Subgroups — C₇×#19645 #13196861

Order 1792.

772 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3228	C₂	1	2	yes	maximal, complemented
2	C₇×#3182	C₂	1	2	yes	maximal, complemented
3	C₇×#3170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3147	C₂	1	1	yes	maximal, characteristic
5	C₇×#1719	C₂	1	2	yes	maximal, complemented
6	C₇×#1714	C₂	1	2	yes	maximal, complemented
7	C₇×#1713	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×#533	C₂²	1	2	yes	—
14	C₇×#532	C₂²	1	2	yes	—
15	C₇×#500	C₂²	1	1	yes	characteristic
16	C₇×#500	C₂²	1	2	yes	—
17	C₇×#500	C₂²	1	2	yes	—
18	C₇×#498	C₂²	1	2	yes	—
19	C₇×#495	—	2	2	no	—
20	C₇×#494	C₂²	1	1	yes	characteristic
21	C₇×#494	C₂²	1	1	yes	characteristic
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×#487	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×#482	—	2	4	no	—
31	C₇×#478	—	2	4	no	—
32	C₇×#468	—	2	4	no	—
33	C₇×#467	—	2	2	no	—
34	C₇×#440	—	2	4	no	—
35	C₄×C₇×Q₁₆	—	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	#19645	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×C₇×#41	—	2	2	no	—
45	C₇×#135	—	4	8	no	—
46	C₂×C₇×Q₁₆	—	2	2	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×M₁₆	D₄	1	2	yes	—
52	C₂²×C₇×C₈	D₄	1	2	yes	—
53	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
54	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
55	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
56	C₇×#127	C₂³	1	2	yes	—
57	C₇×#125	—	2	2	no	—
58	C₇×#124	—	2	2	no	—
59	C₇×C₂²⋊Q₈	—	2	2	no	—
60	C₇×C₂²⋊Q₈	—	2	2	no	—
61	C₇×#121	—	2	2	no	—
62	C₇×#121	—	2	2	no	—
63	C₄×C₇×Q₈	—	2	2	no	—
64	C₄×C₇×Q₈	—	2	2	no	—
65	C₄×C₇×D₄	—	2	2	no	—
66	C₄×C₇×D₄	—	2	2	no	—
67	C₇×#117	—	2	2	no	—
68	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
69	C₇×#107	—	2	2	no	—
70	C₇×#107	—	2	4	no	—
71	C₇×#107	—	2	4	no	—
72	C₇×#106	—	2	4	no	—
73	C₇×C₄⋊C₈	D₄	1	2	yes	—
74	C₇×#103	—	2	4	no	—
75	C₇×#103	—	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	#3228	C₂×C₇	1	2	yes	complemented
89	#3182	C₂×C₇	1	2	yes	complemented
90	#3170	C₂×C₇	1	1	yes	characteristic, complemented
91	#3147	C₂×C₇	1	1	yes	characteristic
92	#1719	C₂×C₇	1	2	yes	complemented
93	#1714	C₂×C₇	1	2	yes	complemented
94	#1713	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	C₂×C₇×Q₈	—	2	2	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	—	4	2	no	—
107	C₂²×C₄×C₇	—	4	2	no	—
108	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
111	C₇×Q₁₆	—	4	4	no	—
112	C₇×Q₁₆	—	4	4	no	—
113	C₇×SD₁₆	—	4	8	no	—
114	C₇×SD₁₆	—	4	8	no	—
115	C₇×D₈	—	4	4	no	—
116	C₇×M₁₆	—	2	4	no	—
117	C₂×C₇×C₈	—	2	4	no	—
118	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
119	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
120	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
121	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
122	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
123	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
124	C₇×C₄⋊C₄	—	2	2	no	—
125	C₇×C₄⋊C₄	—	2	2	no	—
126	C₇×C₄⋊C₄	—	2	2	no	—
127	C₇×C₄⋊C₄	—	2	2	no	—
128	C₇×C₄⋊C₄	—	2	2	no	—
129	C₇×C₄⋊C₄	—	2	2	no	—
130	C₇×C₄⋊C₄	—	4	2	no	—
131	C₇×C₄⋊C₄	—	4	2	no	—
132	C₇×C₂²⋊C₄	—	4	2	no	—
133	C₇×C₂²⋊C₄	—	4	2	no	—
134	C₇×C₂²⋊C₄	—	4	2	no	—
135	C₄²×C₇	—	4	2	no	—
136	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
138	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
139	C₄²×C₇	C₂⁴	1	1	yes	characteristic
140	C₂×#135	—	2	2	no	—
141	#533	C₂²×C₇	1	2	yes	—
142	#532	C₂²×C₇	1	2	yes	—
143	#500	C₂²×C₇	1	1	yes	characteristic
144	#500	C₂²×C₇	1	2	yes	—
145	#500	C₂²×C₇	1	2	yes	—
146	#498	C₂²×C₇	1	2	yes	—
147	#495	—	2	2	no	—
148	#494	C₂²×C₇	1	1	yes	characteristic
149	#494	C₂²×C₇	1	1	yes	characteristic
150	#493	C₂²×C₇	1	1	yes	characteristic
151	#492	C₂²×C₇	1	2	yes	—
152	#492	C₂²×C₇	1	2	yes	—
153	#487	C₂²×C₇	1	2	yes	—
154	#487	C₂²×C₇	1	2	yes	—
155	#486	C₂²×C₇	1	2	yes	—
156	#486	C₂²×C₇	1	2	yes	—
157	#486	C₂²×C₇	1	2	yes	—
158	#482	—	2	4	no	—
159	#478	—	2	4	no	—
160	#468	—	2	4	no	—
161	#467	—	2	2	no	—
162	#440	—	2	4	no	—
163	C₄×Q₁₆	—	2	2	no	—
164	#431	C₂²×C₇	1	1	yes	characteristic
165	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
167	#322	C₂²×C₇	1	1	yes	characteristic
168	#322	C₂²×C₇	1	1	yes	characteristic
169	C₈²	C₂²×C₇	1	1	yes	characteristic
170	C₈²	C₂²×C₇	1	1	yes	characteristic
171	C₂³×C₇	—	4	2	no	—
172	C₂³×C₇	#127	1	1	yes	characteristic
173	C₇×Q₈	—	4	4	no	—
174	C₇×Q₈	—	4	4	no	—
175	C₇×Q₈	—	4	4	no	—
176	C₇×D₄	—	4	4	no	—
177	C₇×D₄	—	4	4	no	—
178	C₇×D₄	—	4	4	no	—
179	C₇×D₄	—	8	4	no	—
180	C₂×C₄×C₇	—	2	2	no	—
181	C₂×C₄×C₇	—	2	2	no	—
182	C₂×C₄×C₇	—	2	2	no	—
183	C₂×C₄×C₇	—	4	2	no	—
184	C₂×C₄×C₇	—	4	2	no	—
185	C₂×C₄×C₇	—	4	2	no	—
186	C₂×C₄×C₇	—	8	4	no	—
187	C₂×C₄×C₇	#127	1	1	yes	characteristic
188	C₂×C₄×C₇	#127	1	1	yes	characteristic
189	C₂×C₄×C₇	#127	1	1	yes	characteristic
190	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
193	C₇×C₈	—	2	2	no	—
194	C₇×C₈	—	2	2	no	—
195	C₇×C₈	—	2	2	no	—
196	C₇×C₈	—	2	2	no	—
197	C₇×C₈	#135	1	4	yes	complemented
198	C₇×C₈	AGL(1,ℤ/8)	1	4	yes	—
199	C₂×#41	—	2	2	no	—
200	#135	—	4	8	no	—
201	C₂×Q₁₆	—	2	2	no	—
202	C₂×Q₁₆	—	2	2	no	—
203	C₂×SD₁₆	—	2	4	no	—
204	C₂×SD₁₆	—	2	4	no	—
205	C₂×D₈	—	2	2	no	—
206	C₂×M₁₆	C₇×D₄	1	2	yes	—
207	C₂²×C₈	C₇×D₄	1	2	yes	—
208	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
209	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
210	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
211	#127	C₂³×C₇	1	2	yes	—
212	#125	—	2	2	no	—
213	#124	—	2	2	no	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₂²⋊Q₈	—	2	2	no	—
216	#121	—	2	2	no	—
217	#121	—	2	2	no	—
218	C₄×Q₈	—	2	2	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×D₄	—	2	2	no	—
221	C₄×D₄	—	2	2	no	—
222	#117	—	2	2	no	—
223	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
224	#107	—	2	2	no	—
225	#107	—	2	4	no	—
226	#107	—	2	4	no	—
227	#106	—	2	4	no	—
228	C₄⋊C₈	C₇×D₄	1	2	yes	—
229	#103	—	2	4	no	—
230	#103	—	2	4	no	—
231	#102	—	2	4	no	—
232	#102	—	2	4	no	—
233	#102	—	2	4	no	—
234	C₂²⋊C₈	C₇×D₄	1	2	yes	—
235	#97	C₇×D₄	1	2	yes	—
236	C₄×C₈	C₇×D₄	1	2	yes	—
237	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
238	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
239	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
240	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
241	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
243	C₂²×C₇	—	2	1	no	—
244	C₂²×C₇	—	2	1	no	—
245	C₂²×C₇	—	2	1	no	—
246	C₂²×C₇	—	8	2	no	—
247	C₂²×C₇	—	8	2	no	—
248	C₂²×C₇	—	8	2	no	—
249	C₂²×C₇	C₂×#127	1	1	yes	characteristic
250	C₄×C₇	—	2	1	no	—
251	C₄×C₇	—	2	1	no	—
252	C₄×C₇	—	2	1	no	—
253	C₄×C₇	—	8	2	no	—
254	C₄×C₇	—	8	2	no	—
255	C₄×C₇	—	8	2	no	—
256	C₄×C₇	C₂×#135	1	2	yes	—
257	C₄×C₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
258	C₄×C₇	#578	1	2	yes	—
259	#41	—	4	8	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	4	2	no	—
267	C₂²×C₄	—	4	2	no	—
268	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
271	Q₁₆	—	4	4	no	—
272	Q₁₆	—	4	4	no	—
273	SD₁₆	—	4	8	no	—
274	SD₁₆	—	4	8	no	—
275	D₈	—	4	4	no	—
276	M₁₆	—	2	4	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
279	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
280	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
281	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
282	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
283	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	4	2	no	—
291	C₄⋊C₄	—	4	2	no	—
292	C₂²⋊C₄	—	4	2	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₄²	—	4	2	no	—
296	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
297	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
298	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
299	C₄²	C₂⁴×C₇	1	1	yes	characteristic
300	C₂×C₇	—	4	1	no	—
301	C₂×C₇	—	16	2	no	—
302	C₂×C₇	#3147	1	1	yes	characteristic
303	C₂×C₇	#3151	1	1	yes	characteristic
304	C₂×C₇	#3156	1	1	yes	characteristic
305	C₂³	—	4	2	no	—
306	C₂³	C₇×#127	1	1	yes	characteristic
307	Q₈	—	4	4	no	—
308	Q₈	—	4	4	no	—
309	Q₈	—	4	4	no	—
310	D₄	—	4	4	no	—
311	D₄	—	4	4	no	—
312	D₄	—	4	4	no	—
313	D₄	—	8	4	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	8	4	no	—
321	C₂×C₄	C₇×#127	1	1	yes	characteristic
322	C₂×C₄	C₇×#127	1	1	yes	characteristic
323	C₂×C₄	C₇×#127	1	1	yes	characteristic
324	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
325	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
326	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
327	C₈	—	2	2	no	—
328	C₈	—	2	2	no	—
329	C₈	—	2	2	no	—
330	C₈	—	2	2	no	—
331	C₈	C₇×#135	1	4	yes	complemented
332	C₈	C₇×AGL(1,ℤ/8)	1	4	yes	—
333	C₇	#19645	1	1	yes	characteristic, direct product factor, Sylow
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	C₂×C₇×#127	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	8	2	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	2	no	—
347	C₄	C₂×C₇×#135	1	2	yes	—
348	C₄	C₂×C₇×AGL(1,ℤ/8)	1	2	yes	—
349	C₄	C₇×#578	1	2	yes	—
350	C₂	—	4	1	no	—
351	C₂	—	16	2	no	—
352	C₂	C₇×#3147	1	1	yes	characteristic
353	C₂	C₇×#3151	1	1	yes	characteristic
354	C₂	C₇×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#3147	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×#3228	C₂	semidirect
2. C₇×#3182	C₂	semidirect
3. C₇×#3170	C₂	semidirect
5. C₇×#1719	C₂	semidirect
6. C₇×#1714	C₂	semidirect
7. C₇×#1713	C₂	semidirect
8. C₇×#1709	C₂	semidirect
9. C₇×#1707	C₂	semidirect
10. C₇×#1706	C₂	semidirect
11. C₇×C₈×M₁₆	C₂	semidirect
43. #19645	C₇	direct (N × H)
88. #3228	C₂×C₇	semidirect
89. #3182	C₂×C₇	semidirect
90. #3170	C₂×C₇	semidirect
92. #1719	C₂×C₇	semidirect
93. #1714	C₂×C₇	semidirect
94. #1713	C₂×C₇	semidirect
95. #1709	C₂×C₇	semidirect
96. #1707	C₂×C₇	semidirect
97. #1706	C₂×C₇	semidirect
98. C₈×M₁₆	C₂×C₇	semidirect
197. C₇×C₈	#135	semidirect
331. C₈	C₇×#135	semidirect
333. C₇	#19645	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×#3228	2
C₇×#3182	2
C₇×#3170	2
C₇×#3147	2
C₇×#1719	2
C₇×#1714	2
C₇×#1713	2
C₇×#1709	2
C₇×#1707	2
C₇×#1706	2
C₇×C₈×M₁₆	2
#19645	7

Subgroups — C7×#19645 #13196861

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19645 #13196861