	—	4	8	no	—
163	C₇×D₄	—	4	2	no	—
164	C₇×D₄	—	4	2	no	—
165	C₇×D₄	—	8	4	no	—
166	C₇×D₄	—	8	4	no	—
167	C₇×D₄	—	8	4	no	—
168	C₇×D₄	—	8	4	no	—
169	C₂×C₄×C₇	—	2	2	no	—
170	C₂×C₄×C₇	—	2	2	no	—
171	C₂×C₄×C₇	—	2	2	no	—
172	C₂×C₄×C₇	—	4	2	no	—
173	C₂×C₄×C₇	—	4	2	no	—
174	C₂×C₄×C₇	#121	1	1	yes	characteristic
175	C₂×C₄×C₇	#121	1	1	yes	characteristic
176	C₂×C₄×C₇	#121	1	1	yes	characteristic
177	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
178	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
179	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
182	C₇×C₈	—	2	1	no	—
183	C₇×C₈	—	2	1	no	—
184	C₇×C₈	—	4	2	no	—
185	C₇×C₈	—	4	4	no	—
186	C₇×C₈	C₂×D₈	1	2	yes	—
187	C₇×C₈	C₂×D₈	1	2	yes	—
188	C₂×#41	—	2	1	no	—
189	C₂²×D₄	—	2	2	no	—
190	C₂²×D₄	—	2	2	no	—
191	C₂²×D₄	—	2	2	no	—
192	C₂×D₈	—	2	8	no	—
193	C₂×D₈	—	2	2	no	—
194	C₂×D₈	—	2	2	no	—
195	C₂×D₈	—	2	2	no	—
196	C₂×D₈	—	2	2	no	—
197	C₂×D₈	—	4	4	no	—
198	C₂×D₈	—	4	4	no	—
199	#131	—	4	4	no	—
200	C₂×M₁₆	—	2	1	no	—
201	C₂×M₁₆	—	2	2	no	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
204	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
205	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
206	#127	—	2	2	no	—
207	#127	—	2	2	no	—
208	#127	—	2	2	no	—
209	#127	C₇×D₄	1	1	yes	characteristic
210	#127	C₂³×C₇	1	4	yes	—
211	#124	—	2	1	no	—
212	#121	—	2	1	no	—
213	#121	C₇×D₄	1	2	yes	—
214	C₄×D₄	—	2	1	no	—
215	C₄×D₄	C₂³×C₇	1	2	yes	—
216	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
217	C₄⋊C₈	—	2	1	no	—
218	C₄⋊C₈	C₂³×C₇	1	2	yes	—
219	C₄⋊C₈	C₂³×C₇	1	4	yes	—
220	#102	—	2	8	no	—
221	C₂²⋊C₈	—	2	2	no	—
222	C₂²⋊C₈	—	2	2	no	—
223	C₂²⋊C₈	—	2	2	no	—
224	C₄×C₈	—	2	2	no	—
225	C₄×C₈	C₇×D₄	1	1	yes	characteristic
226	C₄×C₈	C₇×D₄	1	1	yes	characteristic
227	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
228	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
229	C₂²×C₇	—	2	1	no	—
230	C₂²×C₇	—	2	1	no	—
231	C₂²×C₇	—	4	2	no	—
232	C₂²×C₇	—	4	2	no	—
233	C₂²×C₇	—	4	2	no	—
234	C₂²×C₇	—	8	4	no	—
235	C₂²×C₇	—	8	4	no	—
236	C₂²×C₇	—	8	4	no	—
237	C₂²×C₇	—	16	8	no	—
238	C₂²×C₇	#459	1	2	yes	—
239	C₂²×C₇	C₂×#121	1	1	yes	characteristic
240	C₄×C₇	—	2	1	no	—
241	C₄×C₇	—	2	1	no	—
242	C₄×C₇	—	4	2	no	—
243	C₄×C₇	#459	1	2	yes	—
244	C₄×C₇	#472	1	2	yes	—
245	C₄×C₇	C₂²×D₈	1	1	yes	characteristic
246	C₄×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
247	C₄×C₇	#576	1	2	yes	—
248	C₂⁴	—	4	2	no	—
249	#41	—	4	2	no	—
250	#41	—	4	2	no	—
251	C₂×Q₈	—	2	1	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	2	1	no	—
255	C₂×D₄	—	2	4	no	—
256	C₂×D₄	—	2	4	no	—
257	C₂×D₄	—	2	4	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	4	4	no	—
262	C₂×D₄	—	4	4	no	—
263	C₂×D₄	—	8	4	no	—
264	C₂×D₄	—	8	4	no	—
265	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
269	C₂²×C₄	C₇×#41	1	1	yes	characteristic
270	D₈	—	4	4	no	—
271	D₈	—	4	4	no	—
272	D₈	—	4	4	no	—
273	D₈	—	4	4	no	—
274	D₈	—	8	8	no	—
275	M₁₆	—	4	2	no	—
276	M₁₆	—	4	4	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	4	4	no	—
281	C₂×C₈	C₇×D₈	1	4	yes	complemented
282	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
283	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
284	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
285	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
286	C₄⋊C₄	—	2	1	no	—
287	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
288	C₂²⋊C₄	—	2	2	no	—
289	C₂²⋊C₄	—	2	2	no	—
290	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
291	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
292	C₄²	C₇×#41	1	1	yes	characteristic
293	C₄²	C₂⁴×C₇	1	1	yes	characteristic
294	C₂×C₇	—	2	2	no	—
295	C₂×C₇	—	8	2	no	—
296	C₂×C₇	—	16	4	no	—
297	C₂×C₇	C₂×#459	1	1	yes	characteristic
298	C₂×C₇	#3058	1	1	yes	characteristic
299	C₂×C₇	C₂×#472	1	1	yes	characteristic
300	C₂³	—	2	2	no	—
301	C₂³	—	4	4	no	—
302	C₂³	—	8	4	no	—
303	C₂³	—	8	4	no	—
304	C₂³	—	8	4	no	—
305	C₂³	C₇×#121	1	1	yes	characteristic
306	Q₈	—	4	2	no	—
307	D₄	—	2	4	no	—
308	D₄	—	4	4	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	4	no	—
311	D₄	—	4	4	no	—
312	D₄	—	4	8	no	—
313	D₄	—	4	2	no	—
314	D₄	—	8	4	no	—
315	D₄	—	8	4	no	—
316	D₄	—	8	4	no	—
317	D₄	—	8	4	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	C₇×#121	1	1	yes	characteristic
324	C₂×C₄	C₇×#121	1	1	yes	characteristic
325	C₂×C₄	C₇×#121	1	1	yes	characteristic
326	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
327	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
328	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
329	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
330	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	4	2	no	—
334	C₈	—	4	4	no	—
335	C₈	C₂×C₇×D₈	1	2	yes	—
336	C₈	C₂×C₇×D₈	1	2	yes	—
337	C₇	#19187	1	1	yes	characteristic, direct product factor, Sylow
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	8	4	no	—
344	C₂²	—	8	4	no	—
345	C₂²	—	8	4	no	—
346	C₂²	—	16	8	no	—
347	C₂²	C₇×#459	1	2	yes	—
348	C₂²	C₂×C₇×#121	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	4	2	no	—
352	C₄	C₇×#459	1	2	yes	—
353	C₄	C₇×#472	1	2	yes	—
354	C₄	C₂²×C₇×D₈	1	1	yes	characteristic
355	C₄	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
356	C₄	C₇×#576	1	2	yes	—
357	C₂	—	2	2	no	—
358	C₂	—	8	2	no	—
359	C₂	—	16	4	no	—
360	C₂	C₂×C₇×#459	1	1	yes	characteristic
361	C₂	C₇×#3058	1	1	yes	characteristic
362	C₂	C₂×C₇×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
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×#3205	C₂	semidirect
2. C₂×C₇×#493	C₂	semidirect
3. C₇×#2931	C₂	semidirect
4. C₇×#1687	C₂	semidirect
5. C₇×#1667	C₂	semidirect
6. C₇×#1564	C₂	semidirect
7. C₇×#1545	C₂	semidirect
19. C₇×#459	C₂²	semidirect
22. C₇×#436	C₂²	semidirect
24. C₇×C₈×D₄	C₂²	semidirect
29. C₇×#335	C₂²	semidirect
30. C₇×#334	C₂²	semidirect
31. C₇×#331	C₂²	semidirect
32. #19187	C₇	direct (N × H)
74. #3205	C₂×C₇	semidirect
75. C₂×#493	C₂×C₇	semidirect
76. #2931	C₂×C₇	semidirect
77. #1687	C₂×C₇	semidirect
78. #1667	C₂×C₇	semidirect
79. #1564	C₂×C₇	semidirect
80. #1545	C₂×C₇	semidirect
114. C₂×C₇×C₈	D₈	semidirect
138. #459	C₂²×C₇	semidirect
141. #436	C₂²×C₇	semidirect
143. C₈×D₄	C₂²×C₇	semidirect
148. #335	C₂²×C₇	semidirect
149. #334	C₂²×C₇	semidirect
150. #331	C₂²×C₇	semidirect
281. C₂×C₈	C₇×D₈	semidirect
337. C₇	#19187	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×#3205	2
C₂×C₇×#493	2
C₇×#2931	2
C₇×#1687	2
C₇×#1667	2
C₇×#1564	2
C₇×#1545	2
#19187	7

Subgroups — C7×#19187 #13196403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19187 #13196403