Subgroups — #13043637

Order 1280.

490 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10615059	C₂	1	1	yes	maximal, characteristic
2	#10604448	C₂	1	8	yes	maximal, complemented
3	#10604367	C₂	1	2	yes	maximal
4	#10604366	C₂	1	2	yes	maximal
5	#10604150	C₂	1	1	yes	maximal, characteristic
6	C₂×#64859	C₂	1	1	yes	maximal, characteristic
7	C₅×#533	C₄	1	1	yes	characteristic
8	C₂×#3998	C₂²	1	2	yes	—
9	C₂²×#674	C₂²	1	1	yes	characteristic
10	C₂²×#674	C₂²	1	1	yes	characteristic
11	#65313	C₄	1	4	yes	—
12	Q₈×#168	C₂²	1	4	yes	—
13	#65288	C₄	1	2	yes	—
14	#65286	C₂²	1	2	yes	—
15	#65213	C₄	1	1	yes	characteristic
16	C₂×C₄×#168	C₂²	1	1	yes	characteristic
17	#64859	C₂²	1	4	yes	—
18	#64859	C₂²	1	4	yes	—
19	C₄⋊#674	C₂²	1	8	yes	—
20	C₄⋊#674	C₂²	1	4	yes	—
21	C₄×#674	C₂²	1	4	yes	—
22	#33475	—	5	1	no	maximal, Sylow
23	C₅×C₄⋊Q₈	C₈	1	2	yes	—
24	C₅×#125	C₈	1	2	yes	—
25	C₅×C₂²⋊Q₈	C₈	1	4	yes	—
26	C₄×C₅×Q₈	C₂×C₄	1	4	yes	—
27	C₅×#117	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂×#681	C₂×C₄	1	2	yes	—
30	C₂²×#168	C₂×C₄	1	1	yes	characteristic
31	C₂²×#168	C₂³	1	1	yes	characteristic
32	#3998	—	2	4	no	—
33	C₂×#674	—	2	2	no	—
34	C₂×#674	Q₈	1	4	yes	complemented
35	C₂×#674	C₂³	1	4	yes	—
36	C₂×#674	C₂³	1	2	yes	—
37	C₂×#674	C₂³	1	2	yes	—
38	C₄⋊#168	C₂×C₄	1	8	yes	—
39	C₄⋊#168	C₂×C₄	1	4	yes	—
40	#3935	C₂×C₄	1	4	yes	—
41	C₄×#168	C₂×C₄	1	2	yes	—
42	C₄×#168	C₂³	1	4	yes	—
43	C₄×#168	C₂³	1	2	yes	—
44	#2960	—	5	1	no	—
45	#2183	—	5	8	no	—
46	#2151	—	5	2	no	—
47	#2150	—	5	1	no	—
48	#2108	—	5	2	no	—
49	C₂×#346	—	5	1	no	—
50	C₂×C₅×Q₈	C₂×C₈	1	4	yes	—
51	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₅×C₄⋊C₄	C₂×C₈	1	4	yes	—
54	C₅×C₄⋊C₄	C₂×C₈	1	8	yes	—
55	C₅×C₄⋊C₄	C₂×C₈	1	4	yes	—
56	C₅×C₂²⋊C₄	C₂×C₈	1	4	yes	—
57	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
58	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
59	C₄²×C₅	C₂²×C₄	1	2	yes	—
60	C₄²×C₅	C₂²×C₄	1	4	yes	—
61	#681	—	2	4	no	—
62	C₂×#168	—	2	2	no	—
63	C₂×#168	C₂²×C₄	1	2	yes	—
64	C₂×#168	C₂²×C₄	1	4	yes	—
65	C₂×#168	C₂²×C₄	1	1	yes	characteristic
66	C₂×#168	C₂×Q₈	1	2	yes	—
67	C₂×#168	#41	1	2	yes	—
68	C₂×#168	C₂⁴	1	1	yes	characteristic
69	#674	—	2	2	no	—
70	#674	—	2	4	no	—
71	#674	C₂×Q₈	1	4	yes	—
72	#533	AGL(1,𝔽₅)	1	1	yes	characteristic
73	C₂×M₃₂	—	5	2	no	—
74	C₂²×C₁₆	—	5	1	no	—
75	C₂²×C₁₆	—	5	1	no	—
76	#446	—	5	4	no	—
77	C₈×Q₈	—	5	4	no	—
78	#424	—	5	2	no	—
79	#423	—	5	1	no	—
80	#405	—	5	2	no	—
81	C₂×C₄×C₈	—	5	1	no	—
82	C₄⋊C₁₆	—	5	4	no	—
83	C₄⋊C₁₆	—	5	8	no	—
84	#346	—	5	4	no	—
85	#346	—	5	4	no	—
86	C₄×C₁₆	—	5	4	no	—
87	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
88	C₅×Q₈	—	4	4	no	—
89	C₂×C₄×C₅	—	4	2	no	—
90	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
91	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
92	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
93	C₂×C₄×C₅	C₂²×C₈	1	8	yes	—
94	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₅	#131	1	2	yes	—
96	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
97	#168	—	2	2	no	—
98	#168	—	2	4	no	—
99	#168	C₄×Q₈	1	2	yes	—
100	#168	C₂²×Q₈	1	1	yes	characteristic
101	#168	C₂×#41	1	1	yes	characteristic
102	C₂×M₁₆	—	5	2	no	—
103	C₂²×C₈	—	5	1	no	—
104	C₂²×C₈	—	5	1	no	—
105	C₄⋊Q₈	#170	1	2	yes	—
106	#125	#170	1	2	yes	—
107	C₂²⋊Q₈	#170	1	4	yes	—
108	C₄×Q₈	C₂×AGL(1,𝔽₅)	1	4	yes	—
109	#117	C₂×AGL(1,𝔽₅)	1	2	yes	—
110	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
111	M₃₂	—	10	4	no	—
112	C₂×C₁₆	—	5	2	no	—
113	C₂×C₁₆	—	5	4	no	—
114	C₂×C₁₆	—	5	4	no	—
115	C₂×C₁₆	—	5	2	no	—
116	C₂×C₁₆	—	10	2	no	—
117	C₄⋊C₈	—	5	4	no	—
118	C₄⋊C₈	—	5	8	no	—
119	#97	—	5	4	no	—
120	C₄×C₈	—	5	4	no	—
121	C₄×C₈	—	5	2	no	—
122	C₄×C₈	—	5	2	no	—
123	C₂²×C₅	—	2	2	no	—
124	C₂²×C₅	C₈×Q₈	1	2	yes	—
125	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
126	C₄×C₅	—	2	2	no	—
127	C₄×C₅	—	2	2	no	—
128	C₄×C₅	—	2	8	no	—
129	C₄×C₅	C₈×Q₈	1	2	yes	—
130	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
131	C₄×C₅	C₂×#131	1	1	yes	characteristic
132	C₂×Q₈	C₂×#170	1	4	yes	—
133	C₂²×C₄	C₂×#170	1	2	yes	—
134	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
135	M₁₆	—	10	4	no	—
136	C₂×C₈	—	5	2	no	—
137	C₂×C₈	—	5	2	no	—
138	C₂×C₈	—	5	1	no	—
139	C₂×C₈	—	5	1	no	—
140	C₂×C₈	—	5	2	no	—
141	C₂×C₈	—	5	4	no	—
142	C₂×C₈	—	10	2	no	—
143	C₄⋊C₄	C₂×#170	1	4	yes	—
144	C₄⋊C₄	C₂×#170	1	8	yes	—
145	C₄⋊C₄	C₂×#170	1	4	yes	—
146	C₂²⋊C₄	C₂×#170	1	4	yes	—
147	C₄²	C₂×#170	1	1	yes	characteristic
148	C₄²	C₂×#170	1	1	yes	characteristic
149	C₄²	C₂²×AGL(1,𝔽₅)	1	2	yes	—
150	C₄²	C₂²×AGL(1,𝔽₅)	1	4	yes	—
151	C₁₆	—	5	4	no	—
152	C₁₆	—	10	2	no	—
153	C₁₆	—	10	4	no	—
154	C₂×C₅	—	2	2	no	—
155	C₂×C₅	C₂×C₈×Q₈	1	1	yes	characteristic
156	C₂×C₅	#3407	1	2	yes	—
157	C₂³	C₂²×#170	1	1	yes	characteristic
158	Q₈	—	4	4	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	#4133	1	2	yes	—
161	C₂×C₄	Q₈×AGL(1,𝔽₅)	1	2	yes	—
162	C₂×C₄	C₂²×#170	1	2	yes	—
163	C₂×C₄	C₂²×#170	1	2	yes	—
164	C₂×C₄	C₂²×#170	1	8	yes	—
165	C₂×C₄	C₂²×#170	1	1	yes	characteristic
166	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
167	C₈	—	5	1	no	—
168	C₈	—	5	1	no	—
169	C₈	—	5	2	no	—
170	C₈	—	10	2	no	—
171	C₈	—	10	4	no	—
172	C₅	#33475	1	1	yes	characteristic, complemented, Sylow
173	C₂²	—	2	2	no	—
174	C₂²	Q₈×#170	1	2	yes	—
175	C₂²	C₂³×#170	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	2	8	no	—
179	C₄	Q₈×#170	1	2	yes	—
180	C₄	C₂×#4133	1	1	yes	characteristic
181	C₄	C₂×Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	C₂×Q₈×#170	1	1	yes	characteristic
184	C₂	#10617530	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₅×#533	C₄
prime radical	C₂³×C₅	C₂²×C₈
cosocle	C₂×#168	C₂⁴
hypercenter	#533	AGL(1,𝔽₅)
hypocenter	C₅	#33475
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
2. #10604448	C₂	semidirect
34. C₂×#674	Q₈	semidirect
172. C₅	#33475	semidirect

Maximal subgroups

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

subgroup	index
#10615059	2
#10604448	2
#10604367	2
#10604366	2
#10604150	2
C₂×#64859	2
#33475	5