Subgroups — #29569375

Order 15616.

558 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	#21445565	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519688	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839484	C₂	1	2	yes	maximal
4	C₂×#16839483	C₂	1	2	yes	maximal
5	C₄²×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519688	C₄	1	4	yes	—
7	C₂×C₄×#92418	C₂²	1	2	yes	—
8	C₂×C₄×#92418	C₂²	1	2	yes	—
9	#16839517	C₄	1	4	yes	—
10	#16839484	C₄	1	8	yes	complemented
11	#16839483	C₄	1	8	yes	complemented
12	C₂×C₄×C₈×C₆₁	C₄	1	4	yes	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₄³×C₆₁	C₂²	1	1	yes	characteristic
16	C₂²×#92418	C₂×C₄	1	4	yes	—
17	C₂²×#92418	C₂×C₄	1	4	yes	—
18	#14519688	—	2	8	no	—
19	C₄×#92418	C₂×C₄	1	4	yes	—
20	C₄×#92418	C₂×C₄	1	4	yes	—
21	C₄×#92418	C₂×C₄	1	4	yes	—
22	C₄×#92418	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₆₁	—	2	8	no	—
30	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₆₁	C₄²	1	4	yes	—
33	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₆₁	—	2	4	no	—
37	C₂×C₈×C₆₁	C₄²	1	4	yes	—
38	C₂×C₈×C₆₁	C₄²	1	4	yes	—
39	C₂×C₈×C₆₁	M₁₆	1	8	yes	—
40	C₄²×C₆₁	—	2	4	no	—
41	C₄²×C₆₁	—	2	4	no	—
42	C₄²×C₆₁	M₁₆	1	8	yes	—
43	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
44	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
45	C₂×#92418	—	2	8	no	—
46	C₂×#92418	—	2	8	no	—
47	C₂×#92418	C₄²	1	8	yes	—
48	C₂×#92418	C₄²	1	8	yes	—
49	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₆₁	—	2	4	no	—
51	C₂×C₄×C₆₁	—	2	4	no	—
52	C₂×C₄×C₆₁	#97	1	8	yes	—
53	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
55	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
56	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
57	C₂×C₄×C₆₁	C₂×M₁₆	1	4	yes	—
58	C₂×C₄×C₆₁	C₂×M₁₆	1	4	yes	—
59	C₂×C₄×C₆₁	#131	1	4	yes	—
60	C₈×C₆₁	—	2	4	no	—
61	C₈×C₆₁	#97	1	8	yes	complemented
62	#92418	—	2	8	no	—
63	#92418	—	2	8	no	—
64	#7842	—	61	1	no	maximal, Sylow
65	C₂²×C₆₁	C₄³	1	1	yes	characteristic
66	C₂²×C₆₁	C₂×#97	1	2	yes	—
67	C₂²×C₆₁	C₄×M₁₆	1	2	yes	—
68	C₂²×C₆₁	#405	1	2	yes	—
69	C₄×C₆₁	—	2	8	no	—
70	C₄×C₆₁	C₂×#97	1	2	yes	—
71	C₄×C₆₁	C₄×M₁₆	1	4	yes	—
72	C₄×C₆₁	C₄×M₁₆	1	2	yes	—
73	C₄×C₆₁	#405	1	4	yes	—
74	#1744	—	61	1	no	—
75	C₄×#97	—	61	1	no	—
76	C₄²×C₈	D₆₁	1	1	yes	characteristic
77	C₂×#322	—	61	2	no	—
78	C₂×#322	—	61	2	no	—
79	C₂×C₆₁	#1451	1	4	yes	—
80	C₂×C₆₁	C₄×#97	1	1	yes	characteristic
81	C₂×C₆₁	C₄×#97	1	1	yes	characteristic
82	C₂×C₆₁	#1726	1	1	yes	characteristic
83	C₂×#97	—	61	4	no	—
84	C₂×C₄×C₈	—	61	2	no	—
85	C₂×C₄×C₈	—	61	2	no	—
86	C₂×C₄×C₈	Dic₆₁	1	4	yes	—
87	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
89	C₄³	C₂×D₆₁	1	1	yes	characteristic
90	#336	—	61	4	no	—
91	#322	—	61	8	no	—
92	#322	—	61	8	no	—
93	C₆₁	#7842	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	—	61	4	no	—
95	C₂²×C₈	—	61	4	no	—
96	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
97	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
98	C₂×C₄²	C₄×D₆₁	1	2	yes	—
99	C₂×C₄²	C₄×D₆₁	1	2	yes	—
100	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
101	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
102	#97	—	122	8	no	—
103	C₄×C₈	—	2	8	no	—
104	C₄×C₈	—	61	4	no	—
105	C₄×C₈	—	61	4	no	—
106	C₄×C₈	—	61	4	no	—
107	C₄×C₈	—	61	4	no	—
108	C₄×C₈	C₄×D₆₁	1	4	yes	—
109	C₄×C₈	C₄×D₆₁	1	4	yes	—
110	C₂²×C₄	C₄×Dic₆₁	1	4	yes	—
111	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
113	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	61	8	no	—
116	C₂×C₈	—	61	8	no	—
117	C₂×C₈	—	122	8	no	—
118	C₂×C₈	—	122	8	no	—
119	C₂×C₈	#11758194	1	8	yes	—
120	C₂×C₈	C₄×Dic₆₁	1	4	yes	—
121	C₂×C₈	C₄×Dic₆₁	1	4	yes	—
122	C₄²	—	2	4	no	—
123	C₄²	—	2	4	no	—
124	C₄²	#11758189	1	8	yes	—
125	C₄²	C₂×C₄×D₆₁	1	2	yes	—
126	C₄²	C₂×C₄×D₆₁	1	2	yes	—
127	C₂³	C₂×C₄×Dic₆₁	1	1	yes	characteristic
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	#14519687	1	8	yes	—
131	C₂×C₄	C₄²×D₆₁	1	2	yes	—
132	C₂×C₄	C₄²×D₆₁	1	2	yes	—
133	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
135	C₂×C₄	C₂×#11758189	1	4	yes	—
136	C₂×C₄	C₂×#11758194	1	4	yes	—
137	C₂×C₄	#14519783	1	4	yes	—
138	C₈	—	2	4	no	—
139	C₈	—	122	8	no	—
140	C₈	—	122	8	no	—
141	C₈	#14519688	1	8	yes	complemented
142	C₂²	C₄²×Dic₆₁	1	1	yes	characteristic
143	C₂²	C₂×#14519687	1	2	yes	—
144	C₂²	C₄×#11758194	1	2	yes	—
145	C₂²	#16839717	1	2	yes	—
146	C₄	—	2	8	no	—
147	C₄	C₂×#14519688	1	2	yes	—
148	C₄	C₄×#11758189	1	4	yes	—
149	C₄	C₄×#11758194	1	2	yes	—
150	C₄	#16839716	1	4	yes	—
151	C₂	#21444108	1	4	yes	—
152	C₂	C₄×#14519687	1	1	yes	characteristic
153	C₂	C₄×#14519688	1	1	yes	characteristic
154	C₂	#21445505	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₁
commutator	C₂²×C₆₁	C₄³
Fitting	C₄²×C₈×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₄²
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₄²×C₈	D₆₁
hypocenter	C₆₁	#7842
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
10. #16839484	C₄	semidirect
11. #16839483	C₄	semidirect
61. C₈×C₆₁	#97	semidirect
93. C₆₁	#7842	semidirect
141. C₈	#14519688	semidirect

Maximal subgroups

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

subgroup	index
#21445565	2
C₄×#14519688	2
C₂×#16839484	2
C₂×#16839483	2
C₄²×C₈×C₆₁	2
#7842	61