Subgroups — C₇×#19647 #13196863

Order 1792.

840 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3207	C₂	1	2	yes	maximal, complemented
2	C₇×#3184	C₂	1	2	yes	maximal, complemented
3	C₇×#3169	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3147	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1717	C₂	1	2	yes	maximal, complemented
6	C₇×#1715	C₂	1	2	yes	maximal, complemented
7	C₇×#1712	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1709	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1707	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1706	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#135	—	2	2	no	—
13	C₇×#532	C₂²	1	2	yes	—
14	C₇×#532	C₂²	1	2	yes	—
15	C₇×#500	C₂²	1	1	yes	characteristic
16	C₇×#498	C₂²	1	2	yes	—
17	C₇×#495	—	2	2	no	—
18	C₇×#494	C₂²	1	1	yes	characteristic
19	C₇×#493	C₂²	1	1	yes	characteristic
20	C₇×#493	C₂²	1	2	yes	—
21	C₇×#493	C₂²	1	2	yes	—
22	C₇×#493	C₂²	1	1	yes	characteristic
23	C₇×#492	C₂²	1	2	yes	—
24	C₇×#492	C₂²	1	2	yes	—
25	C₇×#487	C₂²	1	2	yes	—
26	C₇×#486	C₂²	1	2	yes	—
27	C₇×#486	C₂²	1	2	yes	—
28	C₇×#486	C₂²	1	2	yes	—
29	C₇×#486	C₂²	1	2	yes	—
30	C₇×#469	—	2	4	no	—
31	C₇×#466	—	2	2	no	—
32	C₇×#463	—	2	4	no	—
33	C₇×#449	—	2	4	no	—
34	C₇×#442	—	2	4	no	—
35	C₄×C₇×D₈	—	2	2	no	—
36	C₇×#431	C₂²	1	1	yes	characteristic
37	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
39	C₇×#322	C₂²	1	1	yes	characteristic
40	C₇×#322	C₂²	1	1	yes	characteristic
41	C₇×C₈²	C₂²	1	1	yes	characteristic
42	C₇×C₈²	C₂²	1	1	yes	characteristic
43	#19647	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×C₇×#41	—	2	2	no	—
45	C₂×C₇×#41	—	2	2	no	—
46	C₇×#135	—	4	8	no	—
47	C₂×C₇×Q₁₆	—	2	2	no	—
48	C₂×C₇×SD₁₆	—	2	4	no	—
49	C₂×C₇×SD₁₆	—	2	4	no	—
50	C₂×C₇×D₈	—	2	2	no	—
51	C₂×C₇×D₈	—	2	4	no	—
52	C₂×C₇×D₈	—	2	2	no	—
53	C₂×C₇×D₈	—	2	4	no	—
54	C₂×C₇×M₁₆	D₄	1	2	yes	—
55	C₂²×C₇×C₈	D₄	1	2	yes	—
56	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
57	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
58	C₇×#127	C₂³	1	2	yes	—
59	C₇×#127	C₂³	1	2	yes	—
60	C₇×#124	—	2	2	no	—
61	C₇×#124	—	2	2	no	—
62	C₇×#121	—	2	2	no	—
63	C₇×#121	—	2	2	no	—
64	C₇×#121	—	2	2	no	—
65	C₇×#121	—	2	2	no	—
66	C₄×C₇×D₄	—	2	2	no	—
67	C₄×C₇×D₄	—	2	2	no	—
68	C₄×C₇×D₄	—	2	2	no	—
69	C₄×C₇×D₄	—	2	2	no	—
70	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
71	C₇×#107	—	2	2	no	—
72	C₇×#106	—	2	4	no	—
73	C₇×C₄⋊C₈	D₄	1	2	yes	—
74	C₇×#103	—	2	4	no	—
75	C₇×#102	—	2	4	no	—
76	C₇×#102	—	2	4	no	—
77	C₇×#102	—	2	4	no	—
78	C₇×#102	—	2	4	no	—
79	C₇×C₂²⋊C₈	D₄	1	2	yes	—
80	C₇×#97	D₄	1	2	yes	—
81	C₄×C₇×C₈	D₄	1	2	yes	—
82	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
83	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
84	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
85	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
86	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
87	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
88	#3207	C₂×C₇	1	2	yes	complemented
89	#3184	C₂×C₇	1	2	yes	complemented
90	#3169	C₂×C₇	1	1	yes	characteristic, complemented
91	#3147	C₂×C₇	1	1	yes	characteristic, complemented
92	#1717	C₂×C₇	1	2	yes	complemented
93	#1715	C₂×C₇	1	2	yes	complemented
94	#1712	C₂×C₇	1	1	yes	characteristic, complemented
95	#1709	C₂×C₇	1	1	yes	characteristic, complemented
96	#1707	C₂×C₇	1	1	yes	characteristic, complemented
97	#1706	C₂×C₇	1	1	yes	characteristic, complemented
98	C₈×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
99	C₇×#41	—	4	8	no	—
100	C₇×#41	—	8	4	no	—
101	C₂×C₇×Q₈	—	2	2	no	—
102	C₂×C₇×Q₈	—	2	2	no	—
103	C₂×C₇×D₄	—	2	2	no	—
104	C₂×C₇×D₄	—	2	2	no	—
105	C₂×C₇×D₄	—	2	2	no	—
106	C₂×C₇×D₄	—	2	2	no	—
107	C₂×C₇×D₄	—	2	2	no	—
108	C₂×C₇×D₄	—	2	2	no	—
109	C₂×C₇×D₄	—	4	2	no	—
110	C₂×C₇×D₄	—	4	2	no	—
111	C₂²×C₄×C₇	—	4	2	no	—
112	C₂²×C₄×C₇	—	4	2	no	—
113	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
116	C₇×Q₁₆	—	4	4	no	—
117	C₇×SD₁₆	—	4	8	no	—
118	C₇×SD₁₆	—	8	4	no	—
119	C₇×D₈	—	4	8	no	—
120	C₇×D₈	—	4	4	no	—
121	C₇×D₈	—	4	4	no	—
122	C₇×D₈	—	8	4	no	—
123	C₇×M₁₆	—	2	4	no	—
124	C₂×C₇×C₈	—	2	4	no	—
125	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
126	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
127	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
128	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
129	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
130	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
131	C₇×C₄⋊C₄	—	2	2	no	—
132	C₇×C₄⋊C₄	—	2	2	no	—
133	C₇×C₄⋊C₄	—	2	2	no	—
134	C₇×C₄⋊C₄	—	2	2	no	—
135	C₇×C₂²⋊C₄	—	4	2	no	—
136	C₇×C₂²⋊C₄	—	4	2	no	—
137	C₇×C₂²⋊C₄	—	4	2	no	—
138	C₇×C₂²⋊C₄	—	4	2	no	—
139	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
141	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
142	C₄²×C₇	C₂⁴	1	1	yes	characteristic
143	C₂×#135	—	2	2	no	—
144	#532	C₂²×C₇	1	2	yes	—
145	#532	C₂²×C₇	1	2	yes	—
146	#500	C₂²×C₇	1	1	yes	characteristic
147	#498	C₂²×C₇	1	2	yes	—
148	#495	—	2	2	no	—
149	#494	C₂²×C₇	1	1	yes	characteristic
150	#493	C₂²×C₇	1	1	yes	characteristic
151	#493	C₂²×C₇	1	1	yes	characteristic
152	#493	C₂²×C₇	1	2	yes	—
153	#493	C₂²×C₇	1	2	yes	—
154	#492	C₂²×C₇	1	2	yes	—
155	#492	C₂²×C₇	1	2	yes	—
156	#487	C₂²×C₇	1	2	yes	—
157	#486	C₂²×C₇	1	2	yes	—
158	#486	C₂²×C₇	1	2	yes	—
159	#486	C₂²×C₇	1	2	yes	—
160	#486	C₂²×C₇	1	2	yes	—
161	#469	—	2	4	no	—
162	#466	—	2	2	no	—
163	#463	—	2	4	no	—
164	#449	—	2	4	no	—
165	#442	—	2	4	no	—
166	C₄×D₈	—	2	2	no	—
167	#431	C₂²×C₇	1	1	yes	characteristic
168	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
169	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
170	#322	C₂²×C₇	1	1	yes	characteristic
171	#322	C₂²×C₇	1	1	yes	characteristic
172	C₈²	C₂²×C₇	1	1	yes	characteristic
173	C₈²	C₂²×C₇	1	1	yes	characteristic
174	C₂³×C₇	—	4	2	no	—
175	C₂³×C₇	—	4	2	no	—
176	C₂³×C₇	#127	1	1	yes	characteristic
177	C₇×Q₈	—	4	4	no	—
178	C₇×Q₈	—	4	4	no	—
179	C₇×D₄	—	4	4	no	—
180	C₇×D₄	—	4	4	no	—
181	C₇×D₄	—	4	4	no	—
182	C₇×D₄	—	4	4	no	—
183	C₇×D₄	—	4	4	no	—
184	C₇×D₄	—	4	4	no	—
185	C₇×D₄	—	8	4	no	—
186	C₇×D₄	—	8	4	no	—
187	C₂×C₄×C₇	—	2	2	no	—
188	C₂×C₄×C₇	—	2	2	no	—
189	C₂×C₄×C₇	—	2	2	no	—
190	C₂×C₄×C₇	—	4	2	no	—
191	C₂×C₄×C₇	—	4	2	no	—
192	C₂×C₄×C₇	—	8	4	no	—
193	C₂×C₄×C₇	—	8	4	no	—
194	C₂×C₄×C₇	#127	1	1	yes	characteristic
195	C₂×C₄×C₇	#127	1	1	yes	characteristic
196	C₂×C₄×C₇	#127	1	1	yes	characteristic
197	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
200	C₇×C₈	—	2	2	no	—
201	C₇×C₈	—	2	2	no	—
202	C₇×C₈	—	2	2	no	—
203	C₇×C₈	—	2	2	no	—
204	C₇×C₈	#135	1	4	yes	complemented
205	C₇×C₈	AGL(1,ℤ/8)	1	4	yes	—
206	C₂×#41	—	2	2	no	—
207	C₂×#41	—	2	2	no	—
208	#135	—	4	8	no	—
209	C₂×Q₁₆	—	2	2	no	—
210	C₂×SD₁₆	—	2	4	no	—
211	C₂×SD₁₆	—	2	4	no	—
212	C₂×D₈	—	2	2	no	—
213	C₂×D₈	—	2	2	no	—
214	C₂×D₈	—	2	4	no	—
215	C₂×D₈	—	2	4	no	—
216	C₂×M₁₆	C₇×D₄	1	2	yes	—
217	C₂²×C₈	C₇×D₄	1	2	yes	—
218	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
219	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
220	#127	C₂³×C₇	1	2	yes	—
221	#127	C₂³×C₇	1	2	yes	—
222	#124	—	2	2	no	—
223	#124	—	2	2	no	—
224	#121	—	2	2	no	—
225	#121	—	2	2	no	—
226	#121	—	2	2	no	—
227	#121	—	2	2	no	—
228	C₄×D₄	—	2	2	no	—
229	C₄×D₄	—	2	2	no	—
230	C₄×D₄	—	2	2	no	—
231	C₄×D₄	—	2	2	no	—
232	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
233	#107	—	2	2	no	—
234	#106	—	2	4	no	—
235	C₄⋊C₈	C₇×D₄	1	2	yes	—
236	#103	—	2	4	no	—
237	#102	—	2	4	no	—
238	#102	—	2	4	no	—
239	#102	—	2	4	no	—
240	#102	—	2	4	no	—
241	C₂²⋊C₈	C₇×D₄	1	2	yes	—
242	#97	C₇×D₄	1	2	yes	—
243	C₄×C₈	C₇×D₄	1	2	yes	—
244	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
245	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
246	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
247	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
248	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
249	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
250	C₂²×C₇	—	2	1	no	—
251	C₂²×C₇	—	2	1	no	—
252	C₂²×C₇	—	2	1	no	—
253	C₂²×C₇	—	8	2	no	—
254	C₂²×C₇	—	8	2	no	—
255	C₂²×C₇	—	8	2	no	—
256	C₂²×C₇	—	8	2	no	—
257	C₂²×C₇	—	8	2	no	—
258	C₂²×C₇	—	8	2	no	—
259	C₂²×C₇	C₂×#127	1	1	yes	characteristic
260	C₄×C₇	—	2	1	no	—
261	C₄×C₇	—	2	1	no	—
262	C₄×C₇	—	2	1	no	—
263	C₄×C₇	—	8	2	no	—
264	C₄×C₇	—	8	2	no	—
265	C₄×C₇	C₂×#135	1	2	yes	—
266	C₄×C₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
267	C₄×C₇	#576	1	2	yes	—
268	#41	—	4	8	no	—
269	#41	—	8	4	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	2	2	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	—	2	2	no	—
278	C₂×D₄	—	4	2	no	—
279	C₂×D₄	—	4	2	no	—
280	C₂²×C₄	—	4	2	no	—
281	C₂²×C₄	—	4	2	no	—
282	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
283	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
284	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
285	Q₁₆	—	4	4	no	—
286	SD₁₆	—	4	8	no	—
287	SD₁₆	—	8	4	no	—
288	D₈	—	4	8	no	—
289	D₈	—	4	4	no	—
290	D₈	—	4	4	no	—
291	D₈	—	8	4	no	—
292	M₁₆	—	2	4	no	—
293	C₂×C₈	—	2	4	no	—
294	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
295	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
296	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
297	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
298	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
299	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	4	2	no	—
305	C₂²⋊C₄	—	4	2	no	—
306	C₂²⋊C₄	—	4	2	no	—
307	C₂²⋊C₄	—	4	2	no	—
308	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
309	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
310	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
311	C₄²	C₂⁴×C₇	1	1	yes	characteristic
312	C₂×C₇	—	4	1	no	—
313	C₂×C₇	—	16	2	no	—
314	C₂×C₇	—	16	2	no	—
315	C₂×C₇	#3147	1	1	yes	characteristic
316	C₂×C₇	#3151	1	1	yes	characteristic
317	C₂×C₇	#3155	1	1	yes	characteristic
318	C₂³	—	4	2	no	—
319	C₂³	—	4	2	no	—
320	C₂³	C₇×#127	1	1	yes	characteristic
321	Q₈	—	4	4	no	—
322	Q₈	—	4	4	no	—
323	D₄	—	4	4	no	—
324	D₄	—	4	4	no	—
325	D₄	—	4	4	no	—
326	D₄	—	4	4	no	—
327	D₄	—	4	4	no	—
328	D₄	—	4	4	no	—
329	D₄	—	8	4	no	—
330	D₄	—	8	4	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	8	4	no	—
337	C₂×C₄	—	8	4	no	—
338	C₂×C₄	C₇×#127	1	1	yes	characteristic
339	C₂×C₄	C₇×#127	1	1	yes	characteristic
340	C₂×C₄	C₇×#127	1	1	yes	characteristic
341	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
342	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
343	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
344	C₈	—	2	2	no	—
345	C₈	—	2	2	no	—
346	C₈	—	2	2	no	—
347	C₈	—	2	2	no	—
348	C₈	C₇×#135	1	4	yes	complemented
349	C₈	C₇×AGL(1,ℤ/8)	1	4	yes	—
350	C₇	#19647	1	1	yes	characteristic, direct product factor, Sylow
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	8	2	no	—
358	C₂²	—	8	2	no	—
359	C₂²	—	8	2	no	—
360	C₂²	C₂×C₇×#127	1	1	yes	characteristic
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	8	2	no	—
365	C₄	—	8	2	no	—
366	C₄	C₂×C₇×#135	1	2	yes	—
367	C₄	C₂×C₇×AGL(1,ℤ/8)	1	2	yes	—
368	C₄	C₇×#576	1	2	yes	—
369	C₂	—	4	1	no	—
370	C₂	—	16	2	no	—
371	C₂	—	16	2	no	—
372	C₂	C₇×#3147	1	1	yes	characteristic
373	C₂	C₇×#3151	1	1	yes	characteristic
374	C₂	C₇×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
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×#3207	C₂	semidirect
2. C₇×#3184	C₂	semidirect
3. C₇×#3169	C₂	semidirect
4. C₇×#3147	C₂	semidirect
5. C₇×#1717	C₂	semidirect
6. C₇×#1715	C₂	semidirect
7. C₇×#1712	C₂	semidirect
8. C₇×#1709	C₂	semidirect
9. C₇×#1707	C₂	semidirect
10. C₇×#1706	C₂	semidirect
11. C₇×C₈×M₁₆	C₂	semidirect
43. #19647	C₇	direct (N × H)
88. #3207	C₂×C₇	semidirect
89. #3184	C₂×C₇	semidirect
90. #3169	C₂×C₇	semidirect
91. #3147	C₂×C₇	semidirect
92. #1717	C₂×C₇	semidirect
93. #1715	C₂×C₇	semidirect
94. #1712	C₂×C₇	semidirect
95. #1709	C₂×C₇	semidirect
96. #1707	C₂×C₇	semidirect
97. #1706	C₂×C₇	semidirect
98. C₈×M₁₆	C₂×C₇	semidirect
204. C₇×C₈	#135	semidirect
348. C₈	C₇×#135	semidirect
350. C₇	#19647	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×#3207	2
C₇×#3184	2
C₇×#3169	2
C₇×#3147	2
C₇×#1717	2
C₇×#1715	2
C₇×#1712	2
C₇×#1709	2
C₇×#1707	2
C₇×#1706	2
C₇×C₈×M₁₆	2
#19647	7

Subgroups — C7×#19647 #13196863

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19647 #13196863