Subgroups — C₂×#11746732 #14314408

Order 1920.

336 conjugacy classes of subgroups.

The subgroup lattice has 216 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	2	yes	maximal
2	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic
3	#11746732	C₂	1	4	yes	maximal, direct product factor
4	C₂×#64861	—	3	1	no	maximal, Hall
5	C₂×C₃×C₄²×C₅	C₄	1	1	yes	characteristic
6	C₂²×C₃×#168	C₄	1	1	yes	characteristic
7	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
8	#91227	—	2	2	no	—
9	#91227	C₂²	1	4	yes	—
10	C₂×C₃⋊#674	—	2	2	no	—
11	C₃×C₄×#168	C₄	1	2	yes	—
12	C₃×C₄×#168	C₂²	1	2	yes	—
13	C₂×#4463	—	5	1	no	maximal, Hall
14	C₂×#3998	—	3	2	no	—
15	C₂×C₄×#168	D₃	1	1	yes	characteristic
16	#64861	—	3	4	no	—
17	C₂²×C₃×C₄×C₅	—	2	1	no	—
18	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₃×C₄²×C₅	—	2	1	no	—
20	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
21	C₂×C₃×#168	C₈	1	2	yes	—
22	C₂×C₃×#168	C₈	1	2	yes	—
23	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×#168	C₂×C₄	1	2	yes	—
25	C₂×C₃×#168	D₄	1	1	yes	characteristic
26	C₂×C₃×#168	D₄	1	1	yes	characteristic
27	C₂×C₃×#168	Q₈	1	1	yes	characteristic
28	C₂×C₃×#168	Q₈	1	1	yes	characteristic
29	C₂×C₃×#168	C₂³	1	1	yes	characteristic
30	C₃⋊#674	—	2	4	no	—
31	C₂×C₃×C₄×C₈	—	5	1	no	—
32	C₂×#812	—	5	2	no	—
33	#4463	—	5	4	no	—
34	C₂×C₄²×C₅	Dic₃	1	1	yes	characteristic
35	C₂²×#168	Dic₃	1	1	yes	characteristic
36	C₂²×#168	C₂×D₃	1	1	yes	characteristic
37	#3998	—	3	4	no	—
38	#3998	—	6	2	no	—
39	C₂×#674	—	6	2	no	—
40	C₄×#168	Dic₃	1	2	yes	—
41	C₄×#168	C₂×D₃	1	2	yes	—
42	C₂×#364	—	15	1	no	Sylow
43	C₂³×C₃×C₅	M₁₆	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	—	2	4	no	—
45	C₂×C₃×C₄×C₅	—	2	2	no	—
46	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₃×#168	C₄⋊C₄	1	1	yes	characteristic
53	C₃×#168	C₄⋊C₄	1	1	yes	characteristic
54	C₃×#168	C₂×C₈	1	4	yes	—
55	C₃×#168	C₂×D₄	1	1	yes	characteristic
56	C₃×#168	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₃×C₈	—	5	1	no	—
58	C₂²×C₃×C₈	—	5	1	no	—
59	C₂×C₃×C₄²	AGL(1,𝔽₅)	1	1	yes	characteristic
60	C₃×C₄×C₈	—	5	2	no	—
61	C₃×C₄×C₈	—	5	2	no	—
62	#812	—	5	4	no	—
63	#812	—	10	2	no	—
64	C₂×C₃⋊C₁₆	—	10	2	no	—
65	C₂²×C₄×C₅	—	2	1	no	—
66	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
67	C₄²×C₅	—	2	1	no	—
68	C₄²×C₅	C₂×Dic₃	1	2	yes	—
69	C₂×#168	C₃⋊C₈	1	2	yes	—
70	C₂×#168	C₃⋊C₈	1	2	yes	—
71	C₂×#168	Dic₆	1	1	yes	characteristic
72	C₂×#168	Dic₆	1	1	yes	characteristic
73	C₂×#168	D₁₂	1	1	yes	characteristic
74	C₂×#168	D₁₂	1	1	yes	characteristic
75	C₂×#168	C₂×Dic₃	1	2	yes	—
76	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
77	C₂×#168	C₂²×D₃	1	1	yes	characteristic
78	#674	—	6	4	no	—
79	C₂×M₃₂	—	15	2	no	—
80	C₂×C₄×C₈	—	5	1	no	—
81	#364	—	15	4	no	—
82	C₂²×C₃×C₅	—	2	2	no	—
83	C₂²×C₃×C₅	C₄⋊C₈	1	1	yes	characteristic
84	C₂²×C₃×C₅	C₄⋊C₈	1	1	yes	characteristic
85	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
86	C₃×C₄×C₅	—	2	4	no	—
87	C₃×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
88	C₃×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
89	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
91	C₂²×C₃×C₄	—	2	1	no	—
92	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
93	C₂×C₃×C₈	—	5	2	no	—
94	C₂×C₃×C₈	—	5	2	no	—
95	C₂×C₃×C₈	—	5	1	no	—
96	C₂×C₃×C₈	—	5	1	no	—
97	C₂×C₃×C₈	—	5	1	no	—
98	C₂×C₃×C₈	—	5	1	no	—
99	C₂×C₃×C₈	—	5	2	no	—
100	C₂×C₃×C₈	—	5	1	no	—
101	C₂×C₃×C₈	—	5	1	no	—
102	C₃×C₄²	—	2	1	no	—
103	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	2	yes	—
104	C₃⋊C₁₆	—	10	4	no	—
105	C₂³×C₅	#208	1	1	yes	characteristic
106	C₂×C₄×C₅	—	2	4	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
110	C₂×C₄×C₅	#208	1	1	yes	characteristic
111	C₂×C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
112	C₂×C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
114	#168	C₂×C₃⋊C₈	1	4	yes	—
115	#168	C₄⋊Dic₃	1	1	yes	characteristic
116	#168	C₄⋊Dic₃	1	1	yes	characteristic
117	#168	C₂×Dic₆	1	1	yes	characteristic
118	#168	C₂×D₁₂	1	1	yes	characteristic
119	C₂²×C₈	—	5	1	no	—
120	C₂²×C₈	—	5	1	no	—
121	C₂×C₄²	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
122	M₃₂	—	15	4	no	—
123	M₃₂	—	30	2	no	—
124	C₂×C₁₆	—	30	2	no	—
125	C₄×C₈	—	5	2	no	—
126	C₄×C₈	—	5	2	no	—
127	C₂×C₃×C₅	—	2	1	no	—
128	C₂×C₃×C₅	—	2	1	no	—
129	C₂×C₃×C₅	#364	1	2	yes	complemented
130	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
131	C₂³×C₃	#704	1	1	yes	characteristic
132	C₂×C₃×C₄	—	2	4	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	D₅⋊C₈	1	1	yes	characteristic
135	C₂×C₃×C₄	D₅⋊C₈	1	1	yes	characteristic
136	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
137	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
138	C₂×C₃×C₄	#704	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
140	C₃×C₈	—	5	4	no	—
141	C₃×C₈	—	5	1	no	—
142	C₃×C₈	—	5	1	no	—
143	C₃×C₈	—	5	1	no	—
144	C₃×C₈	—	5	1	no	—
145	C₂²×C₅	—	2	2	no	—
146	C₂²×C₅	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
147	C₂²×C₅	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
148	C₂²×C₅	C₂×#208	1	1	yes	characteristic
149	C₄×C₅	—	2	4	no	—
150	C₄×C₅	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
151	C₄×C₅	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
152	C₄×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
153	C₄×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
156	C₂×C₈	—	5	2	no	—
157	C₂×C₈	—	5	2	no	—
158	C₂×C₈	—	5	1	no	—
159	C₂×C₈	—	5	1	no	—
160	C₂×C₈	—	5	1	no	—
161	C₂×C₈	—	5	1	no	—
162	C₂×C₈	—	5	1	no	—
163	C₂×C₈	—	5	2	no	—
164	C₂×C₈	—	5	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
167	C₁₆	—	30	4	no	—
168	C₃×C₅	C₂×#364	1	1	yes	characteristic, complemented, Hall
169	C₂²×C₃	—	2	2	no	—
170	C₂²×C₃	#4004	1	1	yes	characteristic
171	C₂²×C₃	#4004	1	1	yes	characteristic
172	C₂²×C₃	C₂×#704	1	1	yes	characteristic
173	C₃×C₄	—	2	4	no	—
174	C₃×C₄	#4004	1	1	yes	characteristic
175	C₃×C₄	#4004	1	1	yes	characteristic
176	C₃×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
177	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
178	C₂×C₅	—	2	1	no	—
179	C₂×C₅	—	2	1	no	—
180	C₂×C₅	#4463	1	2	yes	complemented
181	C₂×C₅	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
182	C₂³	#6762	1	1	yes	characteristic
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
186	C₂×C₄	D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
187	C₂×C₄	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
188	C₂×C₄	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
189	C₂×C₄	#6762	1	1	yes	characteristic
190	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
191	C₈	—	5	4	no	—
192	C₈	—	5	1	no	—
193	C₈	—	5	1	no	—
194	C₈	—	5	1	no	—
195	C₈	—	5	1	no	—
196	C₂×C₃	—	2	1	no	—
197	C₂×C₃	—	2	1	no	—
198	C₂×C₃	#64861	1	2	yes	complemented
199	C₂×C₃	C₂×#4004	1	1	yes	characteristic
200	C₅	C₂×#4463	1	1	yes	characteristic, complemented, Sylow
201	C₂²	—	2	2	no	—
202	C₂²	#91233	1	1	yes	characteristic
203	C₂²	#91233	1	1	yes	characteristic
204	C₂²	C₂×#6762	1	1	yes	characteristic
205	C₄	—	2	4	no	—
206	C₄	#91233	1	1	yes	characteristic
207	C₄	#91233	1	1	yes	characteristic
208	C₄	C₂×D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
209	C₄	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
210	C₃	C₂×#64861	1	1	yes	characteristic, complemented, Sylow
211	C₂	—	2	1	no	—
212	C₂	—	2	1	no	—
213	C₂	#11746732	1	2	yes	direct product factor
214	C₂	C₂×#91233	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊ΓL(1,𝔽₁₆)
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₂×C₃×C₄²×C₅	C₄
socle	C₂²×C₃×C₅	C₄⋊C₈
prime radical	C₂³×C₃×C₅	M₁₆
cosocle	C₂×C₃×#168	C₂³
hypercenter	C₂×C₄²	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	C₂×#364
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. #11746732	C₂	direct (N × H)
129. C₂×C₃×C₅	#364	semidirect
168. C₃×C₅	C₂×#364	semidirect
180. C₂×C₅	#4463	semidirect
198. C₂×C₃	#64861	semidirect
200. C₅	C₂×#4463	semidirect
210. C₃	C₂×#64861	semidirect
213. C₂	#11746732	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×#91227	2
C₂×C₃×C₄×#168	2
#11746732	2
C₂×#64861	3
C₂×#4463	5

Subgroups — C2×#11746732 #14314408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11746732 #14314408