Subgroups — #11998193

Order 1280.

482 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	C₂×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#10602427	C₂	1	2	yes	maximal
3	C₂×Q₈⋊#168	C₂	1	2	yes	maximal
4	C₂×#64678	C₂	1	1	yes	maximal, characteristic
5	C₂×#64674	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×Q₁₆	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
8	C₄×C₅×Q₁₆	—	2	2	no	—
9	C₄×C₅×Q₁₆	—	2	2	no	—
10	C₂×C₅×#107	C₄	1	1	yes	characteristic
11	C₂×C₅×#103	C₄	1	2	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#168	C₂²	1	2	yes	—
14	C₂×C₄×#168	C₂²	1	1	yes	characteristic
15	C₂×C₄×#168	C₂²	1	1	yes	characteristic
16	Q₈⋊#168	—	2	4	no	—
17	#64744	—	2	4	no	—
18	#64678	—	2	2	no	—
19	#64674	—	2	2	no	—
20	#9662	—	5	1	no	maximal, Sylow
21	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
22	C₂×C₅×Q₁₆	—	2	4	no	—
23	C₂×C₅×Q₁₆	—	2	2	no	—
24	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₅×Q₈	—	2	2	no	—
27	C₄×C₅×Q₈	—	2	2	no	—
28	C₄×C₅×Q₈	—	2	2	no	—
29	C₄×C₅×Q₈	—	2	2	no	—
30	C₂×C₅×C₄⋊C₄	—	2	2	no	—
31	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅	—	2	2	no	—
33	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₂²×#168	—	2	1	no	—
35	C₂²×#168	—	2	2	no	—
36	C₂²×#168	D₄	1	2	yes	—
37	C₅×#107	—	2	2	no	—
38	C₅×#103	—	2	4	no	—
39	C₄×C₅×C₈	D₄	1	2	yes	—
40	C₄×C₅×C₈	D₄	1	2	yes	—
41	C₄⋊#168	—	2	2	no	—
42	C₄⋊#168	—	2	2	no	—
43	C₄×#168	—	2	1	no	—
44	C₄×#168	—	2	1	no	—
45	C₄×#168	—	2	1	no	—
46	C₄×#168	D₄	1	2	yes	—
47	C₂×C₄×Q₁₆	D₅	1	1	yes	characteristic
48	#1984	—	5	2	no	—
49	C₂×#335	—	5	1	no	—
50	C₂×Q₈⋊C₈	—	5	2	no	—
51	C₂×#322	—	5	1	no	—
52	C₂×C₅×Q₈	—	2	2	no	—
53	C₂×C₅×Q₈	—	2	2	no	—
54	C₂×C₅×Q₈	—	2	2	no	—
55	C₂×C₅×Q₈	—	2	2	no	—
56	C₂×C₅×Q₈	—	2	2	no	—
57	C₂×C₅×Q₈	M₁₆	1	4	yes	—
58	C₂²×C₄×C₅	—	2	2	no	—
59	C₂²×C₄×C₅	—	2	2	no	—
60	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	#41	1	1	yes	characteristic
63	C₅×Q₁₆	—	2	8	no	—
64	C₂×C₅×C₈	—	2	2	no	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
67	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
68	C₅×C₄⋊C₄	—	2	2	no	—
69	C₅×C₄⋊C₄	—	2	2	no	—
70	C₅×C₄⋊C₄	—	4	2	no	—
71	C₅×C₄⋊C₄	—	4	2	no	—
72	C₄²×C₅	—	4	2	no	—
73	C₄²×C₅	—	4	2	no	—
74	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅	#41	1	1	yes	characteristic
78	C₂×#168	—	2	2	no	—
79	C₂×#168	—	2	2	no	—
80	C₂×#168	—	2	2	no	—
81	C₂×#168	—	4	2	no	—
82	C₂×#168	—	4	2	no	—
83	C₂×#168	—	4	4	no	—
84	C₂×#168	Q₁₆	1	4	yes	complemented
85	C₂²×Q₁₆	Dic₅	1	1	yes	characteristic
86	C₂×C₄×Q₈	C₂×D₅	1	2	yes	—
87	C₄×Q₁₆	—	2	2	no	—
88	C₄×Q₁₆	—	2	2	no	—
89	C₂×#107	Dic₅	1	1	yes	characteristic
90	C₂×C₄⋊C₈	—	5	2	no	—
91	C₂×#103	Dic₅	1	2	yes	—
92	C₂×C₄×C₈	—	5	1	no	—
93	C₂×C₄×C₈	—	5	1	no	—
94	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
95	#336	—	10	4	no	—
96	#335	—	10	2	no	—
97	Q₈⋊C₈	—	10	4	no	—
98	#322	—	10	2	no	—
99	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
100	C₅×Q₈	—	2	4	no	—
101	C₅×Q₈	—	2	4	no	—
102	C₅×Q₈	—	4	2	no	—
103	C₅×Q₈	—	4	2	no	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	2	4	no	—
106	C₂×C₄×C₅	—	4	2	no	—
107	C₂×C₄×C₅	—	4	2	no	—
108	C₂×C₄×C₅	—	4	2	no	—
109	C₂×C₄×C₅	—	4	4	no	—
110	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅	#121	1	1	yes	characteristic
113	C₂×C₄×C₅	#121	1	1	yes	characteristic
114	C₂×C₄×C₅	#124	1	1	yes	characteristic
115	C₂×C₄×C₅	#127	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
117	C₂×C₄×C₅	#131	1	2	yes	—
118	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
119	C₂×C₄×C₅	#135	1	2	yes	—
120	C₅×C₈	—	2	2	no	—
121	C₅×C₈	—	2	2	no	—
122	#168	—	2	4	no	—
123	#168	—	4	2	no	—
124	#168	—	8	2	no	—
125	C₂²×Q₈	C₂×Dic₅	1	2	yes	—
126	C₂×Q₁₆	—	2	4	no	—
127	C₂×Q₁₆	—	2	2	no	—
128	C₂²×C₈	—	5	2	no	—
129	C₂²×C₈	—	10	1	no	—
130	C₂²×C₈	—	10	2	no	—
131	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
132	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
133	C₄×Q₈	—	2	2	no	—
134	C₄×Q₈	—	2	2	no	—
135	C₄×Q₈	—	2	2	no	—
136	C₄×Q₈	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
139	C₂×C₄²	—	2	2	no	—
140	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
141	#107	—	2	2	no	—
142	C₄⋊C₈	—	10	2	no	—
143	C₄⋊C₈	—	10	2	no	—
144	#103	—	2	4	no	—
145	C₄×C₈	—	5	2	no	—
146	C₄×C₈	—	10	1	no	—
147	C₄×C₈	—	10	1	no	—
148	C₄×C₈	—	10	1	no	—
149	C₄×C₈	C₂²⋊D₅	1	2	yes	—
150	C₄×C₈	C₂²⋊D₅	1	2	yes	—
151	C₂²×C₅	#390	1	1	yes	characteristic
152	C₂²×C₅	#435	1	2	yes	—
153	C₂²×C₅	C₄×Q₁₆	1	2	yes	—
154	C₂²×C₅	#444	1	2	yes	—
155	C₄×C₅	—	2	1	no	—
156	C₄×C₅	—	2	1	no	—
157	C₄×C₅	—	4	4	no	—
158	C₄×C₅	—	8	2	no	—
159	C₄×C₅	#407	1	1	yes	characteristic
160	C₄×C₅	#408	1	1	yes	characteristic
161	C₄×C₅	#435	1	2	yes	—
162	C₄×C₅	#467	1	2	yes	—
163	C₄×C₅	#494	1	1	yes	characteristic
164	C₄×C₅	#495	1	1	yes	characteristic
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	2	2	no	—
169	C₂×Q₈	—	2	2	no	—
170	C₂×Q₈	#681	1	4	yes	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
174	C₂²×C₄	#711	1	1	yes	characteristic
175	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
176	Q₁₆	—	2	8	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	5	4	no	—
180	C₂×C₈	—	10	2	no	—
181	C₂×C₈	—	10	2	no	—
182	C₂×C₈	—	10	2	no	—
183	C₂×C₈	—	20	2	no	—
184	C₂×C₈	—	20	4	no	—
185	C₂×C₈	—	20	2	no	—
186	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
187	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄²	—	4	2	no	—
193	C₄²	—	4	2	no	—
194	C₄²	D₄×D₅	1	1	yes	characteristic
195	C₄²	#711	1	1	yes	characteristic
196	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
197	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
198	C₂×C₅	#1584	1	4	yes	—
199	C₂×C₅	#1955	1	1	yes	characteristic
200	C₂×C₅	#1969	1	1	yes	characteristic
201	C₂×C₅	#1975	1	1	yes	characteristic
202	C₂³	D₄×Dic₅	1	1	yes	characteristic
203	Q₈	—	2	4	no	—
204	Q₈	—	2	4	no	—
205	Q₈	—	4	2	no	—
206	Q₈	—	4	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	D₅×Q₁₆	1	2	yes	—
214	C₂×C₄	#4065	1	2	yes	—
215	C₂×C₄	C₂×#681	1	2	yes	—
216	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
217	C₂×C₄	#4082	1	1	yes	characteristic
218	C₂×C₄	#4084	1	1	yes	characteristic
219	C₂×C₄	#4084	1	1	yes	characteristic
220	C₂×C₄	#4086	1	1	yes	characteristic
221	C₂×C₄	#4094	1	2	yes	—
222	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	2	2	no	—
225	C₈	—	10	4	no	—
226	C₈	—	20	2	no	—
227	C₈	—	40	2	no	—
228	C₅	#9662	1	1	yes	characteristic, complemented, Sylow
229	C₂²	#65300	1	2	yes	—
230	C₂²	Q₁₆×Dic₅	1	2	yes	—
231	C₂²	#65486	1	2	yes	—
232	C₂²	#65510	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	4	no	—
236	C₄	—	8	2	no	—
237	C₄	#65300	1	2	yes	—
238	C₄	#65470	1	1	yes	characteristic
239	C₄	#65477	1	2	yes	—
240	C₄	#65480	1	1	yes	characteristic
241	C₄	#65496	1	1	yes	characteristic
242	C₄	#65523	1	1	yes	characteristic
243	C₂	#10601696	1	4	yes	—
244	C₂	#10602408	1	1	yes	characteristic
245	C₂	#10602859	1	1	yes	characteristic
246	C₂	#10602878	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₄×C₅×Q₁₆	C₂
socle, prime radical	C₂³×C₅	C₄×D₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×C₄×Q₁₆	D₅
hypocenter	C₅	#9662
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
84. C₂×#168	Q₁₆	semidirect
228. C₅	#9662	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×Q₁₆	2
#10602427	2
C₂×Q₈⋊#168	2
C₂×#64678	2
C₂×#64674	2
#9662	5