Subgroups — C₇×#5828 #13061543

Order 1344.

660 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	C₂×C₃×C₇×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₃×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#946	C₂	1	2	yes	maximal, complemented
4	C₇×#938	C₂	1	2	yes	maximal, complemented
5	C₇×#937	C₂	1	2	yes	maximal, complemented
6	C₇×#935	C₂	1	2	yes	maximal, complemented
7	C₄×C₇×#67	C₂	1	4	yes	maximal, complemented
8	C₇×#926	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#525	—	3	1	no	maximal, Hall
10	C₂³×D₃×C₇	C₂²	1	1	yes	characteristic
11	C₃×C₇×#41	—	2	4	no	—
12	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
14	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic, complemented
15	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
16	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	complemented
17	C₂×C₇×#67	C₂²	1	4	yes	—
18	C₂²×C₇×Dic₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₃×C₇	—	2	4	no	—
20	C₂×C₄×D₃×C₇	C₂²	1	2	yes	—
21	C₂×C₄×D₃×C₇	C₂²	1	2	yes	—
22	C₇×#217	C₂²	1	2	yes	—
23	C₇×#217	C₂²	1	4	yes	complemented
24	C₇×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
25	C₇×C₄⋊Dic₃	C₂²	1	2	yes	complemented
26	C₇×Dic₃⋊C₄	C₂²	1	4	yes	complemented
27	C₄×C₇×Dic₃	C₂²	1	2	yes	complemented
28	C₂×C₇×#41	D₃	1	1	yes	characteristic, complemented
29	C₂³×C₄×C₇	—	3	1	no	—
30	C₇×#123	—	3	2	no	—
31	C₇×C₂²⋊Q₈	—	3	2	no	—
32	C₇×#121	—	3	2	no	—
33	C₇×#120	—	3	2	no	—
34	C₄×C₇×D₄	—	3	4	no	—
35	C₇×#117	—	3	1	no	—
36	#5828	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
37	C₂³×C₃×C₇	C₂³	1	2	yes	—
38	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
39	C₂²×D₃×C₇	—	2	2	no	—
40	C₂²×D₃×C₇	—	2	4	no	—
41	C₂²×D₃×C₇	C₂³	1	2	yes	—
42	C₃×C₇×Q₈	—	2	2	no	—
43	C₃×C₇×D₄	—	2	4	no	—
44	C₃×C₇×D₄	—	2	2	no	—
45	C₂×C₃×C₄×C₇	—	2	4	no	—
46	C₂×C₃×C₄×C₇	D₄	1	4	yes	complemented
47	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
48	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
50	C₇×#67	—	2	8	no	—
51	C₂×C₇×Dic₃	—	2	2	no	—
52	C₂×C₇×Dic₃	C₂³	1	4	yes	—
53	C₂×C₇×Dic₃	C₂³	1	2	yes	—
54	C₄×D₃×C₇	—	2	4	no	—
55	C₄×D₃×C₇	—	2	4	no	—
56	C₂⁴×C₇	—	3	1	no	—
57	C₇×#41	—	2	4	no	—
58	C₂×C₇×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
59	C₂×C₇×D₄	—	3	4	no	—
60	C₂×C₇×D₄	C₂×D₃	1	2	yes	—
61	C₂×C₇×D₄	C₂×D₃	1	1	yes	characteristic, complemented
62	C₂²×C₄×C₇	—	3	1	no	—
63	C₂²×C₄×C₇	—	3	2	no	—
64	C₂²×C₄×C₇	—	3	2	no	—
65	C₂²×C₄×C₇	—	6	4	no	—
66	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₂×D₃	1	2	yes	complemented
68	C₇×C₄⋊C₄	—	3	4	no	—
69	C₇×C₄⋊C₄	—	3	2	no	—
70	C₇×C₂²⋊C₄	—	3	2	no	—
71	C₇×C₂²⋊C₄	—	3	4	no	—
72	C₇×C₂²⋊C₄	—	3	4	no	—
73	C₄²×C₇	—	3	2	no	—
74	C₂×C₃×#41	C₂×C₇	1	1	yes	characteristic, complemented
75	C₂²×C₄×D₃	C₂×C₇	1	1	yes	characteristic, complemented
76	#946	C₂×C₇	1	2	yes	complemented
77	#938	C₂×C₇	1	2	yes	complemented
78	#937	C₂×C₇	1	2	yes	complemented
79	#935	C₂×C₇	1	2	yes	complemented
80	C₄×#67	C₂×C₇	1	4	yes	complemented
81	#926	C₂×C₇	1	1	yes	characteristic, complemented
82	C₂²×C₃×C₇	—	2	2	no	—
83	C₂²×C₃×C₇	—	2	2	no	—
84	C₂²×C₃×C₇	—	2	4	no	—
85	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
86	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
87	C₂×D₃×C₇	—	2	8	no	—
88	C₂×D₃×C₇	—	2	2	no	—
89	C₂×D₃×C₇	—	2	2	no	—
90	C₂×D₃×C₇	#41	1	4	yes	complemented
91	C₃×C₄×C₇	—	2	2	no	—
92	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
93	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
94	C₇×Dic₃	—	2	2	no	—
95	C₇×Dic₃	—	2	4	no	—
96	#525	—	3	1	no	Sylow
97	C₂³×C₇	—	3	2	no	—
98	C₂³×C₇	—	6	4	no	—
99	C₂³×C₇	—	6	2	no	—
100	C₂³×C₇	C₂²×D₃	1	2	yes	—
101	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
102	C₇×Q₈	—	2	2	no	—
103	C₇×D₄	—	2	4	no	—
104	C₇×D₄	—	2	2	no	—
105	C₇×D₄	—	6	8	no	—
106	C₂×C₄×C₇	—	2	4	no	—
107	C₂×C₄×C₇	—	3	4	no	—
108	C₂×C₄×C₇	—	3	2	no	—
109	C₂×C₄×C₇	—	6	2	no	—
110	C₂×C₄×C₇	—	6	4	no	—
111	C₂×C₄×C₇	—	6	4	no	—
112	C₂×C₄×C₇	#67	1	4	yes	complemented
113	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
114	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
115	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
116	C₂³×D₃	C₂²×C₇	1	1	yes	characteristic
117	C₃×#41	—	2	4	no	—
118	C₂×C₃×Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
119	C₂×C₃×D₄	C₂²×C₇	1	2	yes	—
120	C₂×C₃×D₄	C₂²×C₇	1	1	yes	characteristic, complemented
121	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
122	C₂²×C₃×C₄	C₂²×C₇	1	2	yes	complemented
123	C₂×#67	C₂²×C₇	1	4	yes	—
124	C₂²×Dic₃	C₂²×C₇	1	1	yes	characteristic
125	C₂×C₄×D₃	—	2	4	no	—
126	C₂×C₄×D₃	C₂²×C₇	1	2	yes	—
127	C₂×C₄×D₃	C₂²×C₇	1	2	yes	—
128	#217	C₂²×C₇	1	2	yes	—
129	#217	C₂²×C₇	1	4	yes	complemented
130	(C₂×D₃)⋊C₄	C₂²×C₇	1	4	yes	—
131	C₄⋊Dic₃	C₂²×C₇	1	2	yes	complemented
132	Dic₃⋊C₄	C₂²×C₇	1	4	yes	complemented
133	C₄×Dic₃	C₂²×C₇	1	2	yes	complemented
134	C₂×C₃×C₇	—	2	2	no	—
135	C₂×C₃×C₇	—	4	2	no	—
136	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₃×C₇	C₂×#41	1	2	yes	—
138	D₃×C₇	—	2	4	no	—
139	C₂×#41	D₃×C₇	1	1	yes	characteristic, complemented
140	C₂³×C₄	—	3	1	no	—
141	#123	—	3	2	no	—
142	C₂²⋊Q₈	—	3	2	no	—
143	#121	—	3	2	no	—
144	#120	—	3	2	no	—
145	C₄×D₄	—	3	4	no	—
146	#117	—	3	1	no	—
147	C₂²×C₇	—	2	2	no	—
148	C₂²×C₇	—	2	4	no	—
149	C₂²×C₇	—	2	2	no	—
150	C₂²×C₇	—	3	4	no	—
151	C₂²×C₇	—	6	8	no	—
152	C₂²×C₇	—	6	2	no	—
153	C₂²×C₇	—	6	2	no	—
154	C₂²×C₇	C₂×#67	1	2	yes	—
155	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
156	C₄×C₇	—	2	2	no	—
157	C₄×C₇	—	6	4	no	—
158	C₄×C₇	—	6	2	no	—
159	C₄×C₇	C₂×#67	1	2	yes	—
160	C₄×C₇	C₂×#67	1	2	yes	—
161	C₂³×C₃	C₂³×C₇	1	2	yes	—
162	C₂³×C₃	C₂³×C₇	1	1	yes	characteristic
163	C₂²×D₃	—	2	2	no	—
164	C₂²×D₃	—	2	4	no	—
165	C₂²×D₃	C₂³×C₇	1	2	yes	—
166	C₃×Q₈	—	2	2	no	—
167	C₃×D₄	—	2	4	no	—
168	C₃×D₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	4	no	—
170	C₂×C₃×C₄	C₇×D₄	1	4	yes	complemented
171	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
172	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
174	#67	—	2	8	no	—
175	C₂×Dic₃	—	2	2	no	—
176	C₂×Dic₃	C₂³×C₇	1	4	yes	—
177	C₂×Dic₃	C₂³×C₇	1	2	yes	—
178	C₄×D₃	—	2	4	no	—
179	C₄×D₃	—	2	4	no	—
180	C₃×C₇	#525	1	1	yes	characteristic, complemented, Hall
181	C₂⁴	—	3	1	no	—
182	#41	—	2	4	no	—
183	C₂×Q₈	C₂×D₃×C₇	1	1	yes	characteristic, complemented
184	C₂×D₄	—	3	4	no	—
185	C₂×D₄	C₂×D₃×C₇	1	2	yes	—
186	C₂×D₄	C₂×D₃×C₇	1	1	yes	characteristic, complemented
187	C₂²×C₄	—	3	1	no	—
188	C₂²×C₄	—	3	2	no	—
189	C₂²×C₄	—	3	2	no	—
190	C₂²×C₄	—	6	4	no	—
191	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₃×C₇	1	2	yes	complemented
193	C₄⋊C₄	—	3	2	no	—
194	C₄⋊C₄	—	3	4	no	—
195	C₂²⋊C₄	—	3	2	no	—
196	C₂²⋊C₄	—	3	4	no	—
197	C₂²⋊C₄	—	3	4	no	—
198	C₄²	—	3	2	no	—
199	C₂×C₇	—	2	2	no	—
200	C₂×C₇	—	4	2	no	—
201	C₂×C₇	—	6	4	no	—
202	C₂×C₇	D₃×#41	1	2	yes	—
203	C₂×C₇	C₂²×#67	1	1	yes	characteristic
204	C₂²×C₃	—	2	2	no	—
205	C₂²×C₃	—	2	2	no	—
206	C₂²×C₃	—	2	4	no	—
207	C₂²×C₃	C₂×C₇×D₄	1	2	yes	—
208	C₂²×C₃	C₂⁴×C₇	1	1	yes	characteristic
209	C₂×D₃	—	2	8	no	—
210	C₂×D₃	—	2	2	no	—
211	C₂×D₃	—	2	2	no	—
212	C₂×D₃	C₇×#41	1	4	yes	complemented
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	C₂×C₇×D₄	1	2	yes	—
215	C₃×C₄	C₂×C₇×D₄	1	2	yes	—
216	Dic₃	—	2	2	no	—
217	Dic₃	—	2	4	no	—
218	C₂³	—	3	2	no	—
219	C₂³	—	6	4	no	—
220	C₂³	—	6	2	no	—
221	C₂³	C₂²×D₃×C₇	1	2	yes	—
222	C₂³	C₂²×D₃×C₇	1	1	yes	characteristic
223	Q₈	—	2	2	no	—
224	D₄	—	2	4	no	—
225	D₄	—	2	2	no	—
226	D₄	—	6	8	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	3	4	no	—
229	C₂×C₄	—	3	2	no	—
230	C₂×C₄	—	6	2	no	—
231	C₂×C₄	—	6	4	no	—
232	C₂×C₄	—	6	4	no	—
233	C₂×C₄	C₇×#67	1	4	yes	complemented
234	C₂×C₄	C₂²×D₃×C₇	1	2	yes	—
235	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
237	C₇	#5828	1	1	yes	characteristic, direct product factor, Sylow
238	C₂×C₃	—	2	2	no	—
239	C₂×C₃	—	4	2	no	—
240	C₂×C₃	C₂²×C₇×D₄	1	1	yes	characteristic
241	C₂×C₃	C₂×C₇×#41	1	2	yes	—
242	D₃	—	2	4	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	3	4	no	—
247	C₂²	—	6	8	no	—
248	C₂²	—	6	2	no	—
249	C₂²	—	6	2	no	—
250	C₂²	C₂×C₇×#67	1	2	yes	—
251	C₂²	C₂³×D₃×C₇	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	6	4	no	—
254	C₄	—	6	2	no	—
255	C₄	C₂×C₇×#67	1	2	yes	—
256	C₄	C₂×C₇×#67	1	2	yes	—
257	C₃	C₇×#525	1	1	yes	characteristic, complemented, Sylow
258	C₂	—	2	2	no	—
259	C₂	—	4	2	no	—
260	C₂	—	6	4	no	—
261	C₂	D₃×C₇×#41	1	2	yes	—
262	C₂	C₂²×C₇×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×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×#525
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×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. C₂²×C₄×D₃×C₇	C₂	semidirect
3. C₇×#946	C₂	semidirect
4. C₇×#938	C₂	semidirect
5. C₇×#937	C₂	semidirect
6. C₇×#935	C₂	semidirect
7. C₄×C₇×#67	C₂	semidirect
8. C₇×#926	C₂	semidirect
12. C₂×C₃×C₇×Q₈	C₂²	semidirect
14. C₂×C₃×C₇×D₄	C₂²	semidirect
16. C₂²×C₃×C₄×C₇	C₂²	semidirect
23. C₇×#217	C₂²	semidirect
25. C₇×C₄⋊Dic₃	C₂²	semidirect
26. C₇×Dic₃⋊C₄	C₂²	semidirect
27. C₄×C₇×Dic₃	C₂²	semidirect
28. C₂×C₇×#41	D₃	semidirect
36. #5828	C₇	direct (N × H)
46. C₂×C₃×C₄×C₇	D₄	semidirect
58. C₂×C₇×Q₈	C₂×D₃	semidirect
61. C₂×C₇×D₄	C₂×D₃	semidirect
67. C₂²×C₄×C₇	C₂×D₃	semidirect
74. C₂×C₃×#41	C₂×C₇	semidirect
75. C₂²×C₄×D₃	C₂×C₇	semidirect
76. #946	C₂×C₇	semidirect
77. #938	C₂×C₇	semidirect
78. #937	C₂×C₇	semidirect
79. #935	C₂×C₇	semidirect
80. C₄×#67	C₂×C₇	semidirect
81. #926	C₂×C₇	semidirect
90. C₂×D₃×C₇	#41	semidirect
112. C₂×C₄×C₇	#67	semidirect
118. C₂×C₃×Q₈	C₂²×C₇	semidirect
120. C₂×C₃×D₄	C₂²×C₇	semidirect
122. C₂²×C₃×C₄	C₂²×C₇	semidirect
129. #217	C₂²×C₇	semidirect
131. C₄⋊Dic₃	C₂²×C₇	semidirect
132. Dic₃⋊C₄	C₂²×C₇	semidirect
133. C₄×Dic₃	C₂²×C₇	semidirect
139. C₂×#41	D₃×C₇	semidirect
170. C₂×C₃×C₄	C₇×D₄	semidirect
180. C₃×C₇	#525	semidirect
183. C₂×Q₈	C₂×D₃×C₇	semidirect
186. C₂×D₄	C₂×D₃×C₇	semidirect
192. C₂²×C₄	C₂×D₃×C₇	semidirect
212. C₂×D₃	C₇×#41	semidirect
233. C₂×C₄	C₇×#67	semidirect
237. C₇	#5828	direct (N × H)
257. C₃	C₇×#525	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
C₂²×C₄×D₃×C₇	2
C₇×#946	2
C₇×#938	2
C₇×#937	2
C₇×#935	2
C₄×C₇×#67	2
C₇×#926	2
C₇×#525	3
#5828	7

Subgroups — C7×#5828 #13061543

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#5828 #13061543