Subgroups — #131×#168 #12064886

Order 1280.

922 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#131	C₂	1	1	yes	maximal, characteristic
2	#41×#168	C₂	1	2	yes	maximal
3	M₁₆×#168	C₂	1	6	yes	maximal, complemented
4	#10600517	C₂	1	3	yes	maximal, complemented
5	C₂×C₈×#168	C₂	1	3	yes	maximal, complemented
6	C₂×C₅×#131	C₄	1	2	yes	—
7	C₄×C₅×#41	C₂²	1	1	yes	characteristic
8	C₅×#405	C₄	1	6	yes	—
9	C₄×C₅×M₁₆	C₂²	1	3	yes	—
10	C₂×C₄×C₅×C₈	C₂²	1	3	yes	—
11	Q₈×#168	C₄	1	4	yes	—
12	D₄×#168	C₄	1	12	yes	—
13	#65214	C₂²	1	6	yes	—
14	C₂×C₄×#168	C₂²	1	6	yes	—
15	#64675	C₂²	1	6	yes	—
16	C₈×#168	C₂²	1	1	yes	characteristic
17	C₈×#168	C₂²	1	3	yes	—
18	C₈×#168	C₂²	1	6	yes	—
19	C₈×#131	—	5	1	no	maximal, Sylow
20	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
21	C₅×#131	C₈	1	16	yes	complemented
22	C₂×C₅×M₁₆	C₂×C₄	1	6	yes	—
23	C₂²×C₅×C₈	C₂×C₄	1	6	yes	—
24	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
25	C₄×C₅×D₄	C₂×C₄	1	6	yes	—
26	C₅×#117	C₂×C₄	1	3	yes	—
27	C₂×C₄²×C₅	C₂³	1	3	yes	—
28	C₂²×#168	C₂×C₄	1	12	yes	—
29	C₅×#97	C₂×C₄	1	6	yes	—
30	C₄×C₅×C₈	C₂×C₄	1	6	yes	—
31	C₄×C₅×C₈	C₂³	1	3	yes	—
32	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₂²⋊#168	C₂×C₄	1	12	yes	—
34	C₄⋊#168	C₂×C₄	1	12	yes	—
35	C₄×#168	C₂×C₄	1	12	yes	—
36	C₄×#168	C₂³	1	2	yes	—
37	C₄×#168	C₂³	1	6	yes	—
38	C₈×#41	—	5	2	no	—
39	C₄×#131	D₅	1	1	yes	characteristic
40	#1450	—	5	3	no	—
41	C₈×M₁₆	—	5	6	no	—
42	C₂×C₈²	—	5	3	no	—
43	C₅×#41	C₂×C₈	1	8	yes	—
44	C₂×C₅×Q₈	C₄²	1	1	yes	characteristic
45	C₂×C₅×D₄	C₄²	1	3	yes	—
46	C₂²×C₄×C₅	C₂²×C₄	1	6	yes	—
47	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
48	C₅×M₁₆	C₂×C₈	1	24	yes	complemented
49	C₂×C₅×C₈	C₂×C₈	1	24	yes	complemented
50	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
51	C₂×C₅×C₈	C₂²×C₄	1	6	yes	—
52	C₅×C₄⋊C₄	C₄²	1	6	yes	—
53	C₅×C₂²⋊C₄	C₄²	1	6	yes	—
54	C₄²×C₅	C₂²×C₄	1	6	yes	—
55	C₄²×C₅	C₂²×C₄	1	3	yes	—
56	C₄²×C₅	C₂⁴	1	1	yes	characteristic
57	C₂×#168	—	2	24	no	—
58	C₂×#168	C₂²×C₄	1	12	yes	—
59	C₂×#168	C₂²×C₄	1	4	yes	—
60	C₂×#131	Dic₅	1	2	yes	—
61	C₄×#41	C₂×D₅	1	1	yes	characteristic
62	C₈×Q₈	—	5	4	no	—
63	C₈×D₄	—	5	12	no	—
64	#431	—	5	6	no	—
65	#405	Dic₅	1	6	yes	—
66	C₄×M₁₆	C₂×D₅	1	3	yes	—
67	C₂×C₄×C₈	—	5	6	no	—
68	C₂×C₄×C₈	C₂×D₅	1	3	yes	—
69	#322	—	5	6	no	—
70	C₈²	—	5	6	no	—
71	C₈²	—	5	1	no	—
72	C₈²	—	5	3	no	—
73	C₂³×C₅	C₂×C₄²	1	3	yes	—
74	C₅×Q₈	C₄×C₈	1	4	yes	—
75	C₅×D₄	C₄×C₈	1	12	yes	—
76	C₂×C₄×C₅	—	2	12	no	—
77	C₂×C₄×C₅	C₂×C₄²	1	6	yes	—
78	C₂×C₄×C₅	C₂×C₄²	1	3	yes	—
79	C₂×C₄×C₅	C₂²×C₈	1	12	yes	—
80	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
81	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
82	C₅×C₈	C₂²×C₈	1	4	yes	—
83	C₅×C₈	C₂²×C₈	1	12	yes	—
84	#168	—	2	12	no	—
85	#168	#131	1	8	yes	direct product factor
86	C₂×#41	C₂×Dic₅	1	1	yes	characteristic
87	#131	#168	1	16	yes	direct product factor
88	C₂×M₁₆	C₂×Dic₅	1	6	yes	—
89	C₂²×C₈	—	5	12	no	—
90	C₂²×C₈	C₂×Dic₅	1	6	yes	—
91	C₄×Q₈	C₄×D₅	1	2	yes	—
92	C₄×D₄	C₄×D₅	1	6	yes	—
93	#117	C₂×Dic₅	1	3	yes	—
94	C₂×C₄²	C₂²×D₅	1	3	yes	—
95	C₄⋊C₈	—	5	12	no	—
96	C₂²⋊C₈	—	5	12	no	—
97	#97	C₂×Dic₅	1	6	yes	—
98	C₄×C₈	—	5	12	no	—
99	C₄×C₈	—	5	2	no	—
100	C₄×C₈	—	5	6	no	—
101	C₄×C₈	C₂×Dic₅	1	6	yes	—
102	C₄×C₈	C₂²×D₅	1	3	yes	—
103	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
104	C₂²×C₅	—	2	6	no	—
105	C₂²×C₅	C₂×C₄×C₈	1	6	yes	—
106	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
107	C₄×C₅	—	2	6	no	—
108	C₄×C₅	C₂×C₄×C₈	1	6	yes	—
109	C₄×C₅	C₂³×C₈	1	2	yes	—
110	C₄×C₅	C₂×#131	1	4	yes	—
111	#41	C₂×#168	1	8	yes	—
112	C₂×Q₈	C₄×Dic₅	1	1	yes	characteristic
113	C₂×D₄	C₄×Dic₅	1	3	yes	—
114	C₂²×C₄	C₂×C₄×D₅	1	6	yes	—
115	C₂²×C₄	C₂²×Dic₅	1	3	yes	—
116	M₁₆	C₂×#168	1	24	yes	complemented
117	C₂×C₈	—	5	4	no	—
118	C₂×C₈	—	5	12	no	—
119	C₂×C₈	—	10	24	no	—
120	C₂×C₈	C₂×#168	1	24	yes	complemented
121	C₂×C₈	C₂²×Dic₅	1	2	yes	—
122	C₂×C₈	C₂²×Dic₅	1	6	yes	—
123	C₄⋊C₄	C₄×Dic₅	1	6	yes	—
124	C₂²⋊C₄	C₄×Dic₅	1	6	yes	—
125	C₄²	C₂×C₄×D₅	1	6	yes	—
126	C₄²	C₂²×Dic₅	1	3	yes	—
127	C₄²	C₂³×D₅	1	1	yes	characteristic
128	C₂×C₅	—	2	6	no	—
129	C₂×C₅	C₂²×C₄×C₈	1	1	yes	characteristic
130	C₂×C₅	C₄×#131	1	2	yes	—
131	C₂³	C₂×C₄×Dic₅	1	3	yes	—
132	Q₈	C₄×#168	1	4	yes	—
133	D₄	C₄×#168	1	12	yes	—
134	C₂×C₄	—	2	12	no	—
135	C₂×C₄	C₂²×#168	1	12	yes	—
136	C₂×C₄	C₂×C₄×Dic₅	1	6	yes	—
137	C₂×C₄	C₂×C₄×Dic₅	1	3	yes	—
138	C₂×C₄	C₂²×C₄×D₅	1	2	yes	—
139	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
140	C₈	—	5	8	no	—
141	C₈	—	10	12	no	—
142	C₈	C₂²×#168	1	4	yes	—
143	C₈	C₂²×#168	1	12	yes	—
144	C₅	C₈×#131	1	1	yes	characteristic, complemented, Sylow
145	C₂²	—	2	6	no	—
146	C₂²	C₂×C₄×#168	1	6	yes	—
147	C₂²	C₂²×C₄×Dic₅	1	1	yes	characteristic
148	C₄	—	2	6	no	—
149	C₄	C₂×C₄×#168	1	6	yes	—
150	C₄	D₅×#131	1	4	yes	—
151	C₄	C₂³×#168	1	2	yes	—
152	C₂	—	2	6	no	—
153	C₂	C₂²×C₄×#168	1	1	yes	characteristic
154	C₂	Dic₅×#131	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₅
commutator	C₂×C₅	C₂²×C₄×C₈
Frattini	C₄²	C₂³×D₅
Fitting	C₄×C₅×#131	C₂
socle	C₂²×C₅	C₂²×C₄²
prime radical	C₂×C₅×#41	C₂×C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₄×#131	D₅
hypocenter	C₅	C₈×#131
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
3. M₁₆×#168	C₂	semidirect
4. #10600517	C₂	semidirect
5. C₂×C₈×#168	C₂	semidirect
21. C₅×#131	C₈	semidirect
48. C₅×M₁₆	C₂×C₈	semidirect
49. C₂×C₅×C₈	C₂×C₈	semidirect
85. #168	#131	direct (N × H)
87. #131	#168	direct (N × H)
116. M₁₆	C₂×#168	semidirect
120. C₂×C₈	C₂×#168	semidirect
144. C₅	C₈×#131	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#131	2
#41×#168	2
M₁₆×#168	2
#10600517	2
C₂×C₈×#168	2
C₈×#131	5