Subgroups — C₃×#69301 #11797590

Order 1152.

311 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4907	C₂	1	2	yes	maximal, complemented
2	C₃×#4700	C₂	1	1	yes	maximal, characteristic
3	C₃²×#364	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4515	C₂	1	2	yes	maximal, complemented
5	C₃×#4461	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2171	—	3	1	no	maximal
7	#69301	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₃×#904	C₄	1	1	yes	characteristic
9	C₃×#901	C₂²	1	2	yes	—
10	C₃²×M₃₂	C₂²	1	2	yes	—
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₃×#819	C₄	1	1	yes	characteristic
13	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₃×#805	C₄	1	2	yes	—
16	C₃×#798	—	2	2	no	—
17	C₃×D₃×C₁₆	—	2	2	no	—
18	C₃×#504	—	3	2	no	—
19	C₃×#443	—	3	1	no	—
20	#4907	C₂×C₃	1	2	yes	complemented
21	#4700	C₂×C₃	1	1	yes	characteristic
22	C₃×#364	—	2	1	no	—
23	C₃×#364	D₃	1	1	yes	characteristic, complemented
24	C₃×#364	C₂×C₃	1	1	yes	characteristic, complemented
25	C₃×#350	—	3	2	no	—
26	C₃×#346	—	3	1	no	—
27	#4515	C₂×C₃	1	2	yes	complemented
28	#4461	C₂×C₃	1	1	yes	characteristic, complemented
29	C₃×#235	C₂×C₄	1	2	yes	—
30	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
32	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
33	C₃×#208	C₂×C₄	1	2	yes	—
34	C₃×Dic₁₂	C₈	1	1	yes	characteristic
35	C₃×D₂₄	C₈	1	1	yes	characteristic
36	C₃×#204	C₈	1	2	yes	—
37	C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
38	C₃×D₃×C₈	—	2	2	no	—
39	C₃²×C₁₆	—	2	2	no	—
40	C₃×C₃⋊C₁₆	—	2	1	no	—
41	C₃×C₃⋊C₁₆	D₄	1	2	yes	—
42	#2171	—	3	1	no	Sylow
43	C₃×#135	—	3	1	no	—
44	C₃×#131	—	3	2	no	—
45	#904	C₃×C₄	1	1	yes	characteristic
46	#901	C₂²×C₃	1	2	yes	—
47	C₃×M₃₂	—	2	2	no	—
48	C₃×M₃₂	—	6	2	no	—
49	C₃×M₃₂	C₂×D₃	1	2	yes	—
50	C₃×M₃₂	C₂²×C₃	1	2	yes	—
51	C₂×C₃×C₁₆	—	3	1	no	—
52	C₂×C₃×C₁₆	—	3	1	no	—
53	C₂×C₃×C₁₆	—	6	2	no	—
54	C₃×#108	—	3	1	no	—
55	C₃×#104	—	3	2	no	—
56	C₃×C₄×C₈	—	2	1	no	—
57	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
58	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
59	#819	C₃×C₄	1	1	yes	characteristic
60	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
62	#805	C₃×C₄	1	2	yes	—
63	#798	—	2	2	no	—
64	D₃×C₁₆	—	2	2	no	—
65	C₂×C₃²×C₄	—	2	1	no	—
66	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
67	C₃×#67	—	2	2	no	—
68	C₃×D₁₂	C₂×C₈	1	2	yes	—
69	C₃×C₄×D₃	—	2	2	no	—
70	C₃×Dic₆	C₂×C₈	1	2	yes	—
71	C₃²×C₈	C₂×C₈	1	1	yes	characteristic
72	C₃²×C₈	C₂×C₈	1	1	yes	characteristic
73	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
74	C₃²×C₈	#41	1	1	yes	characteristic
75	C₃×C₃⋊C₈	—	2	2	no	—
76	#504	—	3	2	no	—
77	#443	—	3	1	no	—
78	#364	C₃×D₃	1	1	yes	characteristic, complemented
79	#350	—	3	2	no	—
80	#346	—	3	1	no	—
81	C₃×#41	—	3	2	no	—
82	#235	C₂×C₃×C₄	1	2	yes	—
83	C₃×Q₁₆	—	3	1	no	—
84	C₃×SD₁₆	—	3	2	no	—
85	C₃×D₈	—	3	1	no	—
86	C₃×M₁₆	—	3	2	no	—
87	C₃×M₁₆	—	6	2	no	—
88	C₂×C₃×C₈	—	2	1	no	—
89	C₂×C₃×C₈	—	2	1	no	—
90	C₂×C₃×C₈	—	6	2	no	—
91	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
92	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
93	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
94	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
95	C₃×C₄²	—	2	1	no	—
96	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
97	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
98	#208	C₂×C₃×C₄	1	2	yes	—
99	Dic₁₂	C₃×C₈	1	1	yes	characteristic
100	D₂₄	C₃×C₈	1	1	yes	characteristic
101	#204	C₃×C₈	1	2	yes	—
102	ΓL(1,𝔽₂₅)	—	2	2	no	—
103	D₃×C₈	—	2	2	no	—
104	C₃×C₁₆	—	2	2	no	—
105	C₃×C₁₆	—	2	2	no	—
106	C₃×C₁₆	—	3	2	no	—
107	C₃×C₁₆	—	4	2	no	—
108	C₃×C₁₆	—	6	1	no	—
109	C₃⋊C₁₆	—	2	1	no	—
110	C₃⋊C₁₆	C₃×D₄	1	2	yes	—
111	C₂²×C₃²	#131	1	1	yes	characteristic
112	C₂×C₃×D₃	—	2	2	no	—
113	C₃²×C₄	—	4	1	no	—
114	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
115	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
116	C₃×Dic₃	—	2	2	no	—
117	#135	—	3	1	no	—
118	#131	—	3	2	no	—
119	M₃₂	—	6	2	no	—
120	M₃₂	C₂×C₃×D₃	1	2	yes	—
121	C₂×C₁₆	—	3	1	no	—
122	C₂×C₁₆	—	3	1	no	—
123	C₂×C₁₆	—	6	2	no	—
124	#108	—	3	1	no	—
125	#104	—	3	2	no	—
126	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
127	C₃×Q₈	—	3	2	no	—
128	C₃×D₄	—	3	2	no	—
129	C₃×D₄	—	6	2	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	4	1	no	—
134	C₂×C₃×C₄	—	6	2	no	—
135	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
137	#67	—	2	2	no	—
138	D₁₂	C₂×C₃×C₈	1	2	yes	—
139	C₄×D₃	—	2	2	no	—
140	Dic₆	C₂×C₃×C₈	1	2	yes	—
141	C₃×C₈	—	2	1	no	—
142	C₃×C₈	—	2	1	no	—
143	C₃×C₈	—	2	1	no	—
144	C₃×C₈	—	2	1	no	—
145	C₃×C₈	—	6	2	no	—
146	C₃×C₈	D₃×C₈	1	1	yes	characteristic
147	C₃×C₈	D₃×C₈	1	1	yes	characteristic
148	C₃×C₈	C₂×C₃×C₈	1	1	yes	characteristic
149	C₃×C₈	C₂×C₃×C₈	1	1	yes	characteristic
150	C₃×C₈	D₃×D₄	1	1	yes	characteristic
151	C₃×C₈	#239	1	1	yes	characteristic
152	C₃×C₈	C₂×C₃×D₄	1	1	yes	characteristic
153	C₃×C₈	C₃×#41	1	1	yes	characteristic
154	C₃⋊C₈	—	2	2	no	—
155	C₂×C₃²	—	2	1	no	—
156	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
157	C₃×D₃	—	4	2	no	—
158	#41	—	3	2	no	—
159	Q₁₆	—	3	1	no	—
160	SD₁₆	—	3	2	no	—
161	D₈	—	3	1	no	—
162	M₁₆	—	3	2	no	—
163	M₁₆	—	6	2	no	—
164	C₂×C₈	—	6	2	no	—
165	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
166	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
167	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
168	C₁₆	—	2	2	no	—
169	C₁₆	—	3	2	no	—
170	C₁₆	—	6	1	no	—
171	C₂²×C₃	—	2	1	no	—
172	C₂²×C₃	—	6	2	no	—
173	C₂²×C₃	#907	1	1	yes	characteristic
174	C₂²×C₃	C₃×#131	1	1	yes	characteristic
175	C₂×D₃	—	2	2	no	—
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	—	2	1	no	—
178	C₃×C₄	—	4	1	no	—
179	C₃×C₄	—	4	1	no	—
180	C₃×C₄	—	4	2	no	—
181	C₃×C₄	—	6	2	no	—
182	C₃×C₄	#893	1	1	yes	characteristic
183	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
184	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
185	C₃×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
186	Dic₃	—	2	2	no	—
187	C₃²	#2171	1	1	yes	characteristic, complemented, Sylow
188	Q₈	—	3	2	no	—
189	D₄	—	3	2	no	—
190	D₄	—	6	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	6	2	no	—
193	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
194	C₈	—	6	2	no	—
195	C₈	C₃×D₃×C₈	1	1	yes	characteristic
196	C₈	C₃×D₃×C₈	1	1	yes	characteristic
197	C₈	C₃×D₃×D₄	1	1	yes	characteristic
198	C₈	C₃×#239	1	1	yes	characteristic
199	C₂×C₃	—	2	1	no	—
200	C₂×C₃	—	2	1	no	—
201	C₂×C₃	—	2	1	no	—
202	C₂×C₃	—	4	1	no	—
203	C₂×C₃	—	12	2	no	—
204	C₂×C₃	#4834	1	1	yes	characteristic
205	C₂×C₃	C₃×C₈×D₄	1	1	yes	characteristic
206	D₃	—	4	2	no	—
207	C₂²	—	6	2	no	—
208	C₂²	C₃×#907	1	1	yes	characteristic
209	C₄	—	4	1	no	—
210	C₄	—	6	2	no	—
211	C₄	C₃×#893	1	1	yes	characteristic
212	C₄	C₂×C₃×D₃×C₈	1	1	yes	characteristic
213	C₃	—	2	1	no	—
214	C₃	#69301	1	1	yes	characteristic, direct product factor
215	C₃	C₃×#2171	1	1	yes	characteristic, complemented
216	C₂	—	2	1	no	—
217	C₂	—	12	2	no	—
218	C₂	C₃×#4834	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	D₃×D₄
commutator	C₃×C₄	C₂²×C₃×C₈
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₃²×#364	C₂
socle	C₂×C₃²	C₈×D₄
prime radical	C₃×#904	C₄
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₃×#364	D₃
hypocenter	C₃	C₃×#2171
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₃×C₈ ◁ C₃×#364

Complements

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

N	H ≅ G/N	type
1. C₃×#4907	C₂	semidirect
3. C₃²×#364	C₂	semidirect
4. C₃×#4515	C₂	semidirect
5. C₃×#4461	C₂	semidirect
7. #69301	C₃	direct (N × H)
20. #4907	C₂×C₃	semidirect
23. C₃×#364	D₃	semidirect
24. C₃×#364	C₂×C₃	semidirect
27. #4515	C₂×C₃	semidirect
28. #4461	C₂×C₃	semidirect
78. #364	C₃×D₃	semidirect
187. C₃²	#2171	semidirect
214. C₃	#69301	direct (N × H)
215. C₃	C₃×#2171	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4907	2
C₃×#4700	2
C₃²×#364	2
C₃×#4515	2
C₃×#4461	2
C₃×#2171	3
#69301	3

Subgroups — C3×#69301 #11797590

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69301 #11797590