Subgroups — C₇×#19105 #13196321

Order 1792.

694 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3101	C₂	1	2	yes	maximal
3	C₂×C₇×C₈×D₄	C₂	1	1	yes	maximal, characteristic
4	C₇×#1675	C₂	1	2	yes	maximal, complemented
5	C₇×#1673	C₂	1	2	yes	maximal, complemented
6	C₇×#1671	C₂	1	4	yes	maximal, complemented
7	C₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1542	C₂	1	2	yes	maximal, complemented
9	C₂³×C₇×C₈	—	2	1	no	—
10	C₇×#533	C₂²	1	2	yes	—
11	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
12	C₇×#500	C₂²	1	1	yes	characteristic, complemented
13	C₇×#500	C₂²	1	1	yes	characteristic, complemented
14	C₇×#499	—	2	1	no	—
15	C₇×#499	—	2	1	no	—
16	C₇×#498	C₂²	1	2	yes	complemented
17	C₇×#483	—	2	2	no	—
18	C₇×#476	C₂²	1	4	yes	—
19	C₇×#461	C₂²	1	4	yes	—
20	C₇×#450	—	2	2	no	—
21	C₄×C₇×SD₁₆	—	2	4	no	—
22	C₇×C₈×D₄	—	2	1	no	—
23	C₇×C₈×D₄	—	2	1	no	—
24	C₇×C₈×D₄	C₂²	1	2	yes	—
25	C₇×C₈×D₄	C₂²	1	2	yes	—
26	C₇×#431	C₂²	1	2	yes	—
27	C₇×#427	—	2	1	no	—
28	C₇×#427	—	2	1	no	—
29	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₇×C₂²⋊C₈	—	2	1	no	—
31	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
32	C₇×#335	C₂²	1	2	yes	complemented
33	C₇×#334	C₂²	1	2	yes	complemented
34	C₇×#333	C₂²	1	4	yes	complemented
35	C₇×#332	C₂²	1	4	yes	complemented
36	#19105	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂²×C₇×D₄	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₇	—	2	1	no	—
39	C₂×C₇×SD₁₆	—	2	4	no	—
40	C₂²×C₇×C₈	—	2	1	no	—
41	C₂²×C₇×C₈	—	2	2	no	—
42	C₂²×C₇×C₈	—	2	2	no	—
43	C₂²×C₇×C₈	—	4	4	no	—
44	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
45	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
46	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₇×#125	—	2	2	no	—
49	C₇×C₂²⋊Q₈	—	2	2	no	—
50	C₇×C₂²⋊Q₈	—	2	2	no	—
51	C₄×C₇×Q₈	—	2	2	no	—
52	C₄×C₇×Q₈	—	2	2	no	—
53	C₄×C₇×D₄	—	2	1	no	—
54	C₄×C₇×D₄	—	2	1	no	—
55	C₄×C₇×D₄	D₄	1	2	yes	—
56	C₄×C₇×D₄	C₂³	1	2	yes	—
57	C₇×#117	—	2	2	no	—
58	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
59	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
60	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
61	C₇×#107	—	2	1	no	—
62	C₇×#107	—	2	1	no	—
63	C₇×#107	—	2	1	no	—
64	C₇×#107	—	2	1	no	—
65	C₇×#106	—	2	2	no	—
66	C₇×#106	—	2	2	no	—
67	C₇×#106	—	2	4	no	—
68	C₇×C₄⋊C₈	—	2	1	no	—
69	C₇×C₄⋊C₈	C₂³	1	4	yes	—
70	C₇×C₄⋊C₈	C₂³	1	2	yes	—
71	C₇×#103	—	2	4	no	—
72	C₇×#102	—	2	4	no	—
73	C₇×C₂²⋊C₈	—	2	2	no	—
74	C₇×C₂²⋊C₈	—	2	2	no	—
75	C₇×C₂²⋊C₈	—	2	2	no	—
76	C₄×C₇×C₈	—	2	2	no	—
77	C₄×C₇×C₈	D₄	1	1	yes	characteristic
78	C₄×C₇×C₈	D₄	1	1	yes	characteristic
79	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
80	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
81	#3160	C₂×C₇	1	1	yes	characteristic, complemented
82	#3101	C₂×C₇	1	2	yes	—
83	C₂×C₈×D₄	C₂×C₇	1	1	yes	characteristic
84	#1675	C₂×C₇	1	2	yes	complemented
85	#1673	C₂×C₇	1	2	yes	complemented
86	#1671	C₂×C₇	1	4	yes	complemented
87	#1564	C₂×C₇	1	1	yes	characteristic, complemented
88	#1542	C₂×C₇	1	2	yes	complemented
89	C₂⁴×C₇	—	2	1	no	—
90	C₂×C₇×Q₈	—	2	2	no	—
91	C₂×C₇×Q₈	—	2	2	no	—
92	C₂×C₇×D₄	—	2	4	no	—
93	C₂×C₇×D₄	—	2	1	no	—
94	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₇	—	2	1	no	—
96	C₂²×C₄×C₇	—	2	2	no	—
97	C₂²×C₄×C₇	—	2	2	no	—
98	C₂²×C₄×C₇	—	4	2	no	—
99	C₂²×C₄×C₇	—	4	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×SD₁₆	—	4	8	no	—
104	C₂×C₇×C₈	—	2	2	no	—
105	C₂×C₇×C₈	—	2	4	no	—
106	C₂×C₇×C₈	—	2	2	no	—
107	C₂×C₇×C₈	—	4	2	no	—
108	C₂×C₇×C₈	—	4	4	no	—
109	C₂×C₇×C₈	—	4	4	no	—
110	C₂×C₇×C₈	SD₁₆	1	4	yes	complemented
111	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
115	C₇×C₄⋊C₄	—	2	1	no	—
116	C₇×C₄⋊C₄	—	2	2	no	—
117	C₇×C₄⋊C₄	—	2	2	no	—
118	C₇×C₄⋊C₄	—	2	2	no	—
119	C₇×C₄⋊C₄	—	2	2	no	—
120	C₇×C₄⋊C₄	—	4	2	no	—
121	C₇×C₄⋊C₄	—	4	2	no	—
122	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
123	C₇×C₂²⋊C₄	—	2	2	no	—
124	C₇×C₂²⋊C₄	—	2	2	no	—
125	C₇×C₂²⋊C₄	—	4	2	no	—
126	C₄²×C₇	—	4	2	no	—
127	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₇	#41	1	1	yes	characteristic
130	C₄²×C₇	C₂⁴	1	1	yes	characteristic
131	C₂³×C₈	—	2	1	no	—
132	#533	C₂²×C₇	1	2	yes	—
133	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
134	#500	C₂²×C₇	1	1	yes	characteristic, complemented
135	#500	C₂²×C₇	1	1	yes	characteristic, complemented
136	#499	—	2	1	no	—
137	#499	—	2	1	no	—
138	#498	C₂²×C₇	1	2	yes	complemented
139	#483	—	2	2	no	—
140	#476	C₂²×C₇	1	4	yes	—
141	#461	C₂²×C₇	1	4	yes	—
142	#450	—	2	2	no	—
143	C₄×SD₁₆	—	2	4	no	—
144	C₈×D₄	—	2	1	no	—
145	C₈×D₄	—	2	1	no	—
146	C₈×D₄	C₂²×C₇	1	2	yes	—
147	C₈×D₄	C₂²×C₇	1	2	yes	—
148	#431	C₂²×C₇	1	2	yes	—
149	#427	—	2	1	no	—
150	#427	—	2	1	no	—
151	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
152	C₂×C₂²⋊C₈	—	2	1	no	—
153	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
154	#335	C₂²×C₇	1	2	yes	complemented
155	#334	C₂²×C₇	1	2	yes	complemented
156	#333	C₂²×C₇	1	4	yes	complemented
157	#332	C₂²×C₇	1	4	yes	complemented
158	C₂³×C₇	—	2	4	no	—
159	C₂³×C₇	—	2	2	no	—
160	C₂³×C₇	—	4	2	no	—
161	C₂³×C₇	—	4	2	no	—
162	C₂³×C₇	#121	1	1	yes	characteristic
163	C₇×Q₈	—	4	4	no	—
164	C₇×Q₈	—	8	2	no	—
165	C₇×D₄	—	2	4	no	—
166	C₇×D₄	—	2	2	no	—
167	C₇×D₄	#135	1	4	yes	—
168	C₂×C₄×C₇	—	2	2	no	—
169	C₂×C₄×C₇	—	2	2	no	—
170	C₂×C₄×C₇	—	4	1	no	—
171	C₂×C₄×C₇	—	4	2	no	—
172	C₂×C₄×C₇	—	4	2	no	—
173	C₂×C₄×C₇	—	4	2	no	—
174	C₂×C₄×C₇	—	4	4	no	—
175	C₂×C₄×C₇	—	4	2	no	—
176	C₂×C₄×C₇	—	4	2	no	—
177	C₂×C₄×C₇	#121	1	1	yes	characteristic
178	C₂×C₄×C₇	#121	1	1	yes	characteristic
179	C₂×C₄×C₇	#121	1	1	yes	characteristic
180	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
181	C₂×C₄×C₇	#137	1	2	yes	—
182	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
185	C₇×C₈	—	2	1	no	—
186	C₇×C₈	—	2	1	no	—
187	C₇×C₈	—	4	2	no	—
188	C₇×C₈	—	4	4	no	—
189	C₇×C₈	C₂×SD₁₆	1	2	yes	—
190	C₇×C₈	C₂×SD₁₆	1	2	yes	—
191	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
192	C₂³×C₄	—	2	1	no	—
193	C₂×SD₁₆	—	2	4	no	—
194	C₂²×C₈	—	2	1	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	4	4	no	—
198	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
199	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
200	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
201	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
202	#125	—	2	2	no	—
203	C₂²⋊Q₈	—	2	2	no	—
204	C₂²⋊Q₈	—	2	2	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×Q₈	—	2	2	no	—
207	C₄×D₄	—	2	1	no	—
208	C₄×D₄	—	2	1	no	—
209	C₄×D₄	C₇×D₄	1	2	yes	—
210	C₄×D₄	C₂³×C₇	1	2	yes	—
211	#117	—	2	2	no	—
212	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
213	C₂×C₂²⋊C₄	—	2	1	no	—
214	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
215	#107	—	2	1	no	—
216	#107	—	2	1	no	—
217	#107	—	2	1	no	—
218	#107	—	2	1	no	—
219	#106	—	2	2	no	—
220	#106	—	2	2	no	—
221	#106	—	2	4	no	—
222	C₄⋊C₈	—	2	1	no	—
223	C₄⋊C₈	C₂³×C₇	1	2	yes	—
224	C₄⋊C₈	C₂³×C₇	1	4	yes	—
225	#103	—	2	4	no	—
226	#102	—	2	4	no	—
227	C₂²⋊C₈	—	2	2	no	—
228	C₂²⋊C₈	—	2	2	no	—
229	C₂²⋊C₈	—	2	2	no	—
230	C₄×C₈	—	2	2	no	—
231	C₄×C₈	C₇×D₄	1	1	yes	characteristic
232	C₄×C₈	C₇×D₄	1	1	yes	characteristic
233	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
234	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
235	C₂²×C₇	—	2	1	no	—
236	C₂²×C₇	—	2	1	no	—
237	C₂²×C₇	—	2	4	no	—
238	C₂²×C₇	—	4	8	no	—
239	C₂²×C₇	—	4	2	no	—
240	C₂²×C₇	—	4	2	no	—
241	C₂²×C₇	#461	1	2	yes	—
242	C₂²×C₇	C₂×#121	1	1	yes	characteristic
243	C₄×C₇	—	2	1	no	—
244	C₄×C₇	—	2	1	no	—
245	C₄×C₇	—	2	1	no	—
246	C₄×C₇	—	4	2	no	—
247	C₄×C₇	—	8	2	no	—
248	C₄×C₇	—	8	2	no	—
249	C₄×C₇	#461	1	2	yes	—
250	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
251	C₄×C₇	C₂×#135	1	2	yes	—
252	C₄×C₇	C₂×#137	1	1	yes	characteristic
253	C₂⁴	—	2	1	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×D₄	—	2	4	no	—
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	4	2	no	—
263	C₂²×C₄	—	4	2	no	—
264	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
266	C₂²×C₄	C₇×#41	1	1	yes	characteristic
267	SD₁₆	—	4	8	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	4	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	4	2	no	—
272	C₂×C₈	—	4	4	no	—
273	C₂×C₈	—	4	4	no	—
274	C₂×C₈	C₇×SD₁₆	1	4	yes	complemented
275	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
276	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
277	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
278	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
287	C₂²⋊C₄	—	2	2	no	—
288	C₂²⋊C₄	—	2	2	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₄²	—	4	2	no	—
291	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
292	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
293	C₄²	C₇×#41	1	1	yes	characteristic
294	C₄²	C₂⁴×C₇	1	1	yes	characteristic
295	C₂×C₇	—	2	2	no	—
296	C₂×C₇	—	4	4	no	—
297	C₂×C₇	C₂×#461	1	1	yes	characteristic
298	C₂×C₇	#3050	1	1	yes	characteristic
299	C₂×C₇	#3067	1	1	yes	characteristic
300	C₂³	—	2	4	no	—
301	C₂³	—	2	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	—	4	2	no	—
304	C₂³	C₇×#121	1	1	yes	characteristic
305	Q₈	—	4	4	no	—
306	Q₈	—	8	2	no	—
307	D₄	—	2	4	no	—
308	D₄	—	2	2	no	—
309	D₄	C₇×#135	1	4	yes	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	C₇×#121	1	1	yes	characteristic
320	C₂×C₄	C₇×#121	1	1	yes	characteristic
321	C₂×C₄	C₇×#121	1	1	yes	characteristic
322	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
323	C₂×C₄	C₇×#137	1	2	yes	—
324	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
325	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
326	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	4	2	no	—
330	C₈	—	4	4	no	—
331	C₈	C₂×C₇×SD₁₆	1	2	yes	—
332	C₈	C₂×C₇×SD₁₆	1	2	yes	—
333	C₇	#19105	1	1	yes	characteristic, direct product factor, Sylow
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	4	no	—
337	C₂²	—	4	8	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	2	no	—
340	C₂²	C₇×#461	1	2	yes	—
341	C₂²	C₂×C₇×#121	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	4	2	no	—
346	C₄	—	8	2	no	—
347	C₄	—	8	2	no	—
348	C₄	C₇×#461	1	2	yes	—
349	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
350	C₄	C₂×C₇×#135	1	2	yes	—
351	C₄	C₂×C₇×#137	1	1	yes	characteristic
352	C₂	—	2	2	no	—
353	C₂	—	4	4	no	—
354	C₂	C₂×C₇×#461	1	1	yes	characteristic
355	C₂	C₇×#3050	1	1	yes	characteristic
356	C₂	C₇×#3067	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
socle	C₂²×C₇	C₂×#121
prime radical	C₂²×C₇×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3160	C₂	semidirect
4. C₇×#1675	C₂	semidirect
5. C₇×#1673	C₂	semidirect
6. C₇×#1671	C₂	semidirect
7. C₇×#1564	C₂	semidirect
8. C₇×#1542	C₂	semidirect
12. C₇×#500	C₂²	semidirect
13. C₇×#500	C₂²	semidirect
16. C₇×#498	C₂²	semidirect
32. C₇×#335	C₂²	semidirect
33. C₇×#334	C₂²	semidirect
34. C₇×#333	C₂²	semidirect
35. C₇×#332	C₂²	semidirect
36. #19105	C₇	direct (N × H)
81. #3160	C₂×C₇	semidirect
84. #1675	C₂×C₇	semidirect
85. #1673	C₂×C₇	semidirect
86. #1671	C₂×C₇	semidirect
87. #1564	C₂×C₇	semidirect
88. #1542	C₂×C₇	semidirect
110. C₂×C₇×C₈	SD₁₆	semidirect
134. #500	C₂²×C₇	semidirect
135. #500	C₂²×C₇	semidirect
138. #498	C₂²×C₇	semidirect
154. #335	C₂²×C₇	semidirect
155. #334	C₂²×C₇	semidirect
156. #333	C₂²×C₇	semidirect
157. #332	C₂²×C₇	semidirect
274. C₂×C₈	C₇×SD₁₆	semidirect
333. C₇	#19105	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×#3160	2
C₇×#3101	2
C₂×C₇×C₈×D₄	2
C₇×#1675	2
C₇×#1673	2
C₇×#1671	2
C₇×#1564	2
C₇×#1542	2
#19105	7

Subgroups — C7×#19105 #13196321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19105 #13196321