Subgroups — C₇×#17824 #13195040

Order 1792.

572 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2989	C₂	1	2	yes	maximal
2	C₇×#2922	C₂	1	1	yes	maximal, characteristic
3	C₇×#1574	C₂	1	2	yes	maximal, complemented
4	C₇×#1573	C₂	1	2	yes	maximal, complemented
5	C₇×#1572	C₂	1	2	yes	maximal, complemented
6	C₇×#1570	C₂	1	2	yes	maximal, complemented
7	C₇×#1569	C₂	1	2	yes	maximal, complemented
8	C₇×#1568	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1566	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₇×#41	C₂²	1	1	yes	characteristic
11	C₇×#446	—	2	2	no	—
12	C₇×#436	—	2	2	no	—
13	C₇×#435	—	2	2	no	—
14	C₇×#435	—	2	2	no	—
15	C₇×#433	C₄	1	4	yes	—
16	C₇×#432	C₂²	1	2	yes	—
17	C₇×#432	C₂²	1	2	yes	—
18	C₇×#431	C₂²	1	2	yes	—
19	C₇×#431	C₂²	1	2	yes	—
20	C₇×#431	C₂²	1	2	yes	—
21	C₇×#424	C₄	1	2	yes	—
22	C₇×#423	C₂²	1	2	yes	—
23	C₇×#423	C₂²	1	2	yes	—
24	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
25	C₇×#408	C₄	1	2	yes	—
26	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
28	C₇×#335	C₂²	1	2	yes	—
29	C₇×#335	C₂²	1	2	yes	—
30	C₇×#335	C₂²	1	2	yes	—
31	C₇×#334	C₂²	1	2	yes	—
32	C₇×#334	C₂²	1	1	yes	characteristic
33	C₇×#334	C₂²	1	1	yes	characteristic
34	C₇×#334	C₂²	1	1	yes	characteristic
35	C₇×#334	C₂²	1	1	yes	characteristic
36	C₇×#334	C₂²	1	2	yes	—
37	C₇×#334	C₂²	1	2	yes	—
38	#17824	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂×C₇×#41	C₂×C₄	1	1	yes	characteristic
40	C₂×C₇×M₁₆	—	2	2	no	—
41	C₂×C₇×M₁₆	—	2	2	no	—
42	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
43	C₂²×C₇×C₈	—	2	2	no	—
44	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
45	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
46	C₄×C₇×Q₈	Q₈	1	1	yes	characteristic
47	C₄×C₇×D₄	D₄	1	1	yes	characteristic
48	C₄×C₇×D₄	D₄	1	2	yes	—
49	C₄×C₇×D₄	Q₈	1	1	yes	characteristic
50	C₄×C₇×D₄	Q₈	1	2	yes	—
51	C₇×#117	C₂×C₄	1	1	yes	characteristic
52	C₇×#117	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₇	C₂³	1	2	yes	—
55	C₇×C₄⋊C₈	—	2	2	no	—
56	C₇×C₄⋊C₈	—	2	2	no	—
57	C₇×C₄⋊C₈	—	2	2	no	—
58	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
59	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
60	C₇×C₄⋊C₈	C₂³	1	2	yes	—
61	C₇×C₄⋊C₈	C₂³	1	2	yes	—
62	C₇×C₄⋊C₈	C₂³	1	2	yes	—
63	C₇×C₄⋊C₈	C₂³	1	2	yes	—
64	C₇×C₂²⋊C₈	—	2	2	no	—
65	C₇×C₂²⋊C₈	—	2	2	no	—
66	C₇×C₂²⋊C₈	—	2	2	no	—
67	C₇×C₂²⋊C₈	C₂×C₄	1	4	yes	—
68	C₇×C₂²⋊C₈	C₂×C₄	1	4	yes	—
69	C₇×#97	—	2	2	no	—
70	C₇×#97	—	2	2	no	—
71	C₇×#97	C₂×C₄	1	2	yes	—
72	C₄×C₇×C₈	—	2	2	no	—
73	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
74	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
75	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
76	C₄×C₇×C₈	C₂³	1	2	yes	—
77	#2989	C₂×C₇	1	2	yes	—
78	#2922	C₂×C₇	1	1	yes	characteristic
79	#1574	C₂×C₇	1	2	yes	complemented
80	#1573	C₂×C₇	1	2	yes	complemented
81	#1572	C₂×C₇	1	2	yes	complemented
82	#1570	C₂×C₇	1	2	yes	complemented
83	#1569	C₂×C₇	1	2	yes	complemented
84	#1568	C₂×C₇	1	1	yes	characteristic, complemented
85	#1566	C₂×C₇	1	1	yes	characteristic, complemented
86	C₇×#41	—	4	2	no	—
87	C₂×C₇×Q₈	C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₇×D₄	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₇×D₄	C₄⋊C₄	1	2	yes	—
90	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
96	C₇×M₁₆	—	4	2	no	—
97	C₇×M₁₆	—	8	2	no	—
98	C₇×M₁₆	—	8	2	no	—
99	C₂×C₇×C₈	—	2	2	no	—
100	C₂×C₇×C₈	—	2	2	no	—
101	C₂×C₇×C₈	—	2	2	no	—
102	C₂×C₇×C₈	—	2	2	no	—
103	C₂×C₇×C₈	—	4	2	no	—
104	C₂×C₇×C₈	—	8	2	no	—
105	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
106	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
107	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
108	C₇×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
109	C₇×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
110	C₇×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
111	C₇×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
112	C₇×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
113	C₇×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
114	C₇×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
115	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₇	C₂²×C₄	1	2	yes	—
117	C₄²×C₇	C₂×D₄	1	2	yes	—
118	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
120	C₄²×C₇	C₂×Q₈	1	2	yes	—
121	C₄²×C₇	C₂⁴	1	1	yes	characteristic
122	C₄×#41	C₂²×C₇	1	1	yes	characteristic
123	#446	—	2	2	no	—
124	#436	—	2	2	no	—
125	#435	—	2	2	no	—
126	#435	—	2	2	no	—
127	#433	C₄×C₇	1	4	yes	—
128	#432	C₂²×C₇	1	2	yes	—
129	#432	C₂²×C₇	1	2	yes	—
130	#431	C₂²×C₇	1	2	yes	—
131	#431	C₂²×C₇	1	2	yes	—
132	#431	C₂²×C₇	1	2	yes	—
133	#424	C₄×C₇	1	2	yes	—
134	#423	C₂²×C₇	1	2	yes	—
135	#423	C₂²×C₇	1	2	yes	—
136	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
137	#408	C₄×C₇	1	2	yes	—
138	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
140	#335	C₂²×C₇	1	2	yes	—
141	#335	C₂²×C₇	1	2	yes	—
142	#335	C₂²×C₇	1	2	yes	—
143	#334	C₂²×C₇	1	2	yes	—
144	#334	C₂²×C₇	1	1	yes	characteristic
145	#334	C₂²×C₇	1	1	yes	characteristic
146	#334	C₂²×C₇	1	1	yes	characteristic
147	#334	C₂²×C₇	1	1	yes	characteristic
148	#334	C₂²×C₇	1	2	yes	—
149	#334	C₂²×C₇	1	2	yes	—
150	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
151	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₇×Q₈	—	4	1	no	—
153	C₇×D₄	—	4	1	no	—
154	C₇×D₄	—	4	2	no	—
155	C₂×C₄×C₇	—	2	2	no	—
156	C₂×C₄×C₇	—	2	4	no	—
157	C₂×C₄×C₇	—	2	2	no	—
158	C₂×C₄×C₇	—	2	2	no	—
159	C₂×C₄×C₇	—	2	2	no	—
160	C₂×C₄×C₇	—	4	1	no	—
161	C₂×C₄×C₇	—	4	2	no	—
162	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
164	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
166	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
167	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
169	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
171	C₇×C₈	—	2	2	no	—
172	C₇×C₈	—	2	2	no	—
173	C₇×C₈	—	2	4	no	—
174	C₇×C₈	—	4	2	no	—
175	C₇×C₈	—	4	2	no	—
176	C₇×C₈	—	4	2	no	—
177	C₇×C₈	—	4	2	no	—
178	C₂×#41	C₂×C₄×C₇	1	1	yes	characteristic
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	C₂×C₄×C₇	1	2	yes	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
184	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
185	C₄×Q₈	C₇×Q₈	1	1	yes	characteristic
186	C₄×D₄	C₇×D₄	1	1	yes	characteristic
187	C₄×D₄	C₇×D₄	1	2	yes	—
188	C₄×D₄	C₇×Q₈	1	1	yes	characteristic
189	C₄×D₄	C₇×Q₈	1	2	yes	—
190	#117	C₂×C₄×C₇	1	1	yes	characteristic
191	#117	C₂×C₄×C₇	1	2	yes	—
192	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₇	1	2	yes	—
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	—
198	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	—
199	C₄⋊C₈	C₂³×C₇	1	2	yes	—
200	C₄⋊C₈	C₂³×C₇	1	2	yes	—
201	C₄⋊C₈	C₂³×C₇	1	2	yes	—
202	C₄⋊C₈	C₂³×C₇	1	2	yes	—
203	C₂²⋊C₈	—	2	2	no	—
204	C₂²⋊C₈	—	2	2	no	—
205	C₂²⋊C₈	—	2	2	no	—
206	C₂²⋊C₈	C₂×C₄×C₇	1	4	yes	—
207	C₂²⋊C₈	C₂×C₄×C₇	1	4	yes	—
208	#97	—	2	2	no	—
209	#97	—	2	2	no	—
210	#97	C₂×C₄×C₇	1	2	yes	—
211	C₄×C₈	—	2	2	no	—
212	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
213	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
214	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
215	C₄×C₈	C₂³×C₇	1	2	yes	—
216	C₂²×C₇	—	2	1	no	—
217	C₂²×C₇	—	2	2	no	—
218	C₂²×C₇	—	2	2	no	—
219	C₂²×C₇	—	2	1	no	—
220	C₂²×C₇	—	2	2	no	—
221	C₂²×C₇	—	2	1	no	—
222	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
223	C₄×C₇	—	2	1	no	—
224	C₄×C₇	—	2	2	no	—
225	C₄×C₇	—	2	2	no	—
226	C₄×C₇	—	2	1	no	—
227	C₄×C₇	—	2	1	no	—
228	C₄×C₇	—	2	1	no	—
229	C₄×C₇	—	2	2	no	—
230	C₄×C₇	#568	1	2	yes	—
231	C₄×C₇	#577	1	2	yes	—
232	#41	—	4	2	no	—
233	C₂×Q₈	C₇×C₄⋊C₄	1	1	yes	characteristic
234	C₂×D₄	C₇×C₄⋊C₄	1	1	yes	characteristic
235	C₂×D₄	C₇×C₄⋊C₄	1	2	yes	—
236	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
237	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
239	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
242	M₁₆	—	4	2	no	—
243	M₁₆	—	8	2	no	—
244	M₁₆	—	8	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	8	2	no	—
251	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
252	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
253	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
254	C₄⋊C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
255	C₄⋊C₄	C₇×C₄⋊C₄	1	2	yes	—
256	C₄⋊C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
257	C₄⋊C₄	C₇×C₄⋊C₄	1	2	yes	—
258	C₂²⋊C₄	C₇×C₄⋊C₄	1	2	yes	—
259	C₂²⋊C₄	C₇×C₄⋊C₄	1	2	yes	—
260	C₂²⋊C₄	C₇×C₄⋊C₄	1	2	yes	—
261	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
262	C₄²	C₂²×C₄×C₇	1	2	yes	—
263	C₄²	C₂×C₇×D₄	1	2	yes	—
264	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
265	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
266	C₄²	C₂×C₇×Q₈	1	2	yes	—
267	C₄²	C₂⁴×C₇	1	1	yes	characteristic
268	C₂×C₇	—	4	1	no	—
269	C₂×C₇	—	4	2	no	—
270	C₂×C₇	#2905	1	1	yes	characteristic
271	C₂×C₇	#2912	1	1	yes	characteristic
272	C₂×C₇	#2916	1	1	yes	characteristic
273	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
274	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
275	Q₈	—	4	1	no	—
276	D₄	—	4	1	no	—
277	D₄	—	4	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
286	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
287	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
288	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
289	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
290	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
291	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	—	2	2	no	—
296	C₈	—	2	4	no	—
297	C₈	—	4	2	no	—
298	C₈	—	4	2	no	—
299	C₈	—	4	2	no	—
300	C₈	—	4	2	no	—
301	C₇	#17824	1	1	yes	characteristic, direct product factor, Sylow
302	C₂²	—	2	1	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	1	no	—
308	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	2	no	—
316	C₄	C₇×#568	1	2	yes	—
317	C₄	C₇×#577	1	2	yes	—
318	C₂	—	4	1	no	—
319	C₂	—	4	2	no	—
320	C₂	C₇×#2905	1	1	yes	characteristic
321	C₂	C₇×#2912	1	1	yes	characteristic
322	C₂	C₇×#2916	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	C₂²×C₄⋊C₄
prime radical	C₂×C₇×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₄×C₇×#41 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1574	C₂	semidirect
4. C₇×#1573	C₂	semidirect
5. C₇×#1572	C₂	semidirect
6. C₇×#1570	C₂	semidirect
7. C₇×#1569	C₂	semidirect
8. C₇×#1568	C₂	semidirect
9. C₇×#1566	C₂	semidirect
38. #17824	C₇	direct (N × H)
79. #1574	C₂×C₇	semidirect
80. #1573	C₂×C₇	semidirect
81. #1572	C₂×C₇	semidirect
82. #1570	C₂×C₇	semidirect
83. #1569	C₂×C₇	semidirect
84. #1568	C₂×C₇	semidirect
85. #1566	C₂×C₇	semidirect
301. C₇	#17824	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×#2989	2
C₇×#2922	2
C₇×#1574	2
C₇×#1573	2
C₇×#1572	2
C₇×#1570	2
C₇×#1569	2
C₇×#1568	2
C₇×#1566	2
#17824	7

Subgroups — C7×#17824 #13195040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#17824 #13195040