Subgroups — C₂×C₄⋊(C₃⋊#672) #14314299

Order 1920.

368 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊#672	C₂	1	2	yes	maximal
3	C₄⋊(C₃⋊#672)	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊#672	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄²×C₅	C₄	1	1	yes	characteristic
8	C₃×C₄×C₅×C₈	C₄	1	2	yes	—
9	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
10	C₂×C₃⋊#672	—	2	4	no	—
11	C₂×C₃⋊#672	C₂²	1	4	yes	—
12	C₂×C₄⋊(C₃⋊C₁₆)	—	5	1	no	maximal, Hall
13	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic
14	C₂²×#672	—	3	2	no	—
15	C₄⋊#672	—	3	4	no	—
16	C₂²×C₃×C₄×C₅	C₈	1	1	yes	characteristic
17	C₂²×C₃×C₄×C₅	C₈	1	1	yes	characteristic
18	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃×C₅×C₈	—	2	2	no	—
20	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
21	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
23	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
24	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
25	C₃×C₄²×C₅	C₈	1	2	yes	—
26	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
27	C₃⋊#672	—	2	4	no	—
28	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic
29	C₂²×C₃⋊C₁₆	—	5	2	no	—
30	C₄⋊(C₃⋊C₁₆)	—	5	4	no	—
31	C₂²×C₅×C₈	Dic₃	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
33	C₂×C₄²×C₅	Dic₃	1	1	yes	characteristic
34	C₄×C₅×C₈	Dic₃	1	2	yes	—
35	C₄×C₅×C₈	C₂×D₃	1	2	yes	—
36	C₂×#672	—	3	4	no	—
37	C₂×#672	—	6	4	no	—
38	C₂×C₄⋊C₁₆	—	15	1	no	Sylow
39	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
40	C₂×C₃×C₄×C₅	—	2	2	no	—
41	C₂×C₃×C₄×C₅	C₁₆	1	4	yes	complemented
42	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
43	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
44	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
45	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
46	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
47	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
48	C₃×C₅×C₈	—	2	2	no	—
49	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
50	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
51	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₃×C₈	Dic₅	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
54	C₂×C₃×C₄²	Dic₅	1	1	yes	characteristic
55	C₃×C₄×C₈	Dic₅	1	2	yes	—
56	C₃×C₄×C₈	C₂×D₅	1	2	yes	—
57	C₂×C₃⋊C₁₆	—	5	4	no	—
58	C₂×C₃⋊C₁₆	—	10	4	no	—
59	C₂²×C₄×C₅	C₃⋊C₈	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₃⋊C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
62	C₂×C₅×C₈	—	2	2	no	—
63	C₂×C₅×C₈	Dic₆	1	2	yes	—
64	C₂×C₅×C₈	D₁₂	1	2	yes	—
65	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
66	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
68	C₄²×C₅	C₃⋊C₈	1	2	yes	—
69	C₄²×C₅	C₂×Dic₃	1	2	yes	—
70	#672	—	6	4	no	—
71	C₂²×C₁₆	—	15	2	no	—
72	C₂×C₄×C₈	D₁₅	1	1	yes	characteristic
73	C₄⋊C₁₆	—	15	4	no	—
74	C₂²×C₃×C₅	C₄⋊C₈	1	2	yes	—
75	C₂²×C₃×C₅	C₂×C₁₆	1	2	yes	—
76	C₂²×C₃×C₅	M₃₂	1	2	yes	—
77	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
78	C₃×C₄×C₅	—	2	2	no	—
79	C₃×C₄×C₅	C₄⋊C₈	1	2	yes	—
80	C₃×C₄×C₅	C₂×C₁₆	1	4	yes	complemented
81	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
83	C₂²×C₃×C₄	#168	1	1	yes	characteristic
84	C₂²×C₃×C₄	#168	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
86	C₂×C₃×C₈	—	2	2	no	—
87	C₂×C₃×C₈	Dic₁₀	1	2	yes	—
88	C₂×C₃×C₈	D₂₀	1	2	yes	—
89	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
90	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
91	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
92	C₃×C₄²	#168	1	2	yes	—
93	C₃×C₄²	C₂×Dic₅	1	2	yes	—
94	C₃⋊C₁₆	—	10	4	no	—
95	C₂³×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
96	C₂×C₄×C₅	—	2	2	no	—
97	C₂×C₄×C₅	C₃⋊C₁₆	1	4	yes	complemented
98	C₂×C₄×C₅	C₂×C₃⋊C₈	1	2	yes	—
99	C₂×C₄×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
100	C₂×C₄×C₅	C₂×C₃⋊C₈	1	2	yes	—
101	C₂×C₄×C₅	#208	1	2	yes	—
102	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
103	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
104	C₅×C₈	—	2	2	no	—
105	C₅×C₈	C₄⋊Dic₃	1	2	yes	—
106	C₅×C₈	C₂×Dic₆	1	1	yes	characteristic
107	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
108	C₂²×C₈	Dic₁₅	1	1	yes	characteristic
109	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
110	C₂×C₄²	Dic₁₅	1	1	yes	characteristic
111	C₂×C₁₆	—	15	4	no	—
112	C₂×C₁₆	—	30	4	no	—
113	C₄×C₈	Dic₁₅	1	2	yes	—
114	C₄×C₈	C₂×D₁₅	1	2	yes	—
115	C₂×C₃×C₅	C₄⋊C₁₆	1	4	yes	complemented
116	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
117	C₂×C₃×C₅	C₂²×C₁₆	1	1	yes	characteristic
118	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
119	C₂³×C₃	C₂×#168	1	1	yes	characteristic
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	#672	1	4	yes	complemented
122	C₂×C₃×C₄	C₂×#168	1	2	yes	—
123	C₂×C₃×C₄	C₂×#168	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂×#168	1	2	yes	—
125	C₂×C₃×C₄	#681	1	2	yes	—
126	C₂×C₃×C₄	C₄⋊Dic₅	1	2	yes	—
127	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
128	C₃×C₈	—	2	2	no	—
129	C₃×C₈	C₄⋊Dic₅	1	2	yes	—
130	C₃×C₈	C₂×Dic₁₀	1	1	yes	characteristic
131	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
132	C₂²×C₅	C₄⋊(C₃⋊C₈)	1	2	yes	—
133	C₂²×C₅	C₂×C₃⋊C₁₆	1	2	yes	—
134	C₂²×C₅	#812	1	2	yes	—
135	C₂²×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
136	C₄×C₅	—	2	2	no	—
137	C₄×C₅	C₄⋊(C₃⋊C₈)	1	2	yes	—
138	C₄×C₅	C₂×C₃⋊C₁₆	1	4	yes	complemented
139	C₄×C₅	C₂×#208	1	1	yes	characteristic
140	C₄×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
141	C₂²×C₄	C₃⋊#168	1	1	yes	characteristic
142	C₂²×C₄	C₃⋊#168	1	1	yes	characteristic
143	C₂²×C₄	C₂×Dic₁₅	1	1	yes	characteristic
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	Dic₃₀	1	2	yes	—
146	C₂×C₈	D₆₀	1	2	yes	—
147	C₂×C₈	C₂×Dic₁₅	1	2	yes	—
148	C₂×C₈	C₂×Dic₁₅	1	1	yes	characteristic
149	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
150	C₄²	C₃⋊#168	1	2	yes	—
151	C₄²	C₂×Dic₁₅	1	2	yes	—
152	C₁₆	—	30	4	no	—
153	C₃×C₅	C₂×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Hall
154	C₂²×C₃	C₄⋊#168	1	2	yes	—
155	C₂²×C₃	C₂×#672	1	2	yes	—
156	C₂²×C₃	#3944	1	2	yes	—
157	C₂²×C₃	C₂²×#168	1	1	yes	characteristic
158	C₃×C₄	—	2	2	no	—
159	C₃×C₄	C₄⋊#168	1	2	yes	—
160	C₃×C₄	C₂×#672	1	4	yes	complemented
161	C₃×C₄	C₂×#681	1	1	yes	characteristic
162	C₃×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
163	C₂×C₅	C₄⋊(C₃⋊C₁₆)	1	4	yes	complemented
164	C₂×C₅	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
165	C₂×C₅	C₂²×C₃⋊C₁₆	1	1	yes	characteristic
166	C₂×C₅	C₂×#812	1	1	yes	characteristic
167	C₂³	C₂×C₃⋊#168	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	C₃⋊#672	1	4	yes	complemented
170	C₂×C₄	C₂×C₃⋊#168	1	2	yes	—
171	C₂×C₄	C₂×C₃⋊#168	1	1	yes	characteristic
172	C₂×C₄	C₂×C₃⋊#168	1	2	yes	—
173	C₂×C₄	#6710	1	2	yes	—
174	C₂×C₄	C₄⋊Dic₁₅	1	2	yes	—
175	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	C₄⋊Dic₁₅	1	2	yes	—
178	C₈	C₂×Dic₃₀	1	1	yes	characteristic
179	C₈	C₂×D₆₀	1	1	yes	characteristic
180	C₂×C₃	C₄⋊#672	1	4	yes	complemented
181	C₂×C₃	C₂×C₄⋊#168	1	1	yes	characteristic
182	C₂×C₃	C₂²×#672	1	1	yes	characteristic
183	C₂×C₃	C₂×#3944	1	1	yes	characteristic
184	C₅	C₂×C₄⋊(C₃⋊C₁₆)	1	1	yes	characteristic, complemented, Sylow
185	C₂²	C₄⋊(C₃⋊#168)	1	2	yes	—
186	C₂²	C₂×C₃⋊#672	1	2	yes	—
187	C₂²	#91096	1	2	yes	—
188	C₂²	C₂²×C₃⋊#168	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	C₄⋊(C₃⋊#168)	1	2	yes	—
191	C₄	C₂×C₃⋊#672	1	4	yes	complemented
192	C₄	C₂×#6710	1	1	yes	characteristic
193	C₄	C₂×C₄⋊Dic₁₅	1	1	yes	characteristic
194	C₃	C₂×C₄⋊#672	1	1	yes	characteristic, complemented, Sylow
195	C₂	C₄⋊(C₃⋊#672)	1	4	yes	direct product factor
196	C₂	C₂×C₄⋊(C₃⋊#168)	1	1	yes	characteristic
197	C₂	C₂²×C₃⋊#672	1	1	yes	characteristic
198	C₂	C₂×#91096	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₁₅
commutator	C₂×C₃×C₅	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×D₁₅
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×C₈
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂×C₄×C₈	D₁₅
hypocenter	C₃×C₅	C₂×C₄⋊C₁₆
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₃⋊#672)	C₂	direct (N × H)
41. C₂×C₃×C₄×C₅	C₁₆	semidirect
80. C₃×C₄×C₅	C₂×C₁₆	semidirect
97. C₂×C₄×C₅	C₃⋊C₁₆	semidirect
115. C₂×C₃×C₅	C₄⋊C₁₆	semidirect
121. C₂×C₃×C₄	#672	semidirect
138. C₄×C₅	C₂×C₃⋊C₁₆	semidirect
153. C₃×C₅	C₂×C₄⋊C₁₆	semidirect
160. C₃×C₄	C₂×#672	semidirect
163. C₂×C₅	C₄⋊(C₃⋊C₁₆)	semidirect
169. C₂×C₄	C₃⋊#672	semidirect
180. C₂×C₃	C₄⋊#672	semidirect
184. C₅	C₂×C₄⋊(C₃⋊C₁₆)	semidirect
191. C₄	C₂×C₃⋊#672	semidirect
194. C₃	C₂×C₄⋊#672	semidirect
195. C₂	C₄⋊(C₃⋊#672)	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×C₄×C₅×C₈	2
C₂²×C₃⋊#672	2
C₄⋊(C₃⋊#672)	2
C₂×C₄⋊#672	3
C₂×C₄⋊(C₃⋊C₁₆)	5

Subgroups — C2×C4⋊(C3⋊#672) #14314299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊(C₃⋊#672) #14314299