Finite Group Databasebeta

Subgroups — #30595097

Order 15616.

178 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 92 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×#16840694	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14519827	C₂	1	1	yes	maximal, characteristic
3	#21447418	C₂	1	1	yes	maximal, characteristic
4	#16840694	—	2	1	no	—
5	#16840694	C₄	1	2	yes	complemented
6	C₂×#14519827	—	2	1	no	—
7	C₂×#14519827	—	2	1	no	—
8	C₂×#14519827	—	2	1	no	—
9	C₂×#14519827	C₄	1	2	yes	complemented
10	C₂²×#11758185	C₂²	1	1	yes	characteristic
11	#16839717	C₄	1	1	yes	characteristic
12	#16839537	—	2	1	no	—
13	C₁₆×Dic₆₁	—	2	1	no	—
14	C₂×M₃₂×C₆₁	C₄	1	1	yes	characteristic
15	#14519827	—	2	2	no	—
16	#14519827	—	2	2	no	—
17	C₂×#11758194	C₈	1	1	yes	characteristic
18	#14519736	C₈	1	1	yes	characteristic
19	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
20	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
21	C₂×#11758185	D₄	1	1	yes	characteristic
22	C₂×#11758185	D₄	1	1	yes	characteristic
23	C₂×#11758185	D₄	1	1	yes	characteristic
24	C₂×#11758185	Q₈	1	1	yes	characteristic
25	#14519687	—	2	1	no	—
26	C₈×Dic₆₁	—	2	1	no	—
27	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₆₁	—	2	1	no	—
29	M₃₂×C₆₁	C₈	1	2	yes	—
30	C₂×C₁₆×C₆₁	—	2	1	no	—
31	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
33	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₆₁	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₆₁	M₁₆	1	1	yes	characteristic
38	C₂²⋊Dic₆₁	—	2	1	no	—
39	C₄⋊Dic₆₁	C₁₆	1	2	yes	—
40	C₄×Dic₆₁	—	2	1	no	—
41	#11758194	—	2	1	no	—
42	#11758194	C₁₆	1	2	yes	—
43	C₂×#92418	—	2	1	no	—
44	C₁₆×C₆₁	—	2	2	no	—
45	#11758185	C₂²⋊C₄	1	1	yes	characteristic
46	#11758185	C₂²⋊C₄	1	1	yes	characteristic
47	#11758185	C₄⋊C₄	1	1	yes	characteristic
48	#11758185	C₄⋊C₄	1	1	yes	characteristic
49	C₂³×C₆₁	M₃₂	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₆₁	C₂×C₁₆	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂×C₁₆	1	1	yes	characteristic
55	C₂×C₄×C₆₁	M₃₂	1	1	yes	characteristic
56	C₂×Dic₆₁	—	2	2	no	—
57	C₈×C₆₁	#95	1	1	yes	characteristic
58	C₈×C₆₁	#97	1	1	yes	characteristic
59	C₈×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
60	C₈×C₆₁	C₄⋊C₈	1	1	yes	characteristic
61	#92418	—	2	2	no	—
62	#7470	—	61	1	no	maximal, Sylow
63	C₂²×C₆₁	—	2	2	no	—
64	C₂²×C₆₁	#346	1	1	yes	characteristic
65	C₂²×C₆₁	C₂²⋊C₁₆	1	1	yes	characteristic
66	C₂²×C₆₁	C₄⋊C₁₆	1	1	yes	characteristic
67	C₄×C₆₁	#336	1	1	yes	characteristic
68	C₄×C₆₁	C₄×C₁₆	1	1	yes	characteristic
69	C₄×C₆₁	C₂²⋊C₁₆	1	1	yes	characteristic
70	C₄×C₆₁	C₄⋊C₁₆	1	1	yes	characteristic
71	Dic₆₁	—	4	2	no	—
72	C₂×M₆₄	—	61	1	no	—
73	C₂²×C₃₂	—	61	1	no	—
74	#2108	—	61	1	no	—
75	C₂×C₆₁	—	2	1	no	—
76	C₂×C₆₁	—	2	1	no	—
77	C₂×C₆₁	#1374	1	1	yes	characteristic
78	C₂×C₆₁	#1401	1	2	yes	—
79	C₂×M₃₂	#6923	1	1	yes	characteristic
80	C₂²×C₁₆	—	61	1	no	—
81	#405	—	61	1	no	—
82	M₆₄	—	61	2	no	—
83	M₆₄	—	122	1	no	—
84	C₂×C₃₂	—	61	2	no	—
85	C₂×C₃₂	—	122	1	no	—
86	C₂×C₃₂	—	122	1	no	—
87	C₂×C₃₂	—	122	1	no	—
88	#346	—	122	1	no	—
89	C₄×C₁₆	—	122	1	no	—
90	C₆₁	#7470	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₁₆	—	61	1	no	—
92	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
93	#117	—	61	1	no	—
94	M₃₂	—	2	1	no	—
95	M₃₂	#92420	1	2	yes	—
96	C₂×C₁₆	—	2	1	no	—
97	C₂×C₁₆	—	61	1	no	—
98	C₂×C₁₆	—	61	1	no	—
99	C₂×C₁₆	—	61	1	no	—
100	C₂×C₁₆	—	61	1	no	—
101	C₂×C₁₆	—	61	1	no	—
102	C₂×C₁₆	—	61	1	no	—
103	#97	—	122	1	no	—
104	C₄×C₈	—	122	1	no	—
105	C₃₂	—	122	2	no	—
106	C₃₂	—	122	2	no	—
107	C₂²×C₄	D₆₁⋊C₈	1	1	yes	characteristic
108	M₁₆	—	61	2	no	—
109	M₁₆	—	122	1	no	—
110	C₂×C₈	—	122	1	no	—
111	C₂×C₈	#11758213	1	1	yes	characteristic
112	C₂×C₈	C₄×#6923	1	1	yes	characteristic
113	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
114	C₂×C₈	C₂×#92420	1	1	yes	characteristic
115	C₂×C₈	#11758217	1	1	yes	characteristic
116	C₂×C₈	C₂²⋊#6923	1	1	yes	characteristic
117	C₄⋊C₄	—	61	2	no	—
118	C₂²⋊C₄	—	122	1	no	—
119	C₄²	—	122	1	no	—
120	C₁₆	—	2	2	no	—
121	C₁₆	—	61	1	no	—
122	C₁₆	—	61	1	no	—
123	C₁₆	—	61	1	no	—
124	C₁₆	—	61	1	no	—
125	C₂³	#14519852	1	1	yes	characteristic
126	C₂×C₄	—	122	2	no	—
127	C₂×C₄	C₄×#92420	1	1	yes	characteristic
128	C₂×C₄	#14519834	1	1	yes	characteristic
129	C₂×C₄	#14519844	1	1	yes	characteristic
130	C₂×C₄	D₆₁⋊C₁₆	1	1	yes	characteristic
131	C₂×C₄	D₆₁⋊C₁₆	1	1	yes	characteristic
132	C₂×C₄	#14519852	1	1	yes	characteristic
133	C₈	—	122	2	no	—
134	C₈	(C₂×#6923)⋊C₄	1	1	yes	characteristic
135	C₈	#14519832	1	1	yes	characteristic
136	C₈	C₂²⋊#92420	1	1	yes	characteristic
137	C₈	C₄⋊#92420	1	1	yes	characteristic
138	C₂²	—	2	2	no	—
139	C₂²	#16840651	1	1	yes	characteristic
140	C₂²	#16840657	1	1	yes	characteristic
141	C₂²	#16840683	1	1	yes	characteristic
142	C₄	—	244	2	no	—
143	C₄	#16840628	1	1	yes	characteristic
144	C₄	#16840648	1	1	yes	characteristic
145	C₄	#16840657	1	1	yes	characteristic
146	C₄	#16840683	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#21460858	1	1	yes	characteristic
150	C₂	#21460937	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²⋊#6923
commutator	C₄×C₆₁	C₄×C₁₆
Frattini	C₂²×C₈	C₂×#6923
Fitting	C₂×M₃₂×C₆₁	C₄
socle	C₂²×C₆₁	C₂²⋊C₁₆
prime radical	C₂³×C₆₁	M₃₂
cosocle	C₂²×#11758185	C₂²
hypercenter	C₂×M₃₂	#6923
hypocenter	C₆₁	#7470
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
5. #16840694	C₄	semidirect
9. C₂×#14519827	C₄	semidirect
90. C₆₁	#7470	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840694	2
C₂²×#14519827	2
#21447418	2
#7470	61