Subgroups — C₄×C₄⋊(C₃⋊#168) #14296093

Order 1920.

632 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄³×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊#168)	C₂	1	2	yes	maximal
3	C₂×C₄×C₃⋊#168	C₂	1	2	yes	maximal
4	#11746345	C₂	1	2	yes	maximal
5	C₄×C₄⋊#168	—	3	1	no	maximal, Hall
6	C₂×C₃×C₄²×C₅	C₄	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₄	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂²×C₃⋊#168	C₂²	1	4	yes	—
12	C₄⋊(C₃⋊#168)	C₄	1	8	yes	direct product factor
13	C₄×C₃⋊#168	—	2	4	no	—
14	C₄×C₄⋊(C₃⋊C₈)	—	5	1	no	maximal, Hall
15	C₄³×C₅	D₃	1	1	yes	characteristic
16	C₂×C₄⋊#168	—	3	2	no	—
17	C₂×C₄×#168	—	3	2	no	—
18	#64744	—	3	2	no	—
19	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
21	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
23	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₃×C₄²×C₅	—	2	4	no	—
25	C₃×C₄²×C₅	—	2	2	no	—
26	C₃×C₄²×C₅	C₈	1	8	yes	complemented
27	C₃×C₄²×C₅	C₂×C₄	1	4	yes	—
28	C₃×C₄²×C₅	D₄	1	2	yes	—
29	C₃×C₄²×C₅	Q₈	1	2	yes	—
30	C₂×C₃⋊#168	—	2	4	no	—
31	C₂×C₃⋊#168	—	2	4	no	—
32	C₂×C₃⋊#168	C₂×C₄	1	8	yes	—
33	C₃×C₄³	D₅	1	1	yes	characteristic
34	C₂×C₄⋊(C₃⋊C₈)	—	5	2	no	—
35	C₂×C₄×C₃⋊C₈	—	5	2	no	—
36	#4524	—	5	2	no	—
37	C₂×C₄²×C₅	Dic₃	1	2	yes	—
38	C₂×C₄²×C₅	Dic₃	1	2	yes	—
39	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
42	C₂²×#168	—	3	4	no	—
43	C₄⋊#168	—	3	8	no	—
44	C₄×#168	—	6	4	no	—
45	C₄×C₄⋊C₈	—	15	1	no	Sylow
46	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	—	2	2	no	—
48	C₂×C₃×C₄×C₅	—	2	2	no	—
49	C₂×C₃×C₄×C₅	—	2	2	no	—
50	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
52	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
53	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
54	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
55	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
56	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	—
57	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	#41	1	2	yes	—
61	C₃⋊#168	—	2	8	no	—
62	C₂×C₃×C₄²	Dic₅	1	2	yes	—
63	C₂×C₃×C₄²	Dic₅	1	2	yes	—
64	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
65	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
66	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
67	C₂²×C₃⋊C₈	—	5	4	no	—
68	C₄⋊(C₃⋊C₈)	—	5	8	no	—
69	C₄×C₃⋊C₈	—	10	4	no	—
70	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
71	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
74	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
75	C₄²×C₅	—	2	4	no	—
76	C₄²×C₅	—	2	2	no	—
77	C₄²×C₅	C₃⋊C₈	1	8	yes	complemented
78	C₄²×C₅	Dic₆	1	2	yes	—
79	C₄²×C₅	C₄×D₃	1	4	yes	—
80	C₄²×C₅	D₁₂	1	2	yes	—
81	C₂×#168	—	3	8	no	—
82	C₂×#168	—	6	4	no	—
83	C₂×#168	—	6	4	no	—
84	C₂×C₄⋊C₈	—	15	2	no	—
85	C₂×C₄×C₈	—	15	2	no	—
86	C₄³	D₁₅	1	1	yes	characteristic
87	#336	—	15	2	no	—
88	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
89	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₃×C₅	#117	1	1	yes	characteristic
91	C₂²×C₃×C₅	C₂²×C₈	1	2	yes	—
92	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
93	C₃×C₄×C₅	—	2	2	no	—
94	C₃×C₄×C₅	—	2	4	no	—
95	C₃×C₄×C₅	C₄×C₈	1	4	yes	complemented
96	C₃×C₄×C₅	C₄⋊C₈	1	8	yes	complemented
97	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
98	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
99	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
100	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
102	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
103	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
104	C₃×C₄²	—	2	4	no	—
105	C₃×C₄²	—	2	2	no	—
106	C₃×C₄²	#168	1	8	yes	complemented
107	C₃×C₄²	Dic₁₀	1	2	yes	—
108	C₃×C₄²	C₄×D₅	1	4	yes	—
109	C₃×C₄²	D₂₀	1	2	yes	—
110	C₂×C₃⋊C₈	—	5	8	no	—
111	C₂×C₃⋊C₈	—	10	4	no	—
112	C₂×C₃⋊C₈	—	10	4	no	—
113	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	C₂×C₃⋊C₈	1	4	yes	—
118	C₂×C₄×C₅	C₂×C₃⋊C₈	1	4	yes	—
119	C₂×C₄×C₅	C₂×C₃⋊C₈	1	4	yes	—
120	C₂×C₄×C₅	#208	1	4	yes	—
121	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
122	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
123	C₂×C₄×C₅	C₄⋊Dic₃	1	4	yes	—
124	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
126	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
127	C₂×C₄×C₅	#235	1	2	yes	—
128	#168	—	6	8	no	—
129	C₂²×C₈	—	15	4	no	—
130	C₂×C₄²	Dic₁₅	1	2	yes	—
131	C₂×C₄²	Dic₁₅	1	2	yes	—
132	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
133	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
134	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
135	C₄⋊C₈	—	15	8	no	—
136	C₄×C₈	—	30	4	no	—
137	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
139	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
140	C₂×C₃×C₅	C₂×C₄⋊C₈	1	2	yes	—
141	C₂×C₃×C₅	#431	1	2	yes	—
142	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	C₂×#168	1	4	yes	—
147	C₂×C₃×C₄	C₂×#168	1	4	yes	—
148	C₂×C₃×C₄	C₂×#168	1	4	yes	—
149	C₂×C₃×C₄	#681	1	4	yes	—
150	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
151	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
152	C₂×C₃×C₄	C₄⋊Dic₅	1	4	yes	—
153	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
155	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
156	C₂×C₃×C₄	#709	1	2	yes	—
157	C₃⋊C₈	—	10	8	no	—
158	C₂²×C₅	C₂²×C₃⋊C₈	1	2	yes	—
159	C₂²×C₅	C₂×#208	1	2	yes	—
160	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
161	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
162	C₂²×C₅	#926	1	1	yes	characteristic
163	C₄×C₅	—	2	2	no	—
164	C₄×C₅	—	2	4	no	—
165	C₄×C₅	C₄×C₃⋊C₈	1	4	yes	complemented
166	C₄×C₅	C₄⋊(C₃⋊C₈)	1	8	yes	complemented
167	C₄×C₅	C₄×Dic₆	1	2	yes	—
168	C₄×C₅	C₄×D₁₂	1	2	yes	—
169	C₂²×C₄	C₂×Dic₁₅	1	2	yes	—
170	C₂²×C₄	C₂×Dic₁₅	1	1	yes	characteristic
171	C₂²×C₄	C₂×Dic₁₅	1	1	yes	characteristic
172	C₂²×C₄	C₂×Dic₁₅	1	2	yes	—
173	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
174	C₂×C₈	—	15	8	no	—
175	C₂×C₈	—	30	4	no	—
176	C₂×C₈	—	30	4	no	—
177	C₄²	—	2	4	no	—
178	C₄²	—	2	2	no	—
179	C₄²	C₃⋊#168	1	8	yes	complemented
180	C₄²	Dic₃₀	1	2	yes	—
181	C₄²	C₄×D₁₅	1	4	yes	—
182	C₄²	D₆₀	1	2	yes	—
183	C₃×C₅	C₄×C₄⋊C₈	1	1	yes	characteristic, complemented, Hall
184	C₂²×C₃	C₂²×#168	1	2	yes	—
185	C₂²×C₃	C₂×#681	1	2	yes	—
186	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
187	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
188	C₂²×C₃	#4072	1	1	yes	characteristic
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	2	4	no	—
191	C₃×C₄	C₄×#168	1	4	yes	complemented
192	C₃×C₄	C₄⋊#168	1	8	yes	complemented
193	C₃×C₄	C₄×Dic₁₀	1	2	yes	—
194	C₃×C₄	C₄×D₂₀	1	2	yes	—
195	C₂×C₅	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
196	C₂×C₅	C₄×#208	1	1	yes	characteristic
197	C₂×C₅	C₂×C₄⋊(C₃⋊C₈)	1	2	yes	—
198	C₂×C₅	#4925	1	2	yes	—
199	C₂×C₅	C₄×C₄⋊Dic₃	1	1	yes	characteristic
200	C₂³	C₂²×Dic₁₅	1	1	yes	characteristic
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	C₂×C₃⋊#168	1	4	yes	—
205	C₂×C₄	C₂×C₃⋊#168	1	4	yes	—
206	C₂×C₄	C₂×C₃⋊#168	1	4	yes	—
207	C₂×C₄	#6710	1	4	yes	—
208	C₂×C₄	C₄×Dic₁₅	1	2	yes	—
209	C₂×C₄	C₄×Dic₁₅	1	2	yes	—
210	C₂×C₄	C₄⋊Dic₁₅	1	4	yes	—
211	C₂×C₄	C₂×Dic₃₀	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
213	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
214	C₂×C₄	#6817	1	2	yes	—
215	C₈	—	30	8	no	—
216	C₂×C₃	C₂×C₄×#168	1	1	yes	characteristic
217	C₂×C₃	C₄×#681	1	1	yes	characteristic
218	C₂×C₃	C₂×C₄⋊#168	1	2	yes	—
219	C₂×C₃	#65214	1	2	yes	—
220	C₂×C₃	C₄×C₄⋊Dic₅	1	1	yes	characteristic
221	C₅	C₄×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic, complemented, Sylow
222	C₂²	C₂²×C₃⋊#168	1	2	yes	—
223	C₂²	C₂×#6710	1	2	yes	—
224	C₂²	C₂×C₄×Dic₁₅	1	1	yes	characteristic
225	C₂²	C₂×C₄⋊Dic₁₅	1	1	yes	characteristic
226	C₂²	#91815	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	4	no	—
229	C₄	C₄×C₃⋊#168	1	4	yes	complemented
230	C₄	C₄⋊(C₃⋊#168)	1	8	yes	direct product factor
231	C₄	C₄×Dic₃₀	1	2	yes	—
232	C₄	C₄×D₆₀	1	2	yes	—
233	C₃	C₄×C₄⋊#168	1	1	yes	characteristic, complemented, Sylow
234	C₂	C₂×C₄×C₃⋊#168	1	1	yes	characteristic
235	C₂	C₄×#6710	1	1	yes	characteristic
236	C₂	C₂×C₄⋊(C₃⋊#168)	1	2	yes	—
237	C₂	#11751026	1	2	yes	—
238	C₂	C₄×C₄⋊Dic₁₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×D₁₅
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₄³×C₅	C₂
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₄³	D₁₅
hypocenter	C₃×C₅	C₄×C₄⋊C₈
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄² ◁ C₄³

Complements

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

N	H ≅ G/N	type
12. C₄⋊(C₃⋊#168)	C₄	direct (N × H)
26. C₃×C₄²×C₅	C₈	semidirect
77. C₄²×C₅	C₃⋊C₈	semidirect
95. C₃×C₄×C₅	C₄×C₈	semidirect
96. C₃×C₄×C₅	C₄⋊C₈	semidirect
106. C₃×C₄²	#168	semidirect
165. C₄×C₅	C₄×C₃⋊C₈	semidirect
166. C₄×C₅	C₄⋊(C₃⋊C₈)	semidirect
179. C₄²	C₃⋊#168	semidirect
183. C₃×C₅	C₄×C₄⋊C₈	semidirect
191. C₃×C₄	C₄×#168	semidirect
192. C₃×C₄	C₄⋊#168	semidirect
221. C₅	C₄×C₄⋊(C₃⋊C₈)	semidirect
229. C₄	C₄×C₃⋊#168	semidirect
230. C₄	C₄⋊(C₃⋊#168)	direct (N × H)
233. C₃	C₄×C₄⋊#168	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄³×C₅	2
C₂×C₄⋊(C₃⋊#168)	2
C₂×C₄×C₃⋊#168	2
#11746345	2
C₄×C₄⋊#168	3
C₄×C₄⋊(C₃⋊C₈)	5

Subgroups — C4×C4⋊(C3⋊#168) #14296093

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₄⋊(C₃⋊#168) #14296093