Subgroups — C₂×#11746324 #14313819

Order 1920.

464 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₆₀	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#91102	C₂	1	1	yes	maximal, characteristic
4	#11746324	C₂	1	4	yes	maximal, direct product factor
5	C₂×#64723	—	3	1	no	maximal, Hall
6	C₂²×Dic₃₀	—	2	1	no	—
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₅	—	2	1	no	—
9	C₂×Dic₆₀	—	2	1	no	—
10	C₂×Dic₆₀	—	2	2	no	—
11	C₂×Dic₆₀	C₄	1	4	yes	complemented
12	C₂×Dic₆₀	C₂²	1	2	yes	—
13	C₂×C₃×C₅×C₁₆	—	2	2	no	—
14	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
15	#91102	—	2	1	no	—
16	#91102	C₂²	1	2	yes	—
17	C₂×#4503	—	5	1	no	maximal, Hall
18	C₂²×Dic₂₀	—	3	1	no	—
19	C₂²×C₅×C₁₆	D₃	1	1	yes	characteristic
20	C₂×#3950	—	3	1	no	—
21	#64723	—	3	4	no	—
22	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂²×Dic₁₅	—	4	1	no	—
24	C₂²×Dic₁₅	—	4	1	no	—
25	C₂×Dic₃₀	—	2	4	no	—
26	C₂×Dic₃₀	—	2	2	no	—
27	C₂×Dic₃₀	—	4	1	no	—
28	C₂×Dic₃₀	—	4	2	no	—
29	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
32	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₄⋊Dic₁₅	—	2	2	no	—
34	C₄⋊Dic₁₅	—	4	1	no	—
35	Dic₆₀	—	2	4	no	—
36	Dic₆₀	—	2	2	no	—
37	Dic₆₀	C₂×C₄	1	4	yes	complemented
38	C₃×C₅×C₁₆	—	2	2	no	—
39	C₂²×Dic₁₂	—	5	1	no	—
40	C₂²×C₃×C₁₆	D₅	1	1	yes	characteristic
41	C₂×#818	—	5	1	no	—
42	#4503	—	5	4	no	—
43	C₂²×Dic₁₀	—	6	1	no	—
44	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
45	C₂×C₄⋊Dic₅	—	6	1	no	—
46	C₂×Dic₂₀	—	3	4	no	—
47	C₂×Dic₂₀	—	3	2	no	—
48	C₂×Dic₂₀	—	6	1	no	—
49	C₂×Dic₂₀	—	6	2	no	—
50	C₂×C₅×C₁₆	—	2	2	no	—
51	C₂×C₅×C₁₆	C₂×D₃	1	2	yes	—
52	#3950	—	3	2	no	—
53	#3950	—	6	1	no	—
54	C₂×#358	—	15	1	no	Sylow
55	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
57	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
59	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×Dic₁₅	—	4	4	no	—
61	C₂×Dic₁₅	—	4	2	no	—
62	C₂×Dic₁₅	—	4	2	no	—
63	C₂×Dic₁₅	—	8	2	no	—
64	Dic₃₀	—	2	4	no	—
65	Dic₃₀	—	4	4	no	—
66	Dic₃₀	—	4	2	no	—
67	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
68	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₆	—	10	1	no	—
71	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
72	C₂×C₄⋊Dic₃	—	10	1	no	—
73	C₂×Dic₁₂	—	5	4	no	—
74	C₂×Dic₁₂	—	5	2	no	—
75	C₂×Dic₁₂	—	10	1	no	—
76	C₂×Dic₁₂	—	10	2	no	—
77	C₂×C₃×C₁₆	—	2	2	no	—
78	C₂×C₃×C₁₆	C₂×D₅	1	2	yes	—
79	#818	—	5	2	no	—
80	#818	—	10	1	no	—
81	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
82	C₂²×Dic₅	—	12	1	no	—
83	C₂²×Dic₅	—	12	1	no	—
84	C₂×Dic₁₀	—	6	4	no	—
85	C₂×Dic₁₀	—	6	2	no	—
86	C₂×Dic₁₀	—	12	2	no	—
87	C₂×Dic₁₀	—	12	1	no	—
88	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
89	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
90	C₂×C₅×C₈	#67	1	2	yes	—
91	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
92	C₄⋊Dic₅	—	6	2	no	—
93	C₄⋊Dic₅	—	12	1	no	—
94	Dic₂₀	—	3	4	no	—
95	Dic₂₀	—	6	4	no	—
96	Dic₂₀	—	6	2	no	—
97	C₅×C₁₆	—	2	2	no	—
98	C₂²×Q₁₆	—	15	1	no	—
99	C₂²×C₁₆	D₁₅	1	1	yes	characteristic
100	C₂×#107	—	15	1	no	—
101	#358	—	15	4	no	—
102	C₂²×C₃×C₅	#102	1	2	yes	—
103	C₂²×C₃×C₅	SD₃₂	1	2	yes	—
104	C₂²×C₃×C₅	Q₃₂	1	2	yes	—
105	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
106	C₃×C₄×C₅	#102	1	2	yes	—
107	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
109	Dic₁₅	—	4	4	no	—
110	Dic₁₅	—	8	2	no	—
111	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
112	C₂²×Dic₃	—	20	1	no	—
113	C₂²×Dic₃	—	20	1	no	—
114	C₂×Dic₆	—	10	4	no	—
115	C₂×Dic₆	—	10	2	no	—
116	C₂×Dic₆	—	20	2	no	—
117	C₂×Dic₆	—	20	1	no	—
118	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
119	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
120	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
121	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
122	C₄⋊Dic₃	—	10	2	no	—
123	C₄⋊Dic₃	—	20	1	no	—
124	Dic₁₂	—	5	4	no	—
125	Dic₁₂	—	10	4	no	—
126	Dic₁₂	—	10	2	no	—
127	C₃×C₁₆	—	2	2	no	—
128	C₂³×C₅	D₂₄	1	1	yes	characteristic
129	C₂×C₄×C₅	#204	1	2	yes	—
130	C₂×C₄×C₅	D₂₄	1	1	yes	characteristic
131	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
132	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
133	C₂×Dic₅	—	12	4	no	—
134	C₂×Dic₅	—	12	2	no	—
135	C₂×Dic₅	—	12	2	no	—
136	C₂×Dic₅	—	24	2	no	—
137	Dic₁₀	—	6	4	no	—
138	Dic₁₀	—	12	4	no	—
139	Dic₁₀	—	12	2	no	—
140	C₅×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
141	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
142	C₅×C₈	C₂×#67	1	1	yes	characteristic
143	C₂²×Q₈	—	30	1	no	—
144	C₂×Q₁₆	—	15	4	no	—
145	C₂×Q₁₆	—	15	2	no	—
146	C₂×Q₁₆	—	30	1	no	—
147	C₂×Q₁₆	—	30	2	no	—
148	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
149	C₂×C₄⋊C₄	—	30	1	no	—
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	C₂×D₁₅	1	2	yes	—
152	#107	—	15	2	no	—
153	#107	—	30	1	no	—
154	C₂×C₃×C₅	#358	1	4	yes	complemented
155	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
156	C₂×C₃×C₅	C₂×SD₃₂	1	1	yes	characteristic
157	C₂×C₃×C₅	C₂×Q₃₂	1	1	yes	characteristic
158	C₂³×C₃	D₄₀	1	1	yes	characteristic
159	C₂×C₃×C₄	#677	1	2	yes	—
160	C₂×C₃×C₄	D₄₀	1	1	yes	characteristic
161	C₂×C₃×C₄	#685	1	2	yes	—
162	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
163	C₂×Dic₃	—	20	4	no	—
164	C₂×Dic₃	—	20	2	no	—
165	C₂×Dic₃	—	20	2	no	—
166	C₂×Dic₃	—	40	2	no	—
167	Dic₆	—	10	4	no	—
168	Dic₆	—	20	4	no	—
169	Dic₆	—	20	2	no	—
170	C₃×C₈	#685	1	2	yes	—
171	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
172	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
173	C₂²×C₅	#800	1	2	yes	—
174	C₂²×C₅	Dic₂₄	1	2	yes	—
175	C₂²×C₅	#821	1	2	yes	—
176	C₂²×C₅	C₂×D₂₄	1	1	yes	characteristic
177	C₄×C₅	#821	1	2	yes	—
178	C₄×C₅	C₂×#204	1	1	yes	characteristic
179	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
180	Dic₅	—	12	4	no	—
181	Dic₅	—	24	2	no	—
182	C₂×Q₈	—	30	4	no	—
183	C₂×Q₈	—	30	2	no	—
184	C₂×Q₈	—	60	2	no	—
185	C₂×Q₈	—	60	1	no	—
186	C₂²×C₄	—	60	1	no	—
187	C₂²×C₄	—	60	1	no	—
188	C₂²×C₄	D₆₀	1	1	yes	characteristic
189	Q₁₆	—	15	4	no	—
190	Q₁₆	—	30	4	no	—
191	Q₁₆	—	30	2	no	—
192	C₂×C₈	C₄×D₁₅	1	2	yes	—
193	C₂×C₈	D₆₀	1	1	yes	characteristic
194	C₂×C₈	#1220	1	2	yes	—
195	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
196	C₄⋊C₄	—	30	2	no	—
197	C₄⋊C₄	—	60	1	no	—
198	C₁₆	—	2	2	no	—
199	C₃×C₅	C₂×#358	1	1	yes	characteristic, complemented, Hall
200	C₂²×C₃	#3932	1	2	yes	—
201	C₂²×C₃	Dic₄₀	1	2	yes	—
202	C₂²×C₃	#3953	1	2	yes	—
203	C₂²×C₃	C₂×D₄₀	1	1	yes	characteristic
204	C₃×C₄	#3953	1	2	yes	—
205	C₃×C₄	C₂×#677	1	1	yes	characteristic
206	C₃×C₄	C₂×#685	1	1	yes	characteristic
207	Dic₃	—	20	4	no	—
208	Dic₃	—	40	2	no	—
209	C₂×C₅	#4503	1	4	yes	complemented
210	C₂×C₅	C₂×#800	1	1	yes	characteristic
211	C₂×C₅	C₂×Dic₂₄	1	1	yes	characteristic
212	C₂×C₅	C₂×#821	1	1	yes	characteristic
213	C₂³	D₁₂₀	1	1	yes	characteristic
214	Q₈	—	30	4	no	—
215	Q₈	—	60	2	no	—
216	Q₈	—	60	4	no	—
217	C₂×C₄	—	60	2	no	—
218	C₂×C₄	—	60	4	no	—
219	C₂×C₄	—	60	2	no	—
220	C₂×C₄	—	120	2	no	—
221	C₂×C₄	#6706	1	2	yes	—
222	C₂×C₄	D₁₂₀	1	1	yes	characteristic
223	C₂×C₄	#6714	1	2	yes	—
224	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
225	C₈	#6714	1	2	yes	—
226	C₈	C₂×C₄×D₁₅	1	1	yes	characteristic
227	C₈	C₂×#1220	1	1	yes	characteristic
228	C₂×C₃	#64723	1	4	yes	complemented
229	C₂×C₃	C₂×#3932	1	1	yes	characteristic
230	C₂×C₃	C₂×Dic₄₀	1	1	yes	characteristic
231	C₂×C₃	C₂×#3953	1	1	yes	characteristic
232	C₅	C₂×#4503	1	1	yes	characteristic, complemented, Sylow
233	C₂²	#91084	1	2	yes	—
234	C₂²	Dic₁₂₀	1	2	yes	—
235	C₂²	#91105	1	2	yes	—
236	C₂²	C₂×D₁₂₀	1	1	yes	characteristic
237	C₄	—	60	4	no	—
238	C₄	—	120	2	no	—
239	C₄	#91105	1	2	yes	—
240	C₄	C₂×#6706	1	1	yes	characteristic
241	C₄	C₂×#6714	1	1	yes	characteristic
242	C₃	C₂×#64723	1	1	yes	characteristic, complemented, Sylow
243	C₂	#11746324	1	4	yes	direct product factor
244	C₂	C₂×#91084	1	1	yes	characteristic
245	C₂	C₂×Dic₁₂₀	1	1	yes	characteristic
246	C₂	C₂×#91105	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₁₂₀
commutator	C₃×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₅
Fitting	C₂²×C₃×C₅×C₁₆	C₂
socle, prime radical	C₂³×C₃×C₅	D₈
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂²×C₁₆	D₁₅
hypocenter	C₃×C₅	C₂×#358
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
4. #11746324	C₂	direct (N × H)
11. C₂×Dic₆₀	C₄	semidirect
37. Dic₆₀	C₂×C₄	semidirect
154. C₂×C₃×C₅	#358	semidirect
199. C₃×C₅	C₂×#358	semidirect
209. C₂×C₅	#4503	semidirect
228. C₂×C₃	#64723	semidirect
232. C₅	C₂×#4503	semidirect
242. C₃	C₂×#64723	semidirect
243. C₂	#11746324	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₆₀	2
C₂²×C₃×C₅×C₁₆	2
C₂×#91102	2
#11746324	2
C₂×#64723	3
C₂×#4503	5

Subgroups — C2×#11746324 #14313819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11746324 #14313819