Subgroups — C₃×#1925 #72640

Order 384.

290 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#102	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#102	C₂	1	1	yes	maximal, characteristic
4	C₃×#390	C₂	1	1	yes	maximal, characteristic
5	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#340	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#336	C₂	1	1	yes	maximal, characteristic, complemented
8	#1925	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
16	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
17	C₂×C₃×C₄²	—	2	1	no	—
18	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
19	C₃×#107	—	2	2	no	—
20	C₃×#102	—	2	2	no	—
21	C₃×#102	C₄	1	4	yes	complemented
22	C₃×#95	—	2	1	no	—
23	C₂×#107	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×#102	C₂×C₃	1	1	yes	characteristic
25	C₂×#102	C₂×C₃	1	1	yes	characteristic
26	#390	C₂×C₃	1	1	yes	characteristic
27	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
28	#340	C₂×C₃	1	1	yes	characteristic, complemented
29	#336	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂⁴×C₃	—	2	1	no	—
31	C₂×C₃×D₄	—	2	1	no	—
32	C₂×C₃×D₄	—	2	1	no	—
33	C₂×C₃×D₄	—	2	1	no	—
34	C₂×C₃×D₄	—	2	1	no	—
35	C₂×C₃×D₄	—	2	1	no	—
36	C₂×C₃×D₄	C₂×C₄	1	2	yes	—
37	C₂²×C₃×C₄	—	2	1	no	—
38	C₂²×C₃×C₄	—	2	1	no	—
39	C₂²×C₃×C₄	—	2	1	no	—
40	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
41	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
42	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₈	—	2	1	no	—
44	C₂×C₃×C₈	—	2	1	no	—
45	C₂×C₃×C₈	—	2	1	no	—
46	C₂×C₃×C₈	—	2	1	no	—
47	C₂×C₃×C₈	—	2	1	no	—
48	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
49	C₃×C₄⋊C₄	—	2	1	no	—
50	C₃×C₄⋊C₄	—	2	1	no	—
51	C₃×C₄⋊C₄	—	2	1	no	—
52	C₃×C₄⋊C₄	—	2	1	no	—
53	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
54	C₃×C₄⋊C₄	D₄	1	2	yes	—
55	C₃×C₂²⋊C₄	—	4	2	no	—
56	C₃×C₂²⋊C₄	—	4	2	no	—
57	C₃×C₄²	—	2	1	no	—
58	C₃×C₄²	—	2	1	no	—
59	C₃×C₄²	—	4	2	no	—
60	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
61	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
62	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₂²⋊C₄	—	2	1	no	—
67	C₂×C₂²⋊C₄	—	2	1	no	—
68	C₂×C₄²	—	2	1	no	—
69	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
70	#107	—	2	2	no	—
71	#102	—	2	2	no	—
72	#102	C₃×C₄	1	4	yes	complemented
73	#95	—	2	1	no	—
74	C₂³×C₃	—	2	2	no	—
75	C₂³×C₃	—	4	1	no	—
76	C₂³×C₃	—	4	1	no	—
77	C₂³×C₃	—	4	1	no	—
78	C₂³×C₃	—	4	1	no	—
79	C₂³×C₃	—	4	1	no	—
80	C₂³×C₃	—	4	1	no	—
81	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
82	C₃×D₄	—	2	2	no	—
83	C₃×D₄	—	2	2	no	—
84	C₃×D₄	—	4	1	no	—
85	C₃×D₄	—	4	1	no	—
86	C₂×C₃×C₄	—	2	1	no	—
87	C₂×C₃×C₄	—	2	1	no	—
88	C₂×C₃×C₄	—	2	2	no	—
89	C₂×C₃×C₄	—	2	2	no	—
90	C₂×C₃×C₄	—	2	1	no	—
91	C₂×C₃×C₄	—	2	1	no	—
92	C₂×C₃×C₄	—	2	1	no	—
93	C₂×C₃×C₄	—	4	1	no	—
94	C₂×C₃×C₄	—	4	1	no	—
95	C₂×C₃×C₄	—	4	1	no	—
96	C₂×C₃×C₄	—	4	1	no	—
97	C₂×C₃×C₄	—	4	1	no	—
98	C₂×C₃×C₄	—	4	1	no	—
99	C₂×C₃×C₄	—	4	1	no	—
100	C₂×C₃×C₄	D₈	1	2	yes	—
101	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
103	C₂×C₃×C₄	#41	1	1	yes	characteristic
104	C₂×C₃×C₄	#41	1	1	yes	characteristic
105	C₂×C₃×C₄	#41	1	1	yes	characteristic
106	C₂×C₃×C₄	#41	1	1	yes	characteristic
107	C₃×C₈	—	2	2	no	—
108	C₃×C₈	—	4	1	no	—
109	C₂⁴	—	2	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	—	2	1	no	—
114	C₂×D₄	—	2	1	no	—
115	C₂×D₄	C₂×C₃×C₄	1	2	yes	—
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
120	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
121	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	C₂×C₃×C₄	1	2	yes	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
133	C₄⋊C₄	C₃×D₄	1	2	yes	—
134	C₂²⋊C₄	—	4	2	no	—
135	C₂²⋊C₄	—	4	2	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	2	1	no	—
138	C₄²	—	4	2	no	—
139	C₂²×C₃	—	4	2	no	—
140	C₂²×C₃	—	4	2	no	—
141	C₂²×C₃	—	4	2	no	—
142	C₂²×C₃	—	8	1	no	—
143	C₂²×C₃	—	8	1	no	—
144	C₂²×C₃	—	8	1	no	—
145	C₂²×C₃	—	8	1	no	—
146	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
147	C₂²×C₃	#121	1	1	yes	characteristic
148	C₂²×C₃	#123	1	1	yes	characteristic
149	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
150	C₂²×C₃	#135	1	1	yes	characteristic
151	C₂²×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂²×C₃	#137	1	1	yes	characteristic
153	C₃×C₄	—	2	2	no	—
154	C₃×C₄	—	4	2	no	—
155	C₃×C₄	—	4	2	no	—
156	C₃×C₄	—	4	1	no	—
157	C₃×C₄	—	8	1	no	—
158	C₃×C₄	#117	1	1	yes	characteristic
159	C₃×C₄	C₄×D₄	1	1	yes	characteristic
160	C₃×C₄	#124	1	1	yes	characteristic
161	C₃×C₄	#126	1	1	yes	characteristic
162	C₂³	—	2	2	no	—
163	C₂³	—	4	1	no	—
164	C₂³	—	4	1	no	—
165	C₂³	—	4	1	no	—
166	C₂³	—	4	1	no	—
167	C₂³	—	4	1	no	—
168	C₂³	—	4	1	no	—
169	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
170	D₄	—	2	2	no	—
171	D₄	—	2	2	no	—
172	D₄	—	4	1	no	—
173	D₄	—	4	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	4	1	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	4	1	no	—
188	C₂×C₄	C₃×D₈	1	2	yes	—
189	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
190	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
191	C₂×C₄	C₃×#41	1	1	yes	characteristic
192	C₂×C₄	C₃×#41	1	1	yes	characteristic
193	C₂×C₄	C₃×#41	1	1	yes	characteristic
194	C₂×C₄	C₃×#41	1	1	yes	characteristic
195	C₈	—	2	2	no	—
196	C₈	—	4	1	no	—
197	C₂×C₃	—	8	2	no	—
198	C₂×C₃	#388	1	1	yes	characteristic
199	C₂×C₃	C₄×D₈	1	1	yes	characteristic
200	C₂×C₃	#440	1	1	yes	characteristic
201	C₂×C₃	#459	1	1	yes	characteristic
202	C₂×C₃	#464	1	1	yes	characteristic
203	C₂×C₃	#480	1	1	yes	characteristic
204	C₂×C₃	#482	1	1	yes	characteristic
205	C₂²	—	4	2	no	—
206	C₂²	—	4	2	no	—
207	C₂²	—	4	2	no	—
208	C₂²	—	8	1	no	—
209	C₂²	—	8	1	no	—
210	C₂²	—	8	1	no	—
211	C₂²	—	8	1	no	—
212	C₂²	C₃×C₄×D₄	1	1	yes	characteristic
213	C₂²	C₃×#121	1	1	yes	characteristic
214	C₂²	C₃×#123	1	1	yes	characteristic
215	C₂²	C₂×C₃×D₈	1	1	yes	characteristic
216	C₂²	C₃×#135	1	1	yes	characteristic
217	C₂²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
218	C₂²	C₃×#137	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	4	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	8	1	no	—
224	C₄	C₃×#117	1	1	yes	characteristic
225	C₄	C₃×C₄×D₄	1	1	yes	characteristic
226	C₄	C₃×#124	1	1	yes	characteristic
227	C₄	C₃×#126	1	1	yes	characteristic
228	C₃	#1925	1	1	yes	characteristic, direct product factor, Sylow
229	C₂	—	8	2	no	—
230	C₂	C₃×#388	1	1	yes	characteristic
231	C₂	C₃×C₄×D₈	1	1	yes	characteristic
232	C₂	C₃×#440	1	1	yes	characteristic
233	C₂	C₃×#459	1	1	yes	characteristic
234	C₂	C₃×#464	1	1	yes	characteristic
235	C₂	C₃×#480	1	1	yes	characteristic
236	C₂	C₃×#482	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂²×C₄	C₂³×C₃
prime radical	C₂²×C₃×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#107	C₂	semidirect
5. C₃×C₄×C₄⋊C₄	C₂	semidirect
6. C₃×#340	C₂	semidirect
7. C₃×#336	C₂	semidirect
8. #1925	C₃	direct (N × H)
21. C₃×#102	C₄	semidirect
23. C₂×#107	C₂×C₃	semidirect
27. C₄×C₄⋊C₄	C₂×C₃	semidirect
28. #340	C₂×C₃	semidirect
29. #336	C₂×C₃	semidirect
72. #102	C₃×C₄	semidirect
228. C₃	#1925	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#107	2
C₂×C₃×#102	2
C₃×#390	2
C₃×C₄×C₄⋊C₄	2
C₃×#340	2
C₃×#336	2
#1925	3

Subgroups — C3×#1925 #72640

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#1925 #72640