Subgroups — C₇×#17978 #13195194

Order 1792.

606 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3168	C₂	1	1	yes	maximal, characteristic
2	C₇×#2966	C₂	1	2	yes	maximal, complemented
3	C₇×#1584	C₂	1	2	yes	maximal, complemented
4	C₇×#1583	C₂	1	2	yes	maximal, complemented
5	C₇×#1582	C₂	1	2	yes	maximal, complemented
6	C₇×#1581	C₂	1	2	yes	maximal, complemented
7	C₇×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1494	C₂	1	2	yes	maximal, complemented
9	C₂×C₇×#322	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#529	C₂²	1	2	yes	—
11	C₇×#499	C₄	1	1	yes	characteristic
12	C₇×#495	C₄	1	1	yes	characteristic
13	C₇×#488	C₄	1	2	yes	—
14	C₇×#467	C₄	1	1	yes	characteristic
15	C₇×#466	C₄	1	1	yes	characteristic
16	C₇×#465	C₄	1	2	yes	—
17	C₇×#446	C₂²	1	2	yes	—
18	C₇×C₈×Q₈	C₂²	1	2	yes	—
19	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
20	C₄×C₇×SD₁₆	C₂²	1	2	yes	—
21	C₄×C₇×D₈	C₂²	1	1	yes	characteristic
22	C₇×#436	C₂²	1	2	yes	—
23	C₇×#435	—	2	2	no	—
24	C₇×C₈×D₄	—	2	2	no	—
25	C₇×C₈×D₄	C₂²	1	2	yes	—
26	C₇×#433	—	2	2	no	—
27	C₇×#432	C₂²	1	2	yes	—
28	C₇×#405	—	2	2	no	—
29	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
32	C₇×#335	C₂²	1	2	yes	—
33	C₇×#334	C₂²	1	1	yes	characteristic
34	C₇×#334	C₂²	1	1	yes	characteristic
35	C₇×Q₈⋊C₈	C₂²	1	2	yes	—
36	C₇×Q₈⋊C₈	C₂²	1	2	yes	—
37	C₇×D₄⋊C₈	C₂²	1	2	yes	—
38	C₇×D₄⋊C₈	C₂²	1	2	yes	—
39	C₇×#322	C₂²	1	2	yes	—
40	C₇×#322	C₂²	1	2	yes	—
41	#17978	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂×C₇×Q₁₆	C₂×C₄	1	1	yes	characteristic
43	C₂×C₇×SD₁₆	C₂×C₄	1	2	yes	—
44	C₂×C₇×D₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₇×M₁₆	—	2	2	no	—
46	C₂²×C₇×C₈	—	2	1	no	—
47	C₂²×C₇×C₈	—	2	2	no	—
48	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₇×C₈	D₄	1	2	yes	—
51	C₇×#126	—	2	2	no	—
52	C₇×#125	C₂×C₄	1	2	yes	—
53	C₇×#124	C₂×C₄	1	2	yes	—
54	C₇×#123	—	2	2	no	—
55	C₇×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
56	C₇×#121	C₂×C₄	1	2	yes	—
57	C₄×C₇×Q₈	C₂³	1	2	yes	—
58	C₄×C₇×D₄	—	2	2	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	C₂×C₄	1	1	yes	characteristic
63	C₇×#107	C₂×C₄	1	1	yes	characteristic
64	C₇×#106	C₂×C₄	1	2	yes	—
65	C₇×C₄⋊C₈	—	2	2	no	—
66	C₇×C₄⋊C₈	—	2	2	no	—
67	C₇×C₄⋊C₈	C₂³	1	2	yes	—
68	C₇×C₄⋊C₈	C₂³	1	2	yes	—
69	C₇×#103	C₂×C₄	1	2	yes	—
70	C₇×#103	C₂×C₄	1	2	yes	—
71	C₇×#102	C₂×C₄	1	2	yes	—
72	C₇×#102	C₂×C₄	1	2	yes	—
73	C₇×C₂²⋊C₈	—	2	2	no	—
74	C₇×C₂²⋊C₈	—	2	2	no	—
75	C₇×C₂²⋊C₈	—	2	2	no	—
76	C₇×#97	—	2	2	no	—
77	C₇×#97	—	2	2	no	—
78	C₄×C₇×C₈	—	2	2	no	—
79	C₄×C₇×C₈	—	2	1	no	—
80	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
81	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
82	C₄×C₇×C₈	D₄	1	1	yes	characteristic
83	C₄×C₇×C₈	D₄	1	1	yes	characteristic
84	C₄×C₇×C₈	C₂³	1	2	yes	—
85	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
86	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
87	C₄×C₇×C₈	C₂³	1	2	yes	—
88	#3168	C₂×C₇	1	1	yes	characteristic
89	#2966	C₂×C₇	1	2	yes	complemented
90	#1584	C₂×C₇	1	2	yes	complemented
91	#1583	C₂×C₇	1	2	yes	complemented
92	#1582	C₂×C₇	1	2	yes	complemented
93	#1581	C₂×C₇	1	2	yes	complemented
94	#1568	C₂×C₇	1	1	yes	characteristic, complemented
95	#1494	C₂×C₇	1	2	yes	complemented
96	C₂×#322	C₂×C₇	1	1	yes	characteristic, complemented
97	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
98	C₂×C₇×D₄	—	2	2	no	—
99	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₇	—	2	2	no	—
101	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₇	#41	1	1	yes	characteristic
104	C₇×Q₁₆	—	4	1	no	—
105	C₇×SD₁₆	—	4	2	no	—
106	C₇×D₈	—	4	1	no	—
107	C₇×M₁₆	—	4	4	no	—
108	C₂×C₇×C₈	—	2	4	no	—
109	C₂×C₇×C₈	—	2	2	no	—
110	C₂×C₇×C₈	—	2	2	no	—
111	C₂×C₇×C₈	—	2	2	no	—
112	C₂×C₇×C₈	—	4	1	no	—
113	C₂×C₇×C₈	—	4	1	no	—
114	C₂×C₇×C₈	—	4	2	no	—
115	C₂×C₇×C₈	—	4	4	no	—
116	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
121	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
122	C₇×C₄⋊C₄	—	2	2	no	—
123	C₇×C₄⋊C₄	—	2	2	no	—
124	C₇×C₄⋊C₄	—	2	2	no	—
125	C₇×C₄⋊C₄	—	2	2	no	—
126	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
127	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
128	C₇×C₂²⋊C₄	—	2	2	no	—
129	C₇×C₂²⋊C₄	—	2	2	no	—
130	C₇×C₂²⋊C₄	—	2	2	no	—
131	C₇×C₂²⋊C₄	—	2	2	no	—
132	C₇×C₂²⋊C₄	—	2	2	no	—
133	C₄²×C₇	—	2	2	no	—
134	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
135	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₇	#41	1	1	yes	characteristic
137	C₄²×C₇	C₂⁴	1	1	yes	characteristic
138	#529	C₂²×C₇	1	2	yes	—
139	#499	C₄×C₇	1	1	yes	characteristic
140	#495	C₄×C₇	1	1	yes	characteristic
141	#488	C₄×C₇	1	2	yes	—
142	#467	C₄×C₇	1	1	yes	characteristic
143	#466	C₄×C₇	1	1	yes	characteristic
144	#465	C₄×C₇	1	2	yes	—
145	#446	C₂²×C₇	1	2	yes	—
146	C₈×Q₈	C₂²×C₇	1	2	yes	—
147	C₄×Q₁₆	C₂²×C₇	1	1	yes	characteristic
148	C₄×SD₁₆	C₂²×C₇	1	2	yes	—
149	C₄×D₈	C₂²×C₇	1	1	yes	characteristic
150	#436	C₂²×C₇	1	2	yes	—
151	#435	—	2	2	no	—
152	C₈×D₄	—	2	2	no	—
153	C₈×D₄	C₂²×C₇	1	2	yes	—
154	#433	—	2	2	no	—
155	#432	C₂²×C₇	1	2	yes	—
156	#405	—	2	2	no	—
157	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
160	#335	C₂²×C₇	1	2	yes	—
161	#334	C₂²×C₇	1	1	yes	characteristic
162	#334	C₂²×C₇	1	1	yes	characteristic
163	Q₈⋊C₈	C₂²×C₇	1	2	yes	—
164	Q₈⋊C₈	C₂²×C₇	1	2	yes	—
165	D₄⋊C₈	C₂²×C₇	1	2	yes	—
166	D₄⋊C₈	C₂²×C₇	1	2	yes	—
167	#322	C₂²×C₇	1	2	yes	—
168	#322	C₂²×C₇	1	2	yes	—
169	C₂³×C₇	—	2	2	no	—
170	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
171	C₇×Q₈	—	2	2	no	—
172	C₇×Q₈	—	2	2	no	—
173	C₇×D₄	—	2	2	no	—
174	C₇×D₄	—	2	2	no	—
175	C₇×D₄	—	4	4	no	—
176	C₂×C₄×C₇	—	2	2	no	—
177	C₂×C₄×C₇	—	2	1	no	—
178	C₂×C₄×C₇	—	2	1	no	—
179	C₂×C₄×C₇	—	2	2	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	C₄×D₄	1	1	yes	characteristic
186	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
187	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
188	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
189	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
191	C₇×C₈	—	2	1	no	—
192	C₇×C₈	—	2	1	no	—
193	C₇×C₈	—	2	1	no	—
194	C₇×C₈	—	2	1	no	—
195	C₇×C₈	—	2	2	no	—
196	C₇×C₈	—	4	2	no	—
197	C₇×C₈	—	4	2	no	—
198	C₇×C₈	—	4	2	no	—
199	C₇×C₈	#135	1	4	yes	—
200	C₂×Q₁₆	C₂×C₄×C₇	1	1	yes	characteristic
201	C₂×SD₁₆	C₂×C₄×C₇	1	2	yes	—
202	C₂×D₈	C₂×C₄×C₇	1	1	yes	characteristic
203	C₂×M₁₆	—	2	2	no	—
204	C₂²×C₈	—	2	1	no	—
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
207	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
208	C₂²×C₈	C₇×D₄	1	2	yes	—
209	#126	—	2	2	no	—
210	#125	C₂×C₄×C₇	1	2	yes	—
211	#124	C₂×C₄×C₇	1	2	yes	—
212	#123	—	2	2	no	—
213	C₂²⋊Q₈	C₂×C₄×C₇	1	2	yes	—
214	#121	C₂×C₄×C₇	1	2	yes	—
215	C₄×Q₈	C₂³×C₇	1	2	yes	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	C₂³×C₇	1	2	yes	—
218	#117	—	2	2	no	—
219	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
220	#107	C₂×C₄×C₇	1	1	yes	characteristic
221	#107	C₂×C₄×C₇	1	1	yes	characteristic
222	#106	C₂×C₄×C₇	1	2	yes	—
223	C₄⋊C₈	—	2	2	no	—
224	C₄⋊C₈	—	2	2	no	—
225	C₄⋊C₈	C₂³×C₇	1	2	yes	—
226	C₄⋊C₈	C₂³×C₇	1	2	yes	—
227	#103	C₂×C₄×C₇	1	2	yes	—
228	#103	C₂×C₄×C₇	1	2	yes	—
229	#102	C₂×C₄×C₇	1	2	yes	—
230	#102	C₂×C₄×C₇	1	2	yes	—
231	C₂²⋊C₈	—	2	2	no	—
232	C₂²⋊C₈	—	2	2	no	—
233	C₂²⋊C₈	—	2	2	no	—
234	#97	—	2	2	no	—
235	#97	—	2	2	no	—
236	C₄×C₈	—	2	2	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
239	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
240	C₄×C₈	C₇×D₄	1	1	yes	characteristic
241	C₄×C₈	C₇×D₄	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₇	1	2	yes	—
243	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
244	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
245	C₄×C₈	C₂³×C₇	1	2	yes	—
246	C₂²×C₇	—	2	1	no	—
247	C₂²×C₇	—	2	1	no	—
248	C₂²×C₇	—	4	2	no	—
249	C₂²×C₇	—	4	2	no	—
250	C₂²×C₇	—	4	2	no	—
251	C₂²×C₇	#444	1	2	yes	—
252	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
253	C₄×C₇	—	2	1	no	—
254	C₄×C₇	—	2	1	no	—
255	C₄×C₇	—	2	1	no	—
256	C₄×C₇	—	2	1	no	—
257	C₄×C₇	—	4	2	no	—
258	C₄×C₇	—	4	2	no	—
259	C₄×C₇	—	4	2	no	—
260	C₄×C₇	#568	1	1	yes	characteristic
261	C₄×C₇	#568	1	1	yes	characteristic
262	C₄×C₇	C₂×#135	1	2	yes	—
263	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
269	C₂²×C₄	C₇×#41	1	1	yes	characteristic
270	Q₁₆	—	4	1	no	—
271	SD₁₆	—	4	2	no	—
272	D₈	—	4	1	no	—
273	M₁₆	—	4	4	no	—
274	C₂×C₈	—	2	4	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	4	1	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	4	2	no	—
281	C₂×C₈	—	4	4	no	—
282	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
283	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
284	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
285	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
286	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
287	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
293	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
294	C₂²⋊C₄	—	2	2	no	—
295	C₂²⋊C₄	—	2	2	no	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₂²⋊C₄	—	2	2	no	—
298	C₂²⋊C₄	—	2	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
301	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
302	C₄²	C₇×#41	1	1	yes	characteristic
303	C₄²	C₂⁴×C₇	1	1	yes	characteristic
304	C₂×C₇	—	2	2	no	—
305	C₂×C₇	—	8	2	no	—
306	C₂×C₇	#2932	1	1	yes	characteristic
307	C₂×C₇	C₄×#135	1	1	yes	characteristic
308	C₂×C₇	C₂×#444	1	1	yes	characteristic
309	C₂³	—	2	2	no	—
310	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
311	Q₈	—	2	2	no	—
312	Q₈	—	2	2	no	—
313	D₄	—	2	2	no	—
314	D₄	—	2	2	no	—
315	D₄	—	4	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
326	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
327	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
328	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
329	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
330	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	2	2	no	—
336	C₈	—	4	2	no	—
337	C₈	—	4	2	no	—
338	C₈	—	4	2	no	—
339	C₈	C₇×#135	1	4	yes	—
340	C₇	#17978	1	1	yes	characteristic, direct product factor, Sylow
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	4	2	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	C₇×#444	1	2	yes	—
347	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	4	2	no	—
353	C₄	—	4	2	no	—
354	C₄	—	4	2	no	—
355	C₄	C₇×#568	1	1	yes	characteristic
356	C₄	C₇×#568	1	1	yes	characteristic
357	C₄	C₂×C₇×#135	1	2	yes	—
358	C₂	—	2	2	no	—
359	C₂	—	8	2	no	—
360	C₂	C₇×#2932	1	1	yes	characteristic
361	C₂	C₄×C₇×#135	1	1	yes	characteristic
362	C₂	C₂×C₇×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₄²	C₂⁴×C₇
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₇×#466	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2966	C₂	semidirect
3. C₇×#1584	C₂	semidirect
4. C₇×#1583	C₂	semidirect
5. C₇×#1582	C₂	semidirect
6. C₇×#1581	C₂	semidirect
7. C₇×#1568	C₂	semidirect
8. C₇×#1494	C₂	semidirect
9. C₂×C₇×#322	C₂	semidirect
41. #17978	C₇	direct (N × H)
89. #2966	C₂×C₇	semidirect
90. #1584	C₂×C₇	semidirect
91. #1583	C₂×C₇	semidirect
92. #1582	C₂×C₇	semidirect
93. #1581	C₂×C₇	semidirect
94. #1568	C₂×C₇	semidirect
95. #1494	C₂×C₇	semidirect
96. C₂×#322	C₂×C₇	semidirect
340. C₇	#17978	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×#3168	2
C₇×#2966	2
C₇×#1584	2
C₇×#1583	2
C₇×#1582	2
C₇×#1581	2
C₇×#1568	2
C₇×#1494	2
C₂×C₇×#322	2
#17978	7

Subgroups — C7×#17978 #13195194

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#17978 #13195194