Subgroups — #14298715

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₃⋊#170)	C₂	1	2	yes	maximal
2	C₃×C₄²×Dic₅	C₂	1	1	yes	maximal, characteristic
3	#11746789	C₂	1	4	yes	maximal
4	#10604185	—	3	1	no	maximal, Hall
5	C₂²×C₃⋊#170	C₂²	1	4	yes	—
6	C₂×C₃×C₄²×C₅	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄×Dic₅	C₄	1	1	yes	characteristic
8	C₂×C₃×C₄×Dic₅	C₄	1	2	yes	—
9	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	—
10	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
11	C₄⋊(C₃⋊#170)	—	2	4	no	—
12	#69392	—	5	1	no	maximal, Hall
13	C₂×C₄⋊#170	—	3	2	no	—
14	C₄²×Dic₅	D₃	1	1	yes	characteristic
15	#64918	—	3	4	no	—
16	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
17	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
20	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
21	C₂×C₃⋊#170	—	2	4	no	—
22	C₂×C₃⋊#170	—	2	8	no	—
23	C₃×C₄²×C₅	C₈	1	4	yes	complemented
24	C₃×C₄×Dic₅	—	2	2	no	—
25	C₃×C₄×Dic₅	—	2	4	no	—
26	C₃×C₄×Dic₅	—	2	4	no	—
27	C₃×C₄×Dic₅	D₄	1	2	yes	—
28	C₃×C₄×Dic₅	Q₈	1	2	yes	—
29	C₃×C₄³	—	5	1	no	—
30	C₂×C₄⋊(C₃⋊C₈)	—	5	2	no	—
31	#4524	—	5	4	no	—
32	C₂²×#170	—	3	4	no	—
33	C₂×C₄²×C₅	Dic₃	1	1	yes	characteristic
34	C₂×C₄×Dic₅	Dic₃	1	1	yes	characteristic
35	C₂×C₄×Dic₅	Dic₃	1	2	yes	—
36	C₂×C₄×Dic₅	C₂×D₃	1	2	yes	—
37	C₂×C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
38	C₄⋊#170	—	6	4	no	—
39	#1831	—	15	1	no	Sylow
40	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	—	2	2	no	—
42	C₂×C₃×C₄×C₅	—	2	2	no	—
43	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
44	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
45	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
46	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
47	C₂×C₃×Dic₅	—	2	2	no	—
48	C₂×C₃×Dic₅	—	2	2	no	—
49	C₂×C₃×Dic₅	—	2	2	no	—
50	C₂×C₃×Dic₅	C₄⋊C₄	1	2	yes	—
51	C₂×C₃×Dic₅	M₁₆	1	4	yes	—
52	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃×Dic₅	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₃×Dic₅	#41	1	4	yes	—
55	C₃⋊#170	—	4	4	no	—
56	C₂×C₃×C₄²	—	5	1	no	—
57	C₂×C₃×C₄²	—	5	2	no	—
58	C₂×C₃×C₄²	—	5	2	no	—
59	C₂×C₃×C₄²	—	5	1	no	—
60	C₂×C₃×C₄²	AGL(1,𝔽₅)	1	1	yes	characteristic
61	C₂²×C₃⋊C₈	—	5	4	no	—
62	C₄⋊(C₃⋊C₈)	—	10	4	no	—
63	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
64	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
65	C₂²×Dic₅	C₂×Dic₃	1	2	yes	—
66	C₂²×Dic₅	C₂×Dic₃	1	1	yes	characteristic
67	C₂²×Dic₅	C₂²×D₃	1	1	yes	characteristic
68	C₂×#170	—	6	4	no	—
69	C₂×#170	—	6	8	no	—
70	C₄²×C₅	C₃⋊C₈	1	4	yes	complemented
71	C₄×Dic₅	—	2	2	no	—
72	C₄×Dic₅	—	2	4	no	—
73	C₄×Dic₅	—	2	4	no	—
74	C₄×Dic₅	Dic₆	1	2	yes	—
75	C₄×Dic₅	D₁₂	1	2	yes	—
76	C₂×C₄⋊C₈	—	15	2	no	—
77	C₄³	—	5	1	no	—
78	#336	—	15	4	no	—
79	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₃×C₅	#117	1	2	yes	—
81	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
82	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
83	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
84	C₃×C₄×C₅	—	2	4	no	—
85	C₃×C₄×C₅	C₄⋊C₈	1	4	yes	—
86	C₃×Dic₅	—	2	2	no	—
87	C₃×Dic₅	—	2	4	no	—
88	C₃×Dic₅	#124	1	2	yes	—
89	C₃×Dic₅	#125	1	2	yes	—
90	C₂²×C₃×C₄	—	5	1	no	—
91	C₂²×C₃×C₄	—	5	2	no	—
92	C₂²×C₃×C₄	—	5	1	no	—
93	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
94	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
95	C₃×C₄²	—	5	2	no	—
96	C₃×C₄²	—	5	2	no	—
97	C₃×C₄²	—	10	4	no	—
98	C₃×C₄²	—	10	4	no	—
99	C₃×C₄²	—	10	2	no	—
100	C₃×C₄²	#170	1	4	yes	complemented
101	C₂×C₃⋊C₈	—	10	4	no	—
102	C₂×C₃⋊C₈	—	10	8	no	—
103	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	C₂×C₃⋊C₈	1	4	yes	—
107	C₂×C₄×C₅	C₂×C₃⋊C₈	1	2	yes	—
108	C₂×C₄×C₅	#208	1	2	yes	—
109	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
110	C₂×Dic₅	—	2	2	no	—
111	C₂×Dic₅	—	2	2	no	—
112	C₂×Dic₅	—	2	2	no	—
113	C₂×Dic₅	#208	1	4	yes	—
114	C₂×Dic₅	C₄⋊Dic₃	1	2	yes	—
115	C₂×Dic₅	C₂×Dic₆	1	1	yes	characteristic
116	C₂×Dic₅	C₂×D₁₂	1	1	yes	characteristic
117	C₂×Dic₅	#235	1	4	yes	—
118	#170	—	12	4	no	—
119	C₂²×C₈	—	15	4	no	—
120	C₂×C₄²	—	5	1	no	—
121	C₂×C₄²	—	5	2	no	—
122	C₂×C₄²	—	5	2	no	—
123	C₂×C₄²	—	5	1	no	—
124	C₂×C₄²	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
125	C₄⋊C₈	—	30	4	no	—
126	C₂×C₃×C₅	#382	1	1	yes	characteristic
127	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
128	C₂×C₃×C₅	#423	1	1	yes	characteristic
129	C₂×C₃×C₅	#431	1	2	yes	—
130	C₂×C₃×C₅	#432	1	2	yes	—
131	C₂³×C₃	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	5	1	no	—
135	C₂×C₃×C₄	—	5	1	no	—
136	C₂×C₃×C₄	—	5	4	no	—
137	C₂×C₃×C₄	—	5	4	no	—
138	C₂×C₃×C₄	—	5	2	no	—
139	C₂×C₃×C₄	—	10	2	no	—
140	C₂×C₃×C₄	—	10	2	no	—
141	C₂×C₃×C₄	—	10	2	no	—
142	C₂×C₃×C₄	#700	1	2	yes	—
143	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
144	C₂×C₃×C₄	C₂×#170	1	4	yes	—
145	C₂×C₃×C₄	C₂×#170	1	2	yes	—
146	C₃⋊C₈	—	20	4	no	—
147	C₂²×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
148	C₂²×C₅	C₂×#208	1	1	yes	characteristic
149	C₂²×C₅	C₂×#208	1	2	yes	—
150	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
151	C₂²×C₅	#926	1	2	yes	—
152	C₄×C₅	—	2	4	no	—
153	C₄×C₅	C₄⋊(C₃⋊C₈)	1	4	yes	—
154	Dic₅	—	2	2	no	—
155	Dic₅	—	2	4	no	—
156	Dic₅	#870	1	2	yes	—
157	Dic₅	#875	1	2	yes	—
158	C₂²×C₄	—	5	1	no	—
159	C₂²×C₄	—	5	2	no	—
160	C₂²×C₄	—	5	1	no	—
161	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
162	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
163	C₂×C₈	—	30	4	no	—
164	C₂×C₈	—	30	8	no	—
165	C₄²	—	5	2	no	—
166	C₄²	—	5	2	no	—
167	C₄²	—	10	4	no	—
168	C₄²	—	10	4	no	—
169	C₄²	—	10	2	no	—
170	C₄²	C₃⋊#170	1	4	yes	complemented
171	C₃×C₅	#1831	1	1	yes	characteristic, complemented, Hall
172	C₂²×C₃	C₂×#700	1	1	yes	characteristic
173	C₂²×C₃	D₅⋊M₁₆	1	2	yes	—
174	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
175	C₂²×C₃	#4130	1	2	yes	—
176	C₂²×C₃	C₂²×#170	1	1	yes	characteristic
177	C₃×C₄	—	2	4	no	—
178	C₃×C₄	—	5	2	no	—
179	C₃×C₄	—	5	2	no	—
180	C₃×C₄	—	10	2	no	—
181	C₃×C₄	—	10	4	no	—
182	C₃×C₄	C₄⋊#170	1	4	yes	—
183	C₂×C₅	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
184	C₂×C₅	#4924	1	1	yes	characteristic
185	C₂×C₅	#4925	1	2	yes	—
186	C₂×C₅	#4926	1	2	yes	—
187	C₂×C₅	#4936	1	1	yes	characteristic
188	C₂³	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	5	1	no	—
192	C₂×C₄	—	5	1	no	—
193	C₂×C₄	—	5	4	no	—
194	C₂×C₄	—	5	4	no	—
195	C₂×C₄	—	5	2	no	—
196	C₂×C₄	—	10	2	no	—
197	C₂×C₄	—	10	2	no	—
198	C₂×C₄	—	10	2	no	—
199	C₂×C₄	#6758	1	2	yes	—
200	C₂×C₄	C₄⋊ΓL(1,𝔽₁₆)	1	2	yes	—
201	C₂×C₄	C₂×C₃⋊#170	1	4	yes	—
202	C₂×C₄	C₂×C₃⋊#170	1	2	yes	—
203	C₈	—	60	4	no	—
204	C₂×C₃	#65682	1	1	yes	characteristic
205	C₂×C₃	#65683	1	2	yes	—
206	C₂×C₃	#65689	1	1	yes	characteristic
207	C₂×C₃	C₂×C₄⋊#170	1	1	yes	characteristic
208	C₂×C₃	#65748	1	2	yes	—
209	C₅	#69392	1	1	yes	characteristic, complemented, Sylow
210	C₂²	C₂×#6758	1	1	yes	characteristic
211	C₂²	D₅⋊#208	1	2	yes	—
212	C₂²	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
213	C₂²	#91989	1	2	yes	—
214	C₂²	C₂²×C₃⋊#170	1	1	yes	characteristic
215	C₄	—	2	4	no	—
216	C₄	—	5	2	no	—
217	C₄	—	5	2	no	—
218	C₄	—	10	2	no	—
219	C₄	—	10	4	no	—
220	C₄	C₄⋊(C₃⋊#170)	1	4	yes	—
221	C₃	#10604185	1	1	yes	characteristic, complemented, Sylow
222	C₂	#11752341	1	1	yes	characteristic
223	C₂	#11752342	1	2	yes	—
224	C₂	#11752348	1	1	yes	characteristic
225	C₂	C₂×C₄⋊(C₃⋊#170)	1	1	yes	characteristic
226	C₂	#11752407	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×ΓL(1,𝔽₁₆)
commutator	C₂²×C₃×C₅	C₂²×C₈
Fitting	C₂×C₃×C₄²×C₅	C₄
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×Dic₅	C₂³
hypercenter	C₂×C₄²	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1831
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
23. C₃×C₄²×C₅	C₈	semidirect
70. C₄²×C₅	C₃⋊C₈	semidirect
100. C₃×C₄²	#170	semidirect
170. C₄²	C₃⋊#170	semidirect
171. C₃×C₅	#1831	semidirect
209. C₅	#69392	semidirect
221. C₃	#10604185	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₃⋊#170)	2
C₃×C₄²×Dic₅	2
#11746789	2
#10604185	3
#69392	5