Finite Group Databasebeta

Subgroups — #29594410

Order 15616.

172 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 55 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	#21448095	C₂	1	1	yes	maximal, characteristic, complemented
2	#21447996	C₂	1	1	yes	maximal, characteristic
3	#21447992	C₂	1	1	yes	maximal, characteristic
4	#21447655	C₂	1	1	yes	maximal, characteristic
5	#21443894	C₂	1	1	yes	maximal, characteristic, complemented
6	#21443881	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1366	C₂	1	1	yes	maximal, characteristic, complemented
8	#16840180	—	2	1	no	—
9	#16840164	C₂²	1	1	yes	characteristic
10	#16839880	C₂²	1	1	yes	characteristic
11	#16839589	—	2	1	no	—
12	#16839587	—	2	1	no	—
13	#16839585	—	2	1	no	—
14	#16839584	—	2	1	no	—
15	#16839583	C₂²	1	1	yes	characteristic
16	#16839573	—	2	1	no	—
17	#16839571	—	2	1	no	—
18	#16839547	—	2	1	no	—
19	C₄×#11758185	C₂²	1	1	yes	characteristic
20	C₆₁×#499	C₂²	1	1	yes	characteristic
21	C₆₁×#364	C₂²	1	1	yes	characteristic
22	C₆₁×#364	C₂²	1	1	yes	characteristic
23	#14519799	D₄	1	1	yes	characteristic
24	#14519783	—	2	1	no	—
25	#14519769	—	2	1	no	—
26	#14519717	—	2	1	no	—
27	#14519716	—	4	1	no	—
28	C₂×#11758185	D₄	1	1	yes	characteristic
29	C₂×#11758185	D₄	1	1	yes	characteristic
30	C₁₆×D₆₁	—	4	1	no	—
31	#14519713	D₄	1	1	yes	characteristic
32	#14519711	—	2	1	no	—
33	#14519709	—	2	1	no	—
34	#14519705	—	2	1	no	—
35	#125×C₆₁	—	2	1	no	—
36	M₃₂×C₆₁	—	2	1	no	—
37	M₃₂×C₆₁	—	2	1	no	—
38	#107×C₆₁	—	2	1	no	—
39	#106×C₆₁	—	2	1	no	—
40	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
41	#11758222	—	2	1	no	—
42	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₄⋊C₄×C₆₁	—	4	1	no	—
45	C₄⋊C₄×C₆₁	—	8	1	no	—
46	C₄²×C₆₁	#41	1	1	yes	characteristic
47	C₄⋊Dic₆₁	—	4	1	no	—
48	#11758196	—	8	1	no	—
49	#11758194	—	2	1	no	—
50	Dic₂₄₄	Q₁₆	1	1	yes	characteristic
51	D₄₈₈	Q₁₆	1	1	yes	characteristic
52	#11758190	—	2	1	no	—
53	#11758189	—	4	1	no	—
54	C₈×D₆₁	—	4	1	no	—
55	C₁₆×C₆₁	—	4	1	no	—
56	C₁₆×C₆₁	—	4	1	no	—
57	#11758185	—	2	1	no	—
58	#11758185	—	4	1	no	—
59	#11758185	SD₁₆	1	1	yes	characteristic
60	#11758185	SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₆₁	—	2	1	no	—
62	C₂×C₄×C₆₁	—	8	1	no	—
63	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
64	C₂²⋊D₆₁	—	4	1	no	—
65	C₂×Dic₆₁	—	8	1	no	—
66	D₂₄₄	—	2	1	no	—
67	C₄×D₆₁	—	4	1	no	—
68	Dic₁₂₂	—	2	1	no	—
69	C₈×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
70	C₈×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
71	C₈×C₆₁	#135	1	1	yes	characteristic
72	C₈×C₆₁	#137	1	1	yes	characteristic
73	#92418	—	4	1	no	—
74	#12464	—	61	1	no	maximal, Sylow
75	C₂²×C₆₁	#472	1	1	yes	characteristic
76	C₂×D₆₁	—	4	1	no	—
77	C₄×C₆₁	—	4	1	no	—
78	C₄×C₆₁	—	16	1	no	—
79	C₄×C₆₁	#461	1	1	yes	characteristic
80	C₄×C₆₁	#467	1	1	yes	characteristic
81	Dic₆₁	—	4	1	no	—
82	Dic₆₁	—	16	1	no	—
83	#2253	—	61	1	no	—
84	#2231	—	61	1	no	—
85	#2231	—	61	1	no	—
86	#2170	—	61	1	no	—
87	#1373	—	61	1	no	—
88	#1366	—	61	1	no	—
89	#1366	D₆₁	1	1	yes	characteristic, complemented
90	C₂×C₆₁	—	2	1	no	—
91	C₂×C₆₁	#1675	1	1	yes	characteristic
92	D₆₁	—	8	1	no	—
93	#504	—	122	1	no	—
94	#499	—	61	1	no	—
95	#499	C₂×D₆₁	1	1	yes	characteristic
96	#443	—	61	1	no	—
97	#367	—	122	1	no	—
98	#366	—	122	1	no	—
99	#365	—	122	1	no	—
100	#365	—	122	1	no	—
101	#364	—	61	1	no	—
102	#364	C₂×D₆₁	1	1	yes	characteristic
103	#364	C₂×D₆₁	1	1	yes	characteristic
104	#360	—	122	1	no	—
105	#360	—	122	1	no	—
106	#350	—	122	1	no	—
107	C₄×C₁₆	—	61	1	no	—
108	C₆₁	#12464	1	1	yes	characteristic, complemented, Sylow
109	#135	—	61	1	no	—
110	#131	—	122	1	no	—
111	#125	—	2	1	no	—
112	#125	—	122	1	no	—
113	M₃₂	—	2	1	no	—
114	M₃₂	—	2	1	no	—
115	M₃₂	—	122	1	no	—
116	M₃₂	—	244	1	no	—
117	C₂×C₁₆	—	61	1	no	—
118	C₂×C₁₆	—	61	1	no	—
119	C₂×C₁₆	—	244	1	no	—
120	#108	—	61	1	no	—
121	#107	—	2	1	no	—
122	#107	—	122	1	no	—
123	#106	—	2	1	no	—
124	#106	—	122	1	no	—
125	#104	—	122	1	no	—
126	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
127	#41	—	122	1	no	—
128	Q₁₆	—	61	1	no	—
129	SD₁₆	—	122	1	no	—
130	D₈	—	61	1	no	—
131	M₁₆	—	122	1	no	—
132	M₁₆	—	244	1	no	—
133	C₂×C₈	—	244	1	no	—
134	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
135	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
136	C₄⋊C₄	—	4	1	no	—
137	C₄⋊C₄	—	8	1	no	—
138	C₄⋊C₄	—	488	1	no	—
139	C₄⋊C₄	—	244	1	no	—
140	C₄²	#11758222	1	1	yes	characteristic
141	C₁₆	—	4	1	no	—
142	C₁₆	—	4	1	no	—
143	C₁₆	—	61	1	no	—
144	C₁₆	—	61	1	no	—
145	C₁₆	—	122	1	no	—
146	C₁₆	—	244	1	no	—
147	Q₈	—	122	1	no	—
148	D₄	—	122	1	no	—
149	D₄	—	244	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	8	1	no	—
152	C₂×C₄	—	488	1	no	—
153	C₂×C₄	—	244	1	no	—
154	C₂×C₄	#14519750	1	1	yes	characteristic
155	C₈	—	244	1	no	—
156	C₈	SD₁₆×D₆₁	1	1	yes	characteristic
157	C₈	Q₁₆×D₆₁	1	1	yes	characteristic
158	C₈	#14519798	1	1	yes	characteristic
159	C₈	#14519809	1	1	yes	characteristic
160	C₂²	—	244	1	no	—
161	C₂²	#16840031	1	1	yes	characteristic
162	C₄	—	4	1	no	—
163	C₄	—	16	1	no	—
164	C₄	—	976	1	no	—
165	C₄	—	244	1	no	—
166	C₄	#16839966	1	1	yes	characteristic
167	C₄	#16840002	1	1	yes	characteristic
168	C₂	—	2	1	no	—
169	C₂	—	488	1	no	—
170	C₂	#21445217	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21445217
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1366	C₂
socle	C₂×C₆₁	#1675
prime radical	#14519799	D₄
hypercenter	#1366	D₆₁
hypocenter	C₆₁	#12464
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1366

Complements

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

N	H ≅ G/N	type
1. #21448095	C₂	semidirect
5. #21443894	C₂	semidirect
6. #21443881	C₂	semidirect
7. C₆₁×#1366	C₂	semidirect
89. #1366	D₆₁	semidirect
108. C₆₁	#12464	semidirect

Maximal subgroups

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

subgroup	index
#21448095	2
#21447996	2
#21447992	2
#21447655	2
#21443894	2
#21443881	2
C₆₁×#1366	2
#12464	61