Subgroups — C₂×C₄×Dic₃×D₇ #13059461

Order 1344.

996 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×D₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₃×D₇	C₂	1	2	yes	maximal
3	C₂×C₄×Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₇×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Dic₃×Dic₇	C₂	1	2	yes	maximal, complemented
6	C₄×Dic₃×D₇	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₄²×D₇	—	3	1	no	maximal, Hall
8	C₂³×C₃×D₇	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂²×Dic₂₁	C₂²	1	2	yes	—
11	C₂²×C₇×Dic₃	C₂²	1	2	yes	—
12	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄×D₇	C₄	1	8	yes	complemented
14	C₂×C₃×C₄×D₇	C₂²	1	4	yes	—
15	C₂×Dic₃×D₇	C₄	1	16	yes	direct product factor
16	C₂×Dic₃×D₇	C₂²	1	8	yes	—
17	C₄×Dic₂₁	C₂²	1	4	yes	complemented
18	C₄×C₇×Dic₃	C₂²	1	4	yes	complemented
19	Dic₃×Dic₇	C₂²	1	8	yes	complemented
20	C₂²×C₄×D₇	—	3	2	no	—
21	C₂²×C₄×D₇	D₃	1	1	yes	characteristic
22	C₂×C₄²×C₇	—	3	1	no	—
23	C₂×C₄×Dic₇	—	3	1	no	—
24	C₂×C₄×Dic₇	—	3	2	no	—
25	C₄²×D₇	—	3	8	no	—
26	C₂²×C₄×Dic₃	—	7	1	no	maximal, Hall
27	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
28	C₂²×C₃×D₇	C₂×C₄	1	8	yes	—
29	C₂²×C₃×D₇	C₂×C₄	1	4	yes	—
30	C₂²×C₃×D₇	C₂³	1	2	yes	—
31	C₂×C₃×C₄×C₇	C₂×C₄	1	4	yes	complemented
32	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
33	C₂×Dic₂₁	C₂×C₄	1	8	yes	complemented
34	C₂×Dic₂₁	C₂³	1	4	yes	—
35	C₂×C₇×Dic₃	C₂×C₄	1	8	yes	complemented
36	C₂×C₇×Dic₃	C₂³	1	4	yes	—
37	C₂×C₃×Dic₇	C₂×C₄	1	4	yes	complemented
38	C₂×C₃×Dic₇	C₂³	1	2	yes	—
39	C₃×C₄×D₇	C₂×C₄	1	16	yes	complemented
40	Dic₃×D₇	C₂×C₄	1	32	yes	direct product factor
41	C₂³×D₇	C₂×D₃	1	1	yes	characteristic
42	C₂²×C₄×C₇	—	3	2	no	—
43	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
44	C₂²×Dic₇	—	3	2	no	—
45	C₂²×Dic₇	C₂×D₃	1	1	yes	characteristic
46	C₂×C₄×D₇	—	3	16	no	—
47	C₂×C₄×D₇	—	3	8	no	—
48	C₂×C₄×D₇	Dic₃	1	8	yes	direct product factor
49	C₂×C₄×D₇	C₂×D₃	1	4	yes	—
50	C₄²×C₇	—	3	4	no	—
51	C₄×Dic₇	—	3	4	no	—
52	C₄×Dic₇	—	3	8	no	—
53	C₂³×C₃×C₄	—	7	1	no	—
54	C₂³×Dic₃	—	7	2	no	—
55	C₂×C₄×Dic₃	—	7	1	no	—
56	C₂×C₄×Dic₃	—	7	2	no	—
57	C₂×C₄×Dic₃	—	7	8	no	—
58	C₂×C₄×Dic₃	D₇	1	1	yes	characteristic, direct product factor
59	C₂²×C₃×C₇	C₂²×C₄	1	4	yes	—
60	C₂²×C₃×C₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
62	C₂×C₃×D₇	C₄²	1	16	yes	complemented
63	C₂×C₃×D₇	C₂²×C₄	1	8	yes	—
64	C₂×C₃×D₇	C₂²×C₄	1	4	yes	—
65	C₃×C₄×C₇	C₂²×C₄	1	4	yes	complemented
66	Dic₂₁	C₂²×C₄	1	8	yes	complemented
67	C₃×Dic₇	C₂²×C₄	1	4	yes	complemented
68	C₇×Dic₃	C₂²×C₄	1	8	yes	complemented
69	C₂²×C₄²	—	21	1	no	Sylow
70	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
71	C₂²×D₇	C₄×D₃	1	8	yes	—
72	C₂²×D₇	C₂×Dic₃	1	4	yes	—
73	C₂²×D₇	C₂²×D₃	1	2	yes	—
74	C₂×C₄×C₇	—	3	8	no	—
75	C₂×C₄×C₇	—	3	4	no	—
76	C₂×C₄×C₇	C₂×Dic₃	1	4	yes	complemented
77	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
78	C₂×Dic₇	—	3	8	no	—
79	C₂×Dic₇	—	3	4	no	—
80	C₂×Dic₇	C₂×Dic₃	1	4	yes	complemented
81	C₂×Dic₇	C₂²×D₃	1	2	yes	—
82	C₄×D₇	—	3	32	no	—
83	C₄×D₇	C₂×Dic₃	1	16	yes	direct product factor
84	C₂⁴×C₃	—	7	1	no	—
85	C₂²×C₃×C₄	—	7	1	no	—
86	C₂²×C₃×C₄	—	7	8	no	—
87	C₂²×C₃×C₄	—	7	4	no	—
88	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
89	C₂²×Dic₃	—	7	2	no	—
90	C₂²×Dic₃	—	7	16	no	—
91	C₂²×Dic₃	—	7	8	no	—
92	C₂²×Dic₃	C₂×D₇	1	2	yes	—
93	C₄×Dic₃	—	7	4	no	—
94	C₄×Dic₃	—	7	8	no	—
95	C₄×Dic₃	C₂×D₇	1	4	yes	direct product factor
96	C₂×C₃×C₇	C₂×C₄²	1	4	yes	complemented
97	C₂×C₃×C₇	C₂³×C₄	1	2	yes	—
98	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
99	C₃×D₇	C₂×C₄²	1	8	yes	complemented
100	C₂³×C₄	—	7	1	no	—
101	C₂³×C₄	—	21	2	no	—
102	C₂×C₄²	—	3	1	no	—
103	C₂×C₄²	—	21	8	no	—
104	C₂×C₄²	—	21	2	no	—
105	C₂×C₄²	—	21	1	no	—
106	C₂²×C₇	C₂×C₄×D₃	1	4	yes	—
107	C₂²×C₇	C₂²×Dic₃	1	2	yes	—
108	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
109	C₂×D₇	C₄×Dic₃	1	16	yes	direct product factor
110	C₂×D₇	C₂×C₄×D₃	1	8	yes	—
111	C₂×D₇	C₂²×Dic₃	1	4	yes	—
112	C₄×C₇	—	3	8	no	—
113	C₄×C₇	C₂²×Dic₃	1	4	yes	complemented
114	Dic₇	—	3	8	no	—
115	Dic₇	C₂²×Dic₃	1	4	yes	complemented
116	C₂³×C₃	—	7	8	no	—
117	C₂³×C₃	—	7	4	no	—
118	C₂³×C₃	—	7	2	no	—
119	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
120	C₂×C₃×C₄	—	7	4	no	—
121	C₂×C₃×C₄	—	7	2	no	—
122	C₂×C₃×C₄	—	7	16	no	—
123	C₂×C₃×C₄	C₄×D₇	1	4	yes	complemented
124	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
125	C₂×Dic₃	—	7	8	no	—
126	C₂×Dic₃	—	7	4	no	—
127	C₂×Dic₃	—	7	32	no	—
128	C₂×Dic₃	C₄×D₇	1	8	yes	direct product factor
129	C₂×Dic₃	C₂²×D₇	1	4	yes	—
130	C₃×C₇	C₂²×C₄²	1	1	yes	characteristic, complemented, Hall
131	C₂⁴	—	7	1	no	—
132	C₂²×C₄	—	3	2	no	—
133	C₂²×C₄	—	7	1	no	—
134	C₂²×C₄	—	7	8	no	—
135	C₂²×C₄	—	7	4	no	—
136	C₂²×C₄	—	21	8	no	—
137	C₂²×C₄	—	21	16	no	—
138	C₂²×C₄	—	21	2	no	—
139	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
140	C₄²	—	3	4	no	—
141	C₄²	—	21	8	no	—
142	C₄²	—	21	4	no	—
143	C₂×C₇	C₂×C₄×Dic₃	1	4	yes	complemented
144	C₂×C₇	C₂²×C₄×D₃	1	2	yes	—
145	C₂×C₇	C₂³×Dic₃	1	1	yes	characteristic
146	D₇	C₂×C₄×Dic₃	1	8	yes	direct product factor
147	C₂²×C₃	—	7	16	no	—
148	C₂²×C₃	—	7	8	no	—
149	C₂²×C₃	—	7	4	no	—
150	C₂²×C₃	C₂×C₄×D₇	1	4	yes	—
151	C₂²×C₃	C₂×C₄×D₇	1	2	yes	—
152	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
153	C₃×C₄	—	7	4	no	—
154	C₃×C₄	C₂×C₄×D₇	1	4	yes	complemented
155	Dic₃	—	7	8	no	—
156	Dic₃	C₂×C₄×D₇	1	8	yes	direct product factor
157	C₂³	—	7	8	no	—
158	C₂³	—	7	4	no	—
159	C₂³	—	7	2	no	—
160	C₂³	C₂×D₃×D₇	1	1	yes	characteristic
161	C₂×C₄	—	3	8	no	—
162	C₂×C₄	—	3	4	no	—
163	C₂×C₄	—	7	4	no	—
164	C₂×C₄	—	7	2	no	—
165	C₂×C₄	—	7	16	no	—
166	C₂×C₄	—	21	32	no	—
167	C₂×C₄	—	21	8	no	—
168	C₂×C₄	—	21	4	no	—
169	C₂×C₄	Dic₃×D₇	1	4	yes	direct product factor
170	C₂×C₄	C₂×D₃×D₇	1	2	yes	—
171	C₇	C₂²×C₄×Dic₃	1	1	yes	characteristic, complemented, Sylow
172	C₂×C₃	—	7	8	no	—
173	C₂×C₃	C₄²×D₇	1	4	yes	complemented
174	C₂×C₃	C₂²×C₄×D₇	1	2	yes	—
175	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
176	C₂²	—	7	16	no	—
177	C₂²	—	7	8	no	—
178	C₂²	—	7	4	no	—
179	C₂²	C₄×D₃×D₇	1	4	yes	—
180	C₂²	C₂×Dic₃×D₇	1	2	yes	—
181	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
182	C₄	—	3	8	no	—
183	C₄	—	7	4	no	—
184	C₄	—	21	8	no	—
185	C₄	C₂×Dic₃×D₇	1	4	yes	direct product factor
186	C₃	C₂×C₄²×D₇	1	1	yes	characteristic, complemented, Sylow
187	C₂	—	7	8	no	—
188	C₂	C₄×Dic₃×D₇	1	4	yes	direct product factor
189	C₂	C₂×C₄×D₃×D₇	1	2	yes	—
190	C₂	C₂²×Dic₃×D₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	D₃×D₇
commutator, hypocenter	C₃×C₇	C₂²×C₄²
Frattini	C₂²	C₂²×D₃×D₇
Fitting	C₂²×C₃×C₄×C₇	C₂²
socle	C₂³×C₃×C₇	C₂³
prime radical	C₂³×C₃×D₇	C₂²
cosocle	C₂²×C₃×C₇	C₂⁴
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×Dic₂₁	C₂	semidirect
4. C₂×C₄×C₇×Dic₃	C₂	semidirect
5. C₂×Dic₃×Dic₇	C₂	semidirect
6. C₄×Dic₃×D₇	C₂	direct (N × H)
13. C₂×C₃×C₄×D₇	C₄	semidirect
15. C₂×Dic₃×D₇	C₄	direct (N × H)
17. C₄×Dic₂₁	C₂²	semidirect
18. C₄×C₇×Dic₃	C₂²	semidirect
19. Dic₃×Dic₇	C₂²	semidirect
31. C₂×C₃×C₄×C₇	C₂×C₄	semidirect
33. C₂×Dic₂₁	C₂×C₄	semidirect
35. C₂×C₇×Dic₃	C₂×C₄	semidirect
37. C₂×C₃×Dic₇	C₂×C₄	semidirect
39. C₃×C₄×D₇	C₂×C₄	semidirect
40. Dic₃×D₇	C₂×C₄	direct (N × H)
48. C₂×C₄×D₇	Dic₃	direct (N × H)
58. C₂×C₄×Dic₃	D₇	direct (N × H)
62. C₂×C₃×D₇	C₄²	semidirect
65. C₃×C₄×C₇	C₂²×C₄	semidirect
66. Dic₂₁	C₂²×C₄	semidirect
67. C₃×Dic₇	C₂²×C₄	semidirect
68. C₇×Dic₃	C₂²×C₄	semidirect
76. C₂×C₄×C₇	C₂×Dic₃	semidirect
80. C₂×Dic₇	C₂×Dic₃	semidirect
83. C₄×D₇	C₂×Dic₃	direct (N × H)
95. C₄×Dic₃	C₂×D₇	direct (N × H)
96. C₂×C₃×C₇	C₂×C₄²	semidirect
99. C₃×D₇	C₂×C₄²	semidirect
109. C₂×D₇	C₄×Dic₃	direct (N × H)
113. C₄×C₇	C₂²×Dic₃	semidirect
115. Dic₇	C₂²×Dic₃	semidirect
123. C₂×C₃×C₄	C₄×D₇	semidirect
128. C₂×Dic₃	C₄×D₇	direct (N × H)
130. C₃×C₇	C₂²×C₄²	semidirect
143. C₂×C₇	C₂×C₄×Dic₃	semidirect
146. D₇	C₂×C₄×Dic₃	direct (N × H)
154. C₃×C₄	C₂×C₄×D₇	semidirect
156. Dic₃	C₂×C₄×D₇	direct (N × H)
169. C₂×C₄	Dic₃×D₇	direct (N × H)
171. C₇	C₂²×C₄×Dic₃	semidirect
173. C₂×C₃	C₄²×D₇	semidirect
185. C₄	C₂×Dic₃×D₇	direct (N × H)
186. C₃	C₂×C₄²×D₇	semidirect
188. C₂	C₄×Dic₃×D₇	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×C₄×D₇	2
C₂²×Dic₃×D₇	2
C₂×C₄×Dic₂₁	2
C₂×C₄×C₇×Dic₃	2
C₂×Dic₃×Dic₇	2
C₄×Dic₃×D₇	2
C₂×C₄²×D₇	3
C₂²×C₄×Dic₃	7

Subgroups — C2×C4×Dic3×D7 #13059461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×Dic₃×D₇ #13059461