Subgroups — C₂²×#65367 #12968896

Order 1280.

2398 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×#4015	C₂	1	1	yes	maximal, characteristic
3	C₂×#65367	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#3939	C₂	1	4	yes	maximal
5	C₂²×C₄×#168	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₅×Q₈	—	2	1	no	—
7	C₂³×Dic₁₀	—	2	1	no	—
8	C₂×C₅×C₄⋊Q₈	—	2	6	no	—
9	C₂×C₅×C₄⋊Q₈	C₂²	1	12	yes	—
10	C₂²×C₅×C₄⋊C₄	—	2	1	no	—
11	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂²×C₄⋊Dic₅	—	2	1	no	—
14	C₂²×C₄⋊Dic₅	C₂²	1	2	yes	—
15	C₂³×#168	C₂²	1	2	yes	—
16	C₂×#4015	—	2	6	no	—
17	C₂×#4015	C₂²	1	12	yes	—
18	#65367	C₂²	1	64	yes	direct product factor
19	C₂×#3939	C₂²	1	48	yes	—
20	C₂×C₄×#168	C₂²	1	12	yes	—
21	C₂²×#500	—	5	1	no	maximal, Sylow
22	C₂²×C₅×Q₈	—	2	12	no	—
23	C₂²×C₅×Q₈	—	4	8	no	—
24	C₂³×C₄×C₅	—	2	2	no	—
25	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂³×Dic₅	—	2	2	no	—
29	C₂²×Dic₁₀	—	2	12	no	—
30	C₂²×Dic₁₀	—	4	8	no	—
31	C₅×C₄⋊Q₈	—	2	24	no	—
32	C₅×C₄⋊Q₈	C₂³	1	16	yes	—
33	C₂×C₅×C₄⋊C₄	—	2	12	no	—
34	C₂×C₅×C₄⋊C₄	—	2	6	no	—
35	C₂×C₅×C₄⋊C₄	C₂³	1	12	yes	—
36	C₂×C₄²×C₅	D₄	1	6	yes	—
37	C₂×C₄²×C₅	C₂³	1	6	yes	—
38	C₂×C₄⋊Dic₅	—	2	6	no	—
39	C₂×C₄⋊Dic₅	—	2	12	no	—
40	C₂×C₄⋊Dic₅	C₂³	1	12	yes	—
41	C₂²×#168	Q₈	1	16	yes	complemented
42	C₂²×#168	C₂³	1	12	yes	—
43	#4015	—	2	24	no	—
44	#4015	C₂³	1	16	yes	—
45	#3939	C₂³	1	64	yes	—
46	C₄×#168	C₂³	1	16	yes	—
47	C₂²×C₄⋊Q₈	—	5	1	no	—
48	C₂²×C₄⋊Q₈	D₅	1	1	yes	characteristic
49	C₂×#500	—	5	24	no	—
50	C₂²×#107	—	5	4	no	—
51	C₂²×C₄×C₈	—	5	1	no	—
52	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂×C₅×Q₈	—	2	16	no	—
54	C₂×C₅×Q₈	—	4	48	no	—
55	C₂²×C₄×C₅	—	2	12	no	—
56	C₂²×C₄×C₅	—	4	8	no	—
57	C₂²×C₄×C₅	D₈	1	8	yes	—
58	C₂²×C₄×C₅	Q₁₆	1	8	yes	—
59	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
60	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
61	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
62	C₂²×C₄×C₅	C₂×Q₈	1	8	yes	—
63	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
64	C₂²×Dic₅	—	2	12	no	—
65	C₂²×Dic₅	—	4	8	no	—
66	C₂×Dic₁₀	—	2	16	no	—
67	C₂×Dic₁₀	—	4	48	no	—
68	C₅×C₄⋊C₄	—	2	16	no	—
69	C₅×C₄⋊C₄	—	2	12	no	—
70	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
71	C₄²×C₅	C₂×D₄	1	12	yes	—
72	C₄²×C₅	C₂⁴	1	4	yes	—
73	C₄⋊Dic₅	—	2	12	no	—
74	C₄⋊Dic₅	—	2	16	no	—
75	C₄⋊Dic₅	C₂⁴	1	8	yes	—
76	C₂×#168	C₂×Q₈	1	48	yes	complemented
77	C₂×#168	C₂⁴	1	8	yes	—
78	C₂³×Q₈	—	2	1	no	—
79	C₂³×Q₈	—	10	1	no	—
80	C₂³×C₈	—	5	2	no	—
81	C₂×C₄⋊Q₈	—	2	6	no	—
82	C₂×C₄⋊Q₈	—	5	12	no	—
83	C₂×C₄⋊Q₈	—	10	6	no	—
84	C₂×C₄⋊Q₈	C₂×D₅	1	12	yes	—
85	C₂²×C₄⋊C₄	—	2	1	no	—
86	C₂²×C₄⋊C₄	—	5	2	no	—
87	C₂²×C₄⋊C₄	—	10	1	no	—
88	C₂²×C₄⋊C₄	C₂×D₅	1	2	yes	—
89	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
90	#500	—	5	64	no	—
91	C₂×#107	—	5	48	no	—
92	C₂×C₄×C₈	—	5	12	no	—
93	C₂³×C₅	C₄⋊Q₈	1	4	yes	—
94	C₂³×C₅	C₂×D₈	1	4	yes	—
95	C₂³×C₅	C₂×Q₁₆	1	4	yes	—
96	C₂³×C₅	C₂²×D₄	1	3	yes	—
97	C₅×Q₈	—	4	32	no	—
98	C₂×C₄×C₅	—	2	8	no	—
99	C₂×C₄×C₅	—	4	24	no	—
100	C₂×C₄×C₅	C₄⋊Q₈	1	12	yes	—
101	C₂×C₄×C₅	C₂×D₈	1	24	yes	—
102	C₂×C₄×C₅	C₂×Q₁₆	1	24	yes	—
103	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
104	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
105	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
106	C₂×C₄×C₅	C₂²×Q₈	1	12	yes	—
107	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
108	C₂×Dic₅	—	2	8	no	—
109	C₂×Dic₅	—	4	24	no	—
110	Dic₁₀	—	4	32	no	—
111	#168	C₂²×Q₈	1	16	yes	complemented
112	C₂²×Q₈	—	2	12	no	—
113	C₂²×Q₈	—	4	8	no	—
114	C₂²×Q₈	—	10	12	no	—
115	C₂²×Q₈	—	20	8	no	—
116	C₂³×C₄	—	2	2	no	—
117	C₂³×C₄	—	10	2	no	—
118	C₂³×C₄	C₂²⋊D₅	1	1	yes	characteristic
119	C₂³×C₄	C₂²⋊D₅	1	1	yes	characteristic
120	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
121	C₂²×C₈	—	5	16	no	—
122	C₂²×C₈	—	5	12	no	—
123	C₄⋊Q₈	—	2	24	no	—
124	C₄⋊Q₈	—	5	16	no	—
125	C₄⋊Q₈	—	10	24	no	—
126	C₄⋊Q₈	C₂²×D₅	1	16	yes	—
127	C₂×C₄⋊C₄	—	2	12	no	—
128	C₂×C₄⋊C₄	—	2	6	no	—
129	C₂×C₄⋊C₄	—	5	12	no	—
130	C₂×C₄⋊C₄	—	10	6	no	—
131	C₂×C₄⋊C₄	—	10	12	no	—
132	C₂×C₄⋊C₄	C₂²×D₅	1	12	yes	—
133	C₂×C₄²	C₂²⋊D₅	1	6	yes	—
134	C₂×C₄²	C₂²×D₅	1	6	yes	—
135	#107	—	5	64	no	—
136	C₄×C₈	—	5	16	no	—
137	C₂²×C₅	#500	1	16	yes	complemented
138	C₂²×C₅	C₂×C₄⋊Q₈	1	6	yes	—
139	C₂²×C₅	C₂²×D₈	1	6	yes	—
140	C₂²×C₅	C₂²×Q₁₆	1	6	yes	—
141	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
142	C₄×C₅	—	4	8	no	—
143	C₄×C₅	C₂×C₄⋊Q₈	1	6	yes	—
144	C₄×C₅	C₂²×D₈	1	8	yes	—
145	C₄×C₅	C₂²×Q₁₆	1	8	yes	—
146	C₄×C₅	C₂³×Q₈	1	2	yes	—
147	Dic₅	—	4	8	no	—
148	C₂⁴	C₂×C₂²⋊D₅	1	1	yes	characteristic
149	C₂×Q₈	—	2	16	no	—
150	C₂×Q₈	—	4	48	no	—
151	C₂×Q₈	—	10	16	no	—
152	C₂×Q₈	—	20	48	no	—
153	C₂²×C₄	—	2	12	no	—
154	C₂²×C₄	—	4	8	no	—
155	C₂²×C₄	—	10	12	no	—
156	C₂²×C₄	—	20	8	no	—
157	C₂²×C₄	#686	1	8	yes	—
158	C₂²×C₄	#689	1	8	yes	—
159	C₂²×C₄	Q₈×D₅	1	8	yes	—
160	C₂²×C₄	C₂×C₂²⋊D₅	1	6	yes	—
161	C₂²×C₄	C₂×C₂²⋊D₅	1	6	yes	—
162	C₂²×C₄	C₂×C₂²⋊D₅	1	3	yes	—
163	C₂²×C₄	C₂³×D₅	1	3	yes	—
164	C₂×C₈	—	5	48	no	—
165	C₂×C₈	—	5	8	no	—
166	C₄⋊C₄	—	2	16	no	—
167	C₄⋊C₄	—	2	12	no	—
168	C₄⋊C₄	—	5	8	no	—
169	C₄⋊C₄	—	10	12	no	—
170	C₄⋊C₄	—	10	16	no	—
171	C₄⋊C₄	C₂³×D₅	1	8	yes	—
172	C₄²	C₂×C₂²⋊D₅	1	12	yes	—
173	C₄²	C₂³×D₅	1	4	yes	—
174	C₂×C₅	C₂×#500	1	12	yes	complemented
175	C₂×C₅	C₂²×C₄⋊Q₈	1	1	yes	characteristic
176	C₂×C₅	C₂³×D₈	1	1	yes	characteristic
177	C₂×C₅	C₂³×Q₁₆	1	1	yes	characteristic
178	C₂³	C₂×#686	1	4	yes	—
179	C₂³	C₂×#689	1	4	yes	—
180	C₂³	#4090	1	4	yes	—
181	C₂³	C₂²×C₂²⋊D₅	1	3	yes	—
182	Q₈	—	4	32	no	—
183	Q₈	—	20	32	no	—
184	C₂×C₄	—	2	8	no	—
185	C₂×C₄	—	4	24	no	—
186	C₂×C₄	—	10	8	no	—
187	C₂×C₄	—	20	24	no	—
188	C₂×C₄	C₂×#686	1	24	yes	—
189	C₂×C₄	C₂×#689	1	24	yes	—
190	C₂×C₄	#4090	1	12	yes	—
191	C₂×C₄	C₂×Q₈×D₅	1	12	yes	—
192	C₂×C₄	C₂²×C₂²⋊D₅	1	4	yes	—
193	C₂×C₄	C₂²×C₂²⋊D₅	1	4	yes	—
194	C₂×C₄	C₂²×C₂²⋊D₅	1	3	yes	—
195	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
196	C₈	—	5	16	no	—
197	C₅	C₂²×#500	1	1	yes	characteristic, complemented, Sylow
198	C₂²	#65367	1	16	yes	direct product factor
199	C₂²	C₂²×#686	1	6	yes	—
200	C₂²	C₂²×#689	1	6	yes	—
201	C₂²	C₂×#4090	1	6	yes	—
202	C₂²	C₂³×C₂²⋊D₅	1	1	yes	characteristic
203	C₄	—	4	8	no	—
204	C₄	—	20	8	no	—
205	C₄	C₂²×#686	1	8	yes	—
206	C₄	C₂²×#689	1	8	yes	—
207	C₄	C₂×#4090	1	6	yes	—
208	C₄	C₂²×Q₈×D₅	1	2	yes	—
209	C₂	C₂×#65367	1	12	yes	direct product factor
210	C₂	C₂³×#686	1	1	yes	characteristic
211	C₂	C₂³×#689	1	1	yes	characteristic
212	C₂	C₂²×#4090	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂²×C₅×C₄⋊Q₈	C₂
socle, prime radical	C₂⁴×C₅	C₂×D₄
hypercenter	C₂²×C₄⋊Q₈	D₅
hypocenter	C₅	C₂²×#500
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
3. C₂×#65367	C₂	direct (N × H)
18. #65367	C₂²	direct (N × H)
41. C₂²×#168	Q₈	semidirect
76. C₂×#168	C₂×Q₈	semidirect
111. #168	C₂²×Q₈	semidirect
137. C₂²×C₅	#500	semidirect
174. C₂×C₅	C₂×#500	semidirect
197. C₅	C₂²×#500	semidirect
198. C₂²	#65367	direct (N × H)
209. C₂	C₂×#65367	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×C₄⋊Q₈	2
C₂²×#4015	2
C₂×#65367	2
C₂²×#3939	2
C₂²×C₄×#168	2
C₂²×#500	5

Subgroups — C22×#65367 #12968896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#65367 #12968896