Subgroups — C₂×C₃×#66063 #14481730

Order 1920.

2160 conjugacy classes of subgroups.

The subgroup lattice has 552 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×#4092	C₂	1	2	yes	maximal
3	C₂×C₃×#4090	C₂	1	2	yes	maximal, complemented
4	C₃×#66063	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₃×#4042	C₂	1	2	yes	maximal
6	C₂×C₃×#4038	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃×D₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃×#4036	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃×#4035	C₂	1	2	yes	maximal, complemented
10	C₂×C₃×#4034	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×#4018	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃×#4016	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#66063	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
15	C₂²×C₃×C₄×D₅	C₂²	1	2	yes	—
16	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
17	C₂²×C₃×Dic₁₀	C₂²	1	2	yes	—
18	C₃×C₅×C₄⋊Q₈	C₂²	1	8	yes	complemented
19	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
23	C₃×#4092	C₂²	1	16	yes	—
24	C₃×#4090	C₂²	1	16	yes	complemented
25	C₂×C₃×#685	C₂²	1	1	yes	characteristic
26	C₂×C₃×#685	C₂²	1	2	yes	—
27	C₂×C₃×#685	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
29	C₂×C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
30	C₂×C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
31	C₂×C₃×#683	C₂²	1	1	yes	characteristic
32	C₂×C₃×#683	C₂²	1	2	yes	—
33	C₂×C₃×#683	C₂²	1	2	yes	—
34	C₂×C₃×#683	C₂²	1	2	yes	—
35	C₂×C₃×#683	C₂²	1	2	yes	—
36	C₂×C₃×#683	C₂²	1	2	yes	—
37	C₂×C₃×#683	C₂²	1	1	yes	characteristic
38	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	—
39	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
40	C₃×#4042	C₂²	1	16	yes	—
41	C₃×#4038	C₂²	1	8	yes	—
42	C₃×D₅×C₄⋊C₄	C₂²	1	8	yes	—
43	C₃×#4036	C₂²	1	8	yes	complemented
44	C₃×#4035	C₂²	1	16	yes	complemented
45	C₃×#4034	C₂²	1	8	yes	complemented
46	C₃×#4018	C₂²	1	8	yes	—
47	C₃×#4016	C₂²	1	8	yes	complemented
48	C₂×C₃×#544	—	5	1	no	maximal, Hall
49	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂×#4092	C₂×C₃	1	2	yes	—
51	C₂×#4090	C₂×C₃	1	2	yes	complemented
52	#66063	C₂×C₃	1	16	yes	direct product factor
53	C₂×#4042	C₂×C₃	1	2	yes	—
54	C₂×#4038	C₂×C₃	1	1	yes	characteristic
55	C₂×D₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
56	C₂×#4036	C₂×C₃	1	1	yes	characteristic, complemented
57	C₂×#4035	C₂×C₃	1	2	yes	complemented
58	C₂×#4034	C₂×C₃	1	1	yes	characteristic, complemented
59	C₂×#4018	C₂×C₃	1	1	yes	characteristic
60	C₂×#4016	C₂×C₃	1	1	yes	characteristic, complemented
61	C₂³×C₃×D₅	C₂³	1	1	yes	characteristic
62	C₂×C₃×C₅×Q₈	—	4	4	no	—
63	C₂×C₃×C₅×Q₈	C₂³	1	8	yes	—
64	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
65	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
66	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
67	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
68	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
69	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
70	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
71	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
72	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
73	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
74	C₂×C₃×C₄×D₅	—	4	4	no	—
75	C₂×C₃×C₄×D₅	Q₈	1	8	yes	complemented
76	C₂×C₃×C₄×D₅	C₂³	1	8	yes	—
77	C₂×C₃×C₄×D₅	C₂³	1	4	yes	—
78	C₂×C₃×Dic₁₀	—	4	4	no	—
79	C₂×C₃×Dic₁₀	C₂³	1	8	yes	—
80	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
81	C₃×C₅×C₄⋊C₄	C₂³	1	8	yes	—
82	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
83	C₃×C₄²×C₅	C₂³	1	4	yes	—
84	C₃×#685	C₂³	1	4	yes	—
85	C₃×#685	C₂³	1	8	yes	—
86	C₃×#685	C₂³	1	4	yes	—
87	C₃×C₄⋊Dic₅	C₂³	1	8	yes	—
88	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
89	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
90	C₃×#683	C₂³	1	4	yes	—
91	C₃×#683	C₂³	1	8	yes	—
92	C₃×#683	C₂³	1	8	yes	—
93	C₃×#683	C₂³	1	8	yes	—
94	C₃×#683	C₂³	1	8	yes	—
95	C₃×#683	C₂³	1	8	yes	—
96	C₃×#683	C₂³	1	4	yes	—
97	C₃×C₄×Dic₅	C₂³	1	8	yes	—
98	C₃×C₄×Dic₅	C₂³	1	4	yes	—
99	C₃×#544	—	5	16	no	—
100	C₂×C₃×C₄⋊Q₈	—	5	2	no	—
101	C₂×C₃×C₄⋊Q₈	—	5	1	no	—
102	C₂×C₃×C₄⋊Q₈	D₅	1	1	yes	characteristic, complemented
103	C₂×C₃×#125	—	5	2	no	—
104	C₂×C₃×#125	—	5	1	no	—
105	C₂×C₃×#125	—	5	1	no	—
106	C₂×C₃×C₂²⋊Q₈	—	5	2	no	—
107	C₂×C₃×C₂²⋊Q₈	—	5	2	no	—
108	C₂×C₃×#117	—	5	1	no	—
109	C₂×C₃×#117	—	5	1	no	—
110	C₂²×C₃×C₄⋊C₄	—	5	1	no	—
111	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
112	C₂²×C₄×D₅	C₂²×C₃	1	2	yes	—
113	C₂²×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
114	C₂²×Dic₁₀	C₂²×C₃	1	2	yes	—
115	C₅×C₄⋊Q₈	C₂²×C₃	1	8	yes	complemented
116	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
117	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
118	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
119	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
120	#4092	C₂²×C₃	1	16	yes	—
121	#4090	C₂²×C₃	1	16	yes	complemented
122	C₂×#685	C₂²×C₃	1	1	yes	characteristic
123	C₂×#685	C₂²×C₃	1	1	yes	characteristic
124	C₂×#685	C₂²×C₃	1	2	yes	—
125	C₂×C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
126	C₂×C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
127	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
128	C₂×#683	C₂²×C₃	1	2	yes	—
129	C₂×#683	C₂²×C₃	1	2	yes	—
130	C₂×#683	C₂²×C₃	1	1	yes	characteristic
131	C₂×#683	C₂²×C₃	1	1	yes	characteristic
132	C₂×#683	C₂²×C₃	1	2	yes	—
133	C₂×#683	C₂²×C₃	1	2	yes	—
134	C₂×#683	C₂²×C₃	1	2	yes	—
135	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
136	C₂×C₄×Dic₅	C₂²×C₃	1	2	yes	—
137	#4042	C₂²×C₃	1	16	yes	—
138	#4038	C₂²×C₃	1	8	yes	—
139	D₅×C₄⋊C₄	C₂²×C₃	1	8	yes	—
140	#4036	C₂²×C₃	1	8	yes	complemented
141	#4035	C₂²×C₃	1	16	yes	complemented
142	#4034	C₂²×C₃	1	8	yes	complemented
143	#4018	C₂²×C₃	1	8	yes	—
144	#4016	C₂²×C₃	1	8	yes	complemented
145	C₂×#544	—	5	1	no	Sylow
146	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
147	C₂²×C₃×D₅	—	2	2	no	—
148	C₂²×C₃×D₅	—	2	2	no	—
149	C₂²×C₃×D₅	C₂×Q₈	1	4	yes	—
150	C₂²×C₃×D₅	C₂⁴	1	2	yes	—
151	C₃×C₅×Q₈	—	4	8	no	—
152	C₂×C₃×C₄×C₅	—	2	4	no	—
153	C₂×C₃×C₄×C₅	—	2	4	no	—
154	C₂×C₃×C₄×C₅	—	2	2	no	—
155	C₂×C₃×C₄×C₅	—	2	2	no	—
156	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
157	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
158	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
159	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
160	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
161	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
162	C₂×C₃×Dic₅	—	2	2	no	—
163	C₂×C₃×Dic₅	—	2	4	no	—
164	C₂×C₃×Dic₅	—	2	4	no	—
165	C₂×C₃×Dic₅	—	2	2	no	—
166	C₂×C₃×Dic₅	C₂×Q₈	1	4	yes	—
167	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
168	C₂×C₃×Dic₅	C₂⁴	1	2	yes	—
169	C₂×C₃×Dic₅	C₂⁴	1	2	yes	—
170	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
171	C₂×C₃×Dic₅	C₂⁴	1	2	yes	—
172	C₃×C₄×D₅	—	4	8	no	—
173	C₃×C₄×D₅	C₂×Q₈	1	16	yes	complemented
174	C₃×Dic₁₀	—	4	8	no	—
175	C₂²×C₃×Q₈	—	5	2	no	—
176	C₂²×C₃×Q₈	C₂×D₅	1	2	yes	—
177	C₂³×C₃×C₄	—	5	1	no	—
178	C₂³×C₃×C₄	—	5	2	no	—
179	C₃×C₄⋊Q₈	—	5	16	no	—
180	C₃×C₄⋊Q₈	—	5	8	no	—
181	C₃×C₄⋊Q₈	C₂×D₅	1	8	yes	complemented
182	C₃×#125	—	5	16	no	—
183	C₃×#125	—	5	8	no	—
184	C₃×#125	—	5	8	no	—
185	C₃×C₂²⋊Q₈	—	5	16	no	—
186	C₃×C₂²⋊Q₈	—	5	16	no	—
187	C₃×#117	—	5	8	no	—
188	C₃×#117	—	5	8	no	—
189	C₂×C₃×C₄⋊C₄	—	5	2	no	—
190	C₂×C₃×C₄⋊C₄	—	5	1	no	—
191	C₂×C₃×C₄⋊C₄	—	5	1	no	—
192	C₂×C₃×C₄⋊C₄	—	5	1	no	—
193	C₂×C₃×C₄⋊C₄	—	5	1	no	—
194	C₂×C₃×C₄⋊C₄	—	5	2	no	—
195	C₂×C₃×C₄⋊C₄	—	5	2	no	—
196	C₂×C₃×C₄⋊C₄	—	5	2	no	—
197	C₂×C₃×C₄⋊C₄	—	5	2	no	—
198	C₂×C₃×C₄⋊C₄	—	5	2	no	—
199	C₂×C₃×C₄⋊C₄	—	5	8	no	—
200	C₂×C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
201	C₂×C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
202	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
203	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
204	C₂×C₃×C₂²⋊C₄	—	5	1	no	—
205	C₂×C₃×C₂²⋊C₄	—	5	1	no	—
206	C₂×C₃×C₄²	—	5	2	no	—
207	C₂×C₃×C₄²	—	5	1	no	—
208	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
209	C₂³×D₅	C₂³×C₃	1	1	yes	characteristic
210	C₂×C₅×Q₈	—	4	4	no	—
211	C₂×C₅×Q₈	C₂³×C₃	1	8	yes	—
212	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
213	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
214	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
215	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
216	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
217	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
218	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
219	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
220	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
221	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
222	C₂×C₄×D₅	—	4	4	no	—
223	C₂×C₄×D₅	C₃×Q₈	1	8	yes	complemented
224	C₂×C₄×D₅	C₂³×C₃	1	8	yes	—
225	C₂×C₄×D₅	C₂³×C₃	1	4	yes	—
226	C₂×Dic₁₀	—	4	4	no	—
227	C₂×Dic₁₀	C₂³×C₃	1	8	yes	—
228	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
229	C₅×C₄⋊C₄	C₂³×C₃	1	8	yes	—
230	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
231	C₄²×C₅	C₂³×C₃	1	4	yes	—
232	#685	C₂³×C₃	1	4	yes	—
233	#685	C₂³×C₃	1	4	yes	—
234	#685	C₂³×C₃	1	8	yes	—
235	C₄⋊Dic₅	C₂³×C₃	1	4	yes	—
236	C₄⋊Dic₅	C₂³×C₃	1	4	yes	—
237	C₄⋊Dic₅	C₂³×C₃	1	8	yes	—
238	#683	C₂³×C₃	1	8	yes	—
239	#683	C₂³×C₃	1	8	yes	—
240	#683	C₂³×C₃	1	4	yes	—
241	#683	C₂³×C₃	1	4	yes	—
242	#683	C₂³×C₃	1	8	yes	—
243	#683	C₂³×C₃	1	8	yes	—
244	#683	C₂³×C₃	1	8	yes	—
245	C₄×Dic₅	C₂³×C₃	1	4	yes	—
246	C₄×Dic₅	C₂³×C₃	1	8	yes	—
247	#544	—	5	16	no	—
248	C₂×C₄⋊Q₈	—	5	2	no	—
249	C₂×C₄⋊Q₈	—	5	1	no	—
250	C₂×C₄⋊Q₈	C₃×D₅	1	1	yes	characteristic, complemented
251	C₂×#125	—	5	2	no	—
252	C₂×#125	—	5	1	no	—
253	C₂×#125	—	5	1	no	—
254	C₂×C₂²⋊Q₈	—	5	2	no	—
255	C₂×C₂²⋊Q₈	—	5	2	no	—
256	C₂×#117	—	5	1	no	—
257	C₂×#117	—	5	1	no	—
258	C₂²×C₄⋊C₄	—	5	1	no	—
259	C₂²×C₃×C₅	C₂²×Q₈	1	2	yes	—
260	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
261	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
262	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
263	C₂×C₃×D₅	—	2	8	no	—
264	C₂×C₃×D₅	—	2	2	no	—
265	C₂×C₃×D₅	—	2	2	no	—
266	C₂×C₃×D₅	C₂²×Q₈	1	4	yes	—
267	C₃×C₄×C₅	—	2	4	no	—
268	C₃×C₄×C₅	—	2	4	no	—
269	C₃×C₄×C₅	—	2	2	no	—
270	C₃×C₄×C₅	—	2	2	no	—
271	C₃×C₄×C₅	C₂²×Q₈	1	4	yes	—
272	C₃×Dic₅	—	2	2	no	—
273	C₃×Dic₅	—	2	4	no	—
274	C₃×Dic₅	—	2	4	no	—
275	C₃×Dic₅	—	2	2	no	—
276	C₃×Dic₅	C₂²×Q₈	1	4	yes	—
277	C₂⁴×C₃	—	5	1	no	—
278	C₂×C₃×Q₈	—	4	4	no	—
279	C₂×C₃×Q₈	—	5	8	no	—
280	C₂×C₃×Q₈	—	20	4	no	—
281	C₂×C₃×Q₈	C₂²×D₅	1	8	yes	—
282	C₂²×C₃×C₄	—	5	1	no	—
283	C₂²×C₃×C₄	—	5	8	no	—
284	C₂²×C₃×C₄	—	5	4	no	—
285	C₂²×C₃×C₄	—	5	1	no	—
286	C₂²×C₃×C₄	—	5	2	no	—
287	C₂²×C₃×C₄	—	5	1	no	—
288	C₂²×C₃×C₄	—	5	2	no	—
289	C₂²×C₃×C₄	—	5	8	no	—
290	C₂²×C₃×C₄	—	20	4	no	—
291	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
292	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
293	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
294	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
295	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
296	C₃×C₄⋊C₄	—	5	8	no	—
297	C₃×C₄⋊C₄	—	5	4	no	—
298	C₃×C₄⋊C₄	—	5	4	no	—
299	C₃×C₄⋊C₄	—	5	8	no	—
300	C₃×C₄⋊C₄	—	5	8	no	—
301	C₃×C₄⋊C₄	—	5	8	no	—
302	C₃×C₄⋊C₄	—	5	8	no	—
303	C₃×C₄⋊C₄	—	5	4	no	—
304	C₃×C₄⋊C₄	—	5	4	no	—
305	C₃×C₄⋊C₄	—	5	8	no	—
306	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
307	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
308	C₃×C₄⋊C₄	C₂²×D₅	1	8	yes	—
309	C₃×C₂²⋊C₄	—	5	8	no	—
310	C₃×C₂²⋊C₄	—	5	4	no	—
311	C₃×C₂²⋊C₄	—	5	4	no	—
312	C₃×C₄²	—	5	8	no	—
313	C₃×C₄²	—	5	4	no	—
314	C₃×C₄²	C₂²×D₅	1	4	yes	—
315	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
316	C₂²×D₅	—	2	2	no	—
317	C₂²×D₅	—	2	2	no	—
318	C₂²×D₅	C₂×C₃×Q₈	1	4	yes	—
319	C₂²×D₅	C₂⁴×C₃	1	2	yes	—
320	C₅×Q₈	—	4	8	no	—
321	C₂×C₄×C₅	—	2	4	no	—
322	C₂×C₄×C₅	—	2	4	no	—
323	C₂×C₄×C₅	—	2	2	no	—
324	C₂×C₄×C₅	—	2	2	no	—
325	C₂×C₄×C₅	C₂×C₃×Q₈	1	4	yes	—
326	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
327	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
328	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
329	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
330	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
331	C₂×Dic₅	—	2	2	no	—
332	C₂×Dic₅	—	2	4	no	—
333	C₂×Dic₅	—	2	2	no	—
334	C₂×Dic₅	—	2	4	no	—
335	C₂×Dic₅	C₂×C₃×Q₈	1	4	yes	—
336	C₂×Dic₅	C₂⁴×C₃	1	2	yes	—
337	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
338	C₂×Dic₅	C₂⁴×C₃	1	2	yes	—
339	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
340	C₂×Dic₅	C₂⁴×C₃	1	2	yes	—
341	C₄×D₅	—	4	8	no	—
342	C₄×D₅	C₂×C₃×Q₈	1	16	yes	complemented
343	Dic₁₀	—	4	8	no	—
344	C₂²×Q₈	—	5	2	no	—
345	C₂²×Q₈	C₂×C₃×D₅	1	2	yes	—
346	C₂³×C₄	—	5	1	no	—
347	C₂³×C₄	—	5	2	no	—
348	C₄⋊Q₈	—	5	16	no	—
349	C₄⋊Q₈	—	5	8	no	—
350	C₄⋊Q₈	C₂×C₃×D₅	1	8	yes	complemented
351	#125	—	5	16	no	—
352	#125	—	5	8	no	—
353	#125	—	5	8	no	—
354	C₂²⋊Q₈	—	5	16	no	—
355	C₂²⋊Q₈	—	5	16	no	—
356	#117	—	5	8	no	—
357	#117	—	5	8	no	—
358	C₂×C₄⋊C₄	—	5	2	no	—
359	C₂×C₄⋊C₄	—	5	1	no	—
360	C₂×C₄⋊C₄	—	5	1	no	—
361	C₂×C₄⋊C₄	—	5	1	no	—
362	C₂×C₄⋊C₄	—	5	1	no	—
363	C₂×C₄⋊C₄	—	5	2	no	—
364	C₂×C₄⋊C₄	—	5	2	no	—
365	C₂×C₄⋊C₄	—	5	2	no	—
366	C₂×C₄⋊C₄	—	5	2	no	—
367	C₂×C₄⋊C₄	—	5	2	no	—
368	C₂×C₄⋊C₄	—	5	8	no	—
369	C₂×C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
370	C₂×C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
371	C₂×C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
372	C₂×C₂²⋊C₄	—	5	2	no	—
373	C₂×C₂²⋊C₄	—	5	1	no	—
374	C₂×C₂²⋊C₄	—	5	1	no	—
375	C₂×C₄²	—	5	2	no	—
376	C₂×C₄²	—	5	1	no	—
377	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
378	C₂×C₃×C₅	#544	1	4	yes	complemented
379	C₂×C₃×C₅	C₂³×Q₈	1	1	yes	characteristic
380	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
381	C₂×C₃×C₅	C₂×2₋¹⁺⁴	1	1	yes	characteristic
382	C₃×D₅	—	2	4	no	—
383	C₂³×C₃	—	5	4	no	—
384	C₂³×C₃	—	5	2	no	—
385	C₂³×C₃	—	10	2	no	—
386	C₂³×C₃	—	10	2	no	—
387	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
388	C₃×Q₈	—	4	8	no	—
389	C₃×Q₈	—	20	8	no	—
390	C₂×C₃×C₄	—	2	4	no	—
391	C₂×C₃×C₄	—	2	2	no	—
392	C₂×C₃×C₄	—	2	4	no	—
393	C₂×C₃×C₄	—	2	2	no	—
394	C₂×C₃×C₄	—	5	4	no	—
395	C₂×C₃×C₄	—	5	2	no	—
396	C₂×C₃×C₄	—	5	16	no	—
397	C₂×C₃×C₄	—	5	2	no	—
398	C₂×C₃×C₄	—	5	4	no	—
399	C₂×C₃×C₄	—	5	2	no	—
400	C₂×C₃×C₄	—	5	4	no	—
401	C₂×C₃×C₄	—	10	4	no	—
402	C₂×C₃×C₄	—	10	2	no	—
403	C₂×C₃×C₄	—	10	2	no	—
404	C₂×C₃×C₄	—	10	4	no	—
405	C₂×C₃×C₄	—	20	8	no	—
406	C₂×C₃×C₄	Q₈×D₅	1	4	yes	—
407	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
408	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
409	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
410	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
411	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
412	C₂²×C₅	C₂²×C₃×Q₈	1	2	yes	—
413	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
414	C₂²×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
415	C₂²×C₅	C₂⁵×C₃	1	1	yes	characteristic
416	C₂×D₅	—	2	2	no	—
417	C₂×D₅	—	2	2	no	—
418	C₂×D₅	—	2	8	no	—
419	C₂×D₅	C₂²×C₃×Q₈	1	4	yes	—
420	C₄×C₅	—	2	4	no	—
421	C₄×C₅	—	2	4	no	—
422	C₄×C₅	—	2	2	no	—
423	C₄×C₅	—	2	2	no	—
424	C₄×C₅	C₂²×C₃×Q₈	1	4	yes	—
425	Dic₅	—	2	2	no	—
426	Dic₅	—	2	4	no	—
427	Dic₅	—	2	2	no	—
428	Dic₅	—	2	4	no	—
429	Dic₅	C₂²×C₃×Q₈	1	4	yes	—
430	C₂⁴	—	5	1	no	—
431	C₂×Q₈	—	4	4	no	—
432	C₂×Q₈	—	5	8	no	—
433	C₂×Q₈	—	20	4	no	—
434	C₂×Q₈	C₂²×C₃×D₅	1	8	yes	—
435	C₂²×C₄	—	5	1	no	—
436	C₂²×C₄	—	5	8	no	—
437	C₂²×C₄	—	5	4	no	—
438	C₂²×C₄	—	5	1	no	—
439	C₂²×C₄	—	5	2	no	—
440	C₂²×C₄	—	5	1	no	—
441	C₂²×C₄	—	5	2	no	—
442	C₂²×C₄	—	5	8	no	—
443	C₂²×C₄	—	20	4	no	—
444	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
445	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
446	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
447	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
448	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
449	C₄⋊C₄	—	5	8	no	—
450	C₄⋊C₄	—	5	4	no	—
451	C₄⋊C₄	—	5	4	no	—
452	C₄⋊C₄	—	5	4	no	—
453	C₄⋊C₄	—	5	4	no	—
454	C₄⋊C₄	—	5	8	no	—
455	C₄⋊C₄	—	5	8	no	—
456	C₄⋊C₄	—	5	8	no	—
457	C₄⋊C₄	—	5	8	no	—
458	C₄⋊C₄	—	5	8	no	—
459	C₄⋊C₄	C₂²×C₃×D₅	1	4	yes	—
460	C₄⋊C₄	C₂²×C₃×D₅	1	4	yes	—
461	C₄⋊C₄	C₂²×C₃×D₅	1	8	yes	—
462	C₂²⋊C₄	—	5	8	no	—
463	C₂²⋊C₄	—	5	4	no	—
464	C₂²⋊C₄	—	5	4	no	—
465	C₄²	—	5	8	no	—
466	C₄²	—	5	4	no	—
467	C₄²	C₂²×C₃×D₅	1	4	yes	—
468	C₃×C₅	C₂×#544	1	1	yes	characteristic, complemented, Hall
469	C₂²×C₃	—	5	4	no	—
470	C₂²×C₃	—	10	8	no	—
471	C₂²×C₃	—	10	2	no	—
472	C₂²×C₃	—	10	2	no	—
473	C₂²×C₃	#4144	1	2	yes	—
474	C₂²×C₃	C₂×Q₈×D₅	1	2	yes	—
475	C₂²×C₃	#4147	1	2	yes	—
476	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
477	C₃×C₄	—	2	4	no	—
478	C₃×C₄	—	2	2	no	—
479	C₃×C₄	—	2	4	no	—
480	C₃×C₄	—	2	2	no	—
481	C₃×C₄	—	5	4	no	—
482	C₃×C₄	—	10	4	no	—
483	C₃×C₄	—	10	2	no	—
484	C₃×C₄	—	10	2	no	—
485	C₃×C₄	—	10	4	no	—
486	C₃×C₄	C₂×Q₈×D₅	1	4	yes	—
487	C₂×C₅	C₃×#544	1	4	yes	complemented
488	C₂×C₅	C₂³×C₃×Q₈	1	1	yes	characteristic
489	C₂×C₅	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
490	C₂×C₅	C₂×C₃×2₋¹⁺⁴	1	1	yes	characteristic
491	D₅	—	2	4	no	—
492	C₂³	—	5	4	no	—
493	C₂³	—	5	2	no	—
494	C₂³	—	10	2	no	—
495	C₂³	—	10	2	no	—
496	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
497	Q₈	—	4	8	no	—
498	Q₈	—	20	8	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	5	4	no	—
504	C₂×C₄	—	5	2	no	—
505	C₂×C₄	—	5	16	no	—
506	C₂×C₄	—	5	2	no	—
507	C₂×C₄	—	5	4	no	—
508	C₂×C₄	—	5	2	no	—
509	C₂×C₄	—	5	4	no	—
510	C₂×C₄	—	10	4	no	—
511	C₂×C₄	—	10	2	no	—
512	C₂×C₄	—	10	2	no	—
513	C₂×C₄	—	10	4	no	—
514	C₂×C₄	—	20	8	no	—
515	C₂×C₄	C₃×Q₈×D₅	1	4	yes	—
516	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
517	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
518	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
519	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
520	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
521	C₂×C₃	—	10	4	no	—
522	C₂×C₃	#66063	1	4	yes	direct product factor
523	C₂×C₃	C₂×#4144	1	1	yes	characteristic
524	C₂×C₃	C₂²×Q₈×D₅	1	1	yes	characteristic
525	C₂×C₃	C₂×#4147	1	1	yes	characteristic
526	C₅	C₂×C₃×#544	1	1	yes	characteristic, complemented, Sylow
527	C₂²	—	5	4	no	—
528	C₂²	—	10	2	no	—
529	C₂²	—	10	2	no	—
530	C₂²	—	10	8	no	—
531	C₂²	C₃×#4144	1	2	yes	—
532	C₂²	C₂×C₃×Q₈×D₅	1	2	yes	—
533	C₂²	C₃×#4147	1	2	yes	—
534	C₂²	C₂⁴×C₃×D₅	1	1	yes	characteristic
535	C₄	—	2	4	no	—
536	C₄	—	2	2	no	—
537	C₄	—	2	4	no	—
538	C₄	—	2	2	no	—
539	C₄	—	5	4	no	—
540	C₄	—	10	4	no	—
541	C₄	—	10	2	no	—
542	C₄	—	10	2	no	—
543	C₄	—	10	4	no	—
544	C₄	C₂×C₃×Q₈×D₅	1	4	yes	—
545	C₃	C₂×#66063	1	1	yes	characteristic, direct product factor, Sylow
546	C₂	—	10	4	no	—
547	C₂	C₃×#66063	1	4	yes	direct product factor
548	C₂	C₂×C₃×#4144	1	1	yes	characteristic
549	C₂	C₂²×C₃×Q₈×D₅	1	1	yes	characteristic
550	C₂	C₂×C₃×#4147	1	1	yes	characteristic

	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
prime radical	C₂³×C₃×D₅	C₂³
hypercenter	C₂×C₃×C₄⋊Q₈	D₅
hypocenter	C₅	C₂×C₃×#544
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
3. C₂×C₃×#4090	C₂	semidirect
4. C₃×#66063	C₂	direct (N × H)
8. C₂×C₃×#4036	C₂	semidirect
9. C₂×C₃×#4035	C₂	semidirect
10. C₂×C₃×#4034	C₂	semidirect
12. C₂×C₃×#4016	C₂	semidirect
13. C₂×#66063	C₃	direct (N × H)
18. C₃×C₅×C₄⋊Q₈	C₂²	semidirect
24. C₃×#4090	C₂²	semidirect
43. C₃×#4036	C₂²	semidirect
44. C₃×#4035	C₂²	semidirect
45. C₃×#4034	C₂²	semidirect
47. C₃×#4016	C₂²	semidirect
49. C₂×C₅×C₄⋊Q₈	C₂×C₃	semidirect
51. C₂×#4090	C₂×C₃	semidirect
52. #66063	C₂×C₃	direct (N × H)
56. C₂×#4036	C₂×C₃	semidirect
57. C₂×#4035	C₂×C₃	semidirect
58. C₂×#4034	C₂×C₃	semidirect
60. C₂×#4016	C₂×C₃	semidirect
75. C₂×C₃×C₄×D₅	Q₈	semidirect
102. C₂×C₃×C₄⋊Q₈	D₅	semidirect
115. C₅×C₄⋊Q₈	C₂²×C₃	semidirect
121. #4090	C₂²×C₃	semidirect
140. #4036	C₂²×C₃	semidirect
141. #4035	C₂²×C₃	semidirect
142. #4034	C₂²×C₃	semidirect
144. #4016	C₂²×C₃	semidirect
173. C₃×C₄×D₅	C₂×Q₈	semidirect
181. C₃×C₄⋊Q₈	C₂×D₅	semidirect
223. C₂×C₄×D₅	C₃×Q₈	semidirect
250. C₂×C₄⋊Q₈	C₃×D₅	semidirect
342. C₄×D₅	C₂×C₃×Q₈	semidirect
350. C₄⋊Q₈	C₂×C₃×D₅	semidirect
378. C₂×C₃×C₅	#544	semidirect
468. C₃×C₅	C₂×#544	semidirect
487. C₂×C₅	C₃×#544	semidirect
522. C₂×C₃	#66063	direct (N × H)
526. C₅	C₂×C₃×#544	semidirect
545. C₃	C₂×#66063	direct (N × H)
547. C₂	C₃×#66063	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×#4092	2
C₂×C₃×#4090	2
C₃×#66063	2
C₂×C₃×#4042	2
C₂×C₃×#4038	2
C₂×C₃×D₅×C₄⋊C₄	2
C₂×C₃×#4036	2
C₂×C₃×#4035	2
C₂×C₃×#4034	2
C₂×C₃×#4018	2
C₂×C₃×#4016	2
C₂×#66063	3
C₂×C₃×#544	5

Subgroups — C2×C3×#66063 #14481730

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#66063 #14481730