Subgroups — #11998091

Order 1280.

530 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈⋊#168	C₂	1	4	yes	maximal
3	#10600185	C₂	1	1	yes	maximal, characteristic
4	#10600181	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
7	C₅×#391	C₄	1	2	yes	—
8	C₅×#389	—	2	2	no	—
9	C₅×#384	C₄	1	1	yes	characteristic
10	C₄×C₅×C₄⋊C₄	—	2	2	no	—
11	C₄³×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#168	C₂²	1	2	yes	—
13	C₂×C₄×#168	C₂²	1	2	yes	—
14	Q₈⋊#168	—	2	8	no	—
15	#64744	—	2	2	no	—
16	#64744	—	2	2	no	—
17	#9641	—	5	1	no	maximal, Sylow
18	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
19	C₅×C₄⋊Q₈	—	2	2	no	—
20	C₅×C₄⋊Q₈	C₈	1	4	yes	complemented
21	C₄×C₅×Q₈	—	2	4	no	—
22	C₄×C₅×Q₈	—	2	4	no	—
23	C₂×C₅×C₄⋊C₄	—	2	2	no	—
24	C₂×C₅×C₄⋊C₄	—	2	2	no	—
25	C₂×C₅×C₄⋊C₄	—	2	2	no	—
26	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅	—	2	2	no	—
28	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅	D₄	1	2	yes	—
31	C₂×C₄²×C₅	D₄	1	2	yes	—
32	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₂²×#168	—	2	2	no	—
34	C₂²×#168	—	2	2	no	—
35	C₅×#95	—	2	4	no	—
36	C₄⋊#168	—	2	4	no	—
37	C₄×#168	—	2	2	no	—
38	C₄×#168	D₄	1	4	yes	—
39	C₄×C₄⋊Q₈	D₅	1	1	yes	characteristic
40	#1831	—	5	1	no	—
41	#1744	—	5	1	no	—
42	C₂×Q₈⋊C₈	—	5	4	no	—
43	C₂×C₅×Q₈	—	2	2	no	—
44	C₂×C₅×Q₈	—	2	4	no	—
45	C₂×C₅×Q₈	—	2	4	no	—
46	C₂×C₅×Q₈	C₂×C₈	1	4	yes	—
47	C₂²×C₄×C₅	—	2	2	no	—
48	C₂²×C₄×C₅	—	2	2	no	—
49	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₅×C₄⋊C₄	—	2	8	no	—
55	C₅×C₄⋊C₄	—	2	4	no	—
56	C₅×C₄⋊C₄	—	4	4	no	—
57	C₅×C₄⋊C₄	—	4	4	no	—
58	C₄²×C₅	—	2	2	no	—
59	C₄²×C₅	—	2	4	no	—
60	C₄²×C₅	—	4	2	no	—
61	C₄²×C₅	—	4	2	no	—
62	C₄²×C₅	—	4	2	no	—
63	C₄²×C₅	C₂×C₈	1	2	yes	—
64	C₄²×C₅	M₁₆	1	2	yes	—
65	C₄²×C₅	C₂×D₄	1	2	yes	—
66	C₄²×C₅	#41	1	2	yes	—
67	C₂×#168	—	2	4	no	—
68	C₂×#168	—	4	4	no	—
69	C₂×#168	—	4	2	no	—
70	C₂×#168	—	4	4	no	—
71	C₂×C₄⋊Q₈	Dic₅	1	1	yes	characteristic
72	C₂×C₄×Q₈	C₂×D₅	1	2	yes	—
73	C₂×C₄⋊C₈	—	5	2	no	—
74	C₂×C₄×C₈	—	5	2	no	—
75	#391	Dic₅	1	2	yes	—
76	#389	—	2	2	no	—
77	#384	Dic₅	1	1	yes	characteristic
78	C₄×C₄⋊C₄	—	2	2	no	—
79	C₄³	C₂×D₅	1	1	yes	characteristic
80	#336	—	10	2	no	—
81	#336	—	10	2	no	—
82	Q₈⋊C₈	—	10	8	no	—
83	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₅×Q₈	—	2	8	no	—
85	C₅×Q₈	—	4	4	no	—
86	C₂×C₄×C₅	—	2	2	no	—
87	C₂×C₄×C₅	—	2	2	no	—
88	C₂×C₄×C₅	—	2	2	no	—
89	C₂×C₄×C₅	—	2	4	no	—
90	C₂×C₄×C₅	—	2	2	no	—
91	C₂×C₄×C₅	—	2	2	no	—
92	C₂×C₄×C₅	—	2	2	no	—
93	C₂×C₄×C₅	—	4	2	no	—
94	C₂×C₄×C₅	—	4	2	no	—
95	C₂×C₄×C₅	—	4	2	no	—
96	C₂×C₄×C₅	—	4	4	no	—
97	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
98	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₅	#121	1	2	yes	—
100	C₂×C₄×C₅	#124	1	1	yes	characteristic
101	C₂×C₄×C₅	#127	1	1	yes	characteristic
102	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₅	#131	1	2	yes	—
105	C₂×C₄×C₅	#137	1	4	yes	—
106	#168	—	4	4	no	—
107	#168	—	8	2	no	—
108	C₂²×Q₈	C₂×Dic₅	1	2	yes	—
109	C₂²×C₈	—	10	2	no	—
110	C₂²×C₈	—	10	2	no	—
111	C₄⋊Q₈	—	2	2	no	—
112	C₄⋊Q₈	#168	1	4	yes	complemented
113	C₄×Q₈	—	2	4	no	—
114	C₄×Q₈	—	2	4	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
119	C₂×C₄²	—	2	2	no	—
120	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
121	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
122	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
123	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
124	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
125	C₄⋊C₈	—	10	4	no	—
126	C₄×C₈	—	5	4	no	—
127	C₄×C₈	—	10	2	no	—
128	#95	—	2	4	no	—
129	C₂²×C₅	#390	1	1	yes	characteristic
130	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
131	C₂²×C₅	#408	1	1	yes	characteristic
132	C₂²×C₅	#419	1	2	yes	—
133	C₂²×C₅	#421	1	2	yes	—
134	C₄×C₅	—	2	2	no	—
135	C₄×C₅	—	2	4	no	—
136	C₄×C₅	—	4	4	no	—
137	C₄×C₅	—	4	2	no	—
138	C₄×C₅	—	8	2	no	—
139	C₄×C₅	C₈×D₄	1	2	yes	—
140	C₄×C₅	#436	1	2	yes	—
141	C₄×C₅	#491	1	2	yes	—
142	C₄×C₅	#497	1	2	yes	—
143	C₂×Q₈	—	2	2	no	—
144	C₂×Q₈	—	2	4	no	—
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	C₂×#168	1	4	yes	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
150	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
151	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
154	C₂×C₈	—	10	4	no	—
155	C₂×C₈	—	20	4	no	—
156	C₂×C₈	—	20	2	no	—
157	C₂×C₈	—	20	4	no	—
158	C₄⋊C₄	—	2	8	no	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄²	—	2	4	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	—	4	2	no	—
166	C₄²	—	4	2	no	—
167	C₄²	C₂×#168	1	2	yes	—
168	C₄²	#681	1	2	yes	—
169	C₄²	D₄×D₅	1	2	yes	—
170	C₄²	#711	1	2	yes	—
171	C₂×C₅	#1479	1	2	yes	—
172	C₂×C₅	#1497	1	2	yes	—
173	C₂×C₅	#1954	1	1	yes	characteristic
174	C₂×C₅	#1960	1	1	yes	characteristic
175	C₂×C₅	#1963	1	1	yes	characteristic
176	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
177	Q₈	—	2	8	no	—
178	Q₈	—	4	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	C₂²⋊#168	1	4	yes	—
191	C₂×C₄	C₂²×#168	1	1	yes	characteristic
192	C₂×C₄	C₂×#681	1	1	yes	characteristic
193	C₂×C₄	D₄×Dic₅	1	2	yes	—
194	C₂×C₄	#4082	1	1	yes	characteristic
195	C₂×C₄	#4084	1	2	yes	—
196	C₂×C₄	#4086	1	1	yes	characteristic
197	C₂×C₄	#4088	1	4	yes	—
198	C₂×C₄	#4094	1	2	yes	—
199	C₈	—	20	4	no	—
200	C₈	—	40	2	no	—
201	C₅	#9641	1	1	yes	characteristic, complemented, Sylow
202	C₂²	C₂×C₂²⋊#168	1	1	yes	characteristic
203	C₂²	#65510	1	1	yes	characteristic
204	C₂²	#65514	1	2	yes	—
205	C₂²	#65523	1	1	yes	characteristic
206	C₂²	#65525	1	2	yes	—
207	C₄	—	2	2	no	—
208	C₄	—	2	4	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	4	no	—
211	C₄	—	8	2	no	—
212	C₄	D₄×#168	1	2	yes	—
213	C₄	#65301	1	2	yes	—
214	C₄	#65371	1	2	yes	—
215	C₄	#65375	1	2	yes	—
216	C₂	#10602407	1	1	yes	characteristic
217	C₂	#10602414	1	2	yes	—
218	C₂	#10602579	1	1	yes	characteristic
219	C₂	#10602609	1	1	yes	characteristic
220	C₂	#10602620	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₄×C₅×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₄×C₄⋊Q₈	D₅
hypocenter	C₅	#9641
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
20. C₅×C₄⋊Q₈	C₈	semidirect
112. C₄⋊Q₈	#168	semidirect
201. C₅	#9641	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×C₄⋊Q₈	2
C₂×Q₈⋊#168	2
#10600185	2
#10600181	2
#9641	5