Subgroups — C₇×#5220 #13056254

Order 1344.

572 conjugacy classes of subgroups.

The subgroup lattice has 264 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
2	C₂³×C₇×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#217	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×#217	C₂	1	2	yes	maximal, complemented
5	C₇×C₃⋊#95	C₂	1	2	yes	maximal, complemented
6	C₇×#386	—	3	1	no	maximal, Hall
7	C₂⁴×C₃×C₇	C₂²	1	1	yes	characteristic
8	C₂⁴×C₃×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₇×D₄	—	2	2	no	—
10	C₂×C₃×C₇×D₄	—	2	2	no	—
11	C₂×C₃×C₇×D₄	C₄	1	4	yes	complemented
12	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂²×C₇×Dic₃	—	2	2	no	—
14	C₂²×C₇×Dic₃	—	2	4	no	—
15	C₂²×C₇×Dic₃	C₂²	1	2	yes	—
16	C₂²×C₇×Dic₃	C₂²	1	2	yes	—
17	C₇×#217	—	2	2	no	—
18	C₇×#217	—	2	4	no	—
19	C₂²×C₇×D₄	D₃	1	1	yes	characteristic
20	C₂³×C₄×C₇	—	3	1	no	—
21	C₂×C₇×C₂²⋊C₄	—	3	2	no	—
22	C₂×C₇×C₂²⋊C₄	—	3	1	no	—
23	C₇×#95	—	3	2	no	—
24	#5220	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₂³×C₃×C₇	—	2	2	no	—
26	C₂³×C₃×C₇	—	2	2	no	—
27	C₂³×C₃×C₇	—	2	2	no	—
28	C₂³×C₃×C₇	—	2	1	no	—
29	C₂³×C₃×C₇	—	2	1	no	—
30	C₂³×C₃×C₇	—	2	2	no	—
31	C₂³×C₃×C₇	C₂×C₄	1	2	yes	—
32	C₂³×C₃×C₇	C₂×C₄	1	2	yes	complemented
33	C₂³×C₃×C₇	D₄	1	2	yes	—
34	C₂³×C₃×C₇	D₄	1	2	yes	—
35	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
36	C₃×C₇×D₄	—	2	8	no	—
37	C₂×C₃×C₄×C₇	—	2	1	no	—
38	C₂×C₃×C₄×C₇	—	2	1	no	—
39	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	complemented
40	C₂×C₇×Dic₃	—	2	8	no	—
41	C₂×C₇×Dic₃	—	2	2	no	—
42	C₂×C₇×Dic₃	—	2	2	no	—
43	C₂×C₇×Dic₃	—	2	2	no	—
44	C₂×C₇×Dic₃	—	2	2	no	—
45	C₂×C₇×Dic₃	—	2	2	no	—
46	C₂×C₇×Dic₃	D₄	1	4	yes	complemented
47	C₂⁴×C₇	C₂×D₃	1	1	yes	characteristic
48	C₂⁴×C₇	C₂×D₃	1	1	yes	characteristic
49	C₂×C₇×D₄	—	2	2	no	—
50	C₂×C₇×D₄	—	2	2	no	—
51	C₂×C₇×D₄	Dic₃	1	4	yes	complemented
52	C₂²×C₄×C₇	—	3	2	no	—
53	C₂²×C₄×C₇	—	3	2	no	—
54	C₂²×C₄×C₇	—	6	4	no	—
55	C₂²×C₄×C₇	—	6	2	no	—
56	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
57	C₇×C₂²⋊C₄	—	6	4	no	—
58	C₇×C₂²⋊C₄	—	6	2	no	—
59	C₂²×C₃×D₄	C₂×C₇	1	1	yes	characteristic
60	C₂³×Dic₃	C₂×C₇	1	1	yes	characteristic, complemented
61	C₂×#217	C₂×C₇	1	1	yes	characteristic, complemented
62	C₂×#217	C₂×C₇	1	2	yes	complemented
63	C₃⋊#95	C₂×C₇	1	2	yes	complemented
64	C₂²×C₃×C₇	—	2	4	no	—
65	C₂²×C₃×C₇	—	2	2	no	—
66	C₂²×C₃×C₇	—	2	4	no	—
67	C₂²×C₃×C₇	—	2	2	no	—
68	C₂²×C₃×C₇	—	2	2	no	—
69	C₂²×C₃×C₇	—	2	4	no	—
70	C₂²×C₃×C₇	—	4	1	no	—
71	C₂²×C₃×C₇	—	4	2	no	—
72	C₂²×C₃×C₇	—	4	1	no	—
73	C₂²×C₃×C₇	C₂²⋊C₄	1	4	yes	complemented
74	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
76	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₃×C₇	#41	1	2	yes	—
79	C₃×C₄×C₇	—	2	2	no	—
80	C₇×Dic₃	—	2	4	no	—
81	C₇×Dic₃	—	4	2	no	—
82	#386	—	3	1	no	Sylow
83	C₂³×C₇	—	2	2	no	—
84	C₂³×C₇	—	2	2	no	—
85	C₂³×C₇	—	2	2	no	—
86	C₂³×C₇	—	2	2	no	—
87	C₂³×C₇	—	2	1	no	—
88	C₂³×C₇	—	2	1	no	—
89	C₂³×C₇	C₂×Dic₃	1	2	yes	—
90	C₂³×C₇	C₂×Dic₃	1	2	yes	complemented
91	C₂³×C₇	#67	1	2	yes	—
92	C₂³×C₇	#67	1	2	yes	—
93	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
94	C₇×D₄	—	2	8	no	—
95	C₂×C₄×C₇	—	2	1	no	—
96	C₂×C₄×C₇	—	2	1	no	—
97	C₂×C₄×C₇	—	3	4	no	—
98	C₂×C₄×C₇	—	6	8	no	—
99	C₂×C₄×C₇	—	6	2	no	—
100	C₂×C₄×C₇	—	6	2	no	—
101	C₂×C₄×C₇	—	6	2	no	—
102	C₂×C₄×C₇	—	6	2	no	—
103	C₂×C₄×C₇	—	6	2	no	—
104	C₂×C₄×C₇	C₂×Dic₃	1	2	yes	complemented
105	C₂⁴×C₃	C₂²×C₇	1	1	yes	characteristic
106	C₂⁴×C₃	C₂²×C₇	1	1	yes	characteristic
107	C₂×C₃×D₄	—	2	2	no	—
108	C₂×C₃×D₄	—	2	2	no	—
109	C₂×C₃×D₄	C₄×C₇	1	4	yes	complemented
110	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
111	C₂²×Dic₃	—	2	2	no	—
112	C₂²×Dic₃	—	2	4	no	—
113	C₂²×Dic₃	C₂²×C₇	1	2	yes	—
114	C₂²×Dic₃	C₂²×C₇	1	2	yes	—
115	#217	—	2	2	no	—
116	#217	—	2	4	no	—
117	C₂×C₃×C₇	—	2	4	no	—
118	C₂×C₃×C₇	—	4	2	no	—
119	C₂×C₃×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₃×C₇	C₄×D₄	1	2	yes	—
121	C₂×C₃×C₇	#120	1	1	yes	characteristic
122	C₂×C₃×C₇	#121	1	2	yes	—
123	C₂×C₃×C₇	#123	1	1	yes	characteristic
124	C₂²×D₄	D₃×C₇	1	1	yes	characteristic
125	C₂³×C₄	—	3	1	no	—
126	C₂×C₂²⋊C₄	—	3	2	no	—
127	C₂×C₂²⋊C₄	—	3	1	no	—
128	#95	—	3	2	no	—
129	C₂²×C₇	—	2	4	no	—
130	C₂²×C₇	—	2	2	no	—
131	C₂²×C₇	—	2	2	no	—
132	C₂²×C₇	—	2	4	no	—
133	C₂²×C₇	—	2	4	no	—
134	C₂²×C₇	—	2	2	no	—
135	C₂²×C₇	—	4	1	no	—
136	C₂²×C₇	—	4	1	no	—
137	C₂²×C₇	—	4	2	no	—
138	C₂²×C₇	#217	1	4	yes	complemented
139	C₂²×C₇	D₃×D₄	1	2	yes	—
140	C₂²×C₇	#237	1	2	yes	—
141	C₂²×C₇	C₂²×Dic₃	1	1	yes	characteristic
142	C₂²×C₇	C₂×#67	1	1	yes	characteristic
143	C₂²×C₇	C₂×#67	1	1	yes	characteristic
144	C₄×C₇	—	2	2	no	—
145	C₄×C₇	—	6	4	no	—
146	C₄×C₇	—	12	2	no	—
147	C₂³×C₃	—	2	2	no	—
148	C₂³×C₃	—	2	2	no	—
149	C₂³×C₃	—	2	2	no	—
150	C₂³×C₃	—	2	1	no	—
151	C₂³×C₃	—	2	1	no	—
152	C₂³×C₃	—	2	2	no	—
153	C₂³×C₃	C₂×C₄×C₇	1	2	yes	—
154	C₂³×C₃	C₂×C₄×C₇	1	2	yes	complemented
155	C₂³×C₃	C₇×D₄	1	2	yes	—
156	C₂³×C₃	C₇×D₄	1	2	yes	—
157	C₂³×C₃	C₂³×C₇	1	1	yes	characteristic
158	C₃×D₄	—	2	8	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	complemented
162	C₂×Dic₃	—	2	8	no	—
163	C₂×Dic₃	—	2	2	no	—
164	C₂×Dic₃	—	2	2	no	—
165	C₂×Dic₃	—	2	2	no	—
166	C₂×Dic₃	—	2	2	no	—
167	C₂×Dic₃	—	2	2	no	—
168	C₂×Dic₃	C₇×D₄	1	4	yes	complemented
169	C₃×C₇	#386	1	1	yes	characteristic, complemented, Hall
170	C₂⁴	C₂×D₃×C₇	1	1	yes	characteristic
171	C₂⁴	C₂×D₃×C₇	1	1	yes	characteristic
172	C₂×D₄	—	2	2	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	C₇×Dic₃	1	4	yes	complemented
175	C₂²×C₄	—	3	2	no	—
176	C₂²×C₄	—	3	2	no	—
177	C₂²×C₄	—	6	4	no	—
178	C₂²×C₄	—	6	2	no	—
179	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
180	C₂²⋊C₄	—	6	4	no	—
181	C₂²⋊C₄	—	6	2	no	—
182	C₂×C₇	—	2	4	no	—
183	C₂×C₇	—	4	2	no	—
184	C₂×C₇	D₄×Dic₃	1	2	yes	—
185	C₂×C₇	#935	1	1	yes	characteristic
186	C₂×C₇	#937	1	1	yes	characteristic
187	C₂×C₇	#939	1	2	yes	—
188	C₂×C₇	C₂×#217	1	1	yes	characteristic
189	C₂²×C₃	—	2	4	no	—
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	—	2	4	no	—
192	C₂²×C₃	—	2	2	no	—
193	C₂²×C₃	—	2	2	no	—
194	C₂²×C₃	—	2	4	no	—
195	C₂²×C₃	—	4	1	no	—
196	C₂²×C₃	—	4	2	no	—
197	C₂²×C₃	—	4	1	no	—
198	C₂²×C₃	C₇×C₂²⋊C₄	1	4	yes	complemented
199	C₂²×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
200	C₂²×C₃	C₂×C₇×D₄	1	2	yes	—
201	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
202	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
203	C₂²×C₃	C₇×#41	1	2	yes	—
204	C₃×C₄	—	2	2	no	—
205	Dic₃	—	2	4	no	—
206	Dic₃	—	4	2	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	C₂×C₇×Dic₃	1	2	yes	—
214	C₂³	C₂×C₇×Dic₃	1	2	yes	complemented
215	C₂³	C₇×#67	1	2	yes	—
216	C₂³	C₇×#67	1	2	yes	—
217	C₂³	C₂²×D₃×C₇	1	1	yes	characteristic
218	D₄	—	2	8	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	3	4	no	—
222	C₂×C₄	—	6	8	no	—
223	C₂×C₄	—	6	2	no	—
224	C₂×C₄	—	6	2	no	—
225	C₂×C₄	—	6	2	no	—
226	C₂×C₄	—	6	2	no	—
227	C₂×C₄	—	6	2	no	—
228	C₂×C₄	C₂×C₇×Dic₃	1	2	yes	complemented
229	C₇	#5220	1	1	yes	characteristic, direct product factor, Sylow
230	C₂×C₃	—	2	4	no	—
231	C₂×C₃	—	4	2	no	—
232	C₂×C₃	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
233	C₂×C₃	C₄×C₇×D₄	1	2	yes	—
234	C₂×C₃	C₇×#120	1	1	yes	characteristic
235	C₂×C₃	C₇×#121	1	2	yes	—
236	C₂×C₃	C₇×#123	1	1	yes	characteristic
237	C₂²	—	2	4	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	2	no	—
246	C₂²	C₇×#217	1	4	yes	complemented
247	C₂²	D₃×C₇×D₄	1	2	yes	—
248	C₂²	C₇×#237	1	2	yes	—
249	C₂²	C₂²×C₇×Dic₃	1	1	yes	characteristic
250	C₂²	C₂×C₇×#67	1	1	yes	characteristic
251	C₂²	C₂×C₇×#67	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	6	4	no	—
254	C₄	—	12	2	no	—
255	C₃	C₇×#386	1	1	yes	characteristic, complemented, Sylow
256	C₂	—	2	4	no	—
257	C₂	—	4	2	no	—
258	C₂	C₇×D₄×Dic₃	1	2	yes	—
259	C₂	C₇×#935	1	1	yes	characteristic
260	C₂	C₇×#937	1	1	yes	characteristic
261	C₂	C₇×#939	1	2	yes	—
262	C₂	C₂×C₇×#217	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₇	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₇
Frattini	C₂³	C₂²×D₃×C₇
Fitting, prime radical	C₂²×C₃×C₇×D₄	C₂
socle	C₂³×C₃×C₇	C₂³
cosocle	C₂³×C₃	C₂³×C₇
hypercenter	C₂²×C₇×D₄	D₃
hypocenter	C₃	C₇×#386
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₇×D₄

Complements

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

N	H ≅ G/N	type
2. C₂³×C₇×Dic₃	C₂	semidirect
3. C₂×C₇×#217	C₂	semidirect
4. C₂×C₇×#217	C₂	semidirect
5. C₇×C₃⋊#95	C₂	semidirect
11. C₂×C₃×C₇×D₄	C₄	semidirect
24. #5220	C₇	direct (N × H)
32. C₂³×C₃×C₇	C₂×C₄	semidirect
39. C₂×C₃×C₄×C₇	C₂×C₄	semidirect
46. C₂×C₇×Dic₃	D₄	semidirect
51. C₂×C₇×D₄	Dic₃	semidirect
60. C₂³×Dic₃	C₂×C₇	semidirect
61. C₂×#217	C₂×C₇	semidirect
62. C₂×#217	C₂×C₇	semidirect
63. C₃⋊#95	C₂×C₇	semidirect
73. C₂²×C₃×C₇	C₂²⋊C₄	semidirect
90. C₂³×C₇	C₂×Dic₃	semidirect
104. C₂×C₄×C₇	C₂×Dic₃	semidirect
109. C₂×C₃×D₄	C₄×C₇	semidirect
138. C₂²×C₇	#217	semidirect
154. C₂³×C₃	C₂×C₄×C₇	semidirect
161. C₂×C₃×C₄	C₂×C₄×C₇	semidirect
168. C₂×Dic₃	C₇×D₄	semidirect
169. C₃×C₇	#386	semidirect
174. C₂×D₄	C₇×Dic₃	semidirect
198. C₂²×C₃	C₇×C₂²⋊C₄	semidirect
214. C₂³	C₂×C₇×Dic₃	semidirect
228. C₂×C₄	C₂×C₇×Dic₃	semidirect
229. C₇	#5220	direct (N × H)
246. C₂²	C₇×#217	semidirect
255. C₃	C₇×#386	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
C₂³×C₇×Dic₃	2
C₂×C₇×#217	2
C₇×C₃⋊#95	2
C₇×#386	3
#5220	7

Subgroups — C7×#5220 #13056254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#5220 #13056254