Subgroups — #14312856

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×#91790	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91096	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746328	C₂	1	4	yes	maximal, complemented
5	#10602202	—	3	1	no	maximal, Hall
6	C₂×#6817	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×#6710	C₄	1	1	yes	characteristic
9	#91790	—	2	2	no	—
10	#91790	—	2	2	no	—
11	C₂×#6705	C₄	1	2	yes	—
12	C₂×C₈×D₁₅	C₂²	1	2	yes	—
13	C₂×C₃×C₅×C₁₆	—	2	2	no	—
14	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
15	#91096	—	2	2	no	—
16	C₂×C₃⋊#672	C₂²	1	2	yes	—
17	#71409	—	5	1	no	maximal, Hall
18	C₂×#4047	—	3	1	no	—
19	C₂²×C₅×C₁₆	D₃	1	1	yes	characteristic, complemented
20	C₂×#3944	—	3	1	no	—
21	#64727	—	3	4	no	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×#1220	C₈	1	2	yes	—
24	#6817	—	2	2	no	—
25	#6817	—	2	2	no	—
26	C₂×D₆₀	C₈	1	1	yes	characteristic
27	C₂×C₄×D₁₅	C₂×C₄	1	2	yes	—
28	C₂×Dic₃₀	C₈	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
31	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
32	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
33	#6710	—	2	2	no	—
34	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
35	#6705	—	2	4	no	—
36	C₈×D₁₅	—	2	2	no	—
37	C₈×D₁₅	—	2	2	no	—
38	C₃×C₅×C₁₆	—	2	2	no	—
39	C₃⋊#672	—	2	2	no	—
40	C₂×#901	—	5	1	no	—
41	C₂²×C₃×C₁₆	D₅	1	1	yes	characteristic, complemented
42	C₂×#812	—	5	1	no	—
43	(C₂×D₃)⋊C₁₆	—	5	4	no	—
44	C₂×#709	—	3	1	no	—
45	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
46	C₂×#681	—	3	1	no	—
47	#4047	—	6	2	no	—
48	#4047	—	6	2	no	—
49	C₂×#676	—	3	2	no	—
50	C₂×C₈×D₅	—	3	2	no	—
51	C₂×C₅×C₁₆	—	2	2	no	—
52	C₂×C₅×C₁₆	C₂×D₃	1	2	yes	—
53	#3944	—	6	2	no	—
54	C₂×#672	—	3	2	no	—
55	#2113	—	15	1	no	Sylow
56	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
57	C₂²×D₁₅	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	#1220	—	2	4	no	—
63	C₂×Dic₁₅	C₂×C₈	1	2	yes	—
64	D₆₀	—	2	2	no	—
65	C₄×D₁₅	—	2	2	no	—
66	C₄×D₁₅	—	2	2	no	—
67	Dic₃₀	—	2	2	no	—
68	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
69	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
70	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
71	C₃⋊#168	—	2	2	no	—
72	C₂×#235	—	5	1	no	—
73	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
74	C₂×#208	—	5	1	no	—
75	#901	—	10	2	no	—
76	#901	—	10	2	no	—
77	C₂×ΓL(1,𝔽₂₅)	—	5	2	no	—
78	C₂×D₃×C₈	—	5	2	no	—
79	C₂×C₃×C₁₆	—	2	2	no	—
80	C₂×C₃×C₁₆	C₂×D₅	1	2	yes	—
81	#812	—	10	2	no	—
82	C₂×C₃⋊C₁₆	—	5	2	no	—
83	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
84	C₂×C₂²⋊D₅	—	3	2	no	—
85	#709	—	6	2	no	—
86	#709	—	6	2	no	—
87	C₂×D₂₀	—	3	1	no	—
88	C₂×C₄×D₅	—	3	2	no	—
89	C₂×Dic₁₀	—	3	1	no	—
90	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
91	C₂×C₅×C₈	D₁₂	1	2	yes	—
92	C₂×C₅×C₈	#67	1	2	yes	—
93	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
94	#681	—	6	2	no	—
95	C₂×#168	—	3	2	no	—
96	#676	—	6	4	no	—
97	C₈×D₅	—	6	2	no	—
98	C₈×D₅	—	6	2	no	—
99	C₅×C₁₆	—	2	2	no	—
100	#672	—	6	2	no	—
101	C₂×#131	—	15	1	no	—
102	C₂×M₃₂	—	15	1	no	—
103	C₂²×C₁₆	D₁₅	1	1	yes	characteristic, complemented
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×D₁₅	—	2	2	no	—
109	C₂×D₁₅	—	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	Dic₁₅	—	2	2	no	—
114	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
115	C₂×#67	—	5	2	no	—
116	#235	—	10	2	no	—
117	#235	—	10	2	no	—
118	C₂×D₁₂	—	5	1	no	—
119	C₂×C₄×D₃	—	5	2	no	—
120	C₂×Dic₆	—	5	1	no	—
121	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
122	C₂×C₃×C₈	D₂₀	1	2	yes	—
123	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
124	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
125	#208	—	10	2	no	—
126	C₂×C₃⋊C₈	—	5	2	no	—
127	ΓL(1,𝔽₂₅)	—	10	4	no	—
128	D₃×C₈	—	10	2	no	—
129	D₃×C₈	—	10	2	no	—
130	C₃×C₁₆	—	2	2	no	—
131	C₃⋊C₁₆	—	10	2	no	—
132	C₂³×C₅	D₃×C₈	1	1	yes	characteristic
133	C₂²×D₅	—	3	2	no	—
134	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
135	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	2	yes	—
136	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
137	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
138	C₂²⋊D₅	—	6	4	no	—
139	C₂×Dic₅	—	3	2	no	—
140	D₂₀	—	6	2	no	—
141	C₄×D₅	—	6	2	no	—
142	C₄×D₅	—	6	2	no	—
143	Dic₁₀	—	6	2	no	—
144	C₅×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
145	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
146	C₅×C₈	C₂×#67	1	1	yes	characteristic
147	#168	—	6	2	no	—
148	C₂×#41	—	15	1	no	—
149	#131	—	30	2	no	—
150	#131	—	30	2	no	—
151	C₂×M₁₆	—	15	1	no	—
152	C₂×M₁₆	—	15	2	no	—
153	C₂²×C₈	—	15	2	no	—
154	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
155	M₃₂	—	30	2	no	—
156	C₂×C₁₆	—	2	2	no	—
157	C₂×C₁₆	—	15	2	no	—
158	C₂×C₁₆	C₂×D₁₅	1	2	yes	—
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	D₁₅	—	4	2	no	—
163	C₂³×C₃	C₈×D₅	1	1	yes	characteristic
164	C₂²×D₃	—	5	2	no	—
165	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
166	C₂×C₃×C₄	#676	1	2	yes	—
167	C₂×C₃×C₄	#685	1	2	yes	—
168	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
169	#67	—	10	4	no	—
170	C₂×Dic₃	—	5	2	no	—
171	D₁₂	—	10	2	no	—
172	C₄×D₃	—	10	2	no	—
173	C₄×D₃	—	10	2	no	—
174	Dic₆	—	10	2	no	—
175	C₃×C₈	#685	1	2	yes	—
176	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
177	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
178	C₃⋊C₈	—	10	2	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	(C₂×D₃)⋊C₈	1	2	yes	—
181	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
182	C₂×D₅	—	6	4	no	—
183	C₂×D₅	—	6	2	no	—
184	C₄×C₅	(C₂×D₃)⋊C₈	1	2	yes	—
185	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
186	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
187	Dic₅	—	6	2	no	—
188	#41	—	30	2	no	—
189	#41	—	30	2	no	—
190	C₂×Q₈	—	15	1	no	—
191	C₂×D₄	—	15	2	no	—
192	C₂×D₄	—	15	1	no	—
193	C₂²×C₄	—	15	2	no	—
194	C₂²×C₄	C₄×D₁₅	1	1	yes	characteristic
195	M₁₆	—	30	2	no	—
196	M₁₆	—	30	4	no	—
197	C₂×C₈	—	15	2	no	—
198	C₂×C₈	—	30	2	no	—
199	C₂×C₈	—	30	2	no	—
200	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
201	C₂×C₈	D₆₀	1	2	yes	—
202	C₂×C₈	#1220	1	2	yes	—
203	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
204	C₁₆	—	2	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	#3952	1	2	yes	—
209	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
210	C₂×D₃	—	10	2	no	—
211	C₂×D₃	—	10	4	no	—
212	C₃×C₄	#3952	1	2	yes	—
213	C₃×C₄	C₂×#676	1	1	yes	characteristic
214	C₃×C₄	C₂×#685	1	1	yes	characteristic
215	Dic₃	—	10	2	no	—
216	C₂×C₅	—	2	2	no	—
217	C₂×C₅	#4901	1	2	yes	—
218	C₂×C₅	C₂×(C₂×D₃)⋊C₈	1	1	yes	characteristic
219	D₅	—	12	2	no	—
220	C₂³	—	15	2	no	—
221	C₂³	C₈×D₁₅	1	1	yes	characteristic
222	Q₈	—	30	2	no	—
223	D₄	—	30	4	no	—
224	D₄	—	30	2	no	—
225	C₂×C₄	—	15	2	no	—
226	C₂×C₄	—	30	2	no	—
227	C₂×C₄	—	30	2	no	—
228	C₂×C₄	C₈×D₁₅	1	1	yes	characteristic
229	C₂×C₄	#6705	1	2	yes	—
230	C₂×C₄	#6714	1	2	yes	—
231	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
232	C₈	—	30	2	no	—
233	C₈	#6714	1	2	yes	—
234	C₈	C₂×D₆₀	1	1	yes	characteristic
235	C₈	C₂×#1220	1	1	yes	characteristic
236	C₂×C₃	—	2	2	no	—
237	C₂×C₃	#65190	1	2	yes	—
238	C₂×C₃	C₂×#3952	1	1	yes	characteristic
239	D₃	—	20	2	no	—
240	C₅	#71409	1	1	yes	characteristic, complemented, Sylow
241	C₂²	—	2	2	no	—
242	C₂²	—	30	2	no	—
243	C₂²	—	30	4	no	—
244	C₂²	#91104	1	2	yes	—
245	C₂²	C₂×C₈×D₁₅	1	1	yes	characteristic
246	C₄	—	30	2	no	—
247	C₄	#91104	1	2	yes	—
248	C₄	C₂×#6705	1	1	yes	characteristic
249	C₄	C₂×#6714	1	1	yes	characteristic
250	C₃	#10602202	1	1	yes	characteristic, complemented, Sylow
251	C₂	—	2	2	no	—
252	C₂	—	60	2	no	—
253	C₂	#11751002	1	2	yes	—
254	C₂	C₂×#91104	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₅
commutator, socle	C₂²×C₃×C₅	C₂²×C₈
Fitting	C₂²×C₃×C₅×C₁₆	C₂
prime radical	C₂×#6817	C₄
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂²×C₁₆	D₁₅
hypocenter	C₃×C₅	#2113
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₅×C₁₆	C₂	semidirect
3. C₂×#91096	C₂	semidirect
4. #11746328	C₂	semidirect
19. C₂²×C₅×C₁₆	D₃	semidirect
41. C₂²×C₃×C₁₆	D₅	semidirect
103. C₂²×C₁₆	D₁₅	semidirect
206. C₃×C₅	#2113	semidirect
240. C₅	#71409	semidirect
250. C₃	#10602202	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91790	2
C₂²×C₃×C₅×C₁₆	2
C₂×#91096	2
#11746328	2
#10602202	3
#71409	5