Subgroups — C₃×#69806 #11783512

Order 1152.

553 conjugacy classes of subgroups.

The subgroup lattice has 199 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#97	C₂	1	1	yes	maximal, characteristic
2	C₃×#4932	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×#4550	C₂	1	4	yes	maximal
5	C₃×#1746	—	3	1	no	maximal
6	#69806	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃²×C₈	C₂²	1	2	yes	—
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄×Dic₃	C₂²	1	2	yes	—
10	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
11	C₃²×#97	C₄	1	4	yes	complemented
12	C₃×C₃⋊#95	C₄	1	4	yes	—
13	C₃×C₄×C₃⋊C₈	C₄	1	4	yes	complemented
14	C₂×C₃×#97	—	2	1	no	—
15	C₂×C₃×#97	D₃	1	1	yes	characteristic
16	C₂×C₃×#97	C₂×C₃	1	1	yes	characteristic
17	C₂×C₃×C₄×C₈	—	3	1	no	—
18	C₃×#376	—	3	1	no	—
19	#4932	C₂×C₃	1	1	yes	characteristic
20	C₂×C₄×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
21	C₃×#336	—	3	4	no	—
22	#4550	C₂×C₃	1	4	yes	—
23	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
24	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
25	C₂²×C₃×Dic₃	C₂×C₄	1	4	yes	—
26	C₂×C₃²×C₈	—	2	4	no	—
27	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
28	C₃²×C₄²	C₂×C₄	1	2	yes	—
29	C₃²×C₄²	C₂×C₄	1	2	yes	—
30	C₃×C₄×Dic₃	—	2	4	no	—
31	C₂×C₃×C₃⋊C₈	—	2	4	no	—
32	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
33	#1746	—	3	1	no	Sylow
34	C₂²×C₃×C₈	—	2	2	no	—
35	C₂²×C₃×C₈	—	3	2	no	—
36	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
37	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
38	C₂×C₃×C₄²	—	2	1	no	—
39	C₂×C₃×C₄²	—	3	2	no	—
40	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
41	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₄×Dic₃	C₂²×C₃	1	2	yes	—
43	C₂²×C₃⋊C₈	C₂²×C₃	1	2	yes	—
44	C₃×#97	—	2	4	no	—
45	C₃×#97	Dic₃	1	4	yes	complemented
46	C₃×#97	C₃×C₄	1	4	yes	complemented
47	C₃×C₄×C₈	—	3	4	no	—
48	C₃×#95	—	3	4	no	—
49	C₃⋊#95	C₃×C₄	1	4	yes	—
50	C₄×C₃⋊C₈	C₃×C₄	1	4	yes	complemented
51	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃²×C₄	—	2	4	no	—
53	C₂×C₃²×C₄	C₄²	1	4	yes	—
54	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₃²×C₄	#41	1	4	yes	—
57	C₂×C₃×Dic₃	—	2	4	no	—
58	C₂×C₃×Dic₃	—	2	8	no	—
59	C₃²×C₈	—	2	4	no	—
60	C₃×C₃⋊C₈	—	2	4	no	—
61	C₂×#97	C₃×D₃	1	1	yes	characteristic
62	C₂×C₄×C₈	—	3	1	no	—
63	#376	—	3	1	no	—
64	#336	—	3	4	no	—
65	C₂²×C₃×C₄	—	2	2	no	—
66	C₂²×C₃×C₄	—	2	1	no	—
67	C₂²×C₃×C₄	—	3	4	no	—
68	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
69	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
70	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
72	C₂²×Dic₃	C₂×C₃×C₄	1	4	yes	—
73	C₂×C₃×C₈	—	2	8	no	—
74	C₂×C₃×C₈	—	2	4	no	—
75	C₂×C₃×C₈	—	2	4	no	—
76	C₂×C₃×C₈	—	2	4	no	—
77	C₂×C₃×C₈	—	3	4	no	—
78	C₂×C₃×C₈	—	6	4	no	—
79	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
80	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
81	C₃×C₄²	—	2	2	no	—
82	C₃×C₄²	—	2	2	no	—
83	C₃×C₄²	—	6	4	no	—
84	C₃×C₄²	C₄×D₃	1	2	yes	—
85	C₃×C₄²	C₂×Dic₃	1	2	yes	—
86	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
87	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
88	C₄×Dic₃	—	2	4	no	—
89	C₂×C₃⋊C₈	—	2	4	no	—
90	C₂×C₃⋊C₈	C₂×C₃×C₄	1	4	yes	—
91	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
92	C₂²×C₃²	#117	1	2	yes	—
93	C₂²×C₃²	#131	1	4	yes	—
94	C₃²×C₄	—	2	4	no	—
95	C₃²×C₄	#117	1	2	yes	—
96	C₃²×C₄	#117	1	2	yes	—
97	C₃×Dic₃	—	4	4	no	—
98	C₂²×C₈	—	3	2	no	—
99	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
100	C₂×C₄²	—	3	2	no	—
101	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
102	#97	C₃×Dic₃	1	4	yes	complemented
103	C₄×C₈	—	3	4	no	—
104	#95	—	3	4	no	—
105	C₂³×C₃	—	2	1	no	—
106	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
107	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
108	C₂×C₃×C₄	—	2	4	no	—
109	C₂×C₃×C₄	—	2	4	no	—
110	C₂×C₃×C₄	—	2	4	no	—
111	C₂×C₃×C₄	—	2	1	no	—
112	C₂×C₃×C₄	—	2	4	no	—
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	—	4	4	no	—
115	C₂×C₃×C₄	—	6	4	no	—
116	C₂×C₃×C₄	—	6	8	no	—
117	C₂×C₃×C₄	C₄×Dic₃	1	4	yes	—
118	C₂×C₃×C₄	C₃×C₄²	1	4	yes	—
119	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
120	C₂×C₃×C₄	#235	1	4	yes	—
121	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
124	C₂×C₃×C₄	C₃×#41	1	4	yes	—
125	C₂×Dic₃	—	2	4	no	—
126	C₂×Dic₃	—	2	8	no	—
127	C₃×C₈	—	2	8	no	—
128	C₃×C₈	—	2	4	no	—
129	C₃×C₈	—	2	4	no	—
130	C₃×C₈	—	6	4	no	—
131	C₃⋊C₈	—	2	4	no	—
132	C₂×C₃²	#376	1	1	yes	characteristic
133	C₂×C₃²	#405	1	2	yes	—
134	C₂×C₃²	#433	1	4	yes	—
135	C₂²×C₄	—	3	4	no	—
136	C₂²×C₄	C₃×C₄×D₃	1	2	yes	—
137	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	3	4	no	—
140	C₂×C₈	—	6	4	no	—
141	C₂×C₈	C₂×C₃×Dic₃	1	4	yes	—
142	C₄²	—	6	4	no	—
143	C₄²	C₃×C₄×D₃	1	2	yes	—
144	C₄²	C₂×C₃×Dic₃	1	2	yes	—
145	C₂²×C₃	—	2	4	no	—
146	C₂²×C₃	—	2	1	no	—
147	C₂²×C₃	—	2	2	no	—
148	C₂²×C₃	#872	1	2	yes	—
149	C₂²×C₃	#907	1	4	yes	—
150	C₂²×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
151	C₂²×C₃	C₂×C₃×C₄²	1	1	yes	characteristic
152	C₂²×C₃	C₃×#117	1	2	yes	—
153	C₂²×C₃	C₃×#131	1	4	yes	—
154	C₃×C₄	—	2	4	no	—
155	C₃×C₄	—	2	4	no	—
156	C₃×C₄	—	2	2	no	—
157	C₃×C₄	—	2	2	no	—
158	C₃×C₄	—	4	4	no	—
159	C₃×C₄	—	12	4	no	—
160	C₃×C₄	#872	1	2	yes	—
161	C₃×C₄	#926	1	2	yes	—
162	C₃×C₄	C₃×#117	1	2	yes	—
163	C₃×C₄	C₃×#117	1	2	yes	—
164	Dic₃	—	4	4	no	—
165	C₃²	#1746	1	1	yes	characteristic, complemented, Sylow
166	C₂³	C₂×C₃×C₄×D₃	1	1	yes	characteristic
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	6	8	no	—
169	C₂×C₄	—	6	4	no	—
170	C₂×C₄	C₃×C₄×Dic₃	1	4	yes	—
171	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
172	C₂×C₄	C₃×#235	1	4	yes	—
173	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
174	C₈	—	2	4	no	—
175	C₈	—	6	4	no	—
176	C₂×C₃	—	2	1	no	—
177	C₂×C₃	—	2	4	no	—
178	C₂×C₃	—	2	2	no	—
179	C₂×C₃	#4709	1	4	yes	—
180	C₂×C₃	#4932	1	1	yes	characteristic
181	C₂×C₃	#5122	1	2	yes	—
182	C₂×C₃	C₃×#376	1	1	yes	characteristic
183	C₂×C₃	C₃×#405	1	2	yes	—
184	C₂×C₃	C₃×#433	1	4	yes	—
185	C₂²	C₃×#872	1	2	yes	—
186	C₂²	C₃×#907	1	4	yes	—
187	C₂²	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
188	C₄	—	2	4	no	—
189	C₄	—	12	4	no	—
190	C₄	C₃×#872	1	2	yes	—
191	C₄	C₃×#926	1	2	yes	—
192	C₃	—	2	1	no	—
193	C₃	#69806	1	1	yes	characteristic, direct product factor
194	C₃	C₃×#1746	1	1	yes	characteristic, complemented
195	C₂	C₃×#4709	1	4	yes	—
196	C₂	C₃×#4932	1	1	yes	characteristic
197	C₂	C₃×#5122	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×#97	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×#97	D₃
hypocenter	C₃	C₃×#1746
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#97

Complements

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

N	H ≅ G/N	type
6. #69806	C₃	direct (N × H)
11. C₃²×#97	C₄	semidirect
13. C₃×C₄×C₃⋊C₈	C₄	semidirect
45. C₃×#97	Dic₃	semidirect
46. C₃×#97	C₃×C₄	semidirect
50. C₄×C₃⋊C₈	C₃×C₄	semidirect
102. #97	C₃×Dic₃	semidirect
165. C₃²	#1746	semidirect
193. C₃	#69806	direct (N × H)
194. C₃	C₃×#1746	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#97	2
C₃×#4932	2
C₂×C₃×C₄×C₃⋊C₈	2
C₃×#4550	2
C₃×#1746	3
#69806	3

Subgroups — C3×#69806 #11783512

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69806 #11783512