Subgroups — C₇×#19194 #13196410

Order 1792.

638 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3246	C₂	1	2	yes	maximal
2	C₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2931	C₂	1	1	yes	maximal, characteristic
4	C₇×#1695	C₂	1	2	yes	maximal, complemented
5	C₇×#1693	C₂	1	2	yes	maximal, complemented
6	C₇×#1671	C₂	1	4	yes	maximal, complemented
7	C₂×C₇×#334	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1550	C₂	1	2	yes	maximal, complemented
9	C₂×C₇×#131	—	2	1	no	—
10	C₇×#533	C₂²	1	2	yes	—
11	C₇×#532	C₂²	1	1	yes	characteristic
12	C₇×#500	C₂²	1	1	yes	characteristic
13	C₇×#500	C₂²	1	1	yes	characteristic
14	C₇×#499	—	2	1	no	—
15	C₇×#499	—	2	1	no	—
16	C₇×#498	C₂²	1	2	yes	—
17	C₇×#476	C₂²	1	4	yes	—
18	C₇×#475	—	2	2	no	—
19	C₇×#465	—	2	4	no	—
20	C₇×#461	C₂²	1	4	yes	—
21	C₇×#460	—	2	2	no	—
22	C₇×#436	—	2	1	no	—
23	C₇×#436	C₂²	1	2	yes	—
24	C₇×C₈×D₄	—	2	1	no	—
25	C₇×C₈×D₄	C₂²	1	2	yes	—
26	C₇×#427	—	2	1	no	—
27	C₇×#427	—	2	1	no	—
28	C₇×#423	C₂²	1	1	yes	characteristic
29	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
30	C₇×#408	—	2	1	no	—
31	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
32	C₇×#334	C₂²	1	2	yes	—
33	C₇×#334	C₂²	1	2	yes	—
34	C₇×#333	C₂²	1	4	yes	—
35	C₇×#332	C₂²	1	4	yes	—
36	#19194	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×C₇×#41	—	2	1	no	—
38	C₂×C₇×SD₁₆	—	2	4	no	—
39	C₇×#131	—	4	4	no	—
40	C₂×C₇×M₁₆	—	2	1	no	—
41	C₂×C₇×M₁₆	—	2	2	no	—
42	C₂²×C₇×C₈	—	2	2	no	—
43	C₂²×C₇×C₈	—	2	2	no	—
44	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
45	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
46	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
47	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₇×#127	D₄	1	1	yes	characteristic
50	C₇×#125	—	2	2	no	—
51	C₇×#124	—	2	1	no	—
52	C₇×C₂²⋊Q₈	—	2	2	no	—
53	C₇×C₂²⋊Q₈	—	2	2	no	—
54	C₇×#121	—	2	1	no	—
55	C₇×#121	D₄	1	2	yes	—
56	C₄×C₇×Q₈	—	2	2	no	—
57	C₄×C₇×Q₈	—	2	2	no	—
58	C₄×C₇×D₄	—	2	1	no	—
59	C₄×C₇×D₄	C₂³	1	2	yes	—
60	C₇×#117	—	2	2	no	—
61	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
62	C₇×#107	—	2	1	no	—
63	C₇×#107	—	2	1	no	—
64	C₇×#107	—	2	1	no	—
65	C₇×#107	—	2	1	no	—
66	C₇×#106	—	2	2	no	—
67	C₇×#106	—	2	2	no	—
68	C₇×#106	—	2	4	no	—
69	C₇×C₄⋊C₈	—	2	2	no	—
70	C₇×C₄⋊C₈	—	2	1	no	—
71	C₇×C₄⋊C₈	C₂³	1	4	yes	—
72	C₇×C₄⋊C₈	C₂³	1	2	yes	—
73	C₇×#103	—	2	4	no	—
74	C₇×#102	—	2	4	no	—
75	C₇×C₂²⋊C₈	—	2	2	no	—
76	C₇×C₂²⋊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	#3246	C₂×C₇	1	2	yes	—
82	#3160	C₂×C₇	1	1	yes	characteristic, complemented
83	#2931	C₂×C₇	1	1	yes	characteristic
84	#1695	C₂×C₇	1	2	yes	complemented
85	#1693	C₂×C₇	1	2	yes	complemented
86	#1671	C₂×C₇	1	4	yes	complemented
87	C₂×#334	C₂×C₇	1	1	yes	characteristic, complemented
88	#1550	C₂×C₇	1	2	yes	complemented
89	C₇×#41	—	4	2	no	—
90	C₇×#41	—	4	2	no	—
91	C₂×C₇×Q₈	—	2	1	no	—
92	C₂×C₇×Q₈	—	2	2	no	—
93	C₂×C₇×Q₈	—	2	2	no	—
94	C₂×C₇×D₄	—	2	2	no	—
95	C₂×C₇×D₄	—	2	1	no	—
96	C₂×C₇×D₄	—	2	1	no	—
97	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₇	—	2	2	no	—
99	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₇	#41	1	1	yes	characteristic
102	C₇×SD₁₆	—	8	4	no	—
103	C₇×M₁₆	—	4	2	no	—
104	C₇×M₁₆	—	4	4	no	—
105	C₂×C₇×C₈	—	2	2	no	—
106	C₂×C₇×C₈	—	2	2	no	—
107	C₂×C₇×C₈	—	2	4	no	—
108	C₂×C₇×C₈	—	4	4	no	—
109	C₂×C₇×C₈	—	8	2	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	C₂×#131	—	2	1	no	—
132	#533	C₂²×C₇	1	2	yes	—
133	#532	C₂²×C₇	1	1	yes	characteristic
134	#500	C₂²×C₇	1	1	yes	characteristic
135	#500	C₂²×C₇	1	1	yes	characteristic
136	#499	—	2	1	no	—
137	#499	—	2	1	no	—
138	#498	C₂²×C₇	1	2	yes	—
139	#476	C₂²×C₇	1	4	yes	—
140	#475	—	2	2	no	—
141	#465	—	2	4	no	—
142	#461	C₂²×C₇	1	4	yes	—
143	#460	—	2	2	no	—
144	#436	—	2	1	no	—
145	#436	C₂²×C₇	1	2	yes	—
146	C₈×D₄	—	2	1	no	—
147	C₈×D₄	C₂²×C₇	1	2	yes	—
148	#427	—	2	1	no	—
149	#427	—	2	1	no	—
150	#423	C₂²×C₇	1	1	yes	characteristic
151	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
152	#408	—	2	1	no	—
153	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
154	#334	C₂²×C₇	1	2	yes	—
155	#334	C₂²×C₇	1	2	yes	—
156	#333	C₂²×C₇	1	4	yes	—
157	#332	C₂²×C₇	1	4	yes	—
158	C₂³×C₇	—	2	2	no	—
159	C₂³×C₇	#121	1	1	yes	characteristic
160	C₇×Q₈	—	4	2	no	—
161	C₇×Q₈	—	4	2	no	—
162	C₇×Q₈	—	4	2	no	—
163	C₇×Q₈	—	8	2	no	—
164	C₇×D₄	—	2	4	no	—
165	C₇×D₄	—	4	2	no	—
166	C₇×D₄	—	4	4	no	—
167	C₇×D₄	—	4	2	no	—
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	2	no	—
175	C₂×C₄×C₇	#121	1	1	yes	characteristic
176	C₂×C₄×C₇	#121	1	1	yes	characteristic
177	C₂×C₄×C₇	#121	1	1	yes	characteristic
178	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
179	C₂×C₄×C₇	#137	1	2	yes	—
180	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
183	C₇×C₈	—	2	1	no	—
184	C₇×C₈	—	2	1	no	—
185	C₇×C₈	—	4	2	no	—
186	C₇×C₈	—	4	4	no	—
187	C₇×C₈	C₂×SD₁₆	1	2	yes	—
188	C₇×C₈	C₂×SD₁₆	1	2	yes	—
189	C₂×#41	—	2	1	no	—
190	C₂×SD₁₆	—	2	4	no	—
191	#131	—	4	4	no	—
192	C₂×M₁₆	—	2	1	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
197	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
198	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
199	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
200	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
201	#127	C₇×D₄	1	1	yes	characteristic
202	#125	—	2	2	no	—
203	#124	—	2	1	no	—
204	C₂²⋊Q₈	—	2	2	no	—
205	C₂²⋊Q₈	—	2	2	no	—
206	#121	—	2	1	no	—
207	#121	C₇×D₄	1	2	yes	—
208	C₄×Q₈	—	2	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×D₄	—	2	1	no	—
211	C₄×D₄	C₂³×C₇	1	2	yes	—
212	#117	—	2	2	no	—
213	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
214	#107	—	2	1	no	—
215	#107	—	2	1	no	—
216	#107	—	2	1	no	—
217	#107	—	2	1	no	—
218	#106	—	2	2	no	—
219	#106	—	2	2	no	—
220	#106	—	2	4	no	—
221	C₄⋊C₈	—	2	2	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	C₄×C₈	C₇×D₄	1	1	yes	characteristic
230	C₄×C₈	C₇×D₄	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
232	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
233	C₂²×C₇	—	2	1	no	—
234	C₂²×C₇	—	2	1	no	—
235	C₂²×C₇	—	4	2	no	—
236	C₂²×C₇	—	4	2	no	—
237	C₂²×C₇	—	4	2	no	—
238	C₂²×C₇	#461	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	—	2	1	no	—
243	C₄×C₇	—	4	2	no	—
244	C₄×C₇	—	8	2	no	—
245	C₄×C₇	—	8	2	no	—
246	C₄×C₇	#461	1	2	yes	—
247	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
248	C₄×C₇	C₂×#137	1	1	yes	characteristic
249	C₄×C₇	#577	1	2	yes	—
250	#41	—	4	2	no	—
251	#41	—	4	2	no	—
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
262	C₂²×C₄	C₇×#41	1	1	yes	characteristic
263	SD₁₆	—	8	4	no	—
264	M₁₆	—	4	2	no	—
265	M₁₆	—	4	4	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	4	4	no	—
270	C₂×C₈	—	8	2	no	—
271	C₂×C₈	C₇×SD₁₆	1	4	yes	complemented
272	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
273	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
274	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
275	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
284	C₂²⋊C₄	—	2	2	no	—
285	C₂²⋊C₄	—	2	2	no	—
286	C₂²⋊C₄	—	4	2	no	—
287	C₄²	—	4	2	no	—
288	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
289	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
290	C₄²	C₇×#41	1	1	yes	characteristic
291	C₄²	C₂⁴×C₇	1	1	yes	characteristic
292	C₂×C₇	—	2	2	no	—
293	C₂×C₇	—	8	2	no	—
294	C₂×C₇	C₂×#461	1	1	yes	characteristic
295	C₂×C₇	#3060	1	1	yes	characteristic
296	C₂×C₇	#3067	1	1	yes	characteristic
297	C₂³	—	2	2	no	—
298	C₂³	C₇×#121	1	1	yes	characteristic
299	Q₈	—	4	2	no	—
300	Q₈	—	4	2	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	8	2	no	—
303	D₄	—	2	4	no	—
304	D₄	—	4	2	no	—
305	D₄	—	4	4	no	—
306	D₄	—	4	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	C₇×#121	1	1	yes	characteristic
315	C₂×C₄	C₇×#121	1	1	yes	characteristic
316	C₂×C₄	C₇×#121	1	1	yes	characteristic
317	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
318	C₂×C₄	C₇×#137	1	2	yes	—
319	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
320	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
321	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	4	2	no	—
325	C₈	—	4	4	no	—
326	C₈	C₂×C₇×SD₁₆	1	2	yes	—
327	C₈	C₂×C₇×SD₁₆	1	2	yes	—
328	C₇	#19194	1	1	yes	characteristic, direct product factor, Sylow
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	4	2	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	C₇×#461	1	2	yes	—
335	C₂²	C₂×C₇×#121	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	4	2	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	2	no	—
342	C₄	C₇×#461	1	2	yes	—
343	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
344	C₄	C₂×C₇×#137	1	1	yes	characteristic
345	C₄	C₇×#577	1	2	yes	—
346	C₂	—	2	2	no	—
347	C₂	—	8	2	no	—
348	C₂	C₂×C₇×#461	1	1	yes	characteristic
349	C₂	C₇×#3060	1	1	yes	characteristic
350	C₂	C₇×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#532	C₂²
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
2. C₇×#3160	C₂	semidirect
4. C₇×#1695	C₂	semidirect
5. C₇×#1693	C₂	semidirect
6. C₇×#1671	C₂	semidirect
7. C₂×C₇×#334	C₂	semidirect
8. C₇×#1550	C₂	semidirect
36. #19194	C₇	direct (N × H)
82. #3160	C₂×C₇	semidirect
84. #1695	C₂×C₇	semidirect
85. #1693	C₂×C₇	semidirect
86. #1671	C₂×C₇	semidirect
87. C₂×#334	C₂×C₇	semidirect
88. #1550	C₂×C₇	semidirect
110. C₂×C₇×C₈	SD₁₆	semidirect
271. C₂×C₈	C₇×SD₁₆	semidirect
328. C₇	#19194	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×#3246	2
C₇×#3160	2
C₇×#2931	2
C₇×#1695	2
C₇×#1693	2
C₇×#1671	2
C₂×C₇×#334	2
C₇×#1550	2
#19194	7

Subgroups — C7×#19194 #13196410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19194 #13196410