Subgroups — C₇×#19749 #13196965

Order 1792.

822 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×#3185	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3182	C₂	1	2	yes	maximal, complemented
3	C₇×#3151	C₂	1	1	yes	maximal, characteristic
4	C₇×#3138	C₂	1	2	yes	maximal, complemented
5	C₇×#1722	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1719	C₂	1	2	yes	maximal, complemented
7	C₇×#1717	C₂	1	2	yes	maximal, complemented
8	C₇×#1715	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1709	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1708	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1455	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×AGL(1,ℤ/8)	—	2	2	no	—
13	C₇×#533	C₂²	1	2	yes	—
14	C₇×#532	C₂²	1	1	yes	characteristic
15	C₂×C₇×#127	C₂²	1	1	yes	characteristic
16	C₇×#500	C₂²	1	2	yes	complemented
17	C₇×#498	C₂²	1	1	yes	characteristic, complemented
18	C₇×#496	—	2	2	no	—
19	C₇×#494	C₂²	1	1	yes	characteristic, complemented
20	C₇×#493	C₂²	1	1	yes	characteristic
21	C₇×#493	C₂²	1	1	yes	characteristic
22	C₇×#492	C₂²	1	1	yes	characteristic
23	C₇×#492	C₂²	1	1	yes	characteristic
24	C₇×#492	C₂²	1	2	yes	complemented
25	C₇×#490	—	2	4	no	—
26	C₇×#487	C₂²	1	2	yes	complemented
27	C₇×#487	C₂²	1	2	yes	complemented
28	C₇×#486	C₂²	1	2	yes	—
29	C₇×#486	C₂²	1	2	yes	—
30	C₇×#486	C₂²	1	2	yes	—
31	C₇×#486	C₂²	1	2	yes	—
32	C₇×#486	C₂²	1	2	yes	complemented
33	C₇×#470	—	2	2	no	—
34	C₇×#468	—	2	4	no	—
35	C₇×#441	—	2	2	no	—
36	C₇×#440	—	2	4	no	—
37	C₇×#418	—	2	4	no	—
38	C₄×C₇×M₁₆	C₂²	1	2	yes	—
39	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
40	C₇×#322	C₂²	1	1	yes	characteristic, complemented
41	C₇×#322	C₂²	1	2	yes	complemented
42	C₇×C₈²	C₂²	1	1	yes	characteristic, complemented
43	#19749	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×C₇×#41	—	2	1	no	—
45	C₂²×C₇×D₄	—	2	2	no	—
46	C₂²×C₇×D₄	—	2	1	no	—
47	C₇×AGL(1,ℤ/8)	—	4	8	no	—
48	C₂×C₇×Q₁₆	—	2	2	no	—
49	C₂×C₇×SD₁₆	—	2	4	no	—
50	C₂×C₇×SD₁₆	—	2	2	no	—
51	C₂×C₇×SD₁₆	—	2	2	no	—
52	C₂×C₇×D₈	—	2	2	no	—
53	C₂×C₇×D₈	—	2	2	no	—
54	C₂×C₇×M₁₆	D₄	1	2	yes	—
55	C₂×C₇×M₁₆	D₄	1	4	yes	—
56	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
57	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
58	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
59	C₇×#127	—	2	2	no	—
60	C₇×#127	—	2	1	no	—
61	C₇×#127	C₂³	1	1	yes	characteristic
62	C₇×#127	C₂³	1	1	yes	characteristic
63	C₇×#127	C₂³	1	1	yes	characteristic
64	C₇×#125	—	2	2	no	—
65	C₇×#124	—	2	1	no	—
66	C₇×C₂²⋊Q₈	—	2	2	no	—
67	C₇×C₂²⋊Q₈	—	2	2	no	—
68	C₇×#121	—	2	2	no	—
69	C₄×C₇×Q₈	—	2	2	no	—
70	C₄×C₇×Q₈	—	2	2	no	—
71	C₄×C₇×D₄	—	2	2	no	—
72	C₇×#117	—	2	2	no	—
73	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
74	C₇×#107	—	2	4	no	—
75	C₇×#106	—	2	2	no	—
76	C₇×#103	—	2	4	no	—
77	C₇×#103	—	2	4	no	—
78	C₇×#102	—	2	4	no	—
79	C₇×#102	—	2	4	no	—
80	C₇×#102	—	2	4	no	—
81	C₇×#102	—	2	4	no	—
82	C₇×#102	—	2	4	no	—
83	C₇×#97	D₄	1	4	yes	—
84	C₇×#97	D₄	1	2	yes	—
85	C₄×C₇×C₈	C₂³	1	2	yes	—
86	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
87	C₄×C₇×C₈	C₂³	1	2	yes	—
88	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
89	#3185	C₂×C₇	1	1	yes	characteristic, complemented
90	#3182	C₂×C₇	1	2	yes	complemented
91	#3151	C₂×C₇	1	1	yes	characteristic
92	#3138	C₂×C₇	1	2	yes	complemented
93	#1722	C₂×C₇	1	1	yes	characteristic, complemented
94	#1719	C₂×C₇	1	2	yes	complemented
95	#1717	C₂×C₇	1	2	yes	complemented
96	#1715	C₂×C₇	1	1	yes	characteristic, complemented
97	#1709	C₂×C₇	1	1	yes	characteristic, complemented
98	#1708	C₂×C₇	1	1	yes	characteristic, complemented
99	#1455	C₂×C₇	1	1	yes	characteristic, complemented
100	C₂⁴×C₇	—	4	1	no	—
101	C₇×#41	—	4	4	no	—
102	C₂×C₇×Q₈	—	2	2	no	—
103	C₂×C₇×Q₈	—	2	2	no	—
104	C₂×C₇×Q₈	—	2	1	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	1	no	—
109	C₂×C₇×D₄	—	2	1	no	—
110	C₂×C₇×D₄	—	2	1	no	—
111	C₂×C₇×D₄	—	4	4	no	—
112	C₂×C₇×D₄	—	4	8	no	—
113	C₂×C₇×D₄	—	4	1	no	—
114	C₂×C₇×D₄	—	4	2	no	—
115	C₂×C₇×D₄	—	4	1	no	—
116	C₂²×C₄×C₇	—	4	1	no	—
117	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
119	C₇×Q₁₆	—	4	4	no	—
120	C₇×SD₁₆	—	4	8	no	—
121	C₇×SD₁₆	—	4	4	no	—
122	C₇×SD₁₆	—	4	4	no	—
123	C₇×D₈	—	4	4	no	—
124	C₇×D₈	—	4	4	no	—
125	C₇×M₁₆	—	2	4	no	—
126	C₇×M₁₆	—	4	4	no	—
127	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
128	C₂×C₇×C₈	C₂×D₄	1	4	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×C₄⋊C₄	—	2	2	no	—
136	C₇×C₄⋊C₄	—	4	2	no	—
137	C₇×C₄⋊C₄	—	4	2	no	—
138	C₇×C₂²⋊C₄	—	4	2	no	—
139	C₇×C₂²⋊C₄	—	4	2	no	—
140	C₄²×C₇	—	4	2	no	—
141	C₄²×C₇	C₂×D₄	1	2	yes	—
142	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
143	C₄²×C₇	C₂⁴	1	1	yes	characteristic
144	C₂×AGL(1,ℤ/8)	—	2	2	no	—
145	#533	C₂²×C₇	1	2	yes	—
146	#532	C₂²×C₇	1	1	yes	characteristic
147	C₂×#127	C₂²×C₇	1	1	yes	characteristic
148	#500	C₂²×C₇	1	2	yes	complemented
149	#498	C₂²×C₇	1	1	yes	characteristic, complemented
150	#496	—	2	2	no	—
151	#494	C₂²×C₇	1	1	yes	characteristic, complemented
152	#493	C₂²×C₇	1	1	yes	characteristic
153	#493	C₂²×C₇	1	1	yes	characteristic
154	#492	C₂²×C₇	1	1	yes	characteristic
155	#492	C₂²×C₇	1	1	yes	characteristic
156	#492	C₂²×C₇	1	2	yes	complemented
157	#490	—	2	4	no	—
158	#487	C₂²×C₇	1	2	yes	complemented
159	#487	C₂²×C₇	1	2	yes	complemented
160	#486	C₂²×C₇	1	2	yes	—
161	#486	C₂²×C₇	1	2	yes	—
162	#486	C₂²×C₇	1	2	yes	—
163	#486	C₂²×C₇	1	2	yes	—
164	#486	C₂²×C₇	1	2	yes	complemented
165	#470	—	2	2	no	—
166	#468	—	2	4	no	—
167	#441	—	2	2	no	—
168	#440	—	2	4	no	—
169	#418	—	2	4	no	—
170	C₄×M₁₆	C₂²×C₇	1	2	yes	—
171	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
172	#322	C₂²×C₇	1	1	yes	characteristic, complemented
173	#322	C₂²×C₇	1	2	yes	complemented
174	C₈²	C₂²×C₇	1	1	yes	characteristic, complemented
175	C₂³×C₇	—	4	1	no	—
176	C₂³×C₇	—	4	1	no	—
177	C₂³×C₇	—	4	1	no	—
178	C₂³×C₇	—	8	2	no	—
179	C₂³×C₇	—	8	4	no	—
180	C₂³×C₇	#127	1	1	yes	characteristic
181	C₇×Q₈	—	4	2	no	—
182	C₇×Q₈	—	4	4	no	—
183	C₇×Q₈	—	4	4	no	—
184	C₇×D₄	—	4	2	no	—
185	C₇×D₄	—	4	2	no	—
186	C₇×D₄	—	4	2	no	—
187	C₇×D₄	—	4	4	no	—
188	C₇×D₄	—	4	4	no	—
189	C₇×D₄	—	4	4	no	—
190	C₇×D₄	—	8	2	no	—
191	C₇×D₄	—	8	4	no	—
192	C₇×D₄	—	8	2	no	—
193	C₂×C₄×C₇	—	2	2	no	—
194	C₂×C₄×C₇	—	2	4	no	—
195	C₂×C₄×C₇	—	4	2	no	—
196	C₂×C₄×C₇	—	4	2	no	—
197	C₂×C₄×C₇	—	4	1	no	—
198	C₂×C₄×C₇	—	8	2	no	—
199	C₂×C₄×C₇	#127	1	2	yes	—
200	C₂×C₄×C₇	#127	1	1	yes	characteristic
201	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
203	C₇×C₈	—	2	2	no	—
204	C₇×C₈	—	2	4	no	—
205	C₇×C₈	—	2	4	no	—
206	C₇×C₈	AGL(1,ℤ/8)	1	4	yes	—
207	C₂×#41	—	2	1	no	—
208	C₂²×D₄	—	2	1	no	—
209	C₂²×D₄	—	2	2	no	—
210	AGL(1,ℤ/8)	—	4	8	no	—
211	C₂×Q₁₆	—	2	2	no	—
212	C₂×SD₁₆	—	2	2	no	—
213	C₂×SD₁₆	—	2	2	no	—
214	C₂×SD₁₆	—	2	4	no	—
215	C₂×D₈	—	2	2	no	—
216	C₂×D₈	—	2	2	no	—
217	C₂×M₁₆	C₇×D₄	1	2	yes	—
218	C₂×M₁₆	C₇×D₄	1	4	yes	—
219	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
220	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
221	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
222	#127	—	2	2	no	—
223	#127	—	2	1	no	—
224	#127	C₂³×C₇	1	1	yes	characteristic
225	#127	C₂³×C₇	1	1	yes	characteristic
226	#127	C₂³×C₇	1	1	yes	characteristic
227	#125	—	2	2	no	—
228	#124	—	2	1	no	—
229	C₂²⋊Q₈	—	2	2	no	—
230	C₂²⋊Q₈	—	2	2	no	—
231	#121	—	2	2	no	—
232	C₄×Q₈	—	2	2	no	—
233	C₄×Q₈	—	2	2	no	—
234	C₄×D₄	—	2	2	no	—
235	#117	—	2	2	no	—
236	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
237	#107	—	2	4	no	—
238	#106	—	2	2	no	—
239	#103	—	2	4	no	—
240	#103	—	2	4	no	—
241	#102	—	2	4	no	—
242	#102	—	2	4	no	—
243	#102	—	2	4	no	—
244	#102	—	2	4	no	—
245	#102	—	2	4	no	—
246	#97	C₇×D₄	1	4	yes	—
247	#97	C₇×D₄	1	2	yes	—
248	C₄×C₈	C₂³×C₇	1	2	yes	—
249	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
250	C₄×C₈	C₂³×C₇	1	2	yes	—
251	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
252	C₂²×C₇	—	2	1	no	—
253	C₂²×C₇	—	2	2	no	—
254	C₂²×C₇	—	8	1	no	—
255	C₂²×C₇	—	8	1	no	—
256	C₂²×C₇	—	8	1	no	—
257	C₂²×C₇	—	8	2	no	—
258	C₂²×C₇	—	8	2	no	—
259	C₂²×C₇	—	8	2	no	—
260	C₂²×C₇	—	16	4	no	—
261	C₂²×C₇	C₂×#127	1	1	yes	characteristic
262	C₄×C₇	—	2	1	no	—
263	C₄×C₇	—	2	2	no	—
264	C₄×C₇	—	8	1	no	—
265	C₄×C₇	—	8	2	no	—
266	C₄×C₇	—	8	2	no	—
267	C₄×C₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
268	C₄×C₇	#575	1	4	yes	—
269	C₂⁴	—	4	1	no	—
270	#41	—	4	4	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×Q₈	—	2	2	no	—
273	C₂×Q₈	—	2	1	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	—	2	1	no	—
278	C₂×D₄	—	2	1	no	—
279	C₂×D₄	—	2	1	no	—
280	C₂×D₄	—	4	4	no	—
281	C₂×D₄	—	4	8	no	—
282	C₂×D₄	—	4	1	no	—
283	C₂×D₄	—	4	2	no	—
284	C₂×D₄	—	4	1	no	—
285	C₂²×C₄	—	4	1	no	—
286	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
287	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
288	Q₁₆	—	4	4	no	—
289	SD₁₆	—	4	8	no	—
290	SD₁₆	—	4	4	no	—
291	SD₁₆	—	4	4	no	—
292	D₈	—	4	4	no	—
293	D₈	—	4	4	no	—
294	M₁₆	—	2	4	no	—
295	M₁₆	—	4	4	no	—
296	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
297	C₂×C₈	C₂×C₇×D₄	1	4	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	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	4	2	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₂²⋊C₄	—	4	2	no	—
308	C₂²⋊C₄	—	4	2	no	—
309	C₄²	—	4	2	no	—
310	C₄²	C₂×C₇×D₄	1	2	yes	—
311	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
312	C₄²	C₂⁴×C₇	1	1	yes	characteristic
313	C₂×C₇	—	4	1	no	—
314	C₂×C₇	—	16	1	no	—
315	C₂×C₇	—	16	1	no	—
316	C₂×C₇	—	16	1	no	—
317	C₂×C₇	#3151	1	1	yes	characteristic
318	C₂×C₇	#3153	1	2	yes	—
319	C₂³	—	4	1	no	—
320	C₂³	—	4	1	no	—
321	C₂³	—	4	1	no	—
322	C₂³	—	8	2	no	—
323	C₂³	—	8	4	no	—
324	C₂³	C₇×#127	1	1	yes	characteristic
325	Q₈	—	4	2	no	—
326	Q₈	—	4	4	no	—
327	Q₈	—	4	4	no	—
328	D₄	—	4	2	no	—
329	D₄	—	4	2	no	—
330	D₄	—	4	2	no	—
331	D₄	—	4	4	no	—
332	D₄	—	4	4	no	—
333	D₄	—	4	4	no	—
334	D₄	—	8	2	no	—
335	D₄	—	8	4	no	—
336	D₄	—	8	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	8	2	no	—
343	C₂×C₄	C₇×#127	1	2	yes	—
344	C₂×C₄	C₇×#127	1	1	yes	characteristic
345	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
346	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
347	C₈	—	2	2	no	—
348	C₈	—	2	4	no	—
349	C₈	—	2	4	no	—
350	C₈	C₇×AGL(1,ℤ/8)	1	4	yes	—
351	C₇	#19749	1	1	yes	characteristic, direct product factor, Sylow
352	C₂²	—	2	1	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	8	1	no	—
355	C₂²	—	8	1	no	—
356	C₂²	—	8	1	no	—
357	C₂²	—	8	2	no	—
358	C₂²	—	8	2	no	—
359	C₂²	—	8	2	no	—
360	C₂²	—	16	4	no	—
361	C₂²	C₂×C₇×#127	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	2	no	—
364	C₄	—	8	1	no	—
365	C₄	—	8	2	no	—
366	C₄	—	8	2	no	—
367	C₄	C₂×C₇×AGL(1,ℤ/8)	1	2	yes	—
368	C₄	C₇×#575	1	4	yes	—
369	C₂	—	4	1	no	—
370	C₂	—	16	1	no	—
371	C₂	—	16	1	no	—
372	C₂	—	16	1	no	—
373	C₂	C₇×#3151	1	1	yes	characteristic
374	C₂	C₇×#3153	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#3151	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×#3185	C₂	semidirect
2. C₇×#3182	C₂	semidirect
4. C₇×#3138	C₂	semidirect
5. C₇×#1722	C₂	semidirect
6. C₇×#1719	C₂	semidirect
7. C₇×#1717	C₂	semidirect
8. C₇×#1715	C₂	semidirect
9. C₇×#1709	C₂	semidirect
10. C₇×#1708	C₂	semidirect
11. C₇×#1455	C₂	semidirect
16. C₇×#500	C₂²	semidirect
17. C₇×#498	C₂²	semidirect
19. C₇×#494	C₂²	semidirect
24. C₇×#492	C₂²	semidirect
26. C₇×#487	C₂²	semidirect
27. C₇×#487	C₂²	semidirect
32. C₇×#486	C₂²	semidirect
40. C₇×#322	C₂²	semidirect
41. C₇×#322	C₂²	semidirect
42. C₇×C₈²	C₂²	semidirect
43. #19749	C₇	direct (N × H)
89. #3185	C₂×C₇	semidirect
90. #3182	C₂×C₇	semidirect
92. #3138	C₂×C₇	semidirect
93. #1722	C₂×C₇	semidirect
94. #1719	C₂×C₇	semidirect
95. #1717	C₂×C₇	semidirect
96. #1715	C₂×C₇	semidirect
97. #1709	C₂×C₇	semidirect
98. #1708	C₂×C₇	semidirect
99. #1455	C₂×C₇	semidirect
148. #500	C₂²×C₇	semidirect
149. #498	C₂²×C₇	semidirect
151. #494	C₂²×C₇	semidirect
156. #492	C₂²×C₇	semidirect
158. #487	C₂²×C₇	semidirect
159. #487	C₂²×C₇	semidirect
164. #486	C₂²×C₇	semidirect
172. #322	C₂²×C₇	semidirect
173. #322	C₂²×C₇	semidirect
174. C₈²	C₂²×C₇	semidirect
351. C₇	#19749	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×#3185	2
C₇×#3182	2
C₇×#3151	2
C₇×#3138	2
C₇×#1722	2
C₇×#1719	2
C₇×#1717	2
C₇×#1715	2
C₇×#1709	2
C₇×#1708	2
C₇×#1455	2
#19749	7

Subgroups — C7×#19749 #13196965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19749 #13196965