Finite Group Databasebeta

Subgroups — #30607328

Order 15616.

274 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 100 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	#21461760	C₂	1	1	yes	maximal, characteristic
2	#21461759	C₂	1	1	yes	maximal, characteristic
3	#21460879	C₂	1	4	yes	maximal
4	#346×D₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519848	—	2	1	no	—
6	C₂×#14519848	C₂²	1	2	yes	—
7	C₂×#14519848	C₂²	1	2	yes	—
8	C₂×#14519847	—	2	1	no	—
9	#16840767	—	2	1	no	—
10	#16840766	—	2	1	no	—
11	C₂×C₁₆×D₆₁	C₂²	1	2	yes	—
12	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
13	#16839538	C₄	1	1	yes	characteristic
14	#16839537	C₄	1	2	yes	—
15	C₆₁×#346	C₄	1	1	yes	characteristic
16	C₂×#11758213	—	2	2	no	—
17	C₂×#11758213	—	2	2	no	—
18	#14519848	—	2	4	no	—
19	#14519848	—	2	4	no	—
20	#14519848	—	4	2	no	—
21	#14519847	—	4	2	no	—
22	#14519844	—	4	2	no	—
23	C₄⋊#92420	—	4	2	no	—
24	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
25	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
26	C₄²×D₆₁	D₄	1	1	yes	characteristic
27	C₂×#11758185	C₂×C₄	1	2	yes	—
28	C₁₆×D₆₁	—	2	2	no	—
29	C₁₆×D₆₁	—	2	2	no	—
30	C₄×#92418	Q₈	1	1	yes	characteristic
31	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₆₁	Q₈	1	1	yes	characteristic
33	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₆₁	Q₁₆	1	2	yes	—
36	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
37	C₂×#92420	—	4	2	no	—
38	C₂×#92420	—	4	2	no	—
39	#11758213	—	4	2	no	—
40	#11758213	—	4	2	no	—
41	#11758213	—	4	2	no	—
42	#11758213	—	4	2	no	—
43	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
46	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
47	C₄×Dic₆₁	D₈	1	2	yes	—
48	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
49	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
50	C₈×D₆₁	SD₁₆	1	2	yes	—
51	C₈×D₆₁	SD₁₆	1	2	yes	—
52	C₈×D₆₁	#41	1	2	yes	—
53	C₈×D₆₁	#41	1	2	yes	—
54	C₁₆×C₆₁	—	2	2	no	—
55	#11758185	—	2	2	no	—
56	C₂²×D₆₁	C₂×Q₁₆	1	1	yes	characteristic
57	C₂×C₄×C₆₁	#107	1	2	yes	—
58	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×Dic₆₁	#107	1	2	yes	—
60	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
61	C₄×D₆₁	—	4	2	no	—
62	C₄×D₆₁	#124	1	1	yes	characteristic
63	C₄×D₆₁	#124	1	1	yes	characteristic
64	C₄×D₆₁	C₂×SD₁₆	1	1	yes	characteristic
65	C₄×D₆₁	C₂×SD₁₆	1	1	yes	characteristic
66	#92420	—	8	2	no	—
67	#92420	—	8	2	no	—
68	C₈×C₆₁	#106	1	2	yes	—
69	C₈×C₆₁	#117	1	2	yes	—
70	#92418	#106	1	2	yes	—
71	#92418	#125	1	2	yes	—
72	#12892	—	61	1	no	maximal, Sylow
73	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
74	C₂×D₆₁	—	2	2	no	—
75	C₂×D₆₁	#487	1	1	yes	characteristic
76	C₂×D₆₁	#487	1	1	yes	characteristic
77	C₄×C₆₁	—	2	2	no	—
78	C₄×C₆₁	#382	1	1	yes	characteristic
79	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
80	Dic₆₁	—	2	2	no	—
81	Dic₆₁	#486	1	1	yes	characteristic
82	Dic₆₁	#486	1	1	yes	characteristic
83	C₂×#346	—	61	1	no	—
84	#1848	—	61	1	no	—
85	#1848	—	61	1	no	—
86	#1381	—	61	4	no	—
87	C₂×C₆₁	#1834	1	1	yes	characteristic
88	C₂×C₆₁	#2161	1	2	yes	—
89	D₆₁	—	2	1	no	—
90	D₆₁	—	2	1	no	—
91	C₂²×C₁₆	—	61	2	no	—
92	C₂×#108	—	61	2	no	—
93	C₂×#108	—	61	2	no	—
94	C₂×#108	—	122	1	no	—
95	C₂×#108	—	122	1	no	—
96	#423	—	122	1	no	—
97	#423	—	122	1	no	—
98	C₂×C₄×C₈	—	61	1	no	—
99	#346	—	61	2	no	—
100	#346	—	61	1	no	—
101	#346	#6923	1	1	yes	characteristic
102	C₆₁	#12892	1	1	yes	characteristic, complemented, Sylow
103	C₂×M₁₆	—	122	2	no	—
104	C₂×M₁₆	—	122	2	no	—
105	C₂²×C₈	—	61	1	no	—
106	C₂²×C₈	—	61	1	no	—
107	C₂×C₄²	—	61	1	no	—
108	C₂×C₁₆	—	61	2	no	—
109	C₂×C₁₆	—	122	2	no	—
110	C₂×C₁₆	—	122	2	no	—
111	C₂×C₁₆	C₂×#6923	1	2	yes	—
112	#108	—	122	4	no	—
113	#108	—	122	4	no	—
114	#108	—	244	2	no	—
115	#108	—	244	2	no	—
116	C₄⋊C₈	—	244	2	no	—
117	C₄⋊C₈	—	244	2	no	—
118	C₄×C₈	—	61	1	no	—
119	C₄×C₈	—	61	1	no	—
120	C₄×C₈	—	61	1	no	—
121	C₄×C₈	C₂×#6923	1	1	yes	characteristic
122	C₂²×C₄	—	61	1	no	—
123	C₂²×C₄	—	61	2	no	—
124	M₁₆	—	244	2	no	—
125	M₁₆	—	244	2	no	—
126	M₁₆	—	244	2	no	—
127	M₁₆	—	244	2	no	—
128	C₂×C₈	—	61	1	no	—
129	C₂×C₈	—	61	1	no	—
130	C₂×C₈	—	61	2	no	—
131	C₂×C₈	—	61	2	no	—
132	C₂×C₈	—	61	2	no	—
133	C₂×C₈	—	61	2	no	—
134	C₂×C₈	—	244	2	no	—
135	C₂×C₈	—	244	2	no	—
136	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
137	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
138	C₄²	—	61	2	no	—
139	C₄²	—	61	1	no	—
140	C₄²	C₄⋊#6923	1	1	yes	characteristic
141	C₁₆	—	2	2	no	—
142	C₁₆	—	122	2	no	—
143	C₂³	—	61	1	no	—
144	C₂×C₄	—	61	1	no	—
145	C₂×C₄	—	61	1	no	—
146	C₂×C₄	—	61	1	no	—
147	C₂×C₄	—	61	1	no	—
148	C₂×C₄	—	61	1	no	—
149	C₂×C₄	—	61	2	no	—
150	C₂×C₄	—	244	2	no	—
151	C₂×C₄	#14519846	1	2	yes	—
152	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
153	C₈	—	61	2	no	—
154	C₈	—	61	2	no	—
155	C₈	—	488	2	no	—
156	C₈	—	488	2	no	—
157	C₈	#14519845	1	2	yes	—
158	C₈	#14519856	1	2	yes	—
159	C₂²	—	61	1	no	—
160	C₂²	—	61	1	no	—
161	C₂²	—	122	2	no	—
162	C₂²	C₂×#14519846	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	61	1	no	—
165	C₄	—	61	1	no	—
166	C₄	—	122	2	no	—
167	C₄	#16840706	1	1	yes	characteristic
168	C₄	C₂×#14519845	1	1	yes	characteristic
169	C₂	—	122	1	no	—
170	C₂	—	122	1	no	—
171	C₂	#21461740	1	1	yes	characteristic
172	C₂	#21462175	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14519846
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₆₁×#346	C₄
socle	C₂²×C₆₁	C₂×#107
prime radical	C₂²×D₆₁	C₂×Q₁₆
cosocle	C₂×C₈×D₆₁	C₂³
hypercenter	#346	#6923
hypocenter	C₆₁	#12892
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

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

Maximal subgroups

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

subgroup	index
#21461760	2
#21461759	2
#21460879	2
#346×D₆₁	2
#12892	61