Finite Group Databasebeta

Subgroups — #30606129

Order 15616.

350 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462042	C₂	1	1	yes	maximal, characteristic
2	#21462040	C₂	1	2	yes	maximal
3	#21462030	C₂	1	1	yes	maximal, characteristic
4	#21462028	C₂	1	2	yes	maximal
5	#21445573	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519851	C₂²	1	2	yes	—
7	C₂×#14519851	C₂²	1	1	yes	characteristic
8	C₂²×#11758187	C₂²	1	1	yes	characteristic
9	#16840655	C₄	1	4	yes	complemented
10	#16840654	C₄	1	8	yes	complemented
11	#16840651	C₄	1	2	yes	complemented
12	#16840650	C₄	1	4	yes	complemented
13	#16840648	C₄	1	2	yes	complemented
14	C₄×#11758187	C₄	1	4	yes	complemented
15	#16839720	C₂²	1	1	yes	characteristic
16	#16839718	C₂²	1	2	yes	—
17	#16839520	C₄	1	1	yes	characteristic
18	#16839518	C₄	1	2	yes	—
19	C₆₁×H(3,ℤ/4)	C₄	1	1	yes	characteristic
20	#14519851	C₂×C₄	1	4	yes	—
21	#14519851	C₂×C₄	1	4	yes	—
22	#14519851	C₂×C₄	1	4	yes	—
23	C₂×#11758187	C₂×C₄	1	2	yes	—
24	C₂×#11758187	C₂×C₄	1	1	yes	characteristic
25	C₂×#11758187	C₂×C₄	1	2	yes	—
26	C₂×#11758187	C₂×C₄	1	1	yes	characteristic
27	C₂×#11758187	C₂×C₄	1	4	yes	—
28	C₂×#11758187	C₂×C₄	1	2	yes	—
29	C₂×#11758194	C₂×C₄	1	2	yes	—
30	C₂²×#92418	C₂³	1	1	yes	characteristic
31	#14519736	C₂×C₄	1	1	yes	characteristic
32	#14519688	C₂×C₄	1	4	yes	—
33	#14519687	C₂×C₄	1	2	yes	—
34	C₄×#92418	C₂×C₄	1	4	yes	—
35	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
36	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
37	#117×C₆₁	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	M₁₆×C₆₁	—	2	2	no	—
40	C₂×C₈×C₆₁	C₄²	1	2	yes	—
41	C₄⋊C₄×C₆₁	—	2	2	no	—
42	C₂²⋊C₄×C₆₁	—	2	2	no	—
43	C₄²×C₆₁	C₄²	1	4	yes	—
44	C₂²⋊Dic₆₁	—	2	1	no	—
45	C₄⋊Dic₆₁	—	2	1	no	—
46	C₄×Dic₆₁	C₄²	1	2	yes	—
47	#11758194	—	2	2	no	—
48	#11758194	—	2	2	no	—
49	C₂×#92418	C₄²	1	4	yes	—
50	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
51	C₂×#92418	C₂²×C₄	1	2	yes	—
52	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
53	C₂×#92418	C₂²×C₄	1	2	yes	—
54	#11758187	C₄²	1	1	yes	characteristic
55	#11758187	C₄²	1	2	yes	—
56	#11758187	C₄²	1	1	yes	characteristic
57	#11758187	C₄²	1	8	yes	—
58	#11758187	C₄²	1	4	yes	—
59	C₂³×C₆₁	#131	1	1	yes	characteristic
60	C₂×C₄×C₆₁	—	2	4	no	—
61	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
62	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₆₁	#131	1	2	yes	—
64	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
65	C₂×Dic₆₁	—	2	2	no	—
66	C₈×C₆₁	—	2	2	no	—
67	#92418	—	2	4	no	—
68	#92418	C₂×C₄²	1	1	yes	characteristic
69	#92418	C₂×C₄²	1	1	yes	characteristic
70	#92418	C₂×C₄²	1	2	yes	—
71	#12542	—	61	1	no	maximal, Sylow
72	C₂²×C₆₁	—	4	1	no	—
73	C₂²×C₆₁	#405	1	2	yes	—
74	C₂²×C₆₁	#405	1	1	yes	characteristic
75	C₄×C₆₁	—	4	4	no	—
76	C₄×C₆₁	C₄³	1	1	yes	characteristic
77	C₄×C₆₁	#405	1	1	yes	characteristic
78	C₄×C₆₁	#405	1	2	yes	—
79	Dic₆₁	—	4	2	no	—
80	#2110	—	61	2	no	—
81	#2110	—	61	1	no	—
82	#2108	—	61	2	no	—
83	#2108	—	61	1	no	—
84	#1747	—	61	1	no	—
85	C₂×C₆₁	—	2	1	no	—
86	C₂×C₆₁	—	2	2	no	—
87	C₂×C₆₁	#1726	1	1	yes	characteristic
88	C₂×M₃₂	—	61	2	no	—
89	C₂×M₃₂	—	61	1	no	—
90	C₂²×C₁₆	—	61	1	no	—
91	#405	—	61	1	no	—
92	#405	—	61	2	no	—
93	#347	—	61	8	no	—
94	#347	—	61	4	no	—
95	#346	—	61	4	no	—
96	#346	—	61	2	no	—
97	C₄×C₁₆	—	61	4	no	—
98	C₄×C₁₆	—	61	2	no	—
99	#338	—	61	1	no	—
100	H(3,ℤ/4)	—	61	2	no	—
101	H(3,ℤ/4)	#6923	1	1	yes	characteristic
102	C₆₁	#12542	1	1	yes	characteristic, complemented, Sylow
103	C₂×M₁₆	—	61	2	no	—
104	C₂×M₁₆	C₂×#6923	1	1	yes	characteristic
105	C₂²×C₈	—	61	1	no	—
106	#117	—	61	1	no	—
107	#117	C₂×#6923	1	2	yes	—
108	M₃₂	—	61	4	no	—
109	M₃₂	—	61	4	no	—
110	M₃₂	—	61	4	no	—
111	C₂×C₁₆	—	61	2	no	—
112	C₂×C₁₆	—	61	1	no	—
113	C₂×C₁₆	—	61	4	no	—
114	C₂×C₁₆	—	61	2	no	—
115	C₂×C₁₆	—	61	2	no	—
116	C₂×C₁₆	—	61	1	no	—
117	#97	—	61	2	no	—
118	#97	—	61	4	no	—
119	C₄×C₈	—	61	2	no	—
120	C₄×C₈	—	61	4	no	—
121	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
122	M₁₆	—	2	2	no	—
123	M₁₆	—	122	2	no	—
124	M₁₆	—	122	2	no	—
125	C₂×C₈	—	61	1	no	—
126	C₂×C₈	—	61	2	no	—
127	C₂×C₈	—	61	1	no	—
128	C₂×C₈	—	61	2	no	—
129	C₂×C₈	—	61	4	no	—
130	C₂×C₈	C₄×#6923	1	2	yes	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	122	1	no	—
133	C₂²⋊C₄	—	2	2	no	—
134	C₂²⋊C₄	—	122	1	no	—
135	C₄²	—	61	2	no	—
136	C₄²	C₄×#6923	1	4	yes	—
137	C₁₆	—	61	1	no	—
138	C₁₆	—	61	2	no	—
139	C₁₆	—	61	1	no	—
140	C₁₆	—	61	8	no	—
141	C₁₆	—	61	4	no	—
142	C₂³	#14519864	1	1	yes	characteristic
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	122	2	no	—
145	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
146	C₂×C₄	C₂×C₄×#6923	1	2	yes	—
147	C₂×C₄	#14519864	1	2	yes	—
148	C₂×C₄	#14519865	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	61	1	no	—
151	C₈	—	61	1	no	—
152	C₈	—	61	2	no	—
153	C₈	—	122	4	no	—
154	C₂²	—	4	1	no	—
155	C₂²	#16840726	1	2	yes	—
156	C₂²	#16840727	1	1	yes	characteristic
157	C₄	—	4	4	no	—
158	C₄	—	244	2	no	—
159	C₄	C₄²×#6923	1	1	yes	characteristic
160	C₄	#16840726	1	1	yes	characteristic
161	C₄	#16840727	1	2	yes	—
162	C₂	—	2	2	no	—
163	C₂	—	2	1	no	—
164	C₂	#21461476	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄²×#6923
commutator	C₄×C₆₁	C₄³
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₆₁×H(3,ℤ/4)	C₄
socle	C₂×C₆₁	#1726
prime radical	C₂³×C₆₁	#131
cosocle	C₂²×#92418	C₂³
hypercenter	H(3,ℤ/4)	#6923
hypocenter	C₆₁	#12542
LCS	G ▷ C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
9. #16840655	C₄	semidirect
10. #16840654	C₄	semidirect
11. #16840651	C₄	semidirect
12. #16840650	C₄	semidirect
13. #16840648	C₄	semidirect
14. C₄×#11758187	C₄	semidirect
102. C₆₁	#12542	semidirect

Maximal subgroups

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

subgroup	index
#21462042	2
#21462040	2
#21462030	2
#21462028	2
#21445573	2
#12542	61