Subgroups — C₇×C₈×C₉×Dic₃ #72219185

Order 6048.

308 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×C₈×Dic₃	C₃	1	1	yes	maximal, characteristic
2	C₄×C₇×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×C₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₇×C₈×C₉	C₂	1	1	yes	maximal, characteristic
5	C₄×C₇×C₈×C₉	—	3	1	no	maximal
6	C₂×C₃×C₄×C₇×C₉	C₂²	1	1	yes	characteristic
7	C₂×C₇×C₉×Dic₃	C₄	1	2	yes	—
8	C₃×C₇×C₈×C₉	C₄	1	2	yes	complemented
9	C₇×C₉×C₃⋊C₈	C₄	1	2	yes	complemented
10	C₂×C₃²×C₇×C₈	C₂×C₃	1	1	yes	characteristic
11	C₃×C₄×C₇×Dic₃	C₂×C₃	1	1	yes	characteristic
12	C₂×C₃×C₇×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
13	C₂×C₇×C₈×C₉	—	2	1	no	—
14	C₂×C₇×C₈×C₉	—	3	1	no	—
15	C₂×C₇×C₈×C₉	D₃	1	1	yes	characteristic
16	C₄²×C₇×C₉	—	3	1	no	—
17	C₈×C₉×Dic₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂²×C₃×C₇×C₉	C₂×C₄	1	1	yes	characteristic
19	C₃×C₄×C₇×C₉	C₂×C₄	1	1	yes	characteristic
20	C₃×C₄×C₇×C₉	C₂×C₄	1	1	yes	characteristic
21	C₇×C₉×Dic₃	C₈	1	4	yes	direct product factor
22	C₃×C₄×C₇×C₈	—	3	1	no	—
23	C₇×C₈×Dic₃	C₉	1	1	yes	characteristic, direct product factor
24	C₂×C₃²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
25	C₂×C₃×C₇×Dic₃	C₃×C₄	1	2	yes	—
26	C₃²×C₇×C₈	C₃×C₄	1	2	yes	—
27	C₃×C₇×C₃⋊C₈	C₃×C₄	1	2	yes	—
28	C₂×C₄×C₇×C₉	—	2	1	no	—
29	C₂×C₄×C₇×C₉	—	3	2	no	—
30	C₂×C₄×C₇×C₉	C₂×D₃	1	1	yes	characteristic
31	C₇×C₈×C₉	—	2	2	no	—
32	C₇×C₈×C₉	—	3	2	no	—
33	C₇×C₈×C₉	Dic₃	1	2	yes	direct product factor
34	C₂×C₃×C₈×C₉	C₂×C₇	1	1	yes	characteristic
35	C₄×C₉×Dic₃	C₂×C₇	1	1	yes	characteristic
36	C₂×C₉×C₃⋊C₈	C₂×C₇	1	1	yes	characteristic
37	C₂×C₃×C₇×C₉	C₄²	1	1	yes	characteristic
38	C₂×C₃×C₇×C₉	C₂×C₈	1	2	yes	—
39	C₂×C₃×C₇×C₈	—	2	1	no	—
40	C₂×C₃×C₇×C₈	—	3	1	no	—
41	C₂×C₃×C₇×C₈	C₂×C₉	1	1	yes	characteristic
42	C₂×C₃×C₇×C₈	C₃×D₃	1	1	yes	characteristic
43	C₃×C₄²×C₇	—	3	1	no	—
44	C₄×C₇×Dic₃	C₂×C₉	1	1	yes	characteristic
45	C₂×C₇×C₃⋊C₈	C₂×C₉	1	1	yes	characteristic
46	C₃×C₈×Dic₃	C₃×C₇	1	1	yes	characteristic
47	C₄×C₈×C₉	—	3	1	no	—
48	C₂²×C₃²×C₇	C₂×C₃×C₄	1	1	yes	characteristic
49	C₃²×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
50	C₃²×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
51	C₃×C₇×Dic₃	C₃×C₈	1	4	yes	—
52	C₂²×C₇×C₉	—	2	1	no	—
53	C₂²×C₇×C₉	C₄×D₃	1	1	yes	characteristic
54	C₄×C₇×C₉	—	2	1	no	—
55	C₄×C₇×C₉	—	2	1	no	—
56	C₄×C₇×C₉	—	3	4	no	—
57	C₄×C₇×C₉	C₄×D₃	1	1	yes	characteristic
58	C₄×C₇×C₉	C₂×Dic₃	1	1	yes	characteristic
59	C₄×C₇×C₈	—	3	1	no	Hall
60	C₂×C₃×C₄×C₉	C₂²×C₇	1	1	yes	characteristic
61	C₂×C₉×Dic₃	C₄×C₇	1	2	yes	—
62	C₃×C₈×C₉	C₄×C₇	1	2	yes	complemented
63	C₉×C₃⋊C₈	C₄×C₇	1	2	yes	complemented
64	C₃×C₇×C₉	C₄×C₈	1	1	yes	characteristic, complemented, Hall
65	C₂×C₃×C₄×C₇	—	2	1	no	—
66	C₂×C₃×C₄×C₇	—	3	2	no	—
67	C₂×C₃×C₄×C₇	C₂²×C₉	1	1	yes	characteristic
68	C₂×C₃×C₄×C₇	C₂×C₃×D₃	1	1	yes	characteristic
69	C₂×C₇×Dic₃	C₄×C₉	1	2	yes	—
70	C₃×C₇×C₈	—	2	2	no	—
71	C₃×C₇×C₈	—	3	2	no	—
72	C₃×C₇×C₈	C₄×C₉	1	2	yes	complemented
73	C₃×C₇×C₈	C₃×Dic₃	1	2	yes	—
74	C₇×C₃⋊C₈	C₄×C₉	1	2	yes	complemented
75	C₂×C₃²×C₈	C₂×C₃×C₇	1	1	yes	characteristic
76	C₃×C₄×Dic₃	C₂×C₃×C₇	1	1	yes	characteristic
77	C₂×C₃×C₃⋊C₈	C₂×C₃×C₇	1	1	yes	characteristic
78	C₂×C₈×C₉	—	2	1	no	—
79	C₂×C₈×C₉	—	3	1	no	—
80	C₂×C₈×C₉	D₃×C₇	1	1	yes	characteristic
81	C₄²×C₉	—	3	1	no	—
82	C₂×C₃²×C₇	C₃×C₄²	1	1	yes	characteristic
83	C₂×C₃²×C₇	C₂×C₃×C₈	1	2	yes	—
84	C₂×C₇×C₉	—	2	2	no	—
85	C₂×C₇×C₉	—	2	1	no	—
86	C₂×C₇×C₉	D₃×C₈	1	2	yes	—
87	C₂×C₇×C₉	C₄×Dic₃	1	1	yes	characteristic
88	C₂×C₇×C₈	—	3	1	no	—
89	C₂×C₇×C₈	D₃×C₉	1	1	yes	characteristic
90	C₄²×C₇	—	3	1	no	—
91	C₂²×C₃×C₉	C₂×C₄×C₇	1	1	yes	characteristic
92	C₃×C₄×C₉	C₂×C₄×C₇	1	1	yes	characteristic
93	C₃×C₄×C₉	C₂×C₄×C₇	1	1	yes	characteristic
94	C₉×Dic₃	C₇×C₈	1	4	yes	direct product factor
95	C₃×C₄×C₈	—	3	1	no	—
96	C₈×Dic₃	C₇×C₉	1	1	yes	characteristic, direct product factor
97	C₂²×C₃×C₇	—	2	1	no	—
98	C₂²×C₃×C₇	C₂×C₄×C₉	1	1	yes	characteristic
99	C₂²×C₃×C₇	C₃×C₄×D₃	1	1	yes	characteristic
100	C₃×C₄×C₇	—	2	1	no	—
101	C₃×C₄×C₇	—	2	1	no	—
102	C₃×C₄×C₇	—	3	4	no	—
103	C₃×C₄×C₇	C₂×C₄×C₉	1	1	yes	characteristic
104	C₃×C₄×C₇	C₂×C₄×C₉	1	1	yes	characteristic
105	C₃×C₄×C₇	C₃×C₄×D₃	1	1	yes	characteristic
106	C₃×C₄×C₇	C₂×C₃×Dic₃	1	1	yes	characteristic
107	C₇×Dic₃	C₈×C₉	1	4	yes	direct product factor
108	C₂×C₃²×C₄	C₂²×C₃×C₇	1	1	yes	characteristic
109	C₂×C₃×Dic₃	C₃×C₄×C₇	1	2	yes	—
110	C₃²×C₈	C₃×C₄×C₇	1	2	yes	—
111	C₃×C₃⋊C₈	C₃×C₄×C₇	1	2	yes	—
112	C₂×C₄×C₉	—	2	1	no	—
113	C₂×C₄×C₉	—	3	2	no	—
114	C₂×C₄×C₉	C₂×D₃×C₇	1	1	yes	characteristic
115	C₈×C₉	—	2	2	no	—
116	C₈×C₉	—	3	2	no	—
117	C₈×C₉	C₇×Dic₃	1	2	yes	direct product factor
118	C₃²×C₇	C₃×C₄×C₈	1	1	yes	characteristic
119	C₇×C₉	—	2	1	no	—
120	C₇×C₉	C₈×Dic₃	1	1	yes	characteristic, direct product factor
121	C₂×C₄×C₇	—	3	2	no	—
122	C₂×C₄×C₇	C₂×D₃×C₉	1	1	yes	characteristic
123	C₇×C₈	—	3	2	no	—
124	C₇×C₈	C₉×Dic₃	1	2	yes	direct product factor
125	C₂×C₃×C₉	C₄²×C₇	1	1	yes	characteristic
126	C₂×C₃×C₉	C₂×C₇×C₈	1	2	yes	—
127	C₂×C₃×C₈	—	2	1	no	—
128	C₂×C₃×C₈	—	3	1	no	—
129	C₂×C₃×C₈	C₂×C₇×C₉	1	1	yes	characteristic
130	C₂×C₃×C₈	C₃×D₃×C₇	1	1	yes	characteristic
131	C₃×C₄²	—	3	1	no	—
132	C₄×Dic₃	C₂×C₇×C₉	1	1	yes	characteristic
133	C₂×C₃⋊C₈	C₂×C₇×C₉	1	1	yes	characteristic
134	C₂×C₃×C₇	—	2	2	no	—
135	C₂×C₃×C₇	—	2	1	no	—
136	C₂×C₃×C₇	C₄²×C₉	1	1	yes	characteristic
137	C₂×C₃×C₇	C₂×C₈×C₉	1	2	yes	—
138	C₂×C₃×C₇	C₃×D₃×C₈	1	2	yes	—
139	C₂×C₃×C₇	C₃×C₄×Dic₃	1	1	yes	characteristic
140	C₂²×C₃²	C₂×C₃×C₄×C₇	1	1	yes	characteristic
141	C₃²×C₄	C₂×C₃×C₄×C₇	1	1	yes	characteristic
142	C₃²×C₄	C₂×C₃×C₄×C₇	1	1	yes	characteristic
143	C₃×Dic₃	C₃×C₇×C₈	1	4	yes	—
144	C₂²×C₉	—	2	1	no	—
145	C₂²×C₉	C₄×D₃×C₇	1	1	yes	characteristic
146	C₄×C₉	—	2	1	no	—
147	C₄×C₉	—	2	1	no	—
148	C₄×C₉	—	3	4	no	—
149	C₄×C₉	C₄×D₃×C₇	1	1	yes	characteristic
150	C₄×C₉	C₂×C₇×Dic₃	1	1	yes	characteristic
151	C₄×C₈	—	3	1	no	Sylow
152	C₂²×C₇	C₄×D₃×C₉	1	1	yes	characteristic
153	C₄×C₇	—	3	4	no	—
154	C₄×C₇	C₄×D₃×C₉	1	1	yes	characteristic
155	C₄×C₇	C₂×C₉×Dic₃	1	1	yes	characteristic
156	C₃×C₉	C₄×C₇×C₈	1	1	yes	characteristic, complemented, Sylow
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	3	2	no	—
159	C₂×C₃×C₄	C₂²×C₇×C₉	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂×C₃×D₃×C₇	1	1	yes	characteristic
161	C₂×Dic₃	C₄×C₇×C₉	1	2	yes	—
162	C₃×C₈	—	2	2	no	—
163	C₃×C₈	—	3	2	no	—
164	C₃×C₈	C₄×C₇×C₉	1	2	yes	complemented
165	C₃×C₈	C₃×C₇×Dic₃	1	2	yes	—
166	C₃⋊C₈	C₄×C₇×C₉	1	2	yes	complemented
167	C₃×C₇	—	2	1	no	—
168	C₃×C₇	C₄×C₈×C₉	1	1	yes	characteristic, complemented
169	C₃×C₇	C₃×C₈×Dic₃	1	1	yes	characteristic
170	C₂×C₃²	C₃×C₄²×C₇	1	1	yes	characteristic
171	C₂×C₃²	C₂×C₃×C₇×C₈	1	2	yes	—
172	C₂×C₉	—	2	2	no	—
173	C₂×C₉	—	2	1	no	—
174	C₂×C₉	D₃×C₇×C₈	1	2	yes	—
175	C₂×C₉	C₄×C₇×Dic₃	1	1	yes	characteristic
176	C₂×C₈	—	3	1	no	—
177	C₂×C₈	D₃×C₇×C₉	1	1	yes	characteristic
178	C₄²	—	3	1	no	—
179	C₂×C₇	D₃×C₈×C₉	1	2	yes	—
180	C₂×C₇	C₄×C₉×Dic₃	1	1	yes	characteristic
181	C₂²×C₃	—	2	1	no	—
182	C₂²×C₃	C₂×C₄×C₇×C₉	1	1	yes	characteristic
183	C₂²×C₃	C₃×C₄×D₃×C₇	1	1	yes	characteristic
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	3	4	no	—
187	C₃×C₄	C₂×C₄×C₇×C₉	1	1	yes	characteristic
188	C₃×C₄	C₂×C₄×C₇×C₉	1	1	yes	characteristic
189	C₃×C₄	C₃×C₄×D₃×C₇	1	1	yes	characteristic
190	C₃×C₄	C₂×C₃×C₇×Dic₃	1	1	yes	characteristic
191	Dic₃	C₇×C₈×C₉	1	4	yes	direct product factor
192	C₃²	C₃×C₄×C₇×C₈	1	1	yes	characteristic
193	C₉	—	2	1	no	—
194	C₉	C₇×C₈×Dic₃	1	1	yes	characteristic, direct product factor
195	C₂×C₄	—	3	2	no	—
196	C₂×C₄	C₂×D₃×C₇×C₉	1	1	yes	characteristic
197	C₈	—	3	2	no	—
198	C₈	C₇×C₉×Dic₃	1	2	yes	direct product factor
199	C₇	C₈×C₉×Dic₃	1	1	yes	characteristic, direct product factor, Sylow
200	C₂×C₃	—	2	1	no	—
201	C₂×C₃	—	2	2	no	—
202	C₂×C₃	C₄²×C₇×C₉	1	1	yes	characteristic
203	C₂×C₃	C₂×C₇×C₈×C₉	1	2	yes	—
204	C₂×C₃	C₃×D₃×C₇×C₈	1	2	yes	—
205	C₂×C₃	C₃×C₄×C₇×Dic₃	1	1	yes	characteristic
206	C₂²	C₄×D₃×C₇×C₉	1	1	yes	characteristic
207	C₄	—	3	4	no	—
208	C₄	C₄×D₃×C₇×C₉	1	1	yes	characteristic
209	C₄	C₂×C₇×C₉×Dic₃	1	1	yes	characteristic
210	C₃	—	2	1	no	—
211	C₃	C₄×C₇×C₈×C₉	1	1	yes	characteristic, complemented
212	C₃	C₃×C₇×C₈×Dic₃	1	1	yes	characteristic
213	C₂	C₄×C₇×C₉×Dic₃	1	1	yes	characteristic
214	C₂	D₃×C₇×C₈×C₉	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₇×C₈×C₉	D₃
commutator, hypocenter	C₃	C₄×C₇×C₈×C₉
Frattini	C₂×C₃×C₄	C₂×C₃×D₃×C₇
Fitting	C₂×C₃×C₇×C₈×C₉	C₂
socle, prime radical	C₂²×C₃²×C₇	C₂×C₃×C₄
cosocle	C₂×C₃²×C₄	C₂²×C₃×C₇
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₇×C₈×C₉

Complements

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

N	H ≅ G/N	type
8. C₃×C₇×C₈×C₉	C₄	semidirect
9. C₇×C₉×C₃⋊C₈	C₄	semidirect
17. C₈×C₉×Dic₃	C₇	direct (N × H)
21. C₇×C₉×Dic₃	C₈	direct (N × H)
23. C₇×C₈×Dic₃	C₉	direct (N × H)
33. C₇×C₈×C₉	Dic₃	direct (N × H)
62. C₃×C₈×C₉	C₄×C₇	semidirect
63. C₉×C₃⋊C₈	C₄×C₇	semidirect
64. C₃×C₇×C₉	C₄×C₈	semidirect
72. C₃×C₇×C₈	C₄×C₉	semidirect
74. C₇×C₃⋊C₈	C₄×C₉	semidirect
94. C₉×Dic₃	C₇×C₈	direct (N × H)
96. C₈×Dic₃	C₇×C₉	direct (N × H)
107. C₇×Dic₃	C₈×C₉	direct (N × H)
117. C₈×C₉	C₇×Dic₃	direct (N × H)
120. C₇×C₉	C₈×Dic₃	direct (N × H)
124. C₇×C₈	C₉×Dic₃	direct (N × H)
156. C₃×C₉	C₄×C₇×C₈	semidirect
164. C₃×C₈	C₄×C₇×C₉	semidirect
166. C₃⋊C₈	C₄×C₇×C₉	semidirect
168. C₃×C₇	C₄×C₈×C₉	semidirect
191. Dic₃	C₇×C₈×C₉	direct (N × H)
194. C₉	C₇×C₈×Dic₃	direct (N × H)
198. C₈	C₇×C₉×Dic₃	direct (N × H)
199. C₇	C₈×C₉×Dic₃	direct (N × H)
211. C₃	C₄×C₇×C₈×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₇×C₈×Dic₃	3
C₄×C₇×C₉×Dic₃	2
C₂×C₇×C₉×C₃⋊C₈	2
C₂×C₃×C₇×C₈×C₉	2
C₄×C₇×C₈×C₉	3
C₈×C₉×Dic₃	7

Subgroups — C7×C8×C9×Dic3 #72219185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×C₈×C₉×Dic₃ #72219185