Subgroups — Dic₃×#3998 #14312618

Order 1920.

392 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91227	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3998	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₃×#168	C₂	1	1	yes	maximal, characteristic
4	#11746542	C₂	1	2	yes	maximal, complemented
5	Dic₃×#674	C₂	1	2	yes	maximal, complemented
6	C₄×#3998	—	3	1	no	maximal, Hall
7	C₂²×C₃⋊#168	C₄	1	1	yes	characteristic
8	C₂×C₄×C₅×Dic₃	C₄	1	1	yes	characteristic
9	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
10	#91227	C₄	1	4	yes	complemented
11	C₂×C₃⋊#674	C₂²	1	2	yes	—
12	C₃×#3998	C₄	1	4	yes	complemented
13	C₂×C₃×#674	C₂²	1	2	yes	—
14	Dic₃×#168	C₄	1	1	yes	characteristic
15	Dic₃×#168	C₄	1	1	yes	characteristic
16	Dic₃×#168	C₂²	1	1	yes	characteristic
17	Dic₃×#168	C₂²	1	1	yes	characteristic
18	Dic₃×M₃₂	—	5	1	no	maximal, Hall
19	C₂×#3998	—	3	1	no	—
20	C₂×#3998	D₃	1	1	yes	characteristic
21	C₂×C₄×#168	—	3	1	no	—
22	#64859	—	3	2	no	—
23	C₄×#674	—	3	2	no	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×Dic₃	C₈	1	2	yes	—
26	C₂×C₃⋊#168	C₈	1	4	yes	—
27	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
29	C₄×C₅×Dic₃	C₈	1	2	yes	—
30	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×#168	C₂×C₄	1	2	yes	—
34	C₂×C₃×#168	C₂×C₄	1	2	yes	—
35	C₂×C₃×#168	C₂³	1	1	yes	characteristic
36	C₃⋊#674	C₂×C₄	1	4	yes	complemented
37	C₃×#674	C₂×C₄	1	4	yes	complemented
38	C₂×C₃×M₃₂	—	5	1	no	—
39	C₂×C₈×Dic₃	—	5	1	no	—
40	C₂×#812	—	5	1	no	—
41	#4502	—	5	2	no	—
42	Dic₃×C₁₆	—	5	2	no	—
43	C₂×C₄²×C₅	—	3	1	no	—
44	C₂²×#168	—	3	1	no	—
45	C₂²×#168	C₂×D₃	1	1	yes	characteristic
46	#3998	—	3	4	no	—
47	#3998	Dic₃	1	4	yes	direct product factor
48	C₂×#674	—	3	2	no	—
49	C₂×#674	C₂×D₃	1	2	yes	—
50	C₄×#168	—	3	1	no	—
51	C₄×#168	—	3	1	no	—
52	C₄×#168	—	3	1	no	—
53	C₄×#168	—	3	1	no	—
54	C₄×M₃₂	—	15	1	no	Sylow
55	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
57	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
59	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₅×Dic₃	—	2	4	no	—
61	C₂×C₅×Dic₃	C₂×C₈	1	2	yes	—
62	C₂×C₅×Dic₃	C₂×C₈	1	2	yes	—
63	C₃⋊#168	C₂×C₈	1	2	yes	—
64	C₃⋊#168	C₂×C₈	1	2	yes	—
65	C₃×#168	C₄²	1	2	yes	—
66	C₃×#168	C₂²×C₄	1	1	yes	characteristic
67	C₃×#168	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₃×C₈	—	5	1	no	—
69	C₂×C₄×Dic₃	AGL(1,𝔽₅)	1	1	yes	characteristic
70	C₂²×C₃⋊C₈	—	5	1	no	—
71	C₃×M₃₂	—	5	4	no	—
72	C₂×C₃×C₁₆	—	5	2	no	—
73	C₈×Dic₃	—	5	1	no	—
74	C₈×Dic₃	—	5	1	no	—
75	C₈×Dic₃	—	5	1	no	—
76	C₈×Dic₃	—	5	1	no	—
77	#812	—	5	4	no	—
78	C₂×C₃⋊C₁₆	—	5	2	no	—
79	C₂²×C₄×C₅	—	3	2	no	—
80	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
81	C₄²×C₅	—	3	2	no	—
82	C₄²×C₅	—	3	1	no	—
83	C₄²×C₅	—	3	1	no	—
84	C₂×#168	—	3	4	no	—
85	C₂×#168	—	3	1	no	—
86	C₂×#168	—	3	1	no	—
87	C₂×#168	C₄×D₃	1	1	yes	characteristic
88	C₂×#168	C₄×D₃	1	2	yes	—
89	C₂×#168	C₂×Dic₃	1	2	yes	—
90	C₂×#168	C₂²×D₃	1	1	yes	characteristic
91	#674	—	3	4	no	—
92	#674	C₂×Dic₃	1	4	yes	complemented
93	C₂×M₃₂	—	5	1	no	—
94	C₂×M₃₂	—	15	1	no	—
95	C₂×C₄×C₈	—	15	1	no	—
96	#346	—	15	2	no	—
97	C₄×C₁₆	—	15	2	no	—
98	C₂²×C₃×C₅	—	2	2	no	—
99	C₂²×C₃×C₅	C₄×C₈	1	2	yes	—
100	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
101	C₃×C₄×C₅	C₄×C₈	1	2	yes	—
102	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
103	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
104	C₅×Dic₃	—	2	2	no	—
105	C₅×Dic₃	M₃₂	1	4	yes	complemented
106	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
107	C₂²×Dic₃	#170	1	2	yes	—
108	C₂×C₃×C₈	—	5	2	no	—
109	C₂×C₃×C₈	—	5	2	no	—
110	C₂×C₃×C₈	—	5	1	no	—
111	C₂×C₃×C₈	—	5	1	no	—
112	C₄×Dic₃	#170	1	2	yes	—
113	C₄×Dic₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
114	C₄×Dic₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
115	C₂×C₃⋊C₈	—	5	4	no	—
116	C₂×C₃⋊C₈	—	5	1	no	—
117	C₂×C₃⋊C₈	—	5	1	no	—
118	C₃×C₁₆	—	5	4	no	—
119	C₃⋊C₁₆	—	5	4	no	—
120	C₂³×C₅	D₃×C₈	1	1	yes	characteristic
121	C₂×C₄×C₅	—	3	2	no	—
122	C₂×C₄×C₅	—	3	2	no	—
123	C₂×C₄×C₅	—	6	4	no	—
124	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
125	C₂×C₄×C₅	D₃×C₈	1	2	yes	—
126	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
127	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
128	#168	—	3	2	no	—
129	#168	—	3	2	no	—
130	#168	C₄×Dic₃	1	2	yes	—
131	#168	C₂×C₄×D₃	1	1	yes	characteristic
132	#168	C₂²×Dic₃	1	1	yes	characteristic
133	C₂²×C₈	—	5	1	no	—
134	C₂²×C₈	—	15	1	no	—
135	C₂×C₄²	—	3	1	no	—
136	M₃₂	—	5	4	no	—
137	M₃₂	—	15	4	no	—
138	C₂×C₁₆	—	5	2	no	—
139	C₂×C₁₆	—	15	2	no	—
140	C₄×C₈	—	15	1	no	—
141	C₄×C₈	—	15	1	no	—
142	C₄×C₈	—	15	1	no	—
143	C₄×C₈	—	15	1	no	—
144	C₂×C₃×C₅	—	2	2	no	—
145	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
146	C₂×C₃×C₅	C₂×M₃₂	1	2	yes	—
147	C₂³×C₃	C₂×#170	1	1	yes	characteristic
148	C₂×C₃×C₄	D₅⋊C₈	1	2	yes	—
149	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	2	yes	—
150	C₂×C₃×C₄	C₂×#170	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
152	C₂×Dic₃	—	2	4	no	—
153	C₂×Dic₃	C₂×#170	1	2	yes	—
154	C₂×Dic₃	C₂×#170	1	2	yes	—
155	C₃×C₈	—	5	2	no	—
156	C₃×C₈	—	5	1	no	—
157	C₃×C₈	—	5	1	no	—
158	C₃⋊C₈	—	5	2	no	—
159	C₃⋊C₈	—	5	2	no	—
160	C₂²×C₅	—	2	2	no	—
161	C₂²×C₅	C₈×Dic₃	1	2	yes	—
162	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
163	C₄×C₅	—	3	4	no	—
164	C₄×C₅	—	6	2	no	—
165	C₄×C₅	C₈×Dic₃	1	2	yes	—
166	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
167	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
168	C₂²×C₄	—	3	2	no	—
169	C₂²×C₄	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
170	C₂×C₈	—	5	2	no	—
171	C₂×C₈	—	5	2	no	—
172	C₂×C₈	—	5	1	no	—
173	C₂×C₈	—	5	1	no	—
174	C₂×C₈	—	15	4	no	—
175	C₂×C₈	—	15	1	no	—
176	C₂×C₈	—	15	1	no	—
177	C₄²	—	3	2	no	—
178	C₄²	—	3	1	no	—
179	C₄²	—	3	1	no	—
180	C₁₆	—	5	4	no	—
181	C₁₆	—	15	4	no	—
182	C₃×C₅	C₄×M₃₂	1	1	yes	characteristic, complemented, Hall
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	C₄×#170	1	2	yes	—
185	C₂²×C₃	C₂²×#170	1	1	yes	characteristic
186	C₃×C₄	C₄×#170	1	2	yes	—
187	C₃×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
188	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
189	Dic₃	—	2	2	no	—
190	Dic₃	#3998	1	4	yes	direct product factor
191	C₂×C₅	—	2	2	no	—
192	C₂×C₅	D₃×M₃₂	1	2	yes	—
193	C₂×C₅	C₂×C₈×Dic₃	1	1	yes	characteristic
194	C₂³	D₃×#170	1	1	yes	characteristic
195	C₂×C₄	—	3	2	no	—
196	C₂×C₄	—	3	2	no	—
197	C₂×C₄	—	6	4	no	—
198	C₂×C₄	Dic₃×AGL(1,𝔽₅)	1	2	yes	—
199	C₂×C₄	D₃×#170	1	1	yes	characteristic
200	C₂×C₄	#6738	1	2	yes	—
201	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
202	C₈	—	5	1	no	—
203	C₈	—	5	1	no	—
204	C₈	—	5	2	no	—
205	C₈	—	15	2	no	—
206	C₈	—	15	2	no	—
207	C₂×C₃	—	2	2	no	—
208	C₂×C₃	C₂×#3998	1	2	yes	—
209	C₂×C₃	C₂×C₄×#170	1	1	yes	characteristic
210	C₅	Dic₃×M₃₂	1	1	yes	characteristic, complemented, Sylow
211	C₂²	—	2	2	no	—
212	C₂²	Dic₃×#170	1	2	yes	—
213	C₂²	C₂×D₃×#170	1	1	yes	characteristic
214	C₄	—	3	4	no	—
215	C₄	—	6	2	no	—
216	C₄	Dic₃×#170	1	2	yes	—
217	C₄	C₂×Dic₃×AGL(1,𝔽₅)	1	1	yes	characteristic
218	C₄	C₂×#6738	1	1	yes	characteristic
219	C₃	C₄×#3998	1	1	yes	characteristic, complemented, Sylow
220	C₂	—	2	2	no	—
221	C₂	D₃×#3998	1	2	yes	—
222	C₂	C₂×Dic₃×#170	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Fitting	C₂²×C₃×C₄×C₅	C₂×C₄
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂³×C₃×C₅	C₂×C₈
cosocle	C₂×C₃×#168	C₂³
hypercenter	C₂²×C₄	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	C₄×M₃₂
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #11746542	C₂	semidirect
5. Dic₃×#674	C₂	semidirect
10. #91227	C₄	semidirect
12. C₃×#3998	C₄	semidirect
36. C₃⋊#674	C₂×C₄	semidirect
37. C₃×#674	C₂×C₄	semidirect
47. #3998	Dic₃	direct (N × H)
92. #674	C₂×Dic₃	semidirect
105. C₅×Dic₃	M₃₂	semidirect
182. C₃×C₅	C₄×M₃₂	semidirect
190. Dic₃	#3998	direct (N × H)
210. C₅	Dic₃×M₃₂	semidirect
219. C₃	C₄×#3998	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91227	2
C₂×C₃×#3998	2
C₂×Dic₃×#168	2
#11746542	2
Dic₃×#674	2
C₄×#3998	3
Dic₃×M₃₂	5

Subgroups — Dic3×#3998 #14312618

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₃×#3998 #14312618