Subgroups — #11988937

Order 1280.

614 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#10613780	C₂	1	2	yes	maximal
3	#10598152	C₂	1	4	yes	maximal, complemented
4	C₂²×C₅×M₁₆	C₄	1	2	yes	—
5	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₄×C₅×M₁₆	—	2	2	no	—
7	C₄×C₅×M₁₆	—	2	2	no	—
8	C₂×C₅×#97	C₂²	1	2	yes	—
9	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
10	#65496	C₄	1	4	yes	—
11	#65215	—	2	2	no	—
12	#65215	C₄	1	4	yes	—
13	#65214	—	2	2	no	—
14	#65214	C₂²	1	4	yes	—
15	#65213	—	2	2	no	—
16	C₄×#681	—	2	2	no	—
17	#7920	—	5	1	no	maximal, Sylow
18	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂×C₅×M₁₆	—	2	2	no	—
21	C₂×C₅×M₁₆	—	2	2	no	—
22	C₂×C₅×M₁₆	—	2	8	no	—
23	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₅	D₄	1	2	yes	—
29	C₂×C₄²×C₅	D₄	1	2	yes	—
30	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
31	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₂×#681	—	2	4	no	—
34	C₅×#97	—	2	4	no	—
35	C₄×C₅×C₈	—	2	4	no	—
36	C₂²⋊#168	—	2	4	no	—
37	C₂²⋊#168	C₂×C₄	1	8	yes	—
38	C₄⋊#168	—	2	4	no	—
39	C₄⋊#168	—	2	4	no	—
40	#3935	—	2	4	no	—
41	C₄×#168	—	2	4	no	—
42	#2920	—	5	2	no	—
43	C₂×C₄×M₁₆	D₅	1	1	yes	characteristic
44	#1281	—	5	4	no	—
45	C₂⁴×C₅	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₅	—	2	4	no	—
47	C₂²×C₄×C₅	—	4	4	no	—
48	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₄²	1	2	yes	—
50	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₅×M₁₆	—	2	8	no	—
61	C₅×M₁₆	—	2	8	no	—
62	C₂×C₅×C₈	—	2	4	no	—
63	C₂×C₅×C₈	—	2	4	no	—
64	C₂×C₅×C₈	M₁₆	1	8	yes	complemented
65	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₅	C₄⋊C₄	1	2	yes	—
69	C₄²×C₅	C₄⋊C₄	1	2	yes	—
70	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
71	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
72	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₅	C₂×D₄	1	2	yes	—
74	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
75	#681	—	4	8	no	—
76	C₂×#168	—	2	8	no	—
77	C₂²×M₁₆	Dic₅	1	2	yes	—
78	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
79	#432	—	5	4	no	—
80	#432	—	10	2	no	—
81	#431	—	5	4	no	—
82	#431	—	10	2	no	—
83	#423	—	10	2	no	—
84	#407	—	5	4	no	—
85	C₄×M₁₆	—	2	2	no	—
86	C₄×M₁₆	—	2	2	no	—
87	C₄×M₁₆	—	10	2	no	—
88	C₂×#97	C₂×D₅	1	2	yes	—
89	C₂×C₄×C₈	Dic₅	1	2	yes	—
90	C₂³×C₅	—	2	2	no	—
91	C₂³×C₅	—	4	2	no	—
92	C₂³×C₅	#95	1	1	yes	characteristic
93	C₂³×C₅	#95	1	1	yes	characteristic
94	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₅	—	2	2	no	—
96	C₂×C₄×C₅	—	2	4	no	—
97	C₂×C₄×C₅	—	2	4	no	—
98	C₂×C₄×C₅	—	2	4	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	4	2	no	—
104	C₂×C₄×C₅	—	4	2	no	—
105	C₂×C₄×C₅	—	4	2	no	—
106	C₂×C₄×C₅	#95	1	1	yes	characteristic
107	C₂×C₄×C₅	#95	1	1	yes	characteristic
108	C₂×C₄×C₅	#95	1	2	yes	—
109	C₂×C₄×C₅	#95	1	2	yes	—
110	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
115	C₅×C₈	—	2	8	no	—
116	#168	—	4	8	no	—
117	C₂³×C₄	C₄×D₅	1	2	yes	—
118	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
119	C₂×M₁₆	—	2	2	no	—
120	C₂×M₁₆	—	2	2	no	—
121	C₂×M₁₆	—	2	8	no	—
122	C₂×M₁₆	—	10	4	no	—
123	C₂²×C₈	C₂×Dic₅	1	4	yes	—
124	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
125	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
126	C₂×C₄²	C₄×D₅	1	2	yes	—
127	C₂×C₄²	D₂₀	1	1	yes	characteristic
128	C₂×C₄²	D₂₀	1	1	yes	characteristic
129	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
130	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
131	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
132	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
133	C₄⋊C₈	—	10	4	no	—
134	C₄⋊C₈	—	10	4	no	—
135	C₂²⋊C₈	—	5	8	no	—
136	C₂²⋊C₈	—	10	4	no	—
137	#97	—	2	4	no	—
138	#97	—	10	4	no	—
139	C₄×C₈	—	2	4	no	—
140	C₄×C₈	—	10	4	no	—
141	C₂²×C₅	—	2	2	no	—
142	C₂²×C₅	—	2	1	no	—
143	C₂²×C₅	—	2	1	no	—
144	C₂²×C₅	—	2	2	no	—
145	C₂²×C₅	—	2	2	no	—
146	C₂²×C₅	—	4	4	no	—
147	C₂²×C₅	C₂×#95	1	1	yes	characteristic
148	C₂²×C₅	C₄×M₁₆	1	2	yes	—
149	C₄×C₅	—	2	1	no	—
150	C₄×C₅	—	2	1	no	—
151	C₄×C₅	—	2	2	no	—
152	C₄×C₅	—	2	2	no	—
153	C₄×C₅	—	2	2	no	—
154	C₄×C₅	#344	1	4	yes	—
155	C₄×C₅	#407	1	2	yes	—
156	C₄×C₅	#423	1	2	yes	—
157	C₂⁴	C₄×Dic₅	1	1	yes	characteristic
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	4	4	no	—
160	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
161	C₂²×C₄	C₄×Dic₅	1	2	yes	—
162	C₂²×C₄	#683	1	2	yes	—
163	C₂²×C₄	#683	1	2	yes	—
164	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
165	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
166	C₂²×C₄	#685	1	2	yes	—
167	C₂²×C₄	#685	1	2	yes	—
168	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
169	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
171	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
172	M₁₆	—	2	8	no	—
173	M₁₆	—	2	8	no	—
174	M₁₆	—	20	8	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	10	8	no	—
178	C₂×C₈	#681	1	8	yes	complemented
179	C₄²	#683	1	2	yes	—
180	C₄²	#683	1	2	yes	—
181	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
182	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
183	C₄²	#685	1	2	yes	—
184	C₄²	#685	1	2	yes	—
185	C₄²	C₂²⋊Dic₅	1	2	yes	—
186	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
187	C₄²	C₂×D₂₀	1	1	yes	characteristic
188	C₄²	C₂×C₂²⋊D₅	1	2	yes	—
189	C₂×C₅	—	2	2	no	—
190	C₂×C₅	—	4	1	no	—
191	C₂×C₅	—	4	1	no	—
192	C₂×C₅	#1728	1	1	yes	characteristic
193	C₂×C₅	C₂×#344	1	2	yes	—
194	C₂³	—	2	2	no	—
195	C₂³	—	4	2	no	—
196	C₂³	#3963	1	1	yes	characteristic
197	C₂³	#3963	1	1	yes	characteristic
198	C₂³	C₂×C₄×Dic₅	1	1	yes	characteristic
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	#3963	1	1	yes	characteristic
211	C₂×C₄	#3963	1	1	yes	characteristic
212	C₂×C₄	#3963	1	2	yes	—
213	C₂×C₄	#3963	1	2	yes	—
214	C₂×C₄	C₂×#681	1	4	yes	—
215	C₂×C₄	C₂×#683	1	2	yes	—
216	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
217	C₂×C₄	C₂×#685	1	2	yes	—
218	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
219	C₈	—	2	8	no	—
220	C₈	—	20	8	no	—
221	C₅	#7920	1	1	yes	characteristic, complemented, Sylow
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	4	4	no	—
228	C₂²	C₄×#681	1	2	yes	—
229	C₂²	C₂×#3963	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	#64776	1	4	yes	—
236	C₄	#65213	1	2	yes	—
237	C₄	#65496	1	2	yes	—
238	C₂	—	2	2	no	—
239	C₂	—	4	1	no	—
240	C₂	—	4	1	no	—
241	C₂	#10602320	1	1	yes	characteristic
242	C₂	C₂×#64776	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅
commutator	C₂³×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₄×C₅×M₁₆	C₂
socle	C₂²×C₅	C₂×#95
prime radical	C₂⁴×C₅	C₄²
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×C₄×M₁₆	D₅
hypocenter	C₅	#7920
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #10598152	C₂	semidirect
64. C₂×C₅×C₈	M₁₆	semidirect
178. C₂×C₈	#681	semidirect
221. C₅	#7920	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×M₁₆	2
#10613780	2
#10598152	2
#7920	5