Finite Group Databasebeta

Subgroups — #30608522

Order 15616.

178 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 76 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	D₆₁⋊#368	C₂	1	1	yes	maximal, characteristic
2	C₂×#16840690	C₂	1	1	yes	maximal, characteristic
3	#21460983	C₂	1	2	yes	maximal, complemented
4	#21460982	C₂	1	2	yes	maximal, complemented
5	M₆₄×D₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519848	—	2	1	no	—
7	D₆₁⋊#108	—	2	1	no	—
8	#16840691	—	2	1	no	—
9	#16840690	—	2	1	no	—
10	#16840690	C₂²	1	2	yes	—
11	#16840690	C₂²	1	1	yes	characteristic
12	#16840690	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
14	#16839596	C₄	1	1	yes	characteristic
15	#16839595	C₄	1	2	yes	—
16	C₃₂×D₆₁	C₂²	1	2	yes	—
17	C₆₁×M₆₄	C₄	1	1	yes	characteristic
18	C₂×#11758213	—	4	1	no	—
19	D₆₁⋊M₁₆	—	4	1	no	—
20	#14519848	—	2	1	no	—
21	#14519848	—	2	1	no	—
22	#14519848	—	2	2	no	—
23	#14519848	—	4	1	no	—
24	#14519847	—	4	1	no	—
25	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
26	C₂×#11758185	Q₈	1	1	yes	characteristic
27	C₁₆×D₆₁	C₂×C₄	1	1	yes	characteristic
28	C₁₆×D₆₁	D₄	1	1	yes	characteristic
29	C₁₆×D₆₁	Q₈	1	1	yes	characteristic
30	C₁₆×D₆₁	C₂³	1	1	yes	characteristic
31	#14519683	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₃₂×C₆₁	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₆₁	D₈	1	1	yes	characteristic
35	#11758217	—	8	1	no	—
36	C₂×#92420	—	8	1	no	—
37	#11758213	—	4	1	no	—
38	#11758213	—	4	1	no	—
39	#11758213	—	4	2	no	—
40	#11758213	—	8	1	no	—
41	D₆₁⋊C₈	—	8	1	no	—
42	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂×#92418	Q₁₆	1	1	yes	characteristic
44	C₈×D₆₁	C₄⋊C₄	1	1	yes	characteristic
45	C₈×D₆₁	D₈	1	1	yes	characteristic
46	C₈×D₆₁	Q₁₆	1	1	yes	characteristic
47	C₈×D₆₁	C₂×D₄	1	1	yes	characteristic
48	C₁₆×C₆₁	C₄⋊C₄	1	1	yes	characteristic
49	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
50	#11758185	C₄⋊C₄	1	1	yes	characteristic
51	#11758185	C₂×Q₈	1	1	yes	characteristic
52	C₂²×D₆₁	Q₃₂	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
54	C₂×Dic₆₁	D₁₆	1	1	yes	characteristic
55	C₄×D₆₁	#107	1	1	yes	characteristic
56	C₄×D₆₁	D₁₆	1	1	yes	characteristic
57	C₄×D₆₁	Q₃₂	1	1	yes	characteristic
58	C₄×D₆₁	C₂×D₈	1	1	yes	characteristic
59	#92420	—	8	1	no	—
60	#92420	—	8	1	no	—
61	#92420	—	8	2	no	—
62	C₈×C₆₁	#107	1	1	yes	characteristic
63	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
64	#92418	#107	1	1	yes	characteristic
65	#92418	C₂×Q₁₆	1	1	yes	characteristic
66	#13636	—	61	1	no	maximal, Sylow
67	C₂²×C₆₁	#366	1	1	yes	characteristic
68	C₂×D₆₁	—	4	1	no	—
69	C₂×D₆₁	#366	1	1	yes	characteristic
70	C₂×D₆₁	C₂×Q₃₂	1	1	yes	characteristic
71	C₄×C₆₁	#366	1	1	yes	characteristic
72	C₄×C₆₁	C₂×#107	1	1	yes	characteristic
73	Dic₆₁	#366	1	1	yes	characteristic
74	Dic₆₁	C₂×D₁₆	1	1	yes	characteristic
75	C₂×M₆₄	—	61	1	no	—
76	C₂×#368	—	61	1	no	—
77	C₂×#368	—	61	1	no	—
78	#1425	—	61	2	no	—
79	#1425	—	61	2	no	—
80	C₂×C₆₁	—	2	1	no	—
81	C₂×C₆₁	C₂×#366	1	1	yes	characteristic
82	D₆₁	—	2	1	no	—
83	D₆₁	—	2	1	no	—
84	C₂²×C₁₆	—	61	1	no	—
85	C₂×#108	—	122	1	no	—
86	C₂×#108	—	122	1	no	—
87	M₆₄	—	61	1	no	—
88	M₆₄	—	61	2	no	—
89	M₆₄	#6923	1	1	yes	characteristic
90	C₂×C₃₂	—	61	2	no	—
91	#368	—	61	2	no	—
92	#368	—	61	1	no	—
93	#368	—	61	1	no	—
94	#368	—	122	1	no	—
95	#368	—	122	1	no	—
96	C₆₁	#13636	1	1	yes	characteristic, complemented, Sylow
97	C₂×M₁₆	—	244	1	no	—
98	C₂×M₁₆	—	244	1	no	—
99	C₂²×C₈	—	61	1	no	—
100	C₂×C₁₆	—	61	1	no	—
101	C₂×C₁₆	—	61	1	no	—
102	C₂×C₁₆	—	61	1	no	—
103	C₂×C₁₆	—	61	1	no	—
104	C₂×C₁₆	—	61	1	no	—
105	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
106	#108	—	122	1	no	—
107	#108	—	122	1	no	—
108	#108	—	122	2	no	—
109	#108	—	244	1	no	—
110	#108	—	244	1	no	—
111	C₃₂	—	61	2	no	—
112	C₃₂	C₂×#6923	1	2	yes	—
113	C₂²×C₄	—	61	1	no	—
114	M₁₆	—	488	1	no	—
115	M₁₆	—	488	1	no	—
116	M₁₆	—	244	1	no	—
117	M₁₆	—	244	1	no	—
118	M₁₆	—	244	2	no	—
119	C₂×C₈	—	61	1	no	—
120	C₂×C₈	—	61	1	no	—
121	C₂×C₈	—	61	1	no	—
122	C₂×C₈	—	61	1	no	—
123	C₂×C₈	—	61	1	no	—
124	C₂×C₈	—	488	1	no	—
125	C₂×C₈	—	488	1	no	—
126	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
127	C₁₆	—	61	1	no	—
128	C₁₆	—	61	1	no	—
129	C₁₆	C₄⋊#6923	1	1	yes	characteristic
130	C₁₆	C₂²×#6923	1	1	yes	characteristic
131	C₂³	—	61	1	no	—
132	C₂×C₄	—	61	1	no	—
133	C₂×C₄	—	61	1	no	—
134	C₂×C₄	—	61	1	no	—
135	C₂×C₄	—	61	1	no	—
136	C₂×C₄	—	61	1	no	—
137	C₂×C₄	#14519846	1	1	yes	characteristic
138	C₈	—	61	1	no	—
139	C₈	—	61	1	no	—
140	C₈	—	488	1	no	—
141	C₈	—	488	1	no	—
142	C₈	—	488	2	no	—
143	C₈	#14519846	1	1	yes	characteristic
144	C₈	C₂×C₄⋊#6923	1	1	yes	characteristic
145	C₂²	—	61	1	no	—
146	C₂²	—	61	1	no	—
147	C₂²	—	244	1	no	—
148	C₂²	#16840688	1	1	yes	characteristic
149	C₄	—	61	1	no	—
150	C₄	—	61	1	no	—
151	C₄	#16840688	1	1	yes	characteristic
152	C₄	C₂×#14519846	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	122	1	no	—
155	C₂	—	122	1	no	—
156	C₂	C₂×#16840688	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16840688
commutator	C₁₆×C₆₁	C₂²×C₄
Frattini	C₁₆	C₂²×#6923
Fitting	C₆₁×M₆₄	C₄
socle	C₂×C₆₁	C₂×#366
prime radical	C₂²×D₆₁	Q₃₂
cosocle	C₁₆×D₆₁	C₂³
hypercenter	M₆₄	#6923
hypocenter	C₆₁	#13636
LCS	G ▷ C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
3. #21460983	C₂	semidirect
4. #21460982	C₂	semidirect
96. C₆₁	#13636	semidirect

Maximal subgroups

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

subgroup	index
D₆₁⋊#368	2
C₂×#16840690	2
#21460983	2
#21460982	2
M₆₄×D₆₁	2
#13636	61