Subgroups — #14312639

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×#91082	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×D₁₅	C₂	1	1	yes	maximal, characteristic
3	#11750790	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746323	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₃⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599865	—	3	1	no	maximal, Hall
9	C₂×#6705	C₄	1	1	yes	characteristic
10	C₂×C₈×D₁₅	C₂²	1	1	yes	characteristic
11	#91761	C₄	1	1	yes	characteristic
12	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	#91099	C₄	1	1	yes	characteristic
16	C₈×Dic₁₅	C₂²	1	1	yes	characteristic
17	C₂×C₃⋊#672	C₂²	1	1	yes	characteristic
18	C₂×C₃⋊#672	C₂²	1	1	yes	characteristic
19	#91087	C₄	1	1	yes	characteristic
20	#91082	C₄	1	4	yes	complemented
21	C₁₆×D₁₅	C₄	1	4	yes	complemented
22	#69072	—	5	1	no	maximal, Hall
23	C₂×#3930	—	3	1	no	—
24	C₂×D₅×C₁₆	—	3	1	no	—
25	#64978	—	3	1	no	—
26	C₅×#346	D₃	1	1	yes	characteristic, complemented
27	#64722	—	3	1	no	—
28	C₁₆×Dic₅	—	3	1	no	—
29	C₄×#672	—	3	1	no	—
30	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	#6714	C₈	1	2	yes	—
35	#6712	C₈	1	2	yes	—
36	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
39	#6705	C₈	1	4	yes	—
40	C₈×D₁₅	C₂×C₄	1	2	yes	—
41	C₈×D₁₅	C₂×C₄	1	2	yes	—
42	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	—
43	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	—
44	C₃⋊#672	C₂×C₄	1	2	yes	—
45	C₃⋊#672	C₂×C₄	1	2	yes	—
46	C₂×#798	—	5	1	no	—
47	C₂×D₃×C₁₆	—	5	1	no	—
48	#4689	—	5	1	no	—
49	C₃×#346	D₅	1	1	yes	characteristic, complemented
50	#4502	—	5	1	no	—
51	Dic₃×C₁₆	—	5	1	no	—
52	C₄×C₃⋊C₁₆	—	5	1	no	—
53	C₂×#676	—	3	1	no	—
54	C₂×C₈×D₅	—	3	1	no	—
55	#4018	—	3	1	no	—
56	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
57	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
58	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
59	#3947	—	3	1	no	—
60	C₈×Dic₅	—	3	1	no	—
61	C₂×#672	—	3	1	no	—
62	C₂×#672	—	3	1	no	—
63	#3935	—	3	1	no	—
64	#3930	—	3	4	no	—
65	D₅×C₁₆	—	3	4	no	—
66	#2108	—	15	1	no	Sylow
67	C₂²×D₁₅	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×Dic₁₅	C₂×C₈	1	2	yes	—
71	C₂×Dic₁₅	C₂×C₈	1	1	yes	characteristic
72	C₄×D₁₅	C₄²	1	2	yes	—
73	C₄×D₁₅	C₂×C₈	1	2	yes	—
74	C₃×C₅×C₈	C₂×C₈	1	2	yes	—
75	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₃⋊#168	C₄²	1	2	yes	—
78	C₃⋊#168	C₂×C₈	1	2	yes	—
79	C₂×ΓL(1,𝔽₂₅)	—	5	1	no	—
80	C₂×D₃×C₈	—	5	1	no	—
81	#872	—	5	1	no	—
82	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
83	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
84	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
85	#815	—	5	1	no	—
86	C₈×Dic₃	—	5	1	no	—
87	C₂×C₃⋊C₁₆	—	5	1	no	—
88	C₂×C₃⋊C₁₆	—	5	1	no	—
89	#803	—	5	1	no	—
90	#798	—	5	4	no	—
91	D₃×C₁₆	—	5	4	no	—
92	C₂×C₄×D₅	—	3	1	no	—
93	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
94	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
95	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
96	#685	—	3	2	no	—
97	#683	—	3	2	no	—
98	C₄×Dic₅	—	3	1	no	—
99	C₂×#168	—	3	1	no	—
100	C₂×#168	—	3	1	no	—
101	#676	—	3	4	no	—
102	C₈×D₅	—	3	2	no	—
103	C₈×D₅	—	3	2	no	—
104	C₅×C₁₆	C₄×D₃	1	2	yes	—
105	C₅×C₁₆	C₄×D₃	1	2	yes	—
106	#672	—	3	2	no	—
107	#672	—	3	2	no	—
108	C₂×M₃₂	—	15	1	no	—
109	C₂²×C₁₆	—	15	1	no	—
110	#405	—	15	1	no	—
111	#346	—	15	1	no	—
112	#346	D₁₅	1	1	yes	characteristic, complemented
113	C₄×C₁₆	—	15	1	no	—
114	C₄×C₁₆	—	15	1	no	—
115	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
116	C₂×D₁₅	—	2	2	no	—
117	C₂×D₁₅	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	Dic₁₅	—	2	2	no	—
122	Dic₁₅	C₄×C₈	1	2	yes	—
123	C₂×C₄×D₃	—	5	1	no	—
124	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
125	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
126	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
127	(C₂×D₃)⋊C₄	—	5	2	no	—
128	Dic₃⋊C₄	—	5	2	no	—
129	C₄×Dic₃	—	5	1	no	—
130	C₂×C₃⋊C₈	—	5	1	no	—
131	C₂×C₃⋊C₈	—	5	1	no	—
132	ΓL(1,𝔽₂₅)	—	5	4	no	—
133	D₃×C₈	—	5	2	no	—
134	D₃×C₈	—	5	2	no	—
135	C₃×C₁₆	C₄×D₅	1	2	yes	—
136	C₃×C₁₆	C₄×D₅	1	2	yes	—
137	C₃⋊C₁₆	—	5	2	no	—
138	C₃⋊C₁₆	—	5	2	no	—
139	C₂²×D₅	—	3	1	no	—
140	C₂×C₄×C₅	D₃×C₈	1	2	yes	—
141	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
142	C₂×Dic₅	—	3	1	no	—
143	C₂×Dic₅	—	3	2	no	—
144	C₄×D₅	—	3	2	no	—
145	C₄×D₅	—	3	2	no	—
146	C₅×C₈	D₃×C₈	1	2	yes	—
147	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
148	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
149	#168	—	3	2	no	—
150	#168	—	3	2	no	—
151	C₂×M₁₆	—	15	1	no	—
152	C₂²×C₈	—	15	1	no	—
153	#117	—	15	1	no	—
154	M₃₂	—	15	4	no	—
155	C₂×C₁₆	—	15	1	no	—
156	C₂×C₁₆	—	15	1	no	—
157	C₂×C₁₆	—	15	4	no	—
158	C₂×C₁₆	C₂×D₁₅	1	1	yes	characteristic
159	C₂×C₁₆	C₂×D₁₅	1	1	yes	characteristic
160	#97	—	15	1	no	—
161	#97	—	15	1	no	—
162	C₄×C₈	—	15	1	no	—
163	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
164	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
165	C₂×C₃×C₅	#504	1	2	yes	—
166	D₁₅	—	2	2	no	—
167	C₂²×D₃	—	5	1	no	—
168	C₂×C₃×C₄	C₈×D₅	1	2	yes	—
169	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
170	C₂×Dic₃	—	5	1	no	—
171	C₂×Dic₃	—	5	2	no	—
172	C₄×D₃	—	5	2	no	—
173	C₄×D₃	—	5	2	no	—
174	C₃×C₈	C₈×D₅	1	2	yes	—
175	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
176	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
177	C₃⋊C₈	—	5	2	no	—
178	C₃⋊C₈	—	5	2	no	—
179	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
180	C₂×D₅	—	3	2	no	—
181	C₂×D₅	—	6	2	no	—
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
184	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
185	Dic₅	—	3	2	no	—
186	Dic₅	—	6	2	no	—
187	C₂²×C₄	—	15	1	no	—
188	M₁₆	—	15	4	no	—
189	C₂×C₈	—	15	1	no	—
190	C₂×C₈	—	15	1	no	—
191	C₂×C₈	—	15	2	no	—
192	C₂×C₈	—	15	2	no	—
193	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
194	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
195	C₄⋊C₄	—	15	2	no	—
196	C₂²⋊C₄	—	15	2	no	—
197	C₄²	—	15	1	no	—
198	C₄²	C₄×D₁₅	1	1	yes	characteristic
199	C₁₆	—	15	2	no	—
200	C₁₆	—	15	2	no	—
201	C₁₆	C₄×D₁₅	1	2	yes	—
202	C₁₆	C₄×D₁₅	1	2	yes	—
203	C₃×C₅	#2108	1	1	yes	characteristic, complemented, Hall
204	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
205	C₂×D₃	—	5	2	no	—
206	C₂×D₃	—	10	2	no	—
207	C₃×C₄	—	2	2	no	—
208	C₃×C₄	C₄²×D₅	1	1	yes	characteristic
209	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
210	Dic₃	—	5	2	no	—
211	Dic₃	—	10	2	no	—
212	C₂×C₅	C₄×D₃×C₈	1	1	yes	characteristic
213	C₂×C₅	#4907	1	2	yes	—
214	D₅	—	6	2	no	—
215	C₂³	—	15	1	no	—
216	C₂×C₄	—	15	1	no	—
217	C₂×C₄	—	15	2	no	—
218	C₂×C₄	—	15	2	no	—
219	C₂×C₄	—	15	2	no	—
220	C₂×C₄	C₈×D₁₅	1	2	yes	—
221	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
222	C₈	—	15	2	no	—
223	C₈	—	15	2	no	—
224	C₈	C₈×D₁₅	1	2	yes	—
225	C₈	C₂×C₄×D₁₅	1	1	yes	characteristic
226	C₈	C₂×C₄×D₁₅	1	1	yes	characteristic
227	C₂×C₃	C₄×C₈×D₅	1	1	yes	characteristic
228	C₂×C₃	#65196	1	2	yes	—
229	D₃	—	10	2	no	—
230	C₅	#69072	1	1	yes	characteristic, complemented, Sylow
231	C₂²	—	15	2	no	—
232	C₂²	—	30	2	no	—
233	C₂²	C₂×C₈×D₁₅	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	15	2	no	—
236	C₄	—	30	2	no	—
237	C₄	C₄²×D₁₅	1	1	yes	characteristic
238	C₄	C₂×C₈×D₁₅	1	1	yes	characteristic
239	C₃	#10599865	1	1	yes	characteristic, complemented, Sylow
240	C₂	—	30	2	no	—
241	C₂	C₄×C₈×D₁₅	1	1	yes	characteristic
242	C₂	#11751008	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₅
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Fitting	C₃×C₅×#346	C₂
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂²×D₁₅	C₂×C₈
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	#346	D₁₅
hypocenter	C₃×C₅	#2108
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#346	C₂	semidirect
5. #11746323	C₂	semidirect
6. C₁₆×Dic₁₅	C₂	semidirect
7. C₄×C₃⋊#672	C₂	semidirect
20. #91082	C₄	semidirect
21. C₁₆×D₁₅	C₄	semidirect
26. C₅×#346	D₃	semidirect
49. C₃×#346	D₅	semidirect
112. #346	D₁₅	semidirect
203. C₃×C₅	#2108	semidirect
230. C₅	#69072	semidirect
239. C₃	#10599865	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91082	2
C₂×C₁₆×D₁₅	2
#11750790	2
C₃×C₅×#346	2
#11746323	2
C₁₆×Dic₁₅	2
C₄×C₃⋊#672	2
#10599865	3
#69072	5