Subgroups — C₇×#5824 #13061549

Order 1344.

584 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#237	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#946	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#944	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#939	C₂	1	2	yes	maximal, complemented
6	C₇×#935	C₂	1	2	yes	maximal, complemented
7	C₇×D₄×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#929	C₂	1	2	yes	maximal, complemented
9	C₄×C₇×#67	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#924	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₇×Dic₃⋊C₄	C₂	1	2	yes	maximal, complemented
12	C₇×#547	—	3	1	no	maximal, Hall
13	C₃×C₇×#41	—	2	4	no	—
14	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
16	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
17	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
18	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
19	C₂×C₇×#67	C₂²	1	1	yes	characteristic
20	C₂×C₇×#67	C₂²	1	2	yes	—
21	C₂²×C₇×Dic₃	C₂²	1	2	yes	—
22	C₂²×C₇×Dic₃	C₂²	1	2	yes	—
23	C₇×#237	—	2	4	no	—
24	C₂×C₄×D₃×C₇	C₂²	1	1	yes	characteristic
25	C₂×C₇×Dic₆	C₂²	1	1	yes	characteristic
26	C₇×#217	C₂²	1	2	yes	—
27	C₇×#217	C₂²	1	1	yes	characteristic
28	C₇×#217	C₂²	1	2	yes	—
29	C₇×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
30	C₇×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
31	C₇×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
32	C₇×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
33	C₇×Dic₃⋊C₄	C₂²	1	2	yes	—
34	C₇×Dic₃⋊C₄	C₂²	1	2	yes	—
35	C₇×Dic₃⋊C₄	C₂²	1	2	yes	—
36	C₇×Dic₃⋊C₄	C₂²	1	2	yes	—
37	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
38	C₂×C₇×#41	—	3	1	no	—
39	C₂×C₇×#41	D₃	1	1	yes	characteristic, complemented
40	C₇×C₄⋊Q₈	—	3	1	no	—
41	C₇×#123	—	3	2	no	—
42	C₇×#123	—	3	2	no	—
43	C₇×C₂²⋊Q₈	—	3	1	no	—
44	C₇×C₂²⋊Q₈	—	3	1	no	—
45	C₇×#121	—	3	2	no	—
46	C₄×C₇×D₄	—	3	1	no	—
47	C₄×C₇×D₄	—	3	1	no	—
48	C₂×C₇×C₄⋊C₄	—	3	2	no	—
49	#5824	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
50	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
51	C₂³×C₃×C₇	C₂³	1	2	yes	—
52	C₂²×D₃×C₇	C₂³	1	1	yes	characteristic
53	C₃×C₇×Q₈	—	2	2	no	—
54	C₃×C₇×D₄	—	2	4	no	—
55	C₃×C₇×D₄	D₄	1	4	yes	complemented
56	C₂×C₃×C₄×C₇	—	2	4	no	—
57	C₂×C₃×C₄×C₇	—	2	2	no	—
58	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
60	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
61	C₇×#67	—	2	4	no	—
62	C₇×#67	—	2	2	no	—
63	C₂×C₇×Dic₃	—	2	4	no	—
64	C₂×C₇×Dic₃	—	2	4	no	—
65	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
66	C₂×C₇×Dic₃	C₂³	1	2	yes	—
67	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
68	C₂×C₇×Dic₃	C₂³	1	2	yes	—
69	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
70	C₄×D₃×C₇	—	2	2	no	—
71	C₇×Dic₆	—	2	2	no	—
72	C₇×#41	—	2	4	no	—
73	C₇×#41	—	6	4	no	—
74	C₂×C₇×Q₈	—	3	1	no	—
75	C₂×C₇×Q₈	C₂×D₃	1	1	yes	characteristic
76	C₂×C₇×D₄	—	3	1	no	—
77	C₂×C₇×D₄	—	3	2	no	—
78	C₂×C₇×D₄	C₂×D₃	1	2	yes	—
79	C₂×C₇×D₄	C₂×D₃	1	1	yes	characteristic
80	C₂²×C₄×C₇	—	3	2	no	—
81	C₂²×C₄×C₇	—	3	2	no	—
82	C₂²×C₄×C₇	—	3	1	no	—
83	C₂²×C₄×C₇	C₂×D₃	1	2	yes	—
84	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
85	C₇×C₄⋊C₄	—	3	1	no	—
86	C₇×C₄⋊C₄	—	3	2	no	—
87	C₇×C₄⋊C₄	—	3	2	no	—
88	C₇×C₄⋊C₄	—	3	2	no	—
89	C₇×C₄⋊C₄	—	3	2	no	—
90	C₇×C₄⋊C₄	—	3	1	no	—
91	C₇×C₂²⋊C₄	—	3	2	no	—
92	C₇×C₂²⋊C₄	—	3	1	no	—
93	C₇×C₂²⋊C₄	—	3	2	no	—
94	C₇×C₂²⋊C₄	—	3	1	no	—
95	C₇×C₂²⋊C₄	—	3	2	no	—
96	C₄²×C₇	—	3	1	no	—
97	C₂×C₃×#41	C₂×C₇	1	1	yes	characteristic, complemented
98	C₂×#237	C₂×C₇	1	1	yes	characteristic, complemented
99	#946	C₂×C₇	1	1	yes	characteristic, complemented
100	#944	C₂×C₇	1	1	yes	characteristic, complemented
101	#939	C₂×C₇	1	2	yes	complemented
102	#935	C₂×C₇	1	2	yes	complemented
103	D₄×Dic₃	C₂×C₇	1	1	yes	characteristic, complemented
104	#929	C₂×C₇	1	2	yes	complemented
105	C₄×#67	C₂×C₇	1	1	yes	characteristic, complemented
106	#924	C₂×C₇	1	1	yes	characteristic, complemented
107	C₂×Dic₃⋊C₄	C₂×C₇	1	2	yes	complemented
108	C₂²×C₃×C₇	—	2	2	no	—
109	C₂²×C₃×C₇	—	2	2	no	—
110	C₂²×C₃×C₇	—	2	1	no	—
111	C₂²×C₃×C₇	—	2	1	no	—
112	C₂²×C₃×C₇	—	2	1	no	—
113	C₂²×C₃×C₇	C₂×D₄	1	4	yes	—
114	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
115	C₂×D₃×C₇	—	2	1	no	—
116	C₂×D₃×C₇	—	2	1	no	—
117	C₂×D₃×C₇	—	2	1	no	—
118	C₃×C₄×C₇	—	2	1	no	—
119	C₃×C₄×C₇	—	2	2	no	—
120	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
121	C₇×Dic₃	—	2	2	no	—
122	C₇×Dic₃	—	2	1	no	—
123	C₇×Dic₃	—	2	2	no	—
124	C₇×Dic₃	—	2	1	no	—
125	C₇×Dic₃	#41	1	2	yes	complemented
126	#547	—	3	1	no	Sylow
127	C₂³×C₇	—	3	1	no	—
128	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
129	C₂³×C₇	C₂²×D₃	1	2	yes	—
130	C₇×Q₈	—	2	2	no	—
131	C₇×Q₈	—	6	2	no	—
132	C₇×D₄	—	2	4	no	—
133	C₇×D₄	—	6	2	no	—
134	C₇×D₄	—	6	4	no	—
135	C₇×D₄	#67	1	4	yes	complemented
136	C₂×C₄×C₇	—	2	4	no	—
137	C₂×C₄×C₇	—	2	2	no	—
138	C₂×C₄×C₇	—	3	1	no	—
139	C₂×C₄×C₇	—	3	1	no	—
140	C₂×C₄×C₇	—	3	2	no	—
141	C₂×C₄×C₇	—	3	1	no	—
142	C₂×C₄×C₇	—	3	2	no	—
143	C₂×C₄×C₇	—	6	4	no	—
144	C₂×C₄×C₇	—	6	4	no	—
145	C₂×C₄×C₇	—	6	2	no	—
146	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
147	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
148	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
149	C₃×#41	—	2	4	no	—
150	C₂×C₃×Q₈	C₂²×C₇	1	1	yes	characteristic
151	C₂×C₃×D₄	C₂²×C₇	1	2	yes	—
152	C₂×C₃×D₄	C₂²×C₇	1	1	yes	characteristic
153	C₂²×C₃×C₄	C₂²×C₇	1	2	yes	—
154	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
155	C₂×#67	C₂²×C₇	1	1	yes	characteristic
156	C₂×#67	C₂²×C₇	1	2	yes	—
157	C₂²×Dic₃	C₂²×C₇	1	2	yes	—
158	C₂²×Dic₃	C₂²×C₇	1	2	yes	—
159	#237	—	2	4	no	—
160	C₂×C₄×D₃	C₂²×C₇	1	1	yes	characteristic
161	C₂×Dic₆	C₂²×C₇	1	1	yes	characteristic
162	#217	C₂²×C₇	1	2	yes	—
163	#217	C₂²×C₇	1	1	yes	characteristic
164	#217	C₂²×C₇	1	2	yes	—
165	(C₂×D₃)⋊C₄	C₂²×C₇	1	1	yes	characteristic
166	(C₂×D₃)⋊C₄	C₂²×C₇	1	2	yes	—
167	C₄⋊Dic₃	C₂²×C₇	1	1	yes	characteristic
168	Dic₃⋊C₄	C₂²×C₇	1	1	yes	characteristic
169	Dic₃⋊C₄	C₂²×C₇	1	2	yes	—
170	Dic₃⋊C₄	C₂²×C₇	1	2	yes	—
171	Dic₃⋊C₄	C₂²×C₇	1	2	yes	—
172	Dic₃⋊C₄	C₂²×C₇	1	2	yes	—
173	C₄×Dic₃	C₂²×C₇	1	1	yes	characteristic
174	C₂×C₃×C₇	—	2	4	no	—
175	C₂×C₃×C₇	—	4	1	no	—
176	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
178	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
179	D₃×C₇	—	4	1	no	—
180	C₂×#41	—	3	1	no	—
181	C₂×#41	D₃×C₇	1	1	yes	characteristic, complemented
182	C₄⋊Q₈	—	3	1	no	—
183	#123	—	3	2	no	—
184	#123	—	3	2	no	—
185	C₂²⋊Q₈	—	3	1	no	—
186	C₂²⋊Q₈	—	3	1	no	—
187	#121	—	3	2	no	—
188	C₄×D₄	—	3	1	no	—
189	C₄×D₄	—	3	1	no	—
190	C₂×C₄⋊C₄	—	3	2	no	—
191	C₂²×C₇	—	2	2	no	—
192	C₂²×C₇	—	2	2	no	—
193	C₂²×C₇	—	2	1	no	—
194	C₂²×C₇	—	2	1	no	—
195	C₂²×C₇	—	2	1	no	—
196	C₂²×C₇	—	6	1	no	—
197	C₂²×C₇	—	6	1	no	—
198	C₂²×C₇	—	6	1	no	—
199	C₂²×C₇	C₂×#67	1	4	yes	—
200	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
201	C₄×C₇	—	2	1	no	—
202	C₄×C₇	—	2	2	no	—
203	C₄×C₇	—	3	2	no	—
204	C₄×C₇	—	6	2	no	—
205	C₄×C₇	—	6	1	no	—
206	C₄×C₇	—	6	1	no	—
207	C₄×C₇	—	6	2	no	—
208	C₄×C₇	C₂×#67	1	2	yes	—
209	C₂³×C₃	C₂³×C₇	1	1	yes	characteristic
210	C₂³×C₃	C₂³×C₇	1	2	yes	—
211	C₂²×D₃	C₂³×C₇	1	1	yes	characteristic
212	C₃×Q₈	—	2	2	no	—
213	C₃×D₄	—	2	4	no	—
214	C₃×D₄	C₇×D₄	1	4	yes	complemented
215	C₂×C₃×C₄	—	2	4	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
219	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
220	#67	—	2	4	no	—
221	#67	—	2	2	no	—
222	C₂×Dic₃	—	2	4	no	—
223	C₂×Dic₃	—	2	4	no	—
224	C₂×Dic₃	C₂³×C₇	1	1	yes	characteristic
225	C₂×Dic₃	C₂³×C₇	1	2	yes	—
226	C₂×Dic₃	C₂³×C₇	1	1	yes	characteristic
227	C₂×Dic₃	C₂³×C₇	1	2	yes	—
228	C₂×Dic₃	C₂³×C₇	1	1	yes	characteristic
229	C₄×D₃	—	2	2	no	—
230	Dic₆	—	2	2	no	—
231	C₃×C₇	#547	1	1	yes	characteristic, complemented, Hall
232	#41	—	2	4	no	—
233	#41	—	6	4	no	—
234	C₂×Q₈	—	3	1	no	—
235	C₂×Q₈	C₂×D₃×C₇	1	1	yes	characteristic
236	C₂×D₄	—	3	1	no	—
237	C₂×D₄	—	3	2	no	—
238	C₂×D₄	C₂×D₃×C₇	1	2	yes	—
239	C₂×D₄	C₂×D₃×C₇	1	1	yes	characteristic
240	C₂²×C₄	—	3	2	no	—
241	C₂²×C₄	—	3	2	no	—
242	C₂²×C₄	—	3	1	no	—
243	C₂²×C₄	C₂×D₃×C₇	1	2	yes	—
244	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
245	C₄⋊C₄	—	3	1	no	—
246	C₄⋊C₄	—	3	2	no	—
247	C₄⋊C₄	—	3	2	no	—
248	C₄⋊C₄	—	3	2	no	—
249	C₄⋊C₄	—	3	1	no	—
250	C₄⋊C₄	—	3	2	no	—
251	C₂²⋊C₄	—	3	2	no	—
252	C₂²⋊C₄	—	3	1	no	—
253	C₂²⋊C₄	—	3	2	no	—
254	C₂²⋊C₄	—	3	1	no	—
255	C₂²⋊C₄	—	3	2	no	—
256	C₄²	—	3	1	no	—
257	C₂×C₇	—	2	4	no	—
258	C₂×C₇	—	4	1	no	—
259	C₂×C₇	—	12	1	no	—
260	C₂×C₇	D₃×#41	1	1	yes	characteristic
261	C₂×C₇	#1010	1	1	yes	characteristic
262	C₂×C₇	C₂²×#67	1	1	yes	characteristic
263	C₂²×C₃	—	2	2	no	—
264	C₂²×C₃	—	2	2	no	—
265	C₂²×C₃	—	2	1	no	—
266	C₂²×C₃	—	2	1	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	C₂×C₇×D₄	1	4	yes	—
269	C₂²×C₃	C₂⁴×C₇	1	1	yes	characteristic
270	C₂×D₃	—	2	1	no	—
271	C₂×D₃	—	2	1	no	—
272	C₂×D₃	—	2	1	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	2	2	no	—
275	C₃×C₄	C₂×C₇×D₄	1	2	yes	—
276	Dic₃	—	2	2	no	—
277	Dic₃	—	2	1	no	—
278	Dic₃	—	2	2	no	—
279	Dic₃	—	2	1	no	—
280	Dic₃	C₇×#41	1	2	yes	complemented
281	C₂³	—	3	1	no	—
282	C₂³	C₂²×D₃×C₇	1	1	yes	characteristic
283	C₂³	C₂²×D₃×C₇	1	2	yes	—
284	Q₈	—	2	2	no	—
285	Q₈	—	6	2	no	—
286	D₄	—	2	4	no	—
287	D₄	—	6	2	no	—
288	D₄	—	6	4	no	—
289	D₄	C₇×#67	1	4	yes	complemented
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	3	1	no	—
293	C₂×C₄	—	3	1	no	—
294	C₂×C₄	—	3	2	no	—
295	C₂×C₄	—	3	1	no	—
296	C₂×C₄	—	3	2	no	—
297	C₂×C₄	—	6	4	no	—
298	C₂×C₄	—	6	4	no	—
299	C₂×C₄	—	6	2	no	—
300	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
301	C₂×C₄	C₂²×D₃×C₇	1	2	yes	—
302	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
303	C₇	#5824	1	1	yes	characteristic, direct product factor, Sylow
304	C₂×C₃	—	2	4	no	—
305	C₂×C₃	—	4	1	no	—
306	C₂×C₃	C₂²×C₇×D₄	1	1	yes	characteristic
307	C₂×C₃	C₂×C₇×#41	1	1	yes	characteristic
308	C₂×C₃	C₇×2₋¹⁺⁴	1	1	yes	characteristic
309	D₃	—	4	1	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	6	1	no	—
316	C₂²	—	6	1	no	—
317	C₂²	—	6	1	no	—
318	C₂²	C₂×C₇×#67	1	4	yes	—
319	C₂²	C₂³×D₃×C₇	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	2	no	—
322	C₄	—	3	2	no	—
323	C₄	—	6	2	no	—
324	C₄	—	6	1	no	—
325	C₄	—	6	1	no	—
326	C₄	—	6	2	no	—
327	C₄	C₂×C₇×#67	1	2	yes	—
328	C₃	C₇×#547	1	1	yes	characteristic, complemented, Sylow
329	C₂	—	2	4	no	—
330	C₂	—	4	1	no	—
331	C₂	—	12	1	no	—
332	C₂	D₃×C₇×#41	1	1	yes	characteristic
333	C₂	C₇×#1010	1	1	yes	characteristic
334	C₂	C₂²×C₇×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₇
Frattini	C₂²	C₂³×D₃×C₇
Fitting	C₂×C₃×C₇×#41	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₂×C₇×#41	D₃
hypocenter	C₃	C₇×#547
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₇×#41

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₇×#41	C₂	semidirect
2. C₂×C₇×#237	C₂	semidirect
3. C₇×#946	C₂	semidirect
4. C₇×#944	C₂	semidirect
5. C₇×#939	C₂	semidirect
6. C₇×#935	C₂	semidirect
7. C₇×D₄×Dic₃	C₂	semidirect
8. C₇×#929	C₂	semidirect
9. C₄×C₇×#67	C₂	semidirect
10. C₇×#924	C₂	semidirect
11. C₂×C₇×Dic₃⋊C₄	C₂	semidirect
39. C₂×C₇×#41	D₃	semidirect
49. #5824	C₇	direct (N × H)
55. C₃×C₇×D₄	D₄	semidirect
97. C₂×C₃×#41	C₂×C₇	semidirect
98. C₂×#237	C₂×C₇	semidirect
99. #946	C₂×C₇	semidirect
100. #944	C₂×C₇	semidirect
101. #939	C₂×C₇	semidirect
102. #935	C₂×C₇	semidirect
103. D₄×Dic₃	C₂×C₇	semidirect
104. #929	C₂×C₇	semidirect
105. C₄×#67	C₂×C₇	semidirect
106. #924	C₂×C₇	semidirect
107. C₂×Dic₃⋊C₄	C₂×C₇	semidirect
125. C₇×Dic₃	#41	semidirect
135. C₇×D₄	#67	semidirect
181. C₂×#41	D₃×C₇	semidirect
214. C₃×D₄	C₇×D₄	semidirect
231. C₃×C₇	#547	semidirect
280. Dic₃	C₇×#41	semidirect
289. D₄	C₇×#67	semidirect
303. C₇	#5824	direct (N × H)
328. C₃	C₇×#547	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×C₇×#41	2
C₂×C₇×#237	2
C₇×#946	2
C₇×#944	2
C₇×#939	2
C₇×#935	2
C₇×D₄×Dic₃	2
C₇×#929	2
C₄×C₇×#67	2
C₇×#924	2
C₂×C₇×Dic₃⋊C₄	2
C₇×#547	3
#5824	7

Subgroups — C7×#5824 #13061549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#5824 #13061549