Subgroups — C₇×#5804 #13061520

Order 1344.

692 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₇×#67	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#953	C₂	1	2	yes	maximal, complemented
4	C₂×C₇×#217	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#939	C₂	1	4	yes	maximal, complemented
6	C₇×#937	C₂	1	2	yes	maximal, complemented
7	C₇×#935	C₂	1	2	yes	maximal, complemented
8	C₇×#929	C₂	1	2	yes	maximal, complemented
9	C₇×#534	—	3	1	no	maximal, Hall
10	C₂⁴×C₃×C₇	C₂²	1	1	yes	characteristic
11	C₂⁴×C₃×C₇	C₂²	1	1	yes	characteristic, complemented
12	C₂³×D₃×C₇	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₃×C₇×D₄	—	2	4	no	—
14	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
15	C₂×C₃×C₇×D₄	C₂²	1	2	yes	complemented
16	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₇×#67	—	2	4	no	—
18	C₂×C₇×#67	C₂²	1	2	yes	—
19	C₂×C₇×#67	C₂²	1	2	yes	complemented
20	C₂×C₇×#67	C₂²	1	4	yes	complemented
21	C₂²×C₇×Dic₃	C₂²	1	1	yes	characteristic, complemented
22	C₂²×C₇×Dic₃	C₂²	1	2	yes	complemented
23	C₇×#217	C₂²	1	2	yes	complemented
24	C₇×#217	C₂²	1	2	yes	complemented
25	C₇×#217	C₂²	1	4	yes	complemented
26	C₇×(C₂×D₃)⋊C₄	C₂²	1	4	yes	complemented
27	C₇×Dic₃⋊C₄	C₂²	1	4	yes	complemented
28	C₂²×C₇×D₄	—	3	1	no	—
29	C₂²×C₇×D₄	D₃	1	1	yes	characteristic, complemented
30	C₇×#123	—	3	2	no	—
31	C₇×#123	—	3	2	no	—
32	C₇×#121	—	3	4	no	—
33	C₇×#120	—	3	2	no	—
34	C₇×#120	—	3	2	no	—
35	C₂×C₇×C₂²⋊C₄	—	3	1	no	—
36	#5804	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
37	C₂³×C₃×C₇	—	2	2	no	—
38	C₂³×C₃×C₇	—	4	2	no	—
39	C₂³×C₃×C₇	—	4	2	no	—
40	C₂³×C₃×C₇	D₄	1	4	yes	complemented
41	C₂³×C₃×C₇	C₂³	1	2	yes	—
42	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
43	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
44	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
45	C₂²×D₃×C₇	—	2	2	no	—
46	C₂²×D₃×C₇	—	4	2	no	—
47	C₂²×D₃×C₇	C₂³	1	2	yes	—
48	C₃×C₇×D₄	—	2	4	no	—
49	C₃×C₇×D₄	—	2	4	no	—
50	C₂×C₃×C₄×C₇	—	2	2	no	—
51	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
52	C₇×#67	—	2	4	no	—
53	C₇×#67	—	2	4	no	—
54	C₇×#67	—	4	4	no	—
55	C₂×C₇×Dic₃	—	2	2	no	—
56	C₂×C₇×Dic₃	—	4	2	no	—
57	C₂×C₇×Dic₃	C₂³	1	2	yes	—
58	C₂×C₇×Dic₃	C₂³	1	4	yes	—
59	C₂⁴×C₇	—	3	1	no	—
60	C₂⁴×C₇	C₂×D₃	1	1	yes	characteristic
61	C₂⁴×C₇	C₂×D₃	1	1	yes	characteristic, complemented
62	C₂×C₇×D₄	—	2	4	no	—
63	C₂×C₇×D₄	—	3	2	no	—
64	C₂×C₇×D₄	—	3	2	no	—
65	C₂×C₇×D₄	—	3	4	no	—
66	C₂×C₇×D₄	—	6	4	no	—
67	C₂×C₇×D₄	C₂×D₃	1	2	yes	—
68	C₂×C₇×D₄	C₂×D₃	1	2	yes	complemented
69	C₂²×C₄×C₇	—	3	1	no	—
70	C₂²×C₄×C₇	—	3	2	no	—
71	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic, complemented
72	C₇×C₄⋊C₄	—	3	4	no	—
73	C₇×C₂²⋊C₄	—	3	2	no	—
74	C₇×C₂²⋊C₄	—	3	2	no	—
75	C₇×C₂²⋊C₄	—	3	4	no	—
76	C₇×C₂²⋊C₄	—	3	4	no	—
77	C₂²×C₃×D₄	C₂×C₇	1	1	yes	characteristic, complemented
78	C₂²×#67	C₂×C₇	1	1	yes	characteristic, complemented
79	#953	C₂×C₇	1	2	yes	complemented
80	C₂×#217	C₂×C₇	1	1	yes	characteristic, complemented
81	#939	C₂×C₇	1	4	yes	complemented
82	#937	C₂×C₇	1	2	yes	complemented
83	#935	C₂×C₇	1	2	yes	complemented
84	#929	C₂×C₇	1	2	yes	complemented
85	C₂²×C₃×C₇	—	2	2	no	—
86	C₂²×C₃×C₇	—	2	2	no	—
87	C₂²×C₃×C₇	—	2	2	no	—
88	C₂²×C₃×C₇	—	2	2	no	—
89	C₂²×C₃×C₇	—	2	4	no	—
90	C₂²×C₃×C₇	—	4	4	no	—
91	C₂²×C₃×C₇	—	4	4	no	—
92	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
93	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
94	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
95	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
96	C₂×D₃×C₇	—	2	2	no	—
97	C₂×D₃×C₇	—	2	4	no	—
98	C₂×D₃×C₇	—	4	4	no	—
99	C₃×C₄×C₇	—	2	2	no	—
100	C₇×Dic₃	—	2	2	no	—
101	C₇×Dic₃	—	2	4	no	—
102	#534	—	3	1	no	Sylow
103	C₂³×C₇	—	2	2	no	—
104	C₂³×C₇	—	3	2	no	—
105	C₂³×C₇	—	4	2	no	—
106	C₂³×C₇	—	4	2	no	—
107	C₂³×C₇	—	6	2	no	—
108	C₂³×C₇	—	12	2	no	—
109	C₂³×C₇	#67	1	4	yes	complemented
110	C₂³×C₇	C₂²×D₃	1	2	yes	—
111	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
112	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
113	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
114	C₇×D₄	—	2	4	no	—
115	C₇×D₄	—	2	4	no	—
116	C₇×D₄	—	6	4	no	—
117	C₇×D₄	—	6	4	no	—
118	C₇×D₄	—	12	4	no	—
119	C₂×C₄×C₇	—	2	2	no	—
120	C₂×C₄×C₇	—	3	4	no	—
121	C₂×C₄×C₇	—	3	2	no	—
122	C₂×C₄×C₇	—	6	2	no	—
123	C₂×C₄×C₇	—	12	2	no	—
124	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
125	C₂⁴×C₃	C₂²×C₇	1	1	yes	characteristic
126	C₂⁴×C₃	C₂²×C₇	1	1	yes	characteristic, complemented
127	C₂³×D₃	C₂²×C₇	1	1	yes	characteristic, complemented
128	C₂×C₃×D₄	—	2	4	no	—
129	C₂×C₃×D₄	C₂²×C₇	1	2	yes	—
130	C₂×C₃×D₄	C₂²×C₇	1	2	yes	complemented
131	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic, complemented
132	C₂×#67	—	2	4	no	—
133	C₂×#67	C₂²×C₇	1	2	yes	—
134	C₂×#67	C₂²×C₇	1	2	yes	complemented
135	C₂×#67	C₂²×C₇	1	4	yes	complemented
136	C₂²×Dic₃	C₂²×C₇	1	1	yes	characteristic, complemented
137	C₂²×Dic₃	C₂²×C₇	1	2	yes	complemented
138	#217	C₂²×C₇	1	2	yes	complemented
139	#217	C₂²×C₇	1	2	yes	complemented
140	#217	C₂²×C₇	1	4	yes	complemented
141	(C₂×D₃)⋊C₄	C₂²×C₇	1	4	yes	complemented
142	Dic₃⋊C₄	C₂²×C₇	1	4	yes	complemented
143	C₂×C₃×C₇	—	2	2	no	—
144	C₂×C₃×C₇	—	2	2	no	—
145	C₂×C₃×C₇	—	2	2	no	—
146	C₂×C₃×C₇	—	4	2	no	—
147	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₃×C₇	H(5,𝔽₂)	1	2	yes	—
149	D₃×C₇	—	4	2	no	—
150	C₂²×D₄	—	3	1	no	—
151	C₂²×D₄	D₃×C₇	1	1	yes	characteristic, complemented
152	#123	—	3	2	no	—
153	#123	—	3	2	no	—
154	#121	—	3	4	no	—
155	#120	—	3	2	no	—
156	#120	—	3	2	no	—
157	C₂×C₂²⋊C₄	—	3	1	no	—
158	C₂²×C₇	—	2	2	no	—
159	C₂²×C₇	—	2	2	no	—
160	C₂²×C₇	—	2	2	no	—
161	C₂²×C₇	—	2	4	no	—
162	C₂²×C₇	—	2	2	no	—
163	C₂²×C₇	—	4	4	no	—
164	C₂²×C₇	—	4	4	no	—
165	C₂²×C₇	—	6	4	no	—
166	C₂²×C₇	—	6	2	no	—
167	C₂²×C₇	—	12	4	no	—
168	C₂²×C₇	C₂×#67	1	2	yes	—
169	C₂²×C₇	C₂×#67	1	2	yes	—
170	C₂²×C₇	C₂×#67	1	2	yes	—
171	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
172	C₄×C₇	—	2	2	no	—
173	C₄×C₇	—	6	4	no	—
174	C₄×C₇	—	6	2	no	—
175	C₂³×C₃	—	2	2	no	—
176	C₂³×C₃	—	4	2	no	—
177	C₂³×C₃	—	4	2	no	—
178	C₂³×C₃	C₇×D₄	1	4	yes	complemented
179	C₂³×C₃	C₂³×C₇	1	2	yes	—
180	C₂³×C₃	C₂³×C₇	1	1	yes	characteristic
181	C₂³×C₃	C₂³×C₇	1	1	yes	characteristic
182	C₂³×C₃	C₂³×C₇	1	1	yes	characteristic
183	C₂²×D₃	—	2	2	no	—
184	C₂²×D₃	—	4	2	no	—
185	C₂²×D₃	C₂³×C₇	1	2	yes	—
186	C₃×D₄	—	2	4	no	—
187	C₃×D₄	—	2	4	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
190	#67	—	2	4	no	—
191	#67	—	2	4	no	—
192	#67	—	4	4	no	—
193	C₂×Dic₃	—	2	2	no	—
194	C₂×Dic₃	—	4	2	no	—
195	C₂×Dic₃	C₂³×C₇	1	2	yes	—
196	C₂×Dic₃	C₂³×C₇	1	4	yes	—
197	C₃×C₇	#534	1	1	yes	characteristic, complemented, Hall
198	C₂⁴	—	3	1	no	—
199	C₂⁴	C₂×D₃×C₇	1	1	yes	characteristic
200	C₂⁴	C₂×D₃×C₇	1	1	yes	characteristic, complemented
201	C₂×D₄	—	2	4	no	—
202	C₂×D₄	—	3	2	no	—
203	C₂×D₄	—	3	2	no	—
204	C₂×D₄	—	3	4	no	—
205	C₂×D₄	—	6	4	no	—
206	C₂×D₄	C₂×D₃×C₇	1	2	yes	—
207	C₂×D₄	C₂×D₃×C₇	1	2	yes	complemented
208	C₂²×C₄	—	3	1	no	—
209	C₂²×C₄	—	3	2	no	—
210	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic, complemented
211	C₄⋊C₄	—	3	4	no	—
212	C₂²⋊C₄	—	3	2	no	—
213	C₂²⋊C₄	—	3	2	no	—
214	C₂²⋊C₄	—	3	4	no	—
215	C₂²⋊C₄	—	3	4	no	—
216	C₂×C₇	—	2	2	no	—
217	C₂×C₇	—	2	2	no	—
218	C₂×C₇	—	2	2	no	—
219	C₂×C₇	—	4	2	no	—
220	C₂×C₇	—	12	2	no	—
221	C₂×C₇	#1004	1	2	yes	—
222	C₂×C₇	C₂²×#67	1	1	yes	characteristic
223	C₂²×C₃	—	2	2	no	—
224	C₂²×C₃	—	2	2	no	—
225	C₂²×C₃	—	2	2	no	—
226	C₂²×C₃	—	2	2	no	—
227	C₂²×C₃	—	2	4	no	—
228	C₂²×C₃	—	4	4	no	—
229	C₂²×C₃	—	4	4	no	—
230	C₂²×C₃	C₂×C₇×D₄	1	2	yes	—
231	C₂²×C₃	C₂×C₇×D₄	1	2	yes	—
232	C₂²×C₃	C₂×C₇×D₄	1	2	yes	—
233	C₂²×C₃	C₂⁴×C₇	1	1	yes	characteristic
234	C₂×D₃	—	2	2	no	—
235	C₂×D₃	—	2	4	no	—
236	C₂×D₃	—	4	4	no	—
237	C₃×C₄	—	2	2	no	—
238	Dic₃	—	2	2	no	—
239	Dic₃	—	2	4	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	3	2	no	—
242	C₂³	—	4	2	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	6	2	no	—
245	C₂³	—	12	2	no	—
246	C₂³	C₇×#67	1	4	yes	complemented
247	C₂³	C₂²×D₃×C₇	1	2	yes	—
248	C₂³	C₂²×D₃×C₇	1	1	yes	characteristic
249	C₂³	C₂²×D₃×C₇	1	1	yes	characteristic
250	C₂³	C₂²×D₃×C₇	1	1	yes	characteristic
251	D₄	—	2	4	no	—
252	D₄	—	2	4	no	—
253	D₄	—	6	4	no	—
254	D₄	—	6	4	no	—
255	D₄	—	12	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	3	4	no	—
258	C₂×C₄	—	3	2	no	—
259	C₂×C₄	—	6	2	no	—
260	C₂×C₄	—	12	2	no	—
261	C₂×C₄	C₂²×D₃×C₇	1	2	yes	—
262	C₇	#5804	1	1	yes	characteristic, direct product factor, Sylow
263	C₂×C₃	—	2	2	no	—
264	C₂×C₃	—	2	2	no	—
265	C₂×C₃	—	2	2	no	—
266	C₂×C₃	—	4	2	no	—
267	C₂×C₃	C₂²×C₇×D₄	1	1	yes	characteristic
268	C₂×C₃	C₇×H(5,𝔽₂)	1	2	yes	—
269	D₃	—	4	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	4	4	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	6	4	no	—
278	C₂²	—	6	2	no	—
279	C₂²	—	12	4	no	—
280	C₂²	C₂×C₇×#67	1	2	yes	—
281	C₂²	C₂×C₇×#67	1	2	yes	—
282	C₂²	C₂×C₇×#67	1	2	yes	—
283	C₂²	C₂³×D₃×C₇	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	6	4	no	—
286	C₄	—	6	2	no	—
287	C₃	C₇×#534	1	1	yes	characteristic, complemented, Sylow
288	C₂	—	2	2	no	—
289	C₂	—	2	2	no	—
290	C₂	—	2	2	no	—
291	C₂	—	4	2	no	—
292	C₂	—	12	2	no	—
293	C₂	C₇×#1004	1	2	yes	—
294	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	C₂²×C₃×C₇×D₄	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₂²×C₇×D₄	D₃
hypocenter	C₃	C₇×#534
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₇×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₇×D₄	C₂	semidirect
2. C₂²×C₇×#67	C₂	semidirect
3. C₇×#953	C₂	semidirect
4. C₂×C₇×#217	C₂	semidirect
5. C₇×#939	C₂	semidirect
6. C₇×#937	C₂	semidirect
7. C₇×#935	C₂	semidirect
8. C₇×#929	C₂	semidirect
11. C₂⁴×C₃×C₇	C₂²	semidirect
12. C₂³×D₃×C₇	C₂²	semidirect
15. C₂×C₃×C₇×D₄	C₂²	semidirect
16. C₂²×C₃×C₄×C₇	C₂²	semidirect
19. C₂×C₇×#67	C₂²	semidirect
20. C₂×C₇×#67	C₂²	semidirect
21. C₂²×C₇×Dic₃	C₂²	semidirect
22. C₂²×C₇×Dic₃	C₂²	semidirect
23. C₇×#217	C₂²	semidirect
24. C₇×#217	C₂²	semidirect
25. C₇×#217	C₂²	semidirect
26. C₇×(C₂×D₃)⋊C₄	C₂²	semidirect
27. C₇×Dic₃⋊C₄	C₂²	semidirect
29. C₂²×C₇×D₄	D₃	semidirect
36. #5804	C₇	direct (N × H)
40. C₂³×C₃×C₇	D₄	semidirect
61. C₂⁴×C₇	C₂×D₃	semidirect
68. C₂×C₇×D₄	C₂×D₃	semidirect
71. C₂²×C₄×C₇	C₂×D₃	semidirect
77. C₂²×C₃×D₄	C₂×C₇	semidirect
78. C₂²×#67	C₂×C₇	semidirect
79. #953	C₂×C₇	semidirect
80. C₂×#217	C₂×C₇	semidirect
81. #939	C₂×C₇	semidirect
82. #937	C₂×C₇	semidirect
83. #935	C₂×C₇	semidirect
84. #929	C₂×C₇	semidirect
109. C₂³×C₇	#67	semidirect
126. C₂⁴×C₃	C₂²×C₇	semidirect
127. C₂³×D₃	C₂²×C₇	semidirect
130. C₂×C₃×D₄	C₂²×C₇	semidirect
131. C₂²×C₃×C₄	C₂²×C₇	semidirect
134. C₂×#67	C₂²×C₇	semidirect
135. C₂×#67	C₂²×C₇	semidirect
136. C₂²×Dic₃	C₂²×C₇	semidirect
137. C₂²×Dic₃	C₂²×C₇	semidirect
138. #217	C₂²×C₇	semidirect
139. #217	C₂²×C₇	semidirect
140. #217	C₂²×C₇	semidirect
141. (C₂×D₃)⋊C₄	C₂²×C₇	semidirect
142. Dic₃⋊C₄	C₂²×C₇	semidirect
151. C₂²×D₄	D₃×C₇	semidirect
178. C₂³×C₃	C₇×D₄	semidirect
197. C₃×C₇	#534	semidirect
200. C₂⁴	C₂×D₃×C₇	semidirect
207. C₂×D₄	C₂×D₃×C₇	semidirect
210. C₂²×C₄	C₂×D₃×C₇	semidirect
246. C₂³	C₇×#67	semidirect
262. C₇	#5804	direct (N × H)
287. C₃	C₇×#534	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₇×D₄	2
C₂²×C₇×#67	2
C₇×#953	2
C₂×C₇×#217	2
C₇×#939	2
C₇×#937	2
C₇×#935	2
C₇×#929	2
C₇×#534	3
#5804	7

Subgroups — C7×#5804 #13061520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#5804 #13061520