Finite Group Databasebeta

Subgroups — #29596405

Order 15616.

274 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 139 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	C₂×#16839538	C₂	1	1	yes	maximal, characteristic
2	#21445991	C₂	1	1	yes	maximal, characteristic
3	#21443912	C₂	1	4	yes	maximal
4	C₆₁×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11758185	C₂²	1	2	yes	—
6	C₂×#14519713	—	2	1	no	—
7	C₂×#14519713	C₂²	1	2	yes	—
8	#16839786	—	2	1	no	—
9	#16839538	C₄	1	4	yes	—
10	C₂×#108×C₆₁	—	2	1	no	—
11	C₂×#108×C₆₁	C₂²	1	2	yes	—
12	C₆₁×#423	—	2	1	no	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#11758194	—	2	2	no	—
15	C₂×#11758185	—	2	2	no	—
16	C₂×#11758185	—	2	2	no	—
17	C₂×#11758185	C₂×C₄	1	4	yes	—
18	#14519713	—	2	4	no	—
19	#14519713	—	4	2	no	—
20	C₄⋊#92418	—	4	2	no	—
21	C₂×M₁₆×C₆₁	—	2	2	no	—
22	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
25	#108×C₆₁	—	2	4	no	—
26	#108×C₆₁	—	4	2	no	—
27	C₄⋊C₈×C₆₁	—	4	2	no	—
28	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₆₁	Q₈	1	2	yes	—
30	C₂²×C₄×C₆₁	Q₁₆	1	2	yes	—
31	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
32	M₁₆×C₆₁	—	4	2	no	—
33	M₁₆×C₆₁	—	4	2	no	—
34	C₂×C₈×C₆₁	—	4	2	no	—
35	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
37	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
38	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₆₁	#41	1	2	yes	—
41	C₂×C₈×C₆₁	#41	1	2	yes	—
42	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
43	C₄²×C₆₁	D₈	1	2	yes	—
44	#11758194	—	4	2	no	—
45	#11758194	—	4	2	no	—
46	C₂×#92418	—	4	2	no	—
47	#11758185	—	2	4	no	—
48	C₂³×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
49	C₂×C₄×C₆₁	—	4	2	no	—
50	C₂×C₄×C₆₁	#107	1	4	yes	—
51	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
57	C₈×C₆₁	—	8	2	no	—
58	C₈×C₆₁	#106	1	4	yes	—
59	C₈×C₆₁	#117	1	2	yes	—
60	C₈×C₆₁	#125	1	2	yes	—
61	#92418	—	8	2	no	—
62	#12892	—	61	1	no	maximal, Sylow
63	C₂²×C₆₁	—	2	2	no	—
64	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
65	C₂²×C₆₁	#487	1	1	yes	characteristic
66	C₂²×C₆₁	#487	1	1	yes	characteristic
67	C₄×C₆₁	—	2	4	no	—
68	C₄×C₆₁	#382	1	1	yes	characteristic
69	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
70	C₄×C₆₁	#486	1	1	yes	characteristic
71	C₄×C₆₁	#486	1	1	yes	characteristic
72	C₂×#346	—	61	1	no	—
73	#1848	—	61	1	no	—
74	#1848	D₆₁	1	1	yes	characteristic
75	#1381	—	61	4	no	—
76	C₂×C₆₁	—	2	1	no	—
77	C₂×C₆₁	—	2	1	no	—
78	C₂×C₆₁	#1834	1	1	yes	characteristic
79	C₂×C₆₁	#2161	1	2	yes	—
80	C₂²×C₁₆	—	61	2	no	—
81	C₂×#108	—	2	1	no	—
82	C₂×#108	—	61	2	no	—
83	C₂×#108	—	122	1	no	—
84	C₂×#108	C₂×D₆₁	1	2	yes	—
85	#423	—	2	1	no	—
86	#423	—	122	1	no	—
87	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
88	#346	—	61	4	no	—
89	C₆₁	#12892	1	1	yes	characteristic, complemented, Sylow
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	—	122	2	no	—
92	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
93	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
94	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
95	C₂×C₁₆	—	61	4	no	—
96	C₂×C₁₆	—	122	2	no	—
97	C₂×C₁₆	—	122	2	no	—
98	#108	—	2	4	no	—
99	#108	—	4	2	no	—
100	#108	—	122	4	no	—
101	#108	—	244	2	no	—
102	C₄⋊C₈	—	4	2	no	—
103	C₄⋊C₈	—	244	2	no	—
104	C₄×C₈	Dic₁₂₂	1	2	yes	—
105	C₄×C₈	C₄×D₆₁	1	2	yes	—
106	C₂²×C₄	#11758202	1	2	yes	—
107	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
108	M₁₆	—	4	2	no	—
109	M₁₆	—	4	2	no	—
110	M₁₆	—	244	2	no	—
111	M₁₆	—	244	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	244	2	no	—
114	C₂×C₈	#11758196	1	2	yes	—
115	C₂×C₈	#11758200	1	2	yes	—
116	C₂×C₈	#11758201	1	2	yes	—
117	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
118	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
119	C₂×C₈	#11758224	1	2	yes	—
120	C₂×C₈	#11758226	1	2	yes	—
121	C₄²	#11758196	1	2	yes	—
122	C₄²	#11758199	1	2	yes	—
123	C₁₆	—	122	4	no	—
124	C₂³	C₂×#11758202	1	1	yes	characteristic
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	#14519710	1	4	yes	—
127	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
128	C₂×C₄	#14519764	1	1	yes	characteristic
129	C₂×C₄	#14519766	1	1	yes	characteristic
130	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
131	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
132	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
133	C₈	—	8	2	no	—
134	C₈	—	488	2	no	—
135	C₈	#14519708	1	4	yes	—
136	C₈	#14519735	1	2	yes	—
137	C₈	#14519768	1	2	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	C₂×#14519710	1	1	yes	characteristic
140	C₂²	#16840112	1	1	yes	characteristic
141	C₂²	#16840114	1	1	yes	characteristic
142	C₄	—	2	4	no	—
143	C₄	#16839627	1	1	yes	characteristic
144	C₄	C₂×#14519708	1	1	yes	characteristic
145	C₄	#16840108	1	1	yes	characteristic
146	C₄	#16840110	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#21445946	1	1	yes	characteristic
150	C₂	#21447613	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11758199
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1848	C₂
socle	C₂²×C₆₁	C₂×#107
prime radical	C₂³×C₆₁	C₂×Q₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1848	D₆₁
hypocenter	C₆₁	#12892
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
89. C₆₁	#12892	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839538	2
#21445991	2
#21443912	2
C₆₁×#1848	2
#12892	61