Subgroups — C₂×C₃×#66068 #14483926

Order 1920.

2480 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#4093	C₂	1	4	yes	maximal, complemented
3	C₃×#66068	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃×#4041	C₂	1	4	yes	maximal, complemented
5	C₂×C₃×#4038	C₂	1	4	yes	maximal, complemented
6	C₂×C₃×C₄×D₂₀	C₂	1	2	yes	maximal, complemented
7	C₂×#66068	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
9	C₂²×C₃×D₂₀	C₂²	1	2	yes	—
10	C₂²×C₃×D₂₀	C₂²	1	4	yes	—
11	C₂²×C₃×C₄×D₅	C₂²	1	4	yes	—
12	C₃×C₅×C₄⋊Q₈	C₂²	1	8	yes	complemented
13	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₃×#4093	C₂²	1	32	yes	complemented
16	C₂×C₃×#685	C₂²	1	4	yes	—
17	C₂×C₃×#685	C₂²	1	8	yes	—
18	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
19	C₂×C₃×C₄×Dic₅	C₂²	1	4	yes	—
20	C₃×#4041	C₂²	1	32	yes	complemented
21	C₃×#4038	C₂²	1	32	yes	complemented
22	C₃×C₄×D₂₀	C₂²	1	16	yes	complemented
23	C₂×C₃×#555	—	5	1	no	maximal, Hall
24	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×#4093	C₂×C₃	1	4	yes	complemented
26	#66068	C₂×C₃	1	16	yes	direct product factor
27	C₂×#4041	C₂×C₃	1	4	yes	complemented
28	C₂×#4038	C₂×C₃	1	4	yes	complemented
29	C₂×C₄×D₂₀	C₂×C₃	1	2	yes	complemented
30	C₂³×C₃×D₅	C₂³	1	4	yes	—
31	C₂×C₃×C₅×Q₈	—	4	4	no	—
32	C₂×C₃×C₅×Q₈	C₂³	1	8	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
36	C₂²×C₃×Dic₅	C₂³	1	4	yes	—
37	C₂×C₃×D₂₀	—	2	8	no	—
38	C₂×C₃×D₂₀	—	4	8	no	—
39	C₂×C₃×D₂₀	C₂³	1	8	yes	—
40	C₂×C₃×D₂₀	C₂³	1	16	yes	—
41	C₂×C₃×C₄×D₅	—	2	16	no	—
42	C₂×C₃×C₄×D₅	C₂³	1	16	yes	—
43	C₃×C₅×C₄⋊C₄	C₂³	1	16	yes	—
44	C₃×C₄²×C₅	C₂³	1	4	yes	—
45	C₃×#685	C₂³	1	16	yes	—
46	C₃×#685	C₂³	1	32	yes	—
47	C₃×C₄⋊Dic₅	C₂³	1	8	yes	—
48	C₃×C₄×Dic₅	C₂³	1	16	yes	—
49	C₃×#555	—	5	16	no	—
50	C₂×C₃×C₄⋊Q₈	D₅	1	1	yes	characteristic, complemented
51	C₂×C₃×#124	—	5	4	no	—
52	C₂×C₃×#121	—	5	4	no	—
53	C₂×C₃×C₄×D₄	—	5	2	no	—
54	C₂×C₃×#117	—	5	4	no	—
55	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
56	C₂²×D₂₀	C₂²×C₃	1	2	yes	—
57	C₂²×D₂₀	C₂²×C₃	1	4	yes	—
58	C₂²×C₄×D₅	C₂²×C₃	1	4	yes	—
59	C₅×C₄⋊Q₈	C₂²×C₃	1	8	yes	complemented
60	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
61	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
62	#4093	C₂²×C₃	1	32	yes	complemented
63	C₂×#685	C₂²×C₃	1	4	yes	—
64	C₂×#685	C₂²×C₃	1	8	yes	—
65	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
66	C₂×C₄×Dic₅	C₂²×C₃	1	4	yes	—
67	#4041	C₂²×C₃	1	32	yes	complemented
68	#4038	C₂²×C₃	1	32	yes	complemented
69	C₄×D₂₀	C₂²×C₃	1	16	yes	complemented
70	C₂×#555	—	5	1	no	Sylow
71	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
72	C₂²×C₃×D₅	—	2	8	no	—
73	C₂²×C₃×D₅	—	2	8	no	—
74	C₂²×C₃×D₅	—	2	8	no	—
75	C₂²×C₃×D₅	C₂⁴	1	8	yes	—
76	C₃×C₅×Q₈	—	4	8	no	—
77	C₂×C₃×C₄×C₅	—	2	8	no	—
78	C₂×C₃×C₄×C₅	—	2	2	no	—
79	C₂×C₃×C₄×C₅	#41	1	8	yes	—
80	C₂×C₃×C₄×C₅	C₂⁴	1	8	yes	—
81	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
82	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
83	C₂×C₃×Dic₅	—	2	8	no	—
84	C₂×C₃×Dic₅	C₂⁴	1	8	yes	—
85	C₃×D₂₀	—	2	16	no	—
86	C₃×D₂₀	—	4	16	no	—
87	C₃×C₄×D₅	—	2	32	no	—
88	C₂²×C₃×Q₈	C₂×D₅	1	2	yes	—
89	C₂²×C₃×D₄	—	5	2	no	—
90	C₂²×C₃×D₄	—	5	4	no	—
91	C₂³×C₃×C₄	—	5	4	no	—
92	C₃×C₄⋊Q₈	C₂×D₅	1	8	yes	complemented
93	C₃×#124	—	5	32	no	—
94	C₃×#121	—	5	32	no	—
95	C₃×C₄×D₄	—	5	16	no	—
96	C₃×#117	—	5	32	no	—
97	C₂×C₃×C₄⋊C₄	—	5	2	no	—
98	C₂×C₃×C₄⋊C₄	C₂×D₅	1	4	yes	—
99	C₂×C₃×C₂²⋊C₄	—	5	8	no	—
100	C₂×C₃×C₂²⋊C₄	—	5	4	no	—
101	C₂×C₃×C₄²	—	5	4	no	—
102	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
103	C₂³×D₅	C₂³×C₃	1	4	yes	—
104	C₂×C₅×Q₈	—	4	4	no	—
105	C₂×C₅×Q₈	C₂³×C₃	1	8	yes	—
106	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
107	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
108	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
109	C₂²×Dic₅	C₂³×C₃	1	4	yes	—
110	C₂×D₂₀	—	2	8	no	—
111	C₂×D₂₀	—	4	8	no	—
112	C₂×D₂₀	C₂³×C₃	1	8	yes	—
113	C₂×D₂₀	C₂³×C₃	1	16	yes	—
114	C₂×C₄×D₅	—	2	16	no	—
115	C₂×C₄×D₅	C₂³×C₃	1	16	yes	—
116	C₅×C₄⋊C₄	C₂³×C₃	1	16	yes	—
117	C₄²×C₅	C₂³×C₃	1	4	yes	—
118	#685	C₂³×C₃	1	16	yes	—
119	#685	C₂³×C₃	1	32	yes	—
120	C₄⋊Dic₅	C₂³×C₃	1	8	yes	—
121	C₄×Dic₅	C₂³×C₃	1	16	yes	—
122	#555	—	5	16	no	—
123	C₂×C₄⋊Q₈	C₃×D₅	1	1	yes	characteristic, complemented
124	C₂×#124	—	5	4	no	—
125	C₂×#121	—	5	4	no	—
126	C₂×C₄×D₄	—	5	2	no	—
127	C₂×#117	—	5	4	no	—
128	C₂²×C₃×C₅	C₂×#41	1	4	yes	—
129	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
130	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
131	C₂×C₃×D₅	—	2	8	no	—
132	C₂×C₃×D₅	—	2	8	no	—
133	C₂×C₃×D₅	—	2	8	no	—
134	C₂×C₃×D₅	—	4	16	no	—
135	C₃×C₄×C₅	—	2	8	no	—
136	C₃×C₄×C₅	—	2	2	no	—
137	C₃×C₄×C₅	C₂×#41	1	8	yes	—
138	C₃×Dic₅	—	2	8	no	—
139	C₂⁴×C₃	—	5	4	no	—
140	C₂×C₃×Q₈	—	4	4	no	—
141	C₂×C₃×Q₈	C₂²×D₅	1	8	yes	—
142	C₂×C₃×D₄	—	5	8	no	—
143	C₂×C₃×D₄	—	5	16	no	—
144	C₂×C₃×D₄	—	10	8	no	—
145	C₂×C₃×D₄	—	20	8	no	—
146	C₂²×C₃×C₄	—	5	4	no	—
147	C₂²×C₃×C₄	—	5	16	no	—
148	C₂²×C₃×C₄	—	10	16	no	—
149	C₂²×C₃×C₄	C₂²×D₅	1	4	yes	—
150	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
151	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
152	C₃×C₄⋊C₄	—	5	8	no	—
153	C₃×C₄⋊C₄	C₂²×D₅	1	16	yes	—
154	C₃×C₂²⋊C₄	—	5	32	no	—
155	C₃×C₂²⋊C₄	—	5	16	no	—
156	C₃×C₄²	—	5	16	no	—
157	C₃×C₄²	C₂²×D₅	1	4	yes	—
158	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
159	C₂²×D₅	—	2	8	no	—
160	C₂²×D₅	—	2	8	no	—
161	C₂²×D₅	—	2	8	no	—
162	C₂²×D₅	C₂⁴×C₃	1	8	yes	—
163	C₅×Q₈	—	4	8	no	—
164	C₂×C₄×C₅	—	2	8	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	C₃×#41	1	8	yes	—
167	C₂×C₄×C₅	C₂⁴×C₃	1	8	yes	—
168	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
169	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
170	C₂×Dic₅	—	2	8	no	—
171	C₂×Dic₅	C₂⁴×C₃	1	8	yes	—
172	D₂₀	—	2	16	no	—
173	D₂₀	—	4	16	no	—
174	C₄×D₅	—	2	32	no	—
175	C₂²×Q₈	C₂×C₃×D₅	1	2	yes	—
176	C₂²×D₄	—	5	2	no	—
177	C₂²×D₄	—	5	4	no	—
178	C₂³×C₄	—	5	4	no	—
179	C₄⋊Q₈	C₂×C₃×D₅	1	8	yes	complemented
180	#124	—	5	32	no	—
181	#121	—	5	32	no	—
182	C₄×D₄	—	5	16	no	—
183	#117	—	5	32	no	—
184	C₂×C₄⋊C₄	—	5	2	no	—
185	C₂×C₄⋊C₄	C₂×C₃×D₅	1	4	yes	—
186	C₂×C₂²⋊C₄	—	5	8	no	—
187	C₂×C₂²⋊C₄	—	5	4	no	—
188	C₂×C₄²	—	5	4	no	—
189	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
190	C₂×C₃×C₅	#555	1	4	yes	complemented
191	C₂×C₃×C₅	C₂²×#41	1	2	yes	—
192	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
193	C₃×D₅	—	4	8	no	—
194	C₂³×C₃	—	5	8	no	—
195	C₂³×C₃	—	10	8	no	—
196	C₂³×C₃	—	10	8	no	—
197	C₂³×C₃	—	10	8	no	—
198	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
199	C₃×Q₈	—	4	8	no	—
200	C₃×D₄	—	10	16	no	—
201	C₃×D₄	—	20	16	no	—
202	C₂×C₃×C₄	—	2	8	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	5	8	no	—
205	C₂×C₃×C₄	—	10	8	no	—
206	C₂×C₃×C₄	—	10	32	no	—
207	C₂×C₃×C₄	#713	1	8	yes	—
208	C₂×C₃×C₄	C₂³×D₅	1	8	yes	—
209	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
210	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
211	C₂²×C₅	C₂×C₃×#41	1	4	yes	—
212	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
213	C₂²×C₅	C₂⁵×C₃	1	1	yes	characteristic
214	C₂×D₅	—	2	8	no	—
215	C₂×D₅	—	2	8	no	—
216	C₂×D₅	—	2	8	no	—
217	C₂×D₅	—	4	16	no	—
218	C₄×C₅	—	2	8	no	—
219	C₄×C₅	—	2	2	no	—
220	C₄×C₅	C₂×C₃×#41	1	8	yes	—
221	Dic₅	—	2	8	no	—
222	C₂⁴	—	5	4	no	—
223	C₂×Q₈	—	4	4	no	—
224	C₂×Q₈	C₂²×C₃×D₅	1	8	yes	—
225	C₂×D₄	—	5	8	no	—
226	C₂×D₄	—	5	16	no	—
227	C₂×D₄	—	10	8	no	—
228	C₂×D₄	—	20	8	no	—
229	C₂²×C₄	—	5	4	no	—
230	C₂²×C₄	—	5	16	no	—
231	C₂²×C₄	—	10	16	no	—
232	C₂²×C₄	C₂²×C₃×D₅	1	4	yes	—
233	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
234	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
235	C₄⋊C₄	—	5	8	no	—
236	C₄⋊C₄	C₂²×C₃×D₅	1	16	yes	—
237	C₂²⋊C₄	—	5	32	no	—
238	C₂²⋊C₄	—	5	16	no	—
239	C₄²	—	5	16	no	—
240	C₄²	C₂²×C₃×D₅	1	4	yes	—
241	C₃×C₅	C₂×#555	1	1	yes	characteristic, complemented, Hall
242	C₂²×C₃	—	10	8	no	—
243	C₂²×C₃	—	10	8	no	—
244	C₂²×C₃	—	10	8	no	—
245	C₂²×C₃	—	20	16	no	—
246	C₂²×C₃	#4144	1	2	yes	—
247	C₂²×C₃	C₂×#713	1	4	yes	—
248	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
249	C₃×C₄	—	2	8	no	—
250	C₃×C₄	—	2	2	no	—
251	C₃×C₄	—	10	8	no	—
252	C₃×C₄	C₂×#713	1	8	yes	—
253	C₂×C₅	C₃×#555	1	4	yes	complemented
254	C₂×C₅	C₂²×C₃×#41	1	2	yes	—
255	C₂×C₅	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
256	D₅	—	4	8	no	—
257	C₂³	—	5	8	no	—
258	C₂³	—	10	8	no	—
259	C₂³	—	10	8	no	—
260	C₂³	—	10	8	no	—
261	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
262	Q₈	—	4	8	no	—
263	D₄	—	10	16	no	—
264	D₄	—	20	16	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	5	8	no	—
268	C₂×C₄	—	10	8	no	—
269	C₂×C₄	—	10	32	no	—
270	C₂×C₄	C₃×#713	1	8	yes	—
271	C₂×C₄	C₂³×C₃×D₅	1	8	yes	—
272	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
273	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
274	C₂×C₃	—	20	8	no	—
275	C₂×C₃	#66068	1	4	yes	direct product factor
276	C₂×C₃	C₂×#4144	1	1	yes	characteristic
277	C₂×C₃	C₂²×#713	1	2	yes	—
278	C₅	C₂×C₃×#555	1	1	yes	characteristic, complemented, Sylow
279	C₂²	—	10	8	no	—
280	C₂²	—	10	8	no	—
281	C₂²	—	10	8	no	—
282	C₂²	—	20	16	no	—
283	C₂²	C₃×#4144	1	2	yes	—
284	C₂²	C₂×C₃×#713	1	4	yes	—
285	C₂²	C₂⁴×C₃×D₅	1	1	yes	characteristic
286	C₄	—	2	8	no	—
287	C₄	—	2	2	no	—
288	C₄	—	10	8	no	—
289	C₄	C₂×C₃×#713	1	8	yes	—
290	C₃	C₂×#66068	1	1	yes	characteristic, direct product factor, Sylow
291	C₂	—	20	8	no	—
292	C₂	C₃×#66068	1	4	yes	direct product factor
293	C₂	C₂×C₃×#4144	1	1	yes	characteristic
294	C₂	C₂²×C₃×#713	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁵×C₃
Frattini	C₂²	C₂⁴×C₃×D₅
Fitting	C₂×C₃×C₅×C₄⋊Q₈	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂×C₃×C₄⋊Q₈	D₅
hypocenter	C₅	C₂×C₃×#555
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×C₄⋊Q₈	C₂	semidirect
2. C₂×C₃×#4093	C₂	semidirect
3. C₃×#66068	C₂	direct (N × H)
4. C₂×C₃×#4041	C₂	semidirect
5. C₂×C₃×#4038	C₂	semidirect
6. C₂×C₃×C₄×D₂₀	C₂	semidirect
7. C₂×#66068	C₃	direct (N × H)
12. C₃×C₅×C₄⋊Q₈	C₂²	semidirect
15. C₃×#4093	C₂²	semidirect
20. C₃×#4041	C₂²	semidirect
21. C₃×#4038	C₂²	semidirect
22. C₃×C₄×D₂₀	C₂²	semidirect
24. C₂×C₅×C₄⋊Q₈	C₂×C₃	semidirect
25. C₂×#4093	C₂×C₃	semidirect
26. #66068	C₂×C₃	direct (N × H)
27. C₂×#4041	C₂×C₃	semidirect
28. C₂×#4038	C₂×C₃	semidirect
29. C₂×C₄×D₂₀	C₂×C₃	semidirect
50. C₂×C₃×C₄⋊Q₈	D₅	semidirect
59. C₅×C₄⋊Q₈	C₂²×C₃	semidirect
62. #4093	C₂²×C₃	semidirect
67. #4041	C₂²×C₃	semidirect
68. #4038	C₂²×C₃	semidirect
69. C₄×D₂₀	C₂²×C₃	semidirect
92. C₃×C₄⋊Q₈	C₂×D₅	semidirect
123. C₂×C₄⋊Q₈	C₃×D₅	semidirect
179. C₄⋊Q₈	C₂×C₃×D₅	semidirect
190. C₂×C₃×C₅	#555	semidirect
241. C₃×C₅	C₂×#555	semidirect
253. C₂×C₅	C₃×#555	semidirect
275. C₂×C₃	#66068	direct (N × H)
278. C₅	C₂×C₃×#555	semidirect
290. C₃	C₂×#66068	direct (N × H)
292. C₂	C₃×#66068	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₄⋊Q₈	2
C₂×C₃×#4093	2
C₃×#66068	2
C₂×C₃×#4041	2
C₂×C₃×#4038	2
C₂×C₃×C₄×D₂₀	2
C₂×#66068	3
C₂×C₃×#555	5

Subgroups — C2×C3×#66068 #14483926

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#66068 #14483926