Subgroups — C₂×C₃×#6500 #13061143

Order 1344.

1324 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×D₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₂²⋊D₇	C₂	1	2	yes	maximal, complemented
4	C₂×C₃×C₂²⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#6500	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃×(C₂×D₇)⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₂×#6500	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂⁴×C₃×C₇	C₂²	1	1	yes	characteristic
9	C₂⁴×C₃×C₇	C₂²	1	1	yes	characteristic, complemented
10	C₂³×C₃×D₇	—	2	4	no	—
11	C₂³×C₃×D₇	—	2	8	no	—
12	C₂³×C₃×D₇	C₂²	1	2	yes	—
13	C₂³×C₃×D₇	C₂²	1	4	yes	complemented
14	C₂×C₃×C₇×D₄	—	2	4	no	—
15	C₂×C₃×C₇×D₄	C₂²	1	4	yes	complemented
16	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₃×C₂²⋊D₇	—	2	8	no	—
18	C₂×C₃×C₂²⋊D₇	C₂²	1	8	yes	complemented
19	C₂²×C₃×Dic₇	C₂²	1	2	yes	complemented
20	C₃×C₂²⋊Dic₇	C₂²	1	4	yes	complemented
21	C₃×(C₂×D₇)⋊C₄	C₂²	1	8	yes	complemented
22	C₂⁴×D₇	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂²×C₇×D₄	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂²×C₂²⋊D₇	C₂×C₃	1	2	yes	complemented
25	C₂×C₂²⋊Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
26	#6500	C₂×C₃	1	8	yes	direct product factor
27	C₂×(C₂×D₇)⋊C₄	C₂×C₃	1	2	yes	complemented
28	C₂×C₃×#120	—	7	1	no	maximal, Hall
29	C₂³×C₃×C₇	—	2	4	no	—
30	C₂³×C₃×C₇	—	2	4	no	—
31	C₂³×C₃×C₇	—	2	2	no	—
32	C₂³×C₃×C₇	D₄	1	4	yes	complemented
33	C₂³×C₃×C₇	C₂³	1	2	yes	—
34	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
35	C₂³×C₃×C₇	C₂³	1	2	yes	complemented
36	C₂²×C₃×D₇	—	2	32	no	—
37	C₂²×C₃×D₇	—	2	4	no	—
38	C₂²×C₃×D₇	—	2	4	no	—
39	C₂²×C₃×D₇	—	2	8	no	—
40	C₂²×C₃×D₇	—	2	16	no	—
41	C₂²×C₃×D₇	D₄	1	8	yes	complemented
42	C₂²×C₃×D₇	C₂³	1	4	yes	—
43	C₃×C₇×D₄	—	2	8	no	—
44	C₂×C₃×C₄×C₇	—	2	2	no	—
45	C₂×C₃×C₄×C₇	C₂³	1	2	yes	complemented
46	C₃×C₂²⋊D₇	—	2	16	no	—
47	C₂×C₃×Dic₇	—	2	4	no	—
48	C₂×C₃×Dic₇	C₂³	1	4	yes	complemented
49	C₂⁴×C₇	C₂²×C₃	1	1	yes	characteristic
50	C₂⁴×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
51	C₂³×D₇	—	2	4	no	—
52	C₂³×D₇	—	2	8	no	—
53	C₂³×D₇	C₂²×C₃	1	2	yes	—
54	C₂³×D₇	C₂²×C₃	1	4	yes	complemented
55	C₂×C₇×D₄	—	2	4	no	—
56	C₂×C₇×D₄	C₂²×C₃	1	4	yes	complemented
57	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
58	C₂×C₂²⋊D₇	—	2	8	no	—
59	C₂×C₂²⋊D₇	C₂²×C₃	1	8	yes	complemented
60	C₂²×Dic₇	C₂²×C₃	1	2	yes	complemented
61	C₂²⋊Dic₇	C₂²×C₃	1	4	yes	complemented
62	(C₂×D₇)⋊C₄	C₂²×C₃	1	8	yes	complemented
63	C₂⁵×C₃	—	7	1	no	—
64	C₂²×C₃×D₄	—	7	2	no	—
65	C₂²×C₃×D₄	D₇	1	1	yes	characteristic, complemented
66	C₃×#120	—	7	8	no	—
67	C₂×C₃×C₂²⋊C₄	—	7	1	no	—
68	C₂×C₃×C₂²⋊C₄	—	7	2	no	—
69	C₂²×C₃×C₇	—	2	8	no	—
70	C₂²×C₃×C₇	—	2	4	no	—
71	C₂²×C₃×C₇	—	2	4	no	—
72	C₂²×C₃×C₇	—	2	2	no	—
73	C₂²×C₃×C₇	—	4	4	no	—
74	C₂²×C₃×C₇	C₂×D₄	1	4	yes	—
75	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
76	C₂²×C₃×C₇	C₂×D₄	1	4	yes	complemented
77	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
78	C₂×C₃×D₇	—	2	32	no	—
79	C₂×C₃×D₇	—	2	16	no	—
80	C₂×C₃×D₇	—	2	4	no	—
81	C₂×C₃×D₇	—	2	4	no	—
82	C₂×C₃×D₇	C₂×D₄	1	8	yes	complemented
83	C₃×C₄×C₇	—	2	2	no	—
84	C₃×Dic₇	—	2	4	no	—
85	C₂×#120	—	7	1	no	Sylow
86	C₂³×C₇	—	2	4	no	—
87	C₂³×C₇	—	2	4	no	—
88	C₂³×C₇	—	2	2	no	—
89	C₂³×C₇	C₃×D₄	1	4	yes	complemented
90	C₂³×C₇	C₂³×C₃	1	2	yes	—
91	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
92	C₂³×C₇	C₂³×C₃	1	2	yes	complemented
93	C₂²×D₇	—	2	32	no	—
94	C₂²×D₇	—	2	4	no	—
95	C₂²×D₇	—	2	4	no	—
96	C₂²×D₇	—	2	8	no	—
97	C₂²×D₇	—	2	16	no	—
98	C₂²×D₇	C₃×D₄	1	8	yes	complemented
99	C₂²×D₇	C₂³×C₃	1	4	yes	—
100	C₇×D₄	—	2	8	no	—
101	C₂×C₄×C₇	—	2	2	no	—
102	C₂×C₄×C₇	C₂³×C₃	1	2	yes	complemented
103	C₂²⋊D₇	—	2	16	no	—
104	C₂×Dic₇	—	2	4	no	—
105	C₂×Dic₇	C₂³×C₃	1	4	yes	complemented
106	C₂⁴×C₃	—	7	2	no	—
107	C₂⁴×C₃	—	7	4	no	—
108	C₂⁴×C₃	—	14	8	no	—
109	C₂⁴×C₃	—	14	4	no	—
110	C₂⁴×C₃	C₂×D₇	1	1	yes	characteristic
111	C₂⁴×C₃	C₂×D₇	1	1	yes	characteristic, complemented
112	C₂×C₃×D₄	—	2	4	no	—
113	C₂×C₃×D₄	—	7	8	no	—
114	C₂×C₃×D₄	—	14	8	no	—
115	C₂×C₃×D₄	C₂×D₇	1	4	yes	complemented
116	C₂²×C₃×C₄	—	7	2	no	—
117	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic, complemented
118	C₃×C₂²⋊C₄	—	7	4	no	—
119	C₃×C₂²⋊C₄	—	7	8	no	—
120	C₂×C₃×C₇	—	2	4	no	—
121	C₂×C₃×C₇	—	4	2	no	—
122	C₂×C₃×C₇	#120	1	4	yes	complemented
123	C₂×C₃×C₇	C₂²×D₄	1	2	yes	—
124	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
125	C₃×D₇	—	2	8	no	—
126	C₂⁵	—	7	1	no	—
127	C₂²×D₄	—	7	2	no	—
128	C₂²×D₄	C₃×D₇	1	1	yes	characteristic, complemented
129	#120	—	7	8	no	—
130	C₂×C₂²⋊C₄	—	7	1	no	—
131	C₂×C₂²⋊C₄	—	7	2	no	—
132	C₂²×C₇	—	2	8	no	—
133	C₂²×C₇	—	2	4	no	—
134	C₂²×C₇	—	2	4	no	—
135	C₂²×C₇	—	2	2	no	—
136	C₂²×C₇	—	4	4	no	—
137	C₂²×C₇	C₂×C₃×D₄	1	4	yes	—
138	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
139	C₂²×C₇	C₂×C₃×D₄	1	4	yes	complemented
140	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
141	C₂×D₇	—	2	32	no	—
142	C₂×D₇	—	2	16	no	—
143	C₂×D₇	—	2	4	no	—
144	C₂×D₇	—	2	4	no	—
145	C₂×D₇	C₂×C₃×D₄	1	8	yes	complemented
146	C₄×C₇	—	2	2	no	—
147	Dic₇	—	2	4	no	—
148	C₂³×C₃	—	2	4	no	—
149	C₂³×C₃	—	2	4	no	—
150	C₂³×C₃	—	2	2	no	—
151	C₂³×C₃	—	7	8	no	—
152	C₂³×C₃	—	7	4	no	—
153	C₂³×C₃	—	14	16	no	—
154	C₂³×C₃	—	14	32	no	—
155	C₂³×C₃	—	14	4	no	—
156	C₂³×C₃	—	14	4	no	—
157	C₂³×C₃	—	14	8	no	—
158	C₂³×C₃	C₂²⋊D₇	1	4	yes	complemented
159	C₂³×C₃	C₂²×D₇	1	2	yes	—
160	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
161	C₂³×C₃	C₂²×D₇	1	2	yes	complemented
162	C₃×D₄	—	2	8	no	—
163	C₃×D₄	—	14	16	no	—
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	—	7	4	no	—
166	C₂×C₃×C₄	—	14	4	no	—
167	C₂×C₃×C₄	C₂²×D₇	1	2	yes	complemented
168	C₃×C₇	C₂×#120	1	1	yes	characteristic, complemented, Hall
169	C₂⁴	—	7	2	no	—
170	C₂⁴	—	7	4	no	—
171	C₂⁴	—	14	8	no	—
172	C₂⁴	—	14	4	no	—
173	C₂⁴	C₂×C₃×D₇	1	1	yes	characteristic
174	C₂⁴	C₂×C₃×D₇	1	1	yes	characteristic, complemented
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	—	7	8	no	—
177	C₂×D₄	—	14	8	no	—
178	C₂×D₄	C₂×C₃×D₇	1	4	yes	complemented
179	C₂²×C₄	—	7	2	no	—
180	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic, complemented
181	C₂²⋊C₄	—	7	4	no	—
182	C₂²⋊C₄	—	7	8	no	—
183	C₂×C₇	—	2	4	no	—
184	C₂×C₇	—	4	2	no	—
185	C₂×C₇	C₃×#120	1	4	yes	complemented
186	C₂×C₇	C₂²×C₃×D₄	1	2	yes	—
187	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
188	D₇	—	2	8	no	—
189	C₂²×C₃	—	2	8	no	—
190	C₂²×C₃	—	2	4	no	—
191	C₂²×C₃	—	2	4	no	—
192	C₂²×C₃	—	2	2	no	—
193	C₂²×C₃	—	4	4	no	—
194	C₂²×C₃	—	7	8	no	—
195	C₂²×C₃	—	14	32	no	—
196	C₂²×C₃	—	14	16	no	—
197	C₂²×C₃	—	14	4	no	—
198	C₂²×C₃	—	14	4	no	—
199	C₂²×C₃	D₄×D₇	1	4	yes	—
200	C₂²×C₃	C₂×C₂²⋊D₇	1	2	yes	—
201	C₂²×C₃	C₂×C₂²⋊D₇	1	4	yes	complemented
202	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	14	4	no	—
205	C₂³	—	2	4	no	—
206	C₂³	—	2	4	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	7	8	no	—
209	C₂³	—	7	4	no	—
210	C₂³	—	14	16	no	—
211	C₂³	—	14	32	no	—
212	C₂³	—	14	4	no	—
213	C₂³	—	14	4	no	—
214	C₂³	—	14	8	no	—
215	C₂³	C₃×C₂²⋊D₇	1	4	yes	complemented
216	C₂³	C₂²×C₃×D₇	1	2	yes	—
217	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
218	C₂³	C₂²×C₃×D₇	1	2	yes	complemented
219	D₄	—	2	8	no	—
220	D₄	—	14	16	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	7	4	no	—
223	C₂×C₄	—	14	4	no	—
224	C₂×C₄	C₂²×C₃×D₇	1	2	yes	complemented
225	C₇	C₂×C₃×#120	1	1	yes	characteristic, complemented, Sylow
226	C₂×C₃	—	2	4	no	—
227	C₂×C₃	—	4	2	no	—
228	C₂×C₃	—	14	8	no	—
229	C₂×C₃	#6500	1	4	yes	direct product factor
230	C₂×C₃	C₂×D₄×D₇	1	2	yes	—
231	C₂×C₃	C₂²×C₂²⋊D₇	1	1	yes	characteristic
232	C₂²	—	2	8	no	—
233	C₂²	—	2	4	no	—
234	C₂²	—	2	4	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	4	4	no	—
237	C₂²	—	7	8	no	—
238	C₂²	—	14	32	no	—
239	C₂²	—	14	16	no	—
240	C₂²	—	14	4	no	—
241	C₂²	—	14	4	no	—
242	C₂²	C₃×D₄×D₇	1	4	yes	—
243	C₂²	C₂×C₃×C₂²⋊D₇	1	2	yes	—
244	C₂²	C₂×C₃×C₂²⋊D₇	1	4	yes	complemented
245	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	14	4	no	—
248	C₃	C₂×#6500	1	1	yes	characteristic, direct product factor, Sylow
249	C₂	—	2	4	no	—
250	C₂	—	4	2	no	—
251	C₂	—	14	8	no	—
252	C₂	C₃×#6500	1	4	yes	direct product factor
253	C₂	C₂×C₃×D₄×D₇	1	2	yes	—
254	C₂	C₂²×C₃×C₂²⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂²×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₂²×C₃×C₇×D₄	C₂
socle	C₂³×C₃×C₇	C₂³
hypercenter	C₂²×C₃×D₄	D₇
hypocenter	C₇	C₂×C₃×#120
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×D₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₃×D₇	C₂	semidirect
2. C₂²×C₃×C₇×D₄	C₂	semidirect
3. C₂²×C₃×C₂²⋊D₇	C₂	semidirect
4. C₂×C₃×C₂²⋊Dic₇	C₂	semidirect
5. C₃×#6500	C₂	direct (N × H)
6. C₂×C₃×(C₂×D₇)⋊C₄	C₂	semidirect
7. C₂×#6500	C₃	direct (N × H)
9. C₂⁴×C₃×C₇	C₂²	semidirect
13. C₂³×C₃×D₇	C₂²	semidirect
15. C₂×C₃×C₇×D₄	C₂²	semidirect
16. C₂²×C₃×C₄×C₇	C₂²	semidirect
18. C₂×C₃×C₂²⋊D₇	C₂²	semidirect
19. C₂²×C₃×Dic₇	C₂²	semidirect
20. C₃×C₂²⋊Dic₇	C₂²	semidirect
21. C₃×(C₂×D₇)⋊C₄	C₂²	semidirect
22. C₂⁴×D₇	C₂×C₃	semidirect
23. C₂²×C₇×D₄	C₂×C₃	semidirect
24. C₂²×C₂²⋊D₇	C₂×C₃	semidirect
25. C₂×C₂²⋊Dic₇	C₂×C₃	semidirect
26. #6500	C₂×C₃	direct (N × H)
27. C₂×(C₂×D₇)⋊C₄	C₂×C₃	semidirect
32. C₂³×C₃×C₇	D₄	semidirect
35. C₂³×C₃×C₇	C₂³	semidirect
41. C₂²×C₃×D₇	D₄	semidirect
45. C₂×C₃×C₄×C₇	C₂³	semidirect
48. C₂×C₃×Dic₇	C₂³	semidirect
50. C₂⁴×C₇	C₂²×C₃	semidirect
54. C₂³×D₇	C₂²×C₃	semidirect
56. C₂×C₇×D₄	C₂²×C₃	semidirect
57. C₂²×C₄×C₇	C₂²×C₃	semidirect
59. C₂×C₂²⋊D₇	C₂²×C₃	semidirect
60. C₂²×Dic₇	C₂²×C₃	semidirect
61. C₂²⋊Dic₇	C₂²×C₃	semidirect
62. (C₂×D₇)⋊C₄	C₂²×C₃	semidirect
65. C₂²×C₃×D₄	D₇	semidirect
76. C₂²×C₃×C₇	C₂×D₄	semidirect
82. C₂×C₃×D₇	C₂×D₄	semidirect
89. C₂³×C₇	C₃×D₄	semidirect
92. C₂³×C₇	C₂³×C₃	semidirect
98. C₂²×D₇	C₃×D₄	semidirect
102. C₂×C₄×C₇	C₂³×C₃	semidirect
105. C₂×Dic₇	C₂³×C₃	semidirect
111. C₂⁴×C₃	C₂×D₇	semidirect
115. C₂×C₃×D₄	C₂×D₇	semidirect
117. C₂²×C₃×C₄	C₂×D₇	semidirect
122. C₂×C₃×C₇	#120	semidirect
128. C₂²×D₄	C₃×D₇	semidirect
139. C₂²×C₇	C₂×C₃×D₄	semidirect
145. C₂×D₇	C₂×C₃×D₄	semidirect
158. C₂³×C₃	C₂²⋊D₇	semidirect
161. C₂³×C₃	C₂²×D₇	semidirect
167. C₂×C₃×C₄	C₂²×D₇	semidirect
168. C₃×C₇	C₂×#120	semidirect
174. C₂⁴	C₂×C₃×D₇	semidirect
178. C₂×D₄	C₂×C₃×D₇	semidirect
180. C₂²×C₄	C₂×C₃×D₇	semidirect
185. C₂×C₇	C₃×#120	semidirect
201. C₂²×C₃	C₂×C₂²⋊D₇	semidirect
215. C₂³	C₃×C₂²⋊D₇	semidirect
218. C₂³	C₂²×C₃×D₇	semidirect
224. C₂×C₄	C₂²×C₃×D₇	semidirect
225. C₇	C₂×C₃×#120	semidirect
229. C₂×C₃	#6500	direct (N × H)
244. C₂²	C₂×C₃×C₂²⋊D₇	semidirect
248. C₃	C₂×#6500	direct (N × H)
252. C₂	C₃×#6500	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃×D₇	2
C₂²×C₃×C₇×D₄	2
C₂²×C₃×C₂²⋊D₇	2
C₂×C₃×C₂²⋊Dic₇	2
C₃×#6500	2
C₂×C₃×(C₂×D₇)⋊C₄	2
C₂×#6500	3
C₂×C₃×#120	7

Subgroups — C2×C3×#6500 #13061143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#6500 #13061143