Subgroups — #12963693

Order 1280.

1810 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3458	C₂	1	1	yes	maximal, characteristic
2	#10615060	C₂	1	12	yes	maximal, complemented
3	#10615059	C₂	1	12	yes	maximal, complemented
4	#10613935	C₂	1	3	yes	maximal
5	C₂×#65286	C₂	1	3	yes	maximal
6	C₅×#544	C₄	1	12	yes	—
7	C₅×#543	C₄	1	4	yes	—
8	C₅×#533	C₂²	1	12	yes	—
9	C₂×C₅×#117	C₂²	1	3	yes	—
10	C₂²×#681	C₂²	1	3	yes	—
11	C₂³×#168	C₂²	1	1	yes	characteristic
12	#65313	C₂²	1	48	yes	complemented
13	Q₈×#168	C₂²	1	16	yes	complemented
14	#65288	C₂²	1	24	yes	—
15	#65286	C₂²	1	12	yes	—
16	#65286	C₂²	1	12	yes	—
17	#65213	C₂²	1	12	yes	—
18	C₂×#3935	C₂²	1	6	yes	—
19	C₂×C₄×#168	C₂²	1	6	yes	—
20	#60191	—	5	1	no	maximal, Sylow
21	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₅×C₄⋊Q₈	C₂×C₄	1	24	yes	—
23	C₅×#125	C₂×C₄	1	24	yes	—
24	C₅×C₂²⋊Q₈	C₂×C₄	1	48	yes	—
25	C₄×C₅×Q₈	C₂³	1	16	yes	—
26	C₅×#117	C₂×C₄	1	12	yes	—
27	C₅×#117	C₂³	1	12	yes	—
28	C₂×C₅×C₄⋊C₄	C₂×C₄	1	6	yes	—
29	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	6	yes	—
30	C₂×C₄²×C₅	C₂³	1	6	yes	—
31	C₂×#681	—	2	12	no	—
32	C₂×#681	C₂³	1	12	yes	—
33	C₂²×#168	Q₈	1	8	yes	complemented
34	C₂²×#168	C₂³	1	6	yes	—
35	C₂²×#168	C₂³	1	3	yes	—
36	C₂²×#168	C₂³	1	3	yes	—
37	C₄⋊#168	C₂³	1	48	yes	—
38	#3935	C₂³	1	24	yes	—
39	C₄×#168	C₂³	1	12	yes	—
40	C₄×#168	C₂³	1	12	yes	—
41	#3458	D₅	1	1	yes	characteristic
42	#2961	—	5	12	no	—
43	#2960	—	5	12	no	—
44	#2905	—	5	3	no	—
45	C₂×#405	—	5	3	no	—
46	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₅×Q₈	C₂²×C₄	1	16	yes	—
48	C₂²×C₄×C₅	C₂²×C₄	1	12	yes	—
49	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
50	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
51	C₂²×C₄×C₅	#41	1	4	yes	—
52	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
53	C₅×C₄⋊C₄	C₂²×C₄	1	24	yes	—
54	C₅×C₄⋊C₄	C₂²×C₄	1	48	yes	—
55	C₅×C₂²⋊C₄	C₂²×C₄	1	24	yes	—
56	C₄²×C₅	C₂²×C₄	1	12	yes	—
57	C₄²×C₅	C₂⁴	1	12	yes	—
58	#681	—	2	24	no	—
59	C₂×#168	—	2	12	no	—
60	C₂×#168	C₂×Q₈	1	12	yes	—
61	C₂×#168	C₂×Q₈	1	12	yes	—
62	C₂×#168	C₂⁴	1	12	yes	—
63	C₂×#168	C₂⁴	1	1	yes	characteristic
64	C₂×#168	C₂⁴	1	3	yes	—
65	C₂²×M₁₆	—	5	3	no	—
66	C₂³×C₈	—	5	1	no	—
67	#544	Dic₅	1	12	yes	—
68	#543	Dic₅	1	4	yes	—
69	#533	C₂×D₅	1	12	yes	—
70	C₂×#117	C₂×D₅	1	3	yes	—
71	#446	—	5	48	no	—
72	C₈×Q₈	—	5	16	no	—
73	#424	—	5	24	no	—
74	#423	—	5	12	no	—
75	#405	—	5	12	no	—
76	#405	—	5	12	no	—
77	C₂×#97	—	5	6	no	—
78	C₂×C₄×C₈	—	5	6	no	—
79	C₂³×C₅	—	4	2	no	—
80	C₂³×C₅	C₄×Q₈	1	4	yes	—
81	C₂³×C₅	C₂³×C₄	1	3	yes	—
82	C₅×Q₈	—	4	16	no	—
83	C₂×C₄×C₅	—	4	12	no	—
84	C₂×C₄×C₅	C₄×Q₈	1	12	yes	—
85	C₂×C₄×C₅	C₂³×C₄	1	24	yes	—
86	C₂×C₄×C₅	C₂³×C₄	1	3	yes	—
87	C₂×C₄×C₅	C₂²×Q₈	1	6	yes	—
88	C₂×C₄×C₅	C₂×#41	1	6	yes	—
89	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
90	#168	—	2	12	no	—
91	#168	C₂²×Q₈	1	2	yes	—
92	#168	C₂²×Q₈	1	6	yes	—
93	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
94	C₂×M₁₆	—	5	12	no	—
95	C₂×M₁₆	—	10	12	no	—
96	C₂²×C₈	—	5	8	no	—
97	C₂²×C₈	—	5	3	no	—
98	C₂²×C₈	—	5	3	no	—
99	C₂²×C₈	—	5	6	no	—
100	C₄⋊Q₈	C₂×Dic₅	1	24	yes	—
101	#125	C₂×Dic₅	1	24	yes	—
102	C₂²⋊Q₈	C₂×Dic₅	1	48	yes	—
103	C₄×Q₈	C₂²×D₅	1	16	yes	—
104	#117	C₂×Dic₅	1	12	yes	—
105	#117	C₂²×D₅	1	12	yes	—
106	C₂×C₄⋊C₄	C₂×Dic₅	1	6	yes	—
107	C₂×C₂²⋊C₄	C₂×Dic₅	1	6	yes	—
108	C₂×C₄²	C₂²×D₅	1	6	yes	—
109	C₄⋊C₈	—	5	48	no	—
110	#97	—	5	24	no	—
111	C₄×C₈	—	5	12	no	—
112	C₄×C₈	—	5	12	no	—
113	C₂²×C₅	—	2	6	no	—
114	C₂²×C₅	—	4	4	no	—
115	C₂²×C₅	C₂×C₄×Q₈	1	6	yes	—
116	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
117	C₄×C₅	—	2	24	no	—
118	C₄×C₅	C₂×C₄×Q₈	1	6	yes	—
119	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
120	C₄×C₅	C₂²×#41	1	1	yes	characteristic
121	C₂⁴	C₂²×Dic₅	1	1	yes	characteristic
122	C₂×Q₈	C₂²×Dic₅	1	16	yes	—
123	C₂²×C₄	Q₈×D₅	1	4	yes	—
124	C₂²×C₄	#713	1	4	yes	—
125	C₂²×C₄	C₂²×Dic₅	1	12	yes	—
126	C₂²×C₄	C₂²×Dic₅	1	3	yes	—
127	C₂²×C₄	C₂³×D₅	1	3	yes	—
128	M₁₆	—	10	24	no	—
129	C₂×C₈	—	5	12	no	—
130	C₂×C₈	—	5	12	no	—
131	C₂×C₈	—	5	1	no	—
132	C₂×C₈	—	5	3	no	—
133	C₂×C₈	—	5	12	no	—
134	C₂×C₈	—	10	12	no	—
135	C₄⋊C₄	C₂²×Dic₅	1	24	yes	—
136	C₄⋊C₄	C₂²×Dic₅	1	48	yes	—
137	C₂²⋊C₄	C₂²×Dic₅	1	24	yes	—
138	C₄²	C₂²×Dic₅	1	12	yes	—
139	C₄²	C₂³×D₅	1	12	yes	—
140	C₂×C₅	—	2	6	no	—
141	C₂×C₅	C₂²×C₄×Q₈	1	1	yes	characteristic
142	C₂×C₅	#3573	1	2	yes	—
143	C₂³	—	4	2	no	—
144	C₂³	Q₈×Dic₅	1	4	yes	—
145	C₂³	C₂³×Dic₅	1	3	yes	—
146	Q₈	—	4	16	no	—
147	C₂×C₄	—	4	12	no	—
148	C₂×C₄	Q₈×Dic₅	1	12	yes	—
149	C₂×C₄	C₂×Q₈×D₅	1	6	yes	—
150	C₂×C₄	C₂×#713	1	6	yes	—
151	C₂×C₄	C₂³×Dic₅	1	24	yes	—
152	C₂×C₄	C₂³×Dic₅	1	3	yes	—
153	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
154	C₈	—	5	2	no	—
155	C₈	—	5	6	no	—
156	C₈	—	10	12	no	—
157	C₅	#60191	1	1	yes	characteristic, complemented, Sylow
158	C₂²	—	2	6	no	—
159	C₂²	—	4	4	no	—
160	C₂²	C₂×Q₈×Dic₅	1	6	yes	—
161	C₂²	C₂⁴×Dic₅	1	1	yes	characteristic
162	C₄	—	2	24	no	—
163	C₄	C₂×Q₈×Dic₅	1	6	yes	—
164	C₄	C₂²×Q₈×D₅	1	1	yes	characteristic
165	C₄	C₂²×#713	1	1	yes	characteristic
166	C₂	—	2	6	no	—
167	C₂	C₂²×Q₈×Dic₅	1	1	yes	characteristic
168	C₂	#10619197	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₅
commutator, socle	C₂²×C₅	C₂⁴×C₄
Fitting	C₅×#3458	C₂
prime radical	C₂⁴×C₅	C₂²×C₄
cosocle	C₂×C₄×C₅	C₂⁵
hypercenter	#3458	D₅
hypocenter	C₅	#60191
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #3458

Complements

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

N	H ≅ G/N	type
2. #10615060	C₂	semidirect
3. #10615059	C₂	semidirect
12. #65313	C₂²	semidirect
13. Q₈×#168	C₂²	semidirect
33. C₂²×#168	Q₈	semidirect
157. C₅	#60191	semidirect

Maximal subgroups

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

subgroup	index
C₅×#3458	2
#10615060	2
#10615059	2
#10613935	2
C₂×#65286	2
#60191	5