Subgroups — #131×#1055 #15828505

Order 3328.

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	#41×#1055	C₂	1	2	yes	maximal
2	#13113119	C₂	1	3	yes	maximal, complemented
3	M₁₆×#1055	C₂	1	6	yes	maximal, complemented
4	C₂×C₈×#1055	C₂	1	3	yes	maximal, complemented
5	C₄×C₁₃×#131	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#131	C₄	1	2	yes	—
7	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
8	C₁₃×#405	C₄	1	6	yes	—
9	C₄×C₁₃×M₁₆	C₂²	1	3	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
11	Q₈×#1055	C₄	1	4	yes	—
12	D₄×#1055	C₄	1	12	yes	—
13	#11718586	C₂²	1	6	yes	—
14	C₂×C₄×#1055	C₂²	1	6	yes	—
15	#11718047	C₂²	1	6	yes	—
16	C₈×#1055	C₂²	1	1	yes	characteristic
17	C₈×#1055	C₂²	1	6	yes	—
18	C₈×#1055	C₂²	1	3	yes	—
19	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
20	C₁₃×#131	C₈	1	16	yes	complemented
21	C₂×C₁₃×M₁₆	C₂×C₄	1	6	yes	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	6	yes	—
23	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
24	C₄×D₄×C₁₃	C₂×C₄	1	6	yes	—
25	C₁₃×#117	C₂×C₄	1	3	yes	—
26	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
27	C₂²×#1055	C₂×C₄	1	12	yes	—
28	C₁₃×#97	C₂×C₄	1	6	yes	—
29	C₄×C₈×C₁₃	C₂×C₄	1	6	yes	—
30	C₄×C₈×C₁₃	C₂³	1	3	yes	—
31	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²⋊#1055	C₂×C₄	1	12	yes	—
33	C₄⋊#1055	C₂×C₄	1	12	yes	—
34	C₄×#1055	C₂×C₄	1	12	yes	—
35	C₄×#1055	C₂³	1	2	yes	—
36	C₄×#1055	C₂³	1	6	yes	—
37	C₈×#131	—	13	1	no	maximal, Sylow
38	C₁₃×#41	C₂×C₈	1	8	yes	—
39	C₂×Q₈×C₁₃	C₄²	1	1	yes	characteristic
40	C₂×D₄×C₁₃	C₄²	1	3	yes	—
41	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
42	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
43	C₁₃×M₁₆	C₂×C₈	1	24	yes	complemented
44	C₂×C₈×C₁₃	C₂×C₈	1	24	yes	complemented
45	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
46	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
47	C₁₃×C₄⋊C₄	C₄²	1	6	yes	—
48	C₁₃×C₂²⋊C₄	C₄²	1	6	yes	—
49	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
50	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
51	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
52	C₂×#1055	—	2	24	no	—
53	C₂×#1055	C₂²×C₄	1	12	yes	—
54	C₂×#1055	C₂²×C₄	1	4	yes	—
55	C₈×#41	—	13	2	no	—
56	C₄×#131	D₁₃	1	1	yes	characteristic
57	#1450	—	13	3	no	—
58	C₈×M₁₆	—	13	6	no	—
59	C₂×C₈²	—	13	3	no	—
60	C₂³×C₁₃	C₂×C₄²	1	3	yes	—
61	Q₈×C₁₃	C₄×C₈	1	4	yes	—
62	D₄×C₁₃	C₄×C₈	1	12	yes	—
63	C₂×C₄×C₁₃	—	2	12	no	—
64	C₂×C₄×C₁₃	C₂×C₄²	1	6	yes	—
65	C₂×C₄×C₁₃	C₂×C₄²	1	3	yes	—
66	C₂×C₄×C₁₃	C₂²×C₈	1	12	yes	—
67	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
69	C₈×C₁₃	C₂²×C₈	1	4	yes	—
70	C₈×C₁₃	C₂²×C₈	1	12	yes	—
71	#1055	—	2	12	no	—
72	#1055	#131	1	8	yes	direct product factor
73	C₂×#131	Dic₁₃	1	2	yes	—
74	C₄×#41	C₂×D₁₃	1	1	yes	characteristic
75	C₈×Q₈	—	13	4	no	—
76	C₈×D₄	—	13	12	no	—
77	#431	—	13	6	no	—
78	#405	Dic₁₃	1	6	yes	—
79	C₄×M₁₆	C₂×D₁₃	1	3	yes	—
80	C₂×C₄×C₈	—	13	6	no	—
81	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
82	#322	—	13	6	no	—
83	C₈²	—	13	6	no	—
84	C₈²	—	13	1	no	—
85	C₈²	—	13	3	no	—
86	C₂²×C₁₃	—	2	6	no	—
87	C₂²×C₁₃	C₂×C₄×C₈	1	6	yes	—
88	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
89	C₄×C₁₃	—	2	6	no	—
90	C₄×C₁₃	C₂×C₄×C₈	1	6	yes	—
91	C₄×C₁₃	C₂³×C₈	1	2	yes	—
92	C₄×C₁₃	C₂×#131	1	4	yes	—
93	C₂×#41	C₂×Dic₁₃	1	1	yes	characteristic
94	#131	#1055	1	16	yes	direct product factor
95	C₂×M₁₆	C₂×Dic₁₃	1	6	yes	—
96	C₂²×C₈	—	13	12	no	—
97	C₂²×C₈	C₂×Dic₁₃	1	6	yes	—
98	C₄×Q₈	C₄×D₁₃	1	2	yes	—
99	C₄×D₄	C₄×D₁₃	1	6	yes	—
100	#117	C₂×Dic₁₃	1	3	yes	—
101	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
102	C₄⋊C₈	—	13	12	no	—
103	C₂²⋊C₈	—	13	12	no	—
104	#97	C₂×Dic₁₃	1	6	yes	—
105	C₄×C₈	—	13	12	no	—
106	C₄×C₈	—	13	2	no	—
107	C₄×C₈	—	13	6	no	—
108	C₄×C₈	C₂×Dic₁₃	1	6	yes	—
109	C₄×C₈	C₂²×D₁₃	1	3	yes	—
110	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
111	C₂×C₁₃	—	2	6	no	—
112	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
113	C₂×C₁₃	C₄×#131	1	2	yes	—
114	#41	C₂×#1055	1	8	yes	—
115	C₂×Q₈	C₄×Dic₁₃	1	1	yes	characteristic
116	C₂×D₄	C₄×Dic₁₃	1	3	yes	—
117	C₂²×C₄	C₂×C₄×D₁₃	1	6	yes	—
118	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
119	M₁₆	C₂×#1055	1	24	yes	complemented
120	C₂×C₈	—	13	4	no	—
121	C₂×C₈	—	13	12	no	—
122	C₂×C₈	—	26	24	no	—
123	C₂×C₈	C₂×#1055	1	24	yes	complemented
124	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
125	C₂×C₈	C₂²×Dic₁₃	1	6	yes	—
126	C₄⋊C₄	C₄×Dic₁₃	1	6	yes	—
127	C₂²⋊C₄	C₄×Dic₁₃	1	6	yes	—
128	C₄²	C₂×C₄×D₁₃	1	6	yes	—
129	C₄²	C₂²×Dic₁₃	1	3	yes	—
130	C₄²	C₂³×D₁₃	1	1	yes	characteristic
131	C₁₃	C₈×#131	1	1	yes	characteristic, complemented, Sylow
132	C₂³	C₂×C₄×Dic₁₃	1	3	yes	—
133	Q₈	C₄×#1055	1	4	yes	—
134	D₄	C₄×#1055	1	12	yes	—
135	C₂×C₄	—	2	12	no	—
136	C₂×C₄	C₂²×#1055	1	12	yes	—
137	C₂×C₄	C₂×C₄×Dic₁₃	1	6	yes	—
138	C₂×C₄	C₂×C₄×Dic₁₃	1	3	yes	—
139	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
140	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
141	C₈	—	13	8	no	—
142	C₈	—	26	12	no	—
143	C₈	C₂²×#1055	1	4	yes	—
144	C₈	C₂²×#1055	1	12	yes	—
145	C₂²	—	2	6	no	—
146	C₂²	C₂×C₄×#1055	1	6	yes	—
147	C₂²	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
148	C₄	—	2	6	no	—
149	C₄	C₂×C₄×#1055	1	6	yes	—
150	C₄	D₁₃×#131	1	4	yes	—
151	C₄	C₂³×#1055	1	2	yes	—
152	C₂	—	2	6	no	—
153	C₂	C₂²×C₄×#1055	1	1	yes	characteristic
154	C₂	#131×Dic₁₃	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₁₃
commutator	C₂×C₁₃	C₂²×C₄×C₈
Frattini	C₄²	C₂³×D₁₃
Fitting	C₄×C₁₃×#131	C₂
socle	C₂²×C₁₃	C₂²×C₄²
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₄×#131	D₁₃
hypocenter	C₁₃	C₈×#131
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #13113119	C₂	semidirect
3. M₁₆×#1055	C₂	semidirect
4. C₂×C₈×#1055	C₂	semidirect
20. C₁₃×#131	C₈	semidirect
43. C₁₃×M₁₆	C₂×C₈	semidirect
44. C₂×C₈×C₁₃	C₂×C₈	semidirect
72. #1055	#131	direct (N × H)
94. #131	#1055	direct (N × H)
119. M₁₆	C₂×#1055	semidirect
123. C₂×C₈	C₂×#1055	semidirect
131. C₁₃	C₈×#131	semidirect

Maximal subgroups

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

subgroup	index
#41×#1055	2
#13113119	2
M₁₆×#1055	2
C₂×C₈×#1055	2
C₄×C₁₃×#131	2
C₈×#131	13