Subgroups — #30603676

Order 15616.

330 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462025	C₂	1	1	yes	maximal, characteristic
2	#21461022	C₂	1	1	yes	maximal, characteristic
3	#21460730	C₂	1	2	yes	maximal, complemented
4	C₁₆×#92420	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519851	C₄	1	2	yes	—
7	#16840778	C₄	1	2	yes	—
8	#16840651	C₄	1	2	yes	—
9	#16840648	C₂²	1	2	yes	—
10	#16840603	C₄	1	2	yes	—
11	C₈×#92420	C₂²	1	2	yes	—
12	C₂²×#11758185	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
14	C₁₆×Dic₆₁	C₄	1	1	yes	characteristic
15	C₁₆×Dic₆₁	C₄	1	1	yes	characteristic
16	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
17	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₆₁	C₄	1	1	yes	characteristic
19	#14519851	C₈	1	8	yes	complemented
20	C₂×#11758187	C₂×C₄	1	4	yes	—
21	C₄⋊#92420	C₈	1	4	yes	—
22	C₂²⋊#92420	C₈	1	4	yes	—
23	C₄×#92420	C₂×C₄	1	4	yes	—
24	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂×#11758185	C₈	1	4	yes	complemented
27	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
28	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₆₁	C₈	1	4	yes	complemented
35	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
38	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
39	C₂×#92420	C₂×C₈	1	8	yes	—
40	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
44	C₄×Dic₆₁	C₄²	1	1	yes	characteristic
45	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
46	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×#92418	C₄²	1	1	yes	characteristic
48	C₂×#92418	C₂×C₈	1	4	yes	—
49	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
50	#11758187	C₂×C₈	1	8	yes	—
51	C₁₆×C₆₁	C₂×C₈	1	2	yes	—
52	C₁₆×C₆₁	C₂×C₈	1	2	yes	—
53	#11758185	C₂×C₈	1	2	yes	—
54	#11758185	C₂×C₈	1	2	yes	—
55	C₂³×C₆₁	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
57	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
58	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
59	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
60	C₂×Dic₆₁	—	2	4	no	—
61	C₂×Dic₆₁	C₄×C₈	1	2	yes	—
62	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
63	#92420	—	2	8	no	—
64	C₈×C₆₁	C₄×C₈	1	2	yes	—
65	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
66	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
67	#92418	C₄×C₈	1	2	yes	—
68	#92418	C₂²×C₈	1	2	yes	—
69	#11399	—	61	1	no	maximal, Sylow
70	C₂²×C₆₁	—	2	2	no	—
71	C₂²×C₆₁	C₈²	1	2	yes	—
72	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
73	C₄×C₆₁	C₈²	1	2	yes	—
74	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
75	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
76	Dic₆₁	—	2	2	no	—
77	Dic₆₁	#504	1	4	yes	—
78	C₄×M₃₂	—	61	1	no	—
79	C₂×C₄×C₁₆	—	61	1	no	—
80	#1450	—	61	1	no	—
81	#1311	—	61	2	no	—
82	C₈×C₁₆	—	61	2	no	—
83	C₂×C₆₁	—	2	2	no	—
84	C₂×C₆₁	C₂×C₈²	1	1	yes	characteristic
85	C₂×C₆₁	#2108	1	2	yes	—
86	C₂×M₃₂	—	61	2	no	—
87	C₂²×C₁₆	—	61	1	no	—
88	C₂²×C₁₆	#6923	1	1	yes	characteristic
89	#431	—	61	2	no	—
90	C₂×C₄×C₈	—	61	1	no	—
91	#346	—	61	2	no	—
92	C₄×C₁₆	—	61	2	no	—
93	C₄×C₁₆	—	61	1	no	—
94	C₄×C₁₆	—	61	1	no	—
95	C₄×C₁₆	—	61	1	no	—
96	C₄×C₁₆	—	61	1	no	—
97	#322	—	61	2	no	—
98	C₈²	—	61	2	no	—
99	C₆₁	#11399	1	1	yes	characteristic, complemented, Sylow
100	C₂²×C₈	—	61	1	no	—
101	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
102	C₂×C₄²	—	61	1	no	—
103	M₃₂	—	61	8	no	—
104	C₂×C₁₆	—	61	4	no	—
105	C₂×C₁₆	—	61	4	no	—
106	C₂×C₁₆	—	61	1	no	—
107	C₂×C₁₆	—	61	1	no	—
108	C₂×C₁₆	#92420	1	4	yes	complemented
109	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
110	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
111	C₄⋊C₈	—	61	4	no	—
112	C₂²⋊C₈	—	61	4	no	—
113	C₄×C₈	—	61	4	no	—
114	C₄×C₈	—	61	1	no	—
115	C₄×C₈	—	61	1	no	—
116	C₄×C₈	—	61	1	no	—
117	C₄×C₈	—	61	1	no	—
118	C₂²×C₄	—	61	2	no	—
119	C₂²×C₄	C₄×#6923	1	1	yes	characteristic
120	C₂×C₈	—	61	4	no	—
121	C₂×C₈	—	61	1	no	—
122	C₂×C₈	—	61	1	no	—
123	C₂×C₈	—	61	8	no	—
124	C₂×C₈	D₆₁⋊C₈	1	2	yes	—
125	C₂×C₈	C₄×#6923	1	1	yes	characteristic
126	C₂×C₈	C₂×#92420	1	2	yes	—
127	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
128	C₄²	—	61	2	no	—
129	C₄²	—	61	1	no	—
130	C₄²	—	61	1	no	—
131	C₁₆	—	61	8	no	—
132	C₁₆	—	61	2	no	—
133	C₁₆	—	61	2	no	—
134	C₁₆	C₂×#92420	1	2	yes	—
135	C₁₆	C₂×#92420	1	2	yes	—
136	C₂³	C₈×#6923	1	1	yes	characteristic
137	C₂×C₄	—	61	2	no	—
138	C₂×C₄	—	61	2	no	—
139	C₂×C₄	—	122	4	no	—
140	C₂×C₄	C₈×#6923	1	2	yes	—
141	C₂×C₄	C₈×#6923	1	1	yes	characteristic
142	C₂×C₄	C₄×#92420	1	2	yes	—
143	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
144	C₈	—	61	2	no	—
145	C₈	—	61	2	no	—
146	C₈	—	122	8	no	—
147	C₈	C₄×#92420	1	2	yes	—
148	C₈	C₂²×#92420	1	1	yes	characteristic
149	C₈	C₂×D₆₁⋊C₈	1	1	yes	characteristic
150	C₂²	—	2	2	no	—
151	C₂²	C₈×#92420	1	2	yes	—
152	C₂²	C₂×C₈×#6923	1	1	yes	characteristic
153	C₄	—	61	4	no	—
154	C₄	—	122	2	no	—
155	C₄	C₈×#92420	1	2	yes	—
156	C₄	C₂×C₈×#6923	1	1	yes	characteristic
157	C₄	C₂×C₄×#92420	1	1	yes	characteristic
158	C₂	—	2	2	no	—
159	C₂	C₂×C₈×#92420	1	1	yes	characteristic
160	C₂	#21462027	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂×#6923
commutator	C₂×C₆₁	C₂×C₈²
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₂²×C₁₆×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₄×C₈
prime radical	C₂³×C₆₁	C₄×C₈
cosocle	C₈×Dic₆₁	C₂³
hypercenter	C₂²×C₁₆	#6923
hypocenter	C₆₁	#11399
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #21460730	C₂	semidirect
4. C₁₆×#92420	C₂	semidirect
19. #14519851	C₈	semidirect
26. C₂×#11758185	C₈	semidirect
34. C₂×C₁₆×C₆₁	C₈	semidirect
99. C₆₁	#11399	semidirect
108. C₂×C₁₆	#92420	semidirect

Maximal subgroups

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

subgroup	index
#21462025	2
#21461022	2
#21460730	2
C₁₆×#92420	2
C₂×C₁₆×Dic₆₁	2
#11399	61