Subgroups — #14312866

Order 1920.

456 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4053	C₂	1	1	yes	maximal, characteristic
2	C₂×#91227	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊#674	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746758	C₂	1	4	yes	maximal, complemented
5	#10604742	—	3	1	no	maximal, Hall
6	C₂×C₃×#709	C₄	1	1	yes	characteristic
7	C₂×C₃×C₅×M₁₆	C₄	1	1	yes	characteristic
8	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
9	C₃×#4053	—	2	2	no	—
10	C₃×#4053	—	2	2	no	—
11	C₂×C₃×#676	C₄	1	2	yes	—
12	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
13	#91227	—	2	2	no	—
14	C₂×C₃⋊#674	—	2	2	no	—
15	C₂×C₃⋊#674	C₂²	1	2	yes	—
16	C₂×C₃⋊#674	C₂²	1	2	yes	—
17	#72004	—	5	1	no	maximal, Hall
18	C₂×#4053	D₃	1	1	yes	characteristic
19	C₂×#3998	—	3	1	no	—
20	C₂²×#674	—	3	1	no	—
21	#64887	—	3	4	no	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₂²⋊D₅	C₈	1	2	yes	—
24	C₃×#709	—	2	2	no	—
25	C₃×#709	—	2	2	no	—
26	C₂×C₃×D₂₀	C₈	1	1	yes	characteristic
27	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	—
28	C₂×C₃×Dic₁₀	C₈	1	1	yes	characteristic
29	C₃×C₅×M₁₆	—	2	2	no	—
30	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×#168	D₄	1	2	yes	—
33	C₂×C₃×#168	D₄	1	2	yes	—
34	C₂×C₃×#168	C₂³	1	1	yes	characteristic
35	C₃×#676	—	2	4	no	—
36	C₃×C₈×D₅	—	2	2	no	—
37	C₃×C₈×D₅	—	2	2	no	—
38	C₃⋊#674	—	2	2	no	—
39	C₃⋊#674	—	2	2	no	—
40	C₂×C₃×#131	—	5	1	no	—
41	C₂×#812	—	5	1	no	—
42	C₂²×C₃⋊C₁₆	—	5	1	no	—
43	#4549	—	5	4	no	—
44	C₂×#709	Dic₃	1	1	yes	characteristic
45	C₂×C₅×M₁₆	Dic₃	1	1	yes	characteristic
46	C₂²×#168	C₂×D₃	1	1	yes	characteristic
47	#4053	—	2	2	no	—
48	#4053	—	2	2	no	—
49	C₂×#676	Dic₃	1	2	yes	—
50	C₂×C₈×D₅	C₂×D₃	1	2	yes	—
51	#3998	—	6	2	no	—
52	C₂×#674	—	3	2	no	—
53	C₂×#674	—	3	2	no	—
54	C₂×#674	—	6	2	no	—
55	#2113	—	15	1	no	Sylow
56	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₃×D₅	C₂×C₈	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
59	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₃×C₂²⋊D₅	—	2	4	no	—
63	C₂×C₃×Dic₅	C₂×C₈	1	2	yes	—
64	C₃×D₂₀	—	2	2	no	—
65	C₃×C₄×D₅	—	2	2	no	—
66	C₃×C₄×D₅	—	2	2	no	—
67	C₃×Dic₁₀	—	2	2	no	—
68	C₃×C₅×C₈	—	2	2	no	—
69	C₃×#168	C₂²⋊C₄	1	2	yes	—
70	C₃×#168	C₂×D₄	1	1	yes	characteristic
71	C₃×#168	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃×#41	—	5	1	no	—
73	C₃×#131	—	10	2	no	—
74	C₃×#131	—	10	2	no	—
75	C₂×C₃×M₁₆	—	5	2	no	—
76	C₂×C₃×M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
77	C₂²×C₃×C₈	—	5	1	no	—
78	C₂²×C₃×C₈	—	5	2	no	—
79	#812	—	10	2	no	—
80	C₂×C₃⋊C₁₆	—	5	2	no	—
81	C₂×C₃⋊C₁₆	—	5	2	no	—
82	C₂×C₃⋊C₁₆	—	10	2	no	—
83	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
84	C₂×C₂²⋊D₅	C₃⋊C₈	1	2	yes	—
85	#709	—	2	2	no	—
86	#709	—	2	2	no	—
87	C₂×D₂₀	C₃⋊C₈	1	1	yes	characteristic
88	C₂×C₄×D₅	C₂×Dic₃	1	2	yes	—
89	C₂×Dic₁₀	C₃⋊C₈	1	1	yes	characteristic
90	C₅×M₁₆	—	2	2	no	—
91	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
92	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
93	C₂×#168	#67	1	2	yes	—
94	C₂×#168	#67	1	2	yes	—
95	C₂×#168	C₂²×D₃	1	1	yes	characteristic
96	#676	—	2	4	no	—
97	C₈×D₅	—	2	2	no	—
98	C₈×D₅	—	2	2	no	—
99	#674	—	6	2	no	—
100	#674	—	6	2	no	—
101	C₂×#131	—	5	1	no	—
102	C₂×M₃₂	—	15	1	no	—
103	C₂²×C₁₆	—	15	1	no	—
104	C₂²⋊C₁₆	—	15	4	no	—
105	C₂²×C₃×C₅	—	2	2	no	—
106	C₂²×C₃×C₅	C₂²⋊C₈	1	2	yes	—
107	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₃×D₅	—	2	2	no	—
109	C₂×C₃×D₅	—	2	4	no	—
110	C₃×C₄×C₅	C₂²⋊C₈	1	2	yes	—
111	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
113	C₃×Dic₅	—	2	2	no	—
114	C₃×#41	—	10	2	no	—
115	C₃×#41	—	10	2	no	—
116	C₂×C₃×Q₈	—	5	1	no	—
117	C₂×C₃×D₄	—	5	2	no	—
118	C₂×C₃×D₄	—	5	1	no	—
119	C₂²×C₃×C₄	—	5	2	no	—
120	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
121	C₃×M₁₆	—	2	2	no	—
122	C₃×M₁₆	—	10	4	no	—
123	C₂×C₃×C₈	—	5	2	no	—
124	C₂×C₃×C₈	—	5	2	no	—
125	C₂×C₃×C₈	—	5	1	no	—
126	C₂×C₃×C₈	—	5	1	no	—
127	C₂×C₃×C₈	—	10	2	no	—
128	C₂×C₃×C₈	—	10	2	no	—
129	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
130	C₃⋊C₁₆	—	10	2	no	—
131	C₃⋊C₁₆	—	10	2	no	—
132	C₂³×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
133	C₂²×D₅	C₂×C₃⋊C₈	1	2	yes	—
134	C₂×C₄×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
135	C₂×C₄×C₅	#208	1	2	yes	—
136	C₂×C₄×C₅	#217	1	2	yes	—
137	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
138	C₂²⋊D₅	—	2	4	no	—
139	C₂×Dic₅	C₂×C₃⋊C₈	1	2	yes	—
140	D₂₀	—	2	2	no	—
141	C₄×D₅	—	2	2	no	—
142	C₄×D₅	—	2	2	no	—
143	Dic₁₀	—	2	2	no	—
144	C₅×C₈	—	2	2	no	—
145	#168	#217	1	2	yes	—
146	#168	C₂×#67	1	1	yes	characteristic
147	#168	C₂×#67	1	1	yes	characteristic
148	C₂×#41	—	5	1	no	—
149	#131	—	10	2	no	—
150	#131	—	10	2	no	—
151	C₂×M₁₆	—	5	2	no	—
152	C₂×M₁₆	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
153	C₂²×C₈	—	5	1	no	—
154	C₂²×C₈	—	5	2	no	—
155	M₃₂	—	30	2	no	—
156	C₂×C₁₆	—	15	2	no	—
157	C₂×C₁₆	—	15	2	no	—
158	C₂×C₁₆	—	30	2	no	—
159	C₂×C₃×C₅	—	2	2	no	—
160	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
161	C₂×C₃×C₅	#504	1	2	yes	—
162	C₃×D₅	—	4	2	no	—
163	C₂³×C₃	—	5	2	no	—
164	C₂³×C₃	D₅⋊C₈	1	1	yes	characteristic
165	C₃×Q₈	—	10	2	no	—
166	C₃×D₄	—	10	4	no	—
167	C₃×D₄	—	10	2	no	—
168	C₂×C₃×C₄	—	5	2	no	—
169	C₂×C₃×C₄	—	10	2	no	—
170	C₂×C₃×C₄	—	10	2	no	—
171	C₂×C₃×C₄	D₅⋊C₈	1	1	yes	characteristic
172	C₂×C₃×C₄	#700	1	2	yes	—
173	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
174	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
175	C₃×C₈	—	2	2	no	—
176	C₃×C₈	—	5	2	no	—
177	C₃×C₈	—	5	1	no	—
178	C₃×C₈	—	5	1	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	#830	1	2	yes	—
181	C₂²×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
182	C₂×D₅	—	2	4	no	—
183	C₂×D₅	—	2	2	no	—
184	C₄×C₅	#830	1	2	yes	—
185	C₄×C₅	C₂×#208	1	1	yes	characteristic
186	C₄×C₅	C₂×#217	1	1	yes	characteristic
187	Dic₅	—	2	2	no	—
188	#41	—	10	2	no	—
189	#41	—	10	2	no	—
190	C₂×Q₈	—	5	1	no	—
191	C₂×D₄	—	5	2	no	—
192	C₂×D₄	—	5	1	no	—
193	C₂²×C₄	—	5	2	no	—
194	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
195	M₁₆	—	2	2	no	—
196	M₁₆	—	10	4	no	—
197	C₂×C₈	—	5	2	no	—
198	C₂×C₈	—	5	2	no	—
199	C₂×C₈	—	5	1	no	—
200	C₂×C₈	—	5	1	no	—
201	C₂×C₈	—	10	2	no	—
202	C₂×C₈	—	10	2	no	—
203	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
204	C₁₆	—	30	2	no	—
205	C₁₆	—	30	2	no	—
206	C₃×C₅	#2113	1	1	yes	characteristic, complemented, Hall
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	10	4	no	—
209	C₂²×C₃	—	10	2	no	—
210	C₂²×C₃	#4003	1	2	yes	—
211	C₂²×C₃	C₂×D₅⋊C₈	1	1	yes	characteristic
212	C₃×C₄	—	10	2	no	—
213	C₃×C₄	#4003	1	2	yes	—
214	C₃×C₄	C₂×#700	1	1	yes	characteristic
215	C₃×C₄	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
216	C₂×C₅	—	2	2	no	—
217	C₂×C₅	#5140	1	2	yes	—
218	C₂×C₅	C₂×#830	1	1	yes	characteristic
219	D₅	—	4	2	no	—
220	C₂³	—	5	2	no	—
221	C₂³	D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
222	Q₈	—	10	2	no	—
223	D₄	—	10	4	no	—
224	D₄	—	10	2	no	—
225	C₂×C₄	—	5	2	no	—
226	C₂×C₄	—	10	2	no	—
227	C₂×C₄	—	10	2	no	—
228	C₂×C₄	D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
229	C₂×C₄	#6758	1	2	yes	—
230	C₂×C₄	#6763	1	2	yes	—
231	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	—	5	1	no	—
234	C₈	—	5	1	no	—
235	C₈	—	5	2	no	—
236	C₂×C₃	—	2	2	no	—
237	C₂×C₃	—	20	2	no	—
238	C₂×C₃	#65725	1	2	yes	—
239	C₂×C₃	C₂×#4003	1	1	yes	characteristic
240	C₅	#72004	1	1	yes	characteristic, complemented, Sylow
241	C₂²	—	2	2	no	—
242	C₂²	—	10	4	no	—
243	C₂²	—	10	2	no	—
244	C₂²	#91232	1	2	yes	—
245	C₂²	C₂×D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
246	C₄	—	10	2	no	—
247	C₄	#91232	1	2	yes	—
248	C₄	C₂×#6758	1	1	yes	characteristic
249	C₄	C₂×#6763	1	1	yes	characteristic
250	C₃	#10604742	1	1	yes	characteristic, complemented, Sylow
251	C₂	—	2	2	no	—
252	C₂	—	20	2	no	—
253	C₂	#11752384	1	2	yes	—
254	C₂	C₂×#91232	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
commutator, socle	C₂²×C₃×C₅	C₂²×C₈
Fitting	C₂×C₃×C₅×M₁₆	C₄
prime radical	C₂×C₃×#709	C₄
cosocle	C₂×C₃×#168	C₂³
hypercenter	C₂×M₁₆	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#2113
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#91227	C₂	semidirect
3. C₂²×C₃⋊#674	C₂	semidirect
4. #11746758	C₂	semidirect
206. C₃×C₅	#2113	semidirect
240. C₅	#72004	semidirect
250. C₃	#10604742	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#4053	2
C₂×#91227	2
C₂²×C₃⋊#674	2
#11746758	2
#10604742	3
#72004	5