Subgroups — #30623830

Order 15616.

834 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840820	C₂	1	2	yes	maximal
2	#21462030	C₂	1	6	yes	maximal, complemented
3	#21462025	C₂	1	3	yes	maximal, complemented
4	C₂×#16840648	C₂	1	3	yes	maximal, complemented
5	#131×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	#16840820	C₄	1	16	yes	complemented
7	C₂×#14519851	C₂²	1	6	yes	—
8	C₂²×#11758187	C₂²	1	6	yes	—
9	#16840651	C₂²	1	6	yes	—
10	#16840648	C₂²	1	1	yes	characteristic
11	#16840648	C₂²	1	3	yes	—
12	#16840648	C₂²	1	6	yes	—
13	C₂×#14519784	C₂²	1	1	yes	characteristic
14	#41×Dic₆₁	C₄	1	1	yes	characteristic
15	#16839720	C₂²	1	3	yes	—
16	#16839717	C₄	1	3	yes	—
17	M₁₆×Dic₆₁	C₄	1	3	yes	—
18	C₂×C₈×Dic₆₁	C₂²	1	3	yes	—
19	C₂×#131×C₆₁	C₄	1	1	yes	characteristic
20	#14519851	C₂×C₄	1	24	yes	complemented
21	C₂×#11758187	C₂×C₄	1	24	yes	complemented
22	C₂×#11758187	C₂³	1	2	yes	—
23	C₂×#11758187	C₂³	1	6	yes	—
24	#14519784	C₂×C₄	1	8	yes	—
25	C₂×#11758194	C₂×C₄	1	3	yes	—
26	C₂²×#92418	C₂³	1	3	yes	—
27	Q₈×Dic₆₁	C₈	1	2	yes	—
28	D₄×Dic₆₁	C₈	1	6	yes	—
29	#14519736	C₂×C₄	1	3	yes	—
30	C₂×C₄×Dic₆₁	C₂×C₄	1	3	yes	—
31	#14519687	C₂×C₄	1	6	yes	—
32	C₈×Dic₆₁	C₂×C₄	1	3	yes	—
33	C₈×Dic₆₁	C₂×C₄	1	3	yes	—
34	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
35	C₈×Dic₆₁	C₂³	1	3	yes	—
36	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
37	#131×C₆₁	C₈	1	8	yes	—
38	C₂×M₁₆×C₆₁	C₂×C₄	1	3	yes	—
39	C₂²×C₈×C₆₁	C₂×C₄	1	3	yes	—
40	#41×C₆₁	C₄²	1	4	yes	—
41	#41×C₆₁	C₂×C₈	1	4	yes	—
42	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
43	C₂×D₄×C₆₁	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
45	C₂²×Dic₆₁	C₂×C₈	1	6	yes	—
46	M₁₆×C₆₁	C₂×C₈	1	12	yes	—
47	C₂×C₈×C₆₁	C₂×C₈	1	12	yes	—
48	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
49	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²⋊Dic₆₁	C₂×C₈	1	6	yes	—
51	C₄⋊Dic₆₁	C₂×C₈	1	6	yes	—
52	C₄×Dic₆₁	C₂×C₈	1	6	yes	—
53	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
54	C₄×Dic₆₁	C₂²×C₄	1	3	yes	—
55	#11758194	C₄²	1	12	yes	—
56	C₂×#92418	C₂²×C₄	1	3	yes	—
57	C₂×#92418	C₂²×C₄	1	12	yes	—
58	C₂×#92418	C₂⁴	1	1	yes	characteristic
59	#11758187	C₂²×C₄	1	4	yes	—
60	#11758187	C₂²×C₄	1	12	yes	—
61	C₂³×C₆₁	C₂²×C₈	1	3	yes	—
62	Q₈×C₆₁	C₄×C₈	1	4	yes	—
63	D₄×C₆₁	C₄×C₈	1	12	yes	—
64	C₂×C₄×C₆₁	C₂×C₄²	1	6	yes	—
65	C₂×C₄×C₆₁	C₂²×C₈	1	6	yes	—
66	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
67	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
68	C₂×Dic₆₁	—	2	12	no	—
69	C₂×Dic₆₁	C₂²×C₈	1	6	yes	—
70	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
71	C₈×C₆₁	C₂²×C₈	1	2	yes	—
72	C₈×C₆₁	C₂²×C₈	1	6	yes	—
73	#92418	C₂×C₄²	1	6	yes	—
74	#92418	C₂³×C₄	1	2	yes	—
75	C₄×#504	—	61	1	no	maximal, Sylow
76	C₂²×C₆₁	—	2	6	no	—
77	C₂²×C₆₁	C₂×C₄×C₈	1	6	yes	—
78	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
79	C₄×C₆₁	C₂×C₄×C₈	1	6	yes	—
80	C₄×C₆₁	C₂²×C₄²	1	1	yes	characteristic
81	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
82	Dic₆₁	—	2	6	no	—
83	Dic₆₁	#504	1	4	yes	complemented
84	C₂×#504	—	61	2	no	—
85	C₄×#131	—	61	1	no	—
86	#2108	—	61	6	no	—
87	C₄×M₃₂	—	61	3	no	—
88	C₂×C₄×C₁₆	—	61	3	no	—
89	C₂×C₆₁	—	2	6	no	—
90	C₂×C₆₁	C₂²×C₄×C₈	1	1	yes	characteristic
91	C₂×C₆₁	C₂×#504	1	2	yes	—
92	C₂×#131	—	61	1	no	—
93	C₂×#131	#6923	1	1	yes	characteristic
94	C₄×#41	—	61	1	no	—
95	#504	—	61	16	no	—
96	C₂×M₃₂	—	61	6	no	—
97	C₂²×C₁₆	—	61	6	no	—
98	#405	—	61	3	no	—
99	#405	—	61	3	no	—
100	C₄×M₁₆	—	61	3	no	—
101	C₂×C₄×C₈	—	61	3	no	—
102	#346	—	61	6	no	—
103	C₄×C₁₆	—	61	1	no	—
104	C₄×C₁₆	—	61	3	no	—
105	C₄×C₁₆	—	61	6	no	—
106	C₆₁	C₄×#504	1	1	yes	characteristic, complemented, Sylow
107	C₂×#41	C₂×#6923	1	1	yes	characteristic
108	#131	—	61	8	no	—
109	#131	#92420	1	8	yes	—
110	C₂×M₁₆	—	61	3	no	—
111	C₂×M₁₆	C₂×#6923	1	3	yes	—
112	C₂²×C₈	—	61	3	no	—
113	C₂²×C₈	C₂×#6923	1	3	yes	—
114	C₄×Q₈	—	61	2	no	—
115	C₄×D₄	—	61	6	no	—
116	#117	—	61	3	no	—
117	C₂×C₄²	—	61	3	no	—
118	M₃₂	—	61	24	no	—
119	C₂×C₁₆	—	61	24	no	—
120	C₂×C₁₆	—	61	2	no	—
121	C₂×C₁₆	—	61	6	no	—
122	#97	—	61	6	no	—
123	C₄×C₈	—	61	3	no	—
124	C₄×C₈	—	61	3	no	—
125	C₄×C₈	—	61	1	no	—
126	C₄×C₈	—	61	3	no	—
127	#41	C₄×#6923	1	4	yes	—
128	#41	C₂×#92420	1	4	yes	—
129	C₂×Q₈	D₆₁⋊C₈	1	1	yes	characteristic
130	C₂×D₄	D₆₁⋊C₈	1	3	yes	—
131	C₂²×C₄	—	61	6	no	—
132	C₂²×C₄	C₂²×#6923	1	3	yes	—
133	M₁₆	—	61	12	no	—
134	M₁₆	C₂×#92420	1	12	yes	—
135	C₂×C₈	—	61	12	no	—
136	C₂×C₈	—	61	1	no	—
137	C₂×C₈	—	61	3	no	—
138	C₂×C₈	C₂×#92420	1	12	yes	—
139	C₂×C₈	C₂²×#6923	1	3	yes	—
140	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
141	C₄⋊C₄	—	61	6	no	—
142	C₂²⋊C₄	—	61	6	no	—
143	C₄²	—	61	6	no	—
144	C₄²	—	61	1	no	—
145	C₄²	—	61	3	no	—
146	C₁₆	—	61	4	no	—
147	C₁₆	—	61	12	no	—
148	C₂³	C₂×D₆₁⋊C₈	1	3	yes	—
149	Q₈	C₄×#92420	1	4	yes	—
150	D₄	C₄×#92420	1	12	yes	—
151	C₂×C₄	—	61	2	no	—
152	C₂×C₄	—	61	6	no	—
153	C₂×C₄	—	122	12	no	—
154	C₂×C₄	C₂×C₄×#6923	1	6	yes	—
155	C₂×C₄	C₂²×#92420	1	6	yes	—
156	C₂×C₄	C₂×D₆₁⋊C₈	1	3	yes	—
157	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
158	C₈	—	61	6	no	—
159	C₈	—	61	2	no	—
160	C₈	C₂²×#92420	1	2	yes	—
161	C₈	C₂²×#92420	1	6	yes	—
162	C₂²	—	2	6	no	—
163	C₂²	C₂×C₄×#92420	1	6	yes	—
164	C₂²	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
165	C₄	—	61	4	no	—
166	C₄	—	122	6	no	—
167	C₄	C₂×C₄×#92420	1	6	yes	—
168	C₄	C₂²×C₄×#6923	1	1	yes	characteristic
169	C₄	C₂³×#92420	1	1	yes	characteristic
170	C₂	—	2	6	no	—
171	C₂	C₂²×C₄×#92420	1	1	yes	characteristic
172	C₂	D₆₁⋊#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×#6923
commutator	C₂×C₆₁	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₂×#131×C₆₁	C₄
socle	C₂²×C₆₁	C₂³×C₈
prime radical	C₂×#41×C₆₁	C₂×C₄
cosocle	C₂×#92418	C₂⁴
hypercenter	C₂×#131	#6923
hypocenter	C₆₁	C₄×#504
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #21462030	C₂	semidirect
3. #21462025	C₂	semidirect
4. C₂×#16840648	C₂	semidirect
6. #16840820	C₄	semidirect
20. #14519851	C₂×C₄	semidirect
21. C₂×#11758187	C₂×C₄	semidirect
83. Dic₆₁	#504	semidirect
106. C₆₁	C₄×#504	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840820	2
#21462030	2
#21462025	2
C₂×#16840648	2
#131×Dic₆₁	2
C₄×#504	61