Finite Group Databasebeta

Subgroups — #30605851

Order 15616.

210 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 80 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	#21461379	C₂	1	1	yes	maximal, characteristic
2	C₂×#16840623	C₂	1	1	yes	maximal, characteristic
3	#21460845	C₂	1	2	yes	maximal, complemented
4	#21460843	C₂	1	2	yes	maximal, complemented
5	#21447619	C₂	1	1	yes	maximal, characteristic
6	#16840783	—	2	1	no	—
7	C₂×C₄⋊#92420	—	2	1	no	—
8	#16840626	—	2	1	no	—
9	#16840623	—	2	1	no	—
10	#16840623	C₂²	1	1	yes	characteristic
11	#16840623	C₂²	1	2	yes	—
12	#16840623	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
14	#16839580	C₄	1	2	yes	—
15	C₂²⋊#11758185	C₄	1	1	yes	characteristic
16	#16839537	C₂²	1	2	yes	—
17	C₆₁×C₂²⋊C₁₆	C₄	1	1	yes	characteristic
18	C₂²×#92420	—	4	1	no	—
19	C₄⋊#92420	—	2	2	no	—
20	C₄⋊#92420	—	2	1	no	—
21	C₄⋊#92420	—	2	1	no	—
22	C₄⋊#92420	—	4	1	no	—
23	C₂²⋊#92420	—	4	1	no	—
24	#14519832	—	4	1	no	—
25	C₂²×#92418	Q₈	1	1	yes	characteristic
26	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
27	C₂×#11758185	C₂×C₄	1	2	yes	—
28	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₆₁	D₄	1	1	yes	characteristic
30	C₈×Dic₆₁	Q₈	1	1	yes	characteristic
31	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×Dic₆₁	SD₁₆	1	1	yes	characteristic
36	C₂²×Dic₆₁	SD₁₆	1	1	yes	characteristic
37	C₂×#92420	—	4	1	no	—
38	C₂×#92420	—	4	1	no	—
39	C₂×#92420	—	4	2	no	—
40	C₂×#92420	—	8	2	no	—
41	C₂×C₈×C₆₁	—	2	2	no	—
42	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
44	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
45	C₄×Dic₆₁	SD₁₆	1	1	yes	characteristic
46	C₄×Dic₆₁	SD₁₆	1	1	yes	characteristic
47	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×#92418	—	4	1	no	—
49	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
50	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
51	C₁₆×C₆₁	—	2	2	no	—
52	#11758185	—	2	2	no	—
53	C₂³×C₆₁	#106	1	1	yes	characteristic
54	C₂×C₄×C₆₁	—	2	1	no	—
55	C₂×C₄×C₆₁	—	2	1	no	—
56	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
57	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×Dic₆₁	—	2	2	no	—
59	C₂×Dic₆₁	—	4	1	no	—
60	C₂×Dic₆₁	#106	1	1	yes	characteristic
61	C₂×Dic₆₁	#106	1	1	yes	characteristic
62	C₂×Dic₆₁	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×Dic₆₁	C₂×SD₁₆	1	1	yes	characteristic
64	#92420	—	8	1	no	—
65	#92420	—	8	1	no	—
66	#92420	—	8	2	no	—
67	C₈×C₆₁	—	2	1	no	—
68	C₈×C₆₁	#131	1	2	yes	—
69	#92418	—	2	1	no	—
70	#92418	—	2	1	no	—
71	#12028	—	61	1	no	maximal, Sylow
72	C₂²×C₆₁	—	2	1	no	—
73	C₂²×C₆₁	—	2	1	no	—
74	C₂²×C₆₁	#365	1	2	yes	—
75	C₂²×C₆₁	C₂×#106	1	1	yes	characteristic
76	C₄×C₆₁	—	2	1	no	—
77	C₄×C₆₁	#424	1	1	yes	characteristic
78	C₄×C₆₁	#430	1	1	yes	characteristic
79	Dic₆₁	—	2	1	no	—
80	Dic₆₁	—	2	1	no	—
81	Dic₆₁	—	2	1	no	—
82	Dic₆₁	—	2	1	no	—
83	#2163	—	61	1	no	—
84	#1568	—	61	1	no	—
85	C₂×#334	—	61	1	no	—
86	#1368	—	61	2	no	—
87	#1368	—	61	2	no	—
88	C₂×C₆₁	—	2	2	no	—
89	C₂×C₆₁	#1570	1	1	yes	characteristic
90	C₂×C₆₁	C₂×#365	1	1	yes	characteristic
91	C₂×C₆₁	#2155	1	1	yes	characteristic
92	#432	—	122	1	no	—
93	C₂×C₄⋊C₈	—	122	1	no	—
94	C₂×C₄×C₈	—	61	1	no	—
95	C₄⋊C₁₆	—	61	2	no	—
96	C₂²⋊C₁₆	—	61	1	no	—
97	C₂²⋊C₁₆	#6923	1	1	yes	characteristic
98	#346	—	61	2	no	—
99	#335	—	122	1	no	—
100	#334	—	61	2	no	—
101	#334	—	61	1	no	—
102	#334	—	61	1	no	—
103	#334	—	122	1	no	—
104	C₆₁	#12028	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₈	—	61	1	no	—
106	C₂²×C₈	—	244	1	no	—
107	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
108	C₂×C₄²	—	61	1	no	—
109	C₂×C₁₆	—	61	2	no	—
110	C₂×C₁₆	C₂×#6923	1	2	yes	—
111	C₄⋊C₈	—	122	2	no	—
112	C₄⋊C₈	—	122	1	no	—
113	C₄⋊C₈	—	122	1	no	—
114	C₄⋊C₈	—	244	1	no	—
115	C₂²⋊C₈	—	244	1	no	—
116	#97	—	244	1	no	—
117	C₄×C₈	—	61	1	no	—
118	C₄×C₈	—	61	1	no	—
119	C₄×C₈	—	61	1	no	—
120	C₄×C₈	—	61	1	no	—
121	C₂²×C₄	—	61	1	no	—
122	C₂²×C₄	—	61	1	no	—
123	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	61	1	no	—
126	C₂×C₈	—	61	1	no	—
127	C₂×C₈	—	488	2	no	—
128	C₂×C₈	—	244	1	no	—
129	C₂×C₈	—	244	1	no	—
130	C₂×C₈	—	244	1	no	—
131	C₂×C₈	—	244	2	no	—
132	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
133	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
134	C₄²	—	61	1	no	—
135	C₄²	—	61	1	no	—
136	C₄²	—	61	1	no	—
137	C₄²	—	61	1	no	—
138	C₁₆	—	2	2	no	—
139	C₁₆	—	122	2	no	—
140	C₂³	#14519845	1	1	yes	characteristic
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	61	1	no	—
144	C₂×C₄	—	61	1	no	—
145	C₂×C₄	—	61	1	no	—
146	C₂×C₄	—	61	1	no	—
147	C₂×C₄	—	122	2	no	—
148	C₂×C₄	—	244	1	no	—
149	C₂×C₄	#14519845	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	122	1	no	—
153	C₈	—	122	1	no	—
154	C₈	—	488	2	no	—
155	C₈	—	488	1	no	—
156	C₈	—	488	1	no	—
157	C₈	#14519864	1	2	yes	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	#16840686	1	2	yes	—
161	C₂²	C₂×#14519845	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	122	1	no	—
164	C₄	—	122	1	no	—
165	C₄	—	122	1	no	—
166	C₄	—	122	1	no	—
167	C₄	#16840768	1	1	yes	characteristic
168	C₄	#16840777	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	#21461386	1	1	yes	characteristic
171	C₂	C₂×#16840686	1	1	yes	characteristic
172	C₂	#21462167	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14519845
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₆₁×C₂²⋊C₁₆	C₄
socle	C₂²×C₆₁	C₂×#106
prime radical	C₂³×C₆₁	#106
cosocle	C₈×Dic₆₁	C₂³
hypercenter	C₂²⋊C₁₆	#6923
hypocenter	C₆₁	#12028
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #21460845	C₂	semidirect
4. #21460843	C₂	semidirect
104. C₆₁	#12028	semidirect

Maximal subgroups

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

subgroup	index
#21461379	2
C₂×#16840623	2
#21460845	2
#21460843	2
#21447619	2
#12028	61