Subgroups — #14312650

Order 1920.

368 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91219	C₂	1	1	yes	maximal, characteristic
2	C₂×D₅⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₃×#64997	C₂	1	1	yes	maximal, characteristic
4	#11746757	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746756	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746730	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₃⋊#674	C₂	1	1	yes	maximal, characteristic, complemented
8	#10603868	—	3	1	no	maximal, Hall
9	C₂×C₃×#676	C₄	1	1	yes	characteristic
10	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
11	C₃×#4018	C₄	1	1	yes	characteristic
12	C₂×C₃⋊#674	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊#674	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊#674	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊#674	C₂²	1	1	yes	characteristic
16	#91219	C₄	1	4	yes	complemented
17	D₅⋊(C₃⋊C₁₆)	C₄	1	4	yes	complemented
18	C₃×C₅×#97	C₄	1	1	yes	characteristic
19	C₃×#3947	C₄	1	1	yes	characteristic
20	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
21	C₃×C₄×#168	C₂²	1	1	yes	characteristic
22	#69881	—	5	1	no	maximal, Hall
23	C₂×#3990	—	3	1	no	—
24	C₂×D₅⋊C₁₆	—	3	1	no	—
25	#64997	D₃	1	1	yes	characteristic
26	#64886	—	3	1	no	—
27	#64885	—	3	1	no	—
28	#64859	—	3	1	no	—
29	C₄×#674	—	3	1	no	—
30	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₃×#685	C₈	1	2	yes	—
35	C₃×#683	C₈	1	2	yes	—
36	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×#168	C₂³	1	1	yes	characteristic
39	C₃×#676	C₈	1	4	yes	—
40	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
41	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
42	C₃⋊#674	C₂×C₄	1	2	yes	—
43	C₃⋊#674	C₂×C₄	1	2	yes	—
44	C₃⋊#674	C₂×C₄	1	2	yes	—
45	C₃⋊#674	C₂×C₄	1	2	yes	—
46	C₃×#405	—	5	1	no	—
47	C₂×#812	—	5	1	no	—
48	C₂²×C₃⋊C₁₆	—	5	1	no	—
49	#4461	—	5	1	no	—
50	#4461	—	5	1	no	—
51	C₄×C₃⋊C₁₆	—	5	1	no	—
52	C₄×C₃⋊C₁₆	—	5	1	no	—
53	C₂×#676	Dic₃	1	1	yes	characteristic
54	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
55	#4018	Dic₃	1	1	yes	characteristic
56	C₂×#674	—	3	1	no	—
57	C₂×#674	—	3	1	no	—
58	C₂×#674	—	3	1	no	—
59	C₂×#674	—	3	1	no	—
60	#3990	—	3	4	no	—
61	D₅⋊C₁₆	—	3	4	no	—
62	C₅×#97	Dic₃	1	1	yes	characteristic
63	#3947	Dic₃	1	1	yes	characteristic
64	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
65	C₄×#168	C₂×D₃	1	1	yes	characteristic
66	#2108	—	15	1	no	Sylow
67	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
69	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₃×Dic₅	C₂×C₈	1	2	yes	—
71	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
72	C₃×C₄×D₅	C₄²	1	2	yes	—
73	C₃×C₄×D₅	C₂×C₈	1	2	yes	—
74	C₃×C₅×C₈	C₄²	1	2	yes	—
75	C₃×C₅×C₈	C₂×C₈	1	2	yes	—
76	C₃×#168	C₂×C₈	1	2	yes	—
77	C₃×#168	C₂²×C₄	1	1	yes	characteristic
78	C₃×#168	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×M₁₆	—	5	1	no	—
80	C₂²×C₃×C₈	—	5	1	no	—
81	C₃×#117	—	5	1	no	—
82	C₃×#97	—	5	1	no	—
83	C₃×#97	AGL(1,𝔽₅)	1	1	yes	characteristic
84	C₃×C₄×C₈	—	5	1	no	—
85	C₃×C₄×C₈	—	5	1	no	—
86	#812	—	5	4	no	—
87	C₂×C₃⋊C₁₆	—	5	4	no	—
88	C₂×C₃⋊C₁₆	—	5	1	no	—
89	C₂×C₃⋊C₁₆	—	5	1	no	—
90	C₂×C₃⋊C₁₆	—	5	1	no	—
91	C₂×C₃⋊C₁₆	—	5	1	no	—
92	C₂×C₄×D₅	C₂×Dic₃	1	1	yes	characteristic
93	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
94	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
95	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
96	#685	C₃⋊C₈	1	2	yes	—
97	#683	C₃⋊C₈	1	2	yes	—
98	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
99	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
100	C₂×#168	C₂²×D₃	1	1	yes	characteristic
101	#676	C₃⋊C₈	1	4	yes	—
102	C₈×D₅	C₄×D₃	1	2	yes	—
103	C₈×D₅	C₄×D₃	1	2	yes	—
104	#674	—	3	2	no	—
105	#674	—	3	2	no	—
106	#674	—	3	2	no	—
107	#674	—	3	2	no	—
108	C₂×M₃₂	—	15	1	no	—
109	C₂²×C₁₆	—	15	1	no	—
110	#405	—	5	1	no	—
111	#346	—	15	1	no	—
112	#346	—	15	1	no	—
113	C₄×C₁₆	—	15	1	no	—
114	C₄×C₁₆	—	15	1	no	—
115	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₃×D₅	—	2	2	no	—
117	C₂×C₃×D₅	C₄×C₈	1	2	yes	—
118	C₃×C₄×C₅	—	2	2	no	—
119	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
120	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
121	C₃×Dic₅	—	2	2	no	—
122	C₃×Dic₅	C₄×C₈	1	2	yes	—
123	C₂²×C₃×C₄	—	5	1	no	—
124	C₃×M₁₆	—	5	4	no	—
125	C₂×C₃×C₈	—	5	1	no	—
126	C₂×C₃×C₈	—	5	1	no	—
127	C₂×C₃×C₈	—	5	2	no	—
128	C₂×C₃×C₈	—	5	2	no	—
129	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
130	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
131	C₃×C₄⋊C₄	—	5	2	no	—
132	C₃×C₂²⋊C₄	—	5	2	no	—
133	C₃×C₄²	—	5	1	no	—
134	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
135	C₃⋊C₁₆	—	5	2	no	—
136	C₃⋊C₁₆	—	5	2	no	—
137	C₃⋊C₁₆	—	5	2	no	—
138	C₃⋊C₁₆	—	5	2	no	—
139	C₂²×D₅	C₂×C₃⋊C₈	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×C₃⋊C₈	1	2	yes	—
141	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
142	C₂×Dic₅	C₂×C₃⋊C₈	1	2	yes	—
143	C₂×Dic₅	C₂×C₃⋊C₈	1	1	yes	characteristic
144	C₄×D₅	C₂×C₃⋊C₈	1	2	yes	—
145	C₄×D₅	C₄×Dic₃	1	2	yes	—
146	C₅×C₈	C₂×C₃⋊C₈	1	2	yes	—
147	C₅×C₈	C₄×Dic₃	1	2	yes	—
148	#168	C₂×C₃⋊C₈	1	2	yes	—
149	#168	C₂×C₄×D₃	1	1	yes	characteristic
150	#168	C₂×C₄×D₃	1	1	yes	characteristic
151	C₂×M₁₆	—	5	1	no	—
152	C₂²×C₈	—	5	1	no	—
153	#117	—	5	1	no	—
154	M₃₂	—	15	4	no	—
155	C₂×C₁₆	—	15	1	no	—
156	C₂×C₁₆	—	15	1	no	—
157	C₂×C₁₆	—	15	1	no	—
158	C₂×C₁₆	—	15	1	no	—
159	C₂×C₁₆	—	15	4	no	—
160	#97	—	5	1	no	—
161	#97	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
162	C₄×C₈	—	5	1	no	—
163	C₄×C₈	—	5	1	no	—
164	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
165	C₂×C₃×C₅	#504	1	2	yes	—
166	C₃×D₅	—	2	2	no	—
167	C₂³×C₃	—	5	1	no	—
168	C₂×C₃×C₄	—	5	1	no	—
169	C₂×C₃×C₄	—	5	2	no	—
170	C₂×C₃×C₄	—	5	2	no	—
171	C₂×C₃×C₄	—	5	2	no	—
172	C₂×C₃×C₄	D₅⋊C₈	1	2	yes	—
173	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
174	C₃×C₈	—	5	2	no	—
175	C₃×C₈	—	5	1	no	—
176	C₃×C₈	—	5	1	no	—
177	C₃×C₈	D₅⋊C₈	1	2	yes	—
178	C₃×C₈	C₄×AGL(1,𝔽₅)	1	2	yes	—
179	C₂²×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
180	C₂×D₅	—	2	2	no	—
181	C₂×D₅	C₄×C₃⋊C₈	1	2	yes	—
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
184	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
185	Dic₅	—	2	2	no	—
186	Dic₅	C₄×C₃⋊C₈	1	2	yes	—
187	C₂²×C₄	—	5	1	no	—
188	M₁₆	—	5	4	no	—
189	C₂×C₈	—	5	1	no	—
190	C₂×C₈	—	5	1	no	—
191	C₂×C₈	—	5	2	no	—
192	C₂×C₈	—	5	2	no	—
193	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
194	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
195	C₄⋊C₄	—	5	2	no	—
196	C₂²⋊C₄	—	5	2	no	—
197	C₄²	—	5	1	no	—
198	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
199	C₁₆	—	15	2	no	—
200	C₁₆	—	15	2	no	—
201	C₁₆	—	15	2	no	—
202	C₁₆	—	15	2	no	—
203	C₃×C₅	#2108	1	1	yes	characteristic, complemented, Hall
204	C₂²×C₃	—	5	2	no	—
205	C₂²×C₃	—	10	2	no	—
206	C₂²×C₃	C₂×D₅⋊C₈	1	1	yes	characteristic
207	C₃×C₄	—	2	2	no	—
208	C₃×C₄	—	5	2	no	—
209	C₃×C₄	—	10	2	no	—
210	C₃×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
211	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
212	C₂×C₅	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
213	C₂×C₅	#5140	1	2	yes	—
214	D₅	—	2	2	no	—
215	C₂³	—	5	1	no	—
216	C₂×C₄	—	5	1	no	—
217	C₂×C₄	—	5	2	no	—
218	C₂×C₄	—	5	2	no	—
219	C₂×C₄	—	5	2	no	—
220	C₂×C₄	D₅⋊(C₃⋊C₈)	1	2	yes	—
221	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
222	C₈	—	5	2	no	—
223	C₈	—	5	1	no	—
224	C₈	—	5	1	no	—
225	C₈	D₅⋊(C₃⋊C₈)	1	2	yes	—
226	C₈	C₄×ΓL(1,𝔽₁₆)	1	2	yes	—
227	C₂×C₃	—	10	2	no	—
228	C₂×C₃	C₄×D₅⋊C₈	1	1	yes	characteristic
229	C₂×C₃	#65725	1	2	yes	—
230	C₅	#69881	1	1	yes	characteristic, complemented, Sylow
231	C₂²	—	5	2	no	—
232	C₂²	—	10	2	no	—
233	C₂²	C₂×D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	5	2	no	—
236	C₄	—	10	2	no	—
237	C₄	C₂×D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
238	C₄	C₂×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
239	C₃	#10603868	1	1	yes	characteristic, complemented, Sylow
240	C₂	—	10	2	no	—
241	C₂	C₄×D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
242	C₂	#11752384	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Fitting	C₃×C₅×#97	C₄
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂²×C₃×D₅	C₂×C₈
cosocle	C₂×C₃×#168	C₂³
hypercenter	#97	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#2108
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
4. #11746757	C₂	semidirect
5. #11746756	C₂	semidirect
6. #11746730	C₂	semidirect
7. C₄×C₃⋊#674	C₂	semidirect
16. #91219	C₄	semidirect
17. D₅⋊(C₃⋊C₁₆)	C₄	semidirect
203. C₃×C₅	#2108	semidirect
230. C₅	#69881	semidirect
239. C₃	#10603868	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91219	2
C₂×D₅⋊(C₃⋊C₁₆)	2
C₃×#64997	2
#11746757	2
#11746756	2
#11746730	2
C₄×C₃⋊#674	2
#10603868	3
#69881	5