Subgroups — #29588995

Order 15616.

330 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14519717	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11758185	C₂	1	1	yes	maximal, characteristic
3	#21443736	C₂	1	2	yes	maximal, complemented
4	C₈×#11758185	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1450	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519717	C₄	1	2	yes	—
7	C₂²×#11758185	C₄	1	2	yes	—
8	#16839538	C₄	1	2	yes	—
9	C₄×#11758185	C₄	1	2	yes	—
10	C₄×#11758185	C₂²	1	1	yes	characteristic
11	C₄×#11758185	C₂²	1	2	yes	—
12	C₄×#11758185	C₂²	1	1	yes	characteristic
13	C₆₁×#431	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#322	C₄	1	2	yes	—
16	C₈²×C₆₁	C₂²	1	2	yes	—
17	#14519717	C₈	1	8	yes	complemented
18	C₂×#11758185	C₈	1	8	yes	complemented
19	C₂×#11758185	C₂×C₄	1	2	yes	—
20	C₂×#11758185	C₂×C₄	1	2	yes	—
21	C₂×#11758185	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₄⋊C₈×C₆₁	C₈	1	4	yes	—
26	C₂²⋊C₈×C₆₁	C₈	1	4	yes	—
27	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₆₁	C₂×C₈	1	8	yes	—
37	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
38	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
39	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₆₁	C₄²	1	1	yes	characteristic
42	C₄²×C₆₁	C₂×C₈	1	2	yes	—
43	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
44	#11758185	C₂×C₈	1	4	yes	—
45	#11758185	C₂×C₈	1	4	yes	—
46	#11758185	C₂×C₈	1	8	yes	—
47	C₂³×C₆₁	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₆₁	—	2	4	no	—
49	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
50	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
52	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
53	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
55	C₈×C₆₁	—	2	8	no	—
56	C₈×C₆₁	C₄×C₈	1	2	yes	—
57	C₈×C₆₁	C₄×C₈	1	2	yes	—
58	C₈×C₆₁	C₂²×C₈	1	2	yes	—
59	C₈×C₆₁	C₂²×C₈	1	2	yes	—
60	#11399	—	61	1	no	maximal, Sylow
61	C₂²×C₆₁	—	2	2	no	—
62	C₂²×C₆₁	C₈²	1	2	yes	—
63	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
64	C₄×C₆₁	—	2	2	no	—
65	C₄×C₆₁	C₈²	1	2	yes	—
66	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
67	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
68	C₄×C₆₁	#504	1	4	yes	—
69	C₄×M₃₂	—	61	1	no	—
70	C₂×C₄×C₁₆	—	61	1	no	—
71	#1450	D₆₁	1	1	yes	characteristic
72	#1311	—	61	2	no	—
73	C₈×C₁₆	—	61	2	no	—
74	C₂×C₆₁	—	2	2	no	—
75	C₂×C₆₁	C₂×C₈²	1	1	yes	characteristic
76	C₂×C₆₁	#2108	1	2	yes	—
77	C₂×M₃₂	—	61	2	no	—
78	C₂²×C₁₆	—	61	2	no	—
79	#431	Dic₆₁	1	2	yes	—
80	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
81	#346	—	61	2	no	—
82	C₄×C₁₆	—	61	2	no	—
83	C₄×C₁₆	—	61	1	no	—
84	C₄×C₁₆	—	61	2	no	—
85	C₄×C₁₆	—	61	1	no	—
86	#322	Dic₆₁	1	2	yes	—
87	C₈²	C₂×D₆₁	1	2	yes	—
88	C₆₁	#11399	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
90	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
91	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
92	M₃₂	—	61	8	no	—
93	C₂×C₁₆	—	61	8	no	—
94	C₂×C₁₆	—	61	2	no	—
95	C₂×C₁₆	—	61	2	no	—
96	C₂×C₁₆	—	61	4	no	—
97	C₄⋊C₈	#92418	1	4	yes	—
98	C₂²⋊C₈	#92418	1	4	yes	—
99	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
100	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
101	C₄×C₈	C₂×Dic₆₁	1	4	yes	—
102	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
103	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
104	C₂²×C₄	C₂×#92418	1	2	yes	—
105	C₂²×C₄	C₄×Dic₆₁	1	1	yes	characteristic
106	C₂×C₈	C₈×D₆₁	1	4	yes	—
107	C₂×C₈	C₈×D₆₁	1	4	yes	—
108	C₂×C₈	C₂×#92418	1	8	yes	—
109	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
110	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
111	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
112	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
113	C₄²	C₂×#92418	1	2	yes	—
114	C₄²	C₄×Dic₆₁	1	1	yes	characteristic
115	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
116	C₁₆	—	61	4	no	—
117	C₁₆	—	61	4	no	—
118	C₁₆	—	61	8	no	—
119	C₂³	C₄×#92418	1	1	yes	characteristic
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	C₈×Dic₆₁	1	2	yes	—
122	C₂×C₄	C₈×Dic₆₁	1	2	yes	—
123	C₂×C₄	C₄×#92418	1	2	yes	—
124	C₂×C₄	C₄×#92418	1	1	yes	characteristic
125	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
126	C₂×C₄	C₂²×#92418	1	2	yes	—
127	C₈	—	2	8	no	—
128	C₈	C₈×Dic₆₁	1	2	yes	—
129	C₈	C₈×Dic₆₁	1	2	yes	—
130	C₈	C₂×C₈×D₆₁	1	2	yes	—
131	C₈	C₂×C₈×D₆₁	1	2	yes	—
132	C₂²	—	2	2	no	—
133	C₂²	C₈×#92418	1	2	yes	—
134	C₂²	C₂×C₄×#92418	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	C₈×#92418	1	2	yes	—
137	C₄	C₂×C₈×Dic₆₁	1	1	yes	characteristic
138	C₄	C₂×C₈×Dic₆₁	1	1	yes	characteristic
139	C₄	#16840179	1	4	yes	—
140	C₂	—	2	2	no	—
141	C₂	C₂×C₈×#92418	1	1	yes	characteristic
142	C₂	#21447416	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₆₁
commutator	C₂×C₆₁	C₂×C₈²
Fitting	C₆₁×#1450	C₂
socle	C₂²×C₆₁	C₂×C₄×C₈
prime radical	C₂³×C₆₁	C₄×C₈
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	#1450	D₆₁
hypocenter	C₆₁	#11399
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #21443736	C₂	semidirect
4. C₈×#11758185	C₂	semidirect
17. #14519717	C₈	semidirect
18. C₂×#11758185	C₈	semidirect
88. C₆₁	#11399	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14519717	2
C₂×C₄×#11758185	2
#21443736	2
C₈×#11758185	2
C₆₁×#1450	2
#11399	61