Subgroups — C₂×C₃×#6501 #13061144

Order 1344.

852 conjugacy classes of subgroups.

The subgroup lattice has 240 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×C₃×C₂²⋊D₇	C₂	1	2	yes	maximal, complemented
3	C₂²×C₃×C₄×D₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×C₂²⋊Dic₇	C₂	1	2	yes	maximal, complemented
5	C₃×#6501	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#6501	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂⁴×C₃×C₇	C₂²	1	2	yes	—
9	C₂³×C₃×D₇	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₇×D₄	—	2	4	no	—
11	C₂×C₃×C₇×D₄	C₂²	1	4	yes	complemented
12	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₂²⋊D₇	—	2	8	no	—
14	C₂×C₃×C₂²⋊D₇	C₂²	1	8	yes	complemented
15	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
16	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄×D₇	—	2	4	no	—
18	C₂×C₃×C₄×D₇	C₂²	1	4	yes	complemented
19	C₃×C₂²⋊Dic₇	C₂²	1	8	yes	complemented
20	C₃×C₄⋊Dic₇	C₂²	1	4	yes	complemented
21	C₂²×C₇×D₄	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂²×C₂²⋊D₇	C₂×C₃	1	2	yes	complemented
23	C₂²×C₄×D₇	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×C₂²⋊Dic₇	C₂×C₃	1	2	yes	complemented
25	#6501	C₂×C₃	1	8	yes	direct product factor
26	C₂×C₄⋊Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂×C₃×#121	—	7	1	no	maximal, Hall
28	C₂³×C₃×C₇	—	2	4	no	—
29	C₂³×C₃×C₇	—	2	4	no	—
30	C₂³×C₃×C₇	—	2	4	no	—
31	C₂³×C₃×C₇	C₂³	1	4	yes	—
32	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
33	C₂²×C₃×D₇	—	2	2	no	—
34	C₂²×C₃×D₇	—	2	2	no	—
35	C₂²×C₃×D₇	D₄	1	4	yes	complemented
36	C₂²×C₃×D₇	C₂³	1	2	yes	—
37	C₃×C₇×D₄	—	2	8	no	—
38	C₂×C₃×C₄×C₇	D₄	1	4	yes	complemented
39	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
40	C₃×C₂²⋊D₇	—	2	16	no	—
41	C₂×C₃×Dic₇	—	2	2	no	—
42	C₂×C₃×Dic₇	—	2	4	no	—
43	C₂×C₃×Dic₇	C₂³	1	4	yes	—
44	C₂×C₃×Dic₇	C₂³	1	2	yes	—
45	C₃×C₄×D₇	—	2	8	no	—
46	C₂⁴×C₇	C₂²×C₃	1	2	yes	—
47	C₂³×D₇	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₇×D₄	—	2	4	no	—
49	C₂×C₇×D₄	C₂²×C₃	1	4	yes	complemented
50	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₂²⋊D₇	—	2	8	no	—
52	C₂×C₂²⋊D₇	C₂²×C₃	1	8	yes	complemented
53	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
54	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₄×D₇	—	2	4	no	—
56	C₂×C₄×D₇	C₂²×C₃	1	4	yes	complemented
57	C₂²⋊Dic₇	C₂²×C₃	1	8	yes	complemented
58	C₄⋊Dic₇	C₂²×C₃	1	4	yes	complemented
59	C₂²×C₃×D₄	—	7	2	no	—
60	C₂²×C₃×D₄	D₇	1	1	yes	characteristic, complemented
61	C₂³×C₃×C₄	—	7	1	no	—
62	C₃×#121	—	7	8	no	—
63	C₂×C₃×C₄⋊C₄	—	7	1	no	—
64	C₂×C₃×C₂²⋊C₄	—	7	2	no	—
65	C₂²×C₃×C₇	—	2	4	no	—
66	C₂²×C₃×C₇	—	2	4	no	—
67	C₂²×C₃×C₇	—	2	4	no	—
68	C₂²×C₃×C₇	—	4	8	no	—
69	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
70	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
71	C₂²×C₃×C₇	#41	1	2	yes	—
72	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
73	C₂×C₃×D₇	—	2	8	no	—
74	C₂×C₃×D₇	—	2	2	no	—
75	C₂×C₃×D₇	—	2	2	no	—
76	C₂×C₃×D₇	C₂×D₄	1	4	yes	complemented
77	C₃×C₄×C₇	C₂×D₄	1	4	yes	complemented
78	C₃×Dic₇	—	2	2	no	—
79	C₃×Dic₇	—	2	4	no	—
80	C₂×#121	—	7	1	no	Sylow
81	C₂³×C₇	—	2	4	no	—
82	C₂³×C₇	—	2	4	no	—
83	C₂³×C₇	—	2	4	no	—
84	C₂³×C₇	C₂³×C₃	1	4	yes	—
85	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
86	C₂²×D₇	—	2	2	no	—
87	C₂²×D₇	—	2	2	no	—
88	C₂²×D₇	C₃×D₄	1	4	yes	complemented
89	C₂²×D₇	C₂³×C₃	1	2	yes	—
90	C₇×D₄	—	2	8	no	—
91	C₂×C₄×C₇	C₃×D₄	1	4	yes	complemented
92	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
93	C₂²⋊D₇	—	2	16	no	—
94	C₂×Dic₇	—	2	2	no	—
95	C₂×Dic₇	—	2	4	no	—
96	C₂×Dic₇	C₂³×C₃	1	4	yes	—
97	C₂×Dic₇	C₂³×C₃	1	2	yes	—
98	C₄×D₇	—	2	8	no	—
99	C₂⁴×C₃	—	7	1	no	—
100	C₂⁴×C₃	C₂×D₇	1	2	yes	—
101	C₂×C₃×D₄	—	2	4	no	—
102	C₂×C₃×D₄	—	7	8	no	—
103	C₂×C₃×D₄	—	14	8	no	—
104	C₂×C₃×D₄	C₂×D₇	1	4	yes	complemented
105	C₂²×C₃×C₄	—	7	1	no	—
106	C₂²×C₃×C₄	—	7	2	no	—
107	C₂²×C₃×C₄	—	7	4	no	—
108	C₂²×C₃×C₄	—	14	4	no	—
109	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
110	C₃×C₄⋊C₄	—	7	4	no	—
111	C₃×C₂²⋊C₄	—	7	8	no	—
112	C₂×C₃×C₇	—	4	4	no	—
113	C₂×C₃×C₇	#121	1	4	yes	complemented
114	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
117	C₃×D₇	—	2	4	no	—
118	C₂²×D₄	—	7	2	no	—
119	C₂²×D₄	C₃×D₇	1	1	yes	characteristic, complemented
120	C₂³×C₄	—	7	1	no	—
121	#121	—	7	8	no	—
122	C₂×C₄⋊C₄	—	7	1	no	—
123	C₂×C₂²⋊C₄	—	7	2	no	—
124	C₂²×C₇	—	2	4	no	—
125	C₂²×C₇	—	2	4	no	—
126	C₂²×C₇	—	2	4	no	—
127	C₂²×C₇	—	4	8	no	—
128	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
129	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
130	C₂²×C₇	C₃×#41	1	2	yes	—
131	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
132	C₂×D₇	—	2	8	no	—
133	C₂×D₇	—	2	2	no	—
134	C₂×D₇	—	2	2	no	—
135	C₂×D₇	C₂×C₃×D₄	1	4	yes	complemented
136	C₄×C₇	C₂×C₃×D₄	1	4	yes	complemented
137	Dic₇	—	2	2	no	—
138	Dic₇	—	2	4	no	—
139	C₂³×C₃	—	2	4	no	—
140	C₂³×C₃	—	2	4	no	—
141	C₂³×C₃	—	2	4	no	—
142	C₂³×C₃	—	7	4	no	—
143	C₂³×C₃	—	7	2	no	—
144	C₂³×C₃	—	14	2	no	—
145	C₂³×C₃	—	14	2	no	—
146	C₂³×C₃	C₂²×D₇	1	4	yes	—
147	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
148	C₃×D₄	—	2	8	no	—
149	C₃×D₄	—	14	16	no	—
150	C₂×C₃×C₄	—	7	2	no	—
151	C₂×C₃×C₄	—	7	4	no	—
152	C₂×C₃×C₄	—	14	4	no	—
153	C₂×C₃×C₄	—	14	2	no	—
154	C₂×C₃×C₄	—	14	8	no	—
155	C₂×C₃×C₄	C₂²⋊D₇	1	4	yes	complemented
156	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
157	C₃×C₇	C₂×#121	1	1	yes	characteristic, complemented, Hall
158	C₂⁴	—	7	1	no	—
159	C₂⁴	C₂×C₃×D₇	1	2	yes	—
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	—	7	8	no	—
162	C₂×D₄	—	14	8	no	—
163	C₂×D₄	C₂×C₃×D₇	1	4	yes	complemented
164	C₂²×C₄	—	7	1	no	—
165	C₂²×C₄	—	7	2	no	—
166	C₂²×C₄	—	7	4	no	—
167	C₂²×C₄	—	14	4	no	—
168	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
169	C₄⋊C₄	—	7	4	no	—
170	C₂²⋊C₄	—	7	8	no	—
171	C₂×C₇	—	4	4	no	—
172	C₂×C₇	C₃×#121	1	4	yes	complemented
173	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
174	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
175	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
176	D₇	—	2	4	no	—
177	C₂²×C₃	—	2	4	no	—
178	C₂²×C₃	—	2	4	no	—
179	C₂²×C₃	—	2	4	no	—
180	C₂²×C₃	—	4	8	no	—
181	C₂²×C₃	—	7	4	no	—
182	C₂²×C₃	—	14	8	no	—
183	C₂²×C₃	—	14	2	no	—
184	C₂²×C₃	—	14	2	no	—
185	C₂²×C₃	D₄×D₇	1	2	yes	—
186	C₂²×C₃	#1159	1	2	yes	—
187	C₂²×C₃	C₂×C₂²⋊D₇	1	2	yes	—
188	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
189	C₃×C₄	—	14	4	no	—
190	C₃×C₄	—	14	2	no	—
191	C₃×C₄	C₂×C₂²⋊D₇	1	4	yes	complemented
192	C₂³	—	2	4	no	—
193	C₂³	—	2	4	no	—
194	C₂³	—	2	4	no	—
195	C₂³	—	7	4	no	—
196	C₂³	—	7	2	no	—
197	C₂³	—	14	2	no	—
198	C₂³	—	14	2	no	—
199	C₂³	C₂²×C₃×D₇	1	4	yes	—
200	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
201	D₄	—	2	8	no	—
202	D₄	—	14	16	no	—
203	C₂×C₄	—	7	2	no	—
204	C₂×C₄	—	7	4	no	—
205	C₂×C₄	—	14	4	no	—
206	C₂×C₄	—	14	2	no	—
207	C₂×C₄	—	14	8	no	—
208	C₂×C₄	C₃×C₂²⋊D₇	1	4	yes	complemented
209	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
210	C₇	C₂×C₃×#121	1	1	yes	characteristic, complemented, Sylow
211	C₂×C₃	—	4	4	no	—
212	C₂×C₃	—	14	4	no	—
213	C₂×C₃	#6501	1	4	yes	direct product factor
214	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
215	C₂×C₃	C₂×#1159	1	1	yes	characteristic
216	C₂×C₃	C₂²×C₂²⋊D₇	1	1	yes	characteristic
217	C₂²	—	2	4	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	4	8	no	—
221	C₂²	—	7	4	no	—
222	C₂²	—	14	8	no	—
223	C₂²	—	14	2	no	—
224	C₂²	—	14	2	no	—
225	C₂²	C₃×D₄×D₇	1	2	yes	—
226	C₂²	C₃×#1159	1	2	yes	—
227	C₂²	C₂×C₃×C₂²⋊D₇	1	2	yes	—
228	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
229	C₄	—	14	4	no	—
230	C₄	—	14	2	no	—
231	C₄	C₂×C₃×C₂²⋊D₇	1	4	yes	complemented
232	C₃	C₂×#6501	1	1	yes	characteristic, direct product factor, Sylow
233	C₂	—	4	4	no	—
234	C₂	—	14	4	no	—
235	C₂	C₃×#6501	1	4	yes	direct product factor
236	C₂	C₂×C₃×D₄×D₇	1	1	yes	characteristic
237	C₂	C₂×C₃×#1159	1	1	yes	characteristic
238	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×#121
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. C₂²×C₃×C₇×D₄	C₂	semidirect
2. C₂²×C₃×C₂²⋊D₇	C₂	semidirect
3. C₂²×C₃×C₄×D₇	C₂	semidirect
4. C₂×C₃×C₂²⋊Dic₇	C₂	semidirect
5. C₃×#6501	C₂	direct (N × H)
6. C₂×C₃×C₄⋊Dic₇	C₂	semidirect
7. C₂×#6501	C₃	direct (N × H)
11. C₂×C₃×C₇×D₄	C₂²	semidirect
14. C₂×C₃×C₂²⋊D₇	C₂²	semidirect
18. C₂×C₃×C₄×D₇	C₂²	semidirect
19. C₃×C₂²⋊Dic₇	C₂²	semidirect
20. C₃×C₄⋊Dic₇	C₂²	semidirect
21. C₂²×C₇×D₄	C₂×C₃	semidirect
22. C₂²×C₂²⋊D₇	C₂×C₃	semidirect
23. C₂²×C₄×D₇	C₂×C₃	semidirect
24. C₂×C₂²⋊Dic₇	C₂×C₃	semidirect
25. #6501	C₂×C₃	direct (N × H)
26. C₂×C₄⋊Dic₇	C₂×C₃	semidirect
35. C₂²×C₃×D₇	D₄	semidirect
38. C₂×C₃×C₄×C₇	D₄	semidirect
49. C₂×C₇×D₄	C₂²×C₃	semidirect
52. C₂×C₂²⋊D₇	C₂²×C₃	semidirect
56. C₂×C₄×D₇	C₂²×C₃	semidirect
57. C₂²⋊Dic₇	C₂²×C₃	semidirect
58. C₄⋊Dic₇	C₂²×C₃	semidirect
60. C₂²×C₃×D₄	D₇	semidirect
76. C₂×C₃×D₇	C₂×D₄	semidirect
77. C₃×C₄×C₇	C₂×D₄	semidirect
88. C₂²×D₇	C₃×D₄	semidirect
91. C₂×C₄×C₇	C₃×D₄	semidirect
104. C₂×C₃×D₄	C₂×D₇	semidirect
113. C₂×C₃×C₇	#121	semidirect
119. C₂²×D₄	C₃×D₇	semidirect
135. C₂×D₇	C₂×C₃×D₄	semidirect
136. C₄×C₇	C₂×C₃×D₄	semidirect
155. C₂×C₃×C₄	C₂²⋊D₇	semidirect
157. C₃×C₇	C₂×#121	semidirect
163. C₂×D₄	C₂×C₃×D₇	semidirect
172. C₂×C₇	C₃×#121	semidirect
191. C₃×C₄	C₂×C₂²⋊D₇	semidirect
208. C₂×C₄	C₃×C₂²⋊D₇	semidirect
210. C₇	C₂×C₃×#121	semidirect
213. C₂×C₃	#6501	direct (N × H)
231. C₄	C₂×C₃×C₂²⋊D₇	semidirect
232. C₃	C₂×#6501	direct (N × H)
235. C₂	C₃×#6501	direct (N × H)

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×C₃×C₂²⋊D₇	2
C₂²×C₃×C₄×D₇	2
C₂×C₃×C₂²⋊Dic₇	2
C₃×#6501	2
C₂×C₃×C₄⋊Dic₇	2
C₂×#6501	3
C₂×C₃×#121	7

Subgroups — C2×C3×#6501 #13061144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#6501 #13061144