Finite Group Databasebeta

Subgroups — #29600639

Order 15616.

194 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 67 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	#21447705	C₂	1	1	yes	maximal, characteristic, complemented
2	#21447695	C₂	1	1	yes	maximal, characteristic, complemented
3	#21444025	C₂	1	1	yes	maximal, characteristic, complemented
4	#21444009	C₂	1	1	yes	maximal, characteristic, complemented
5	#21444006	C₂	1	1	yes	maximal, characteristic, complemented
6	#21443954	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2263	C₂	1	1	yes	maximal, characteristic
8	#16839879	—	2	1	no	—
9	D₈×Dic₆₁	—	2	1	no	—
10	#16839596	C₂²	1	1	yes	characteristic
11	#16839592	C₂²	1	1	yes	characteristic
12	#16839590	C₂²	1	1	yes	characteristic
13	#16839569	—	2	1	no	—
14	#16839563	—	2	1	no	—
15	#16839537	C₂²	1	1	yes	characteristic
16	C₆₁×#508	C₂²	1	1	yes	characteristic
17	C₂×D₁₆×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×SD₆₄	C₄	1	2	yes	complemented
19	C₆₁×D₃₂	C₄	1	2	yes	complemented
20	C₆₁×M₆₄	C₂²	1	1	yes	characteristic
21	#14519784	—	4	1	no	—
22	D₄×Dic₆₁	—	4	1	no	—
23	C₂×#11758185	D₄	1	1	yes	characteristic
24	#14519713	—	2	1	no	—
25	#14519711	—	2	1	no	—
26	#14519704	—	4	1	no	—
27	#14519698	—	4	1	no	—
28	C₈×Dic₆₁	D₄	1	1	yes	characteristic
29	#14519683	—	2	1	no	—
30	#135×C₆₁	—	2	1	no	—
31	C₂×D₈×C₆₁	—	2	1	no	—
32	Q₃₂×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
33	SD₃₂×C₆₁	—	2	1	no	—
34	D₁₆×C₆₁	—	2	1	no	—
35	D₁₆×C₆₁	C₂×C₄	1	2	yes	—
36	D₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
37	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
38	C₃₂×C₆₁	C₂×C₄	1	2	yes	complemented
39	#41×C₆₁	—	4	1	no	—
40	C₂×D₄×C₆₁	—	4	1	no	—
41	C₂²×Dic₆₁	—	8	1	no	—
42	Q₁₆×C₆₁	—	2	1	no	—
43	SD₁₆×C₆₁	—	4	1	no	—
44	D₈×C₆₁	—	2	2	no	—
45	D₈×C₆₁	—	2	1	no	—
46	D₈×C₆₁	—	4	1	no	—
47	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂²⋊Dic₆₁	—	8	1	no	—
49	C₄⋊Dic₆₁	—	4	1	no	—
50	C₄×Dic₆₁	D₈	1	1	yes	characteristic
51	#11758194	—	4	1	no	—
52	#11758194	—	8	1	no	—
53	C₂×#92418	—	8	1	no	—
54	C₂×#92418	D₈	1	1	yes	characteristic
55	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
56	C₁₆×C₆₁	#41	1	1	yes	characteristic
57	#11758185	—	2	1	no	—
58	C₂³×C₆₁	—	8	1	no	—
59	Q₈×C₆₁	—	4	1	no	—
60	D₄×C₆₁	—	4	2	no	—
61	D₄×C₆₁	—	4	1	no	—
62	D₄×C₆₁	—	8	1	no	—
63	D₄×C₆₁	—	8	1	no	—
64	C₂×C₄×C₆₁	—	8	1	no	—
65	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
66	C₂×Dic₆₁	—	2	1	no	—
67	C₂×Dic₆₁	—	8	1	no	—
68	C₂×Dic₆₁	—	16	2	no	—
69	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
70	C₈×C₆₁	#135	1	1	yes	characteristic
71	#92418	—	2	1	no	—
72	#92418	—	8	1	no	—
73	#13658	—	61	1	no	maximal, Sylow
74	C₂²×C₆₁	—	8	1	no	—
75	C₂²×C₆₁	—	8	2	no	—
76	C₂²×C₆₁	—	16	1	no	—
77	C₂²×C₆₁	—	16	1	no	—
78	C₂²×C₆₁	#509	1	1	yes	characteristic
79	C₄×C₆₁	—	8	1	no	—
80	C₄×C₆₁	C₄×D₈	1	1	yes	characteristic
81	C₄×C₆₁	#509	1	1	yes	characteristic
82	Dic₆₁	—	4	1	no	—
83	Dic₆₁	—	16	1	no	—
84	#2263	D₆₁	1	1	yes	characteristic
85	#2179	—	61	1	no	—
86	#2177	—	61	1	no	—
87	#1426	—	61	1	no	—
88	#1419	—	61	1	no	—
89	#1418	—	61	1	no	—
90	#1398	—	61	1	no	—
91	C₂×C₆₁	—	2	1	no	—
92	C₂×C₆₁	—	16	1	no	—
93	C₂×C₆₁	—	16	2	no	—
94	C₂×C₆₁	#2177	1	1	yes	characteristic
95	#508	C₂×D₆₁	1	1	yes	characteristic
96	C₂×D₁₆	C₂×D₆₁	1	1	yes	characteristic
97	#443	—	122	1	no	—
98	C₄×D₈	—	122	1	no	—
99	SD₆₄	Dic₆₁	1	2	yes	complemented
100	D₃₂	Dic₆₁	1	2	yes	complemented
101	M₆₄	—	61	1	no	—
102	M₆₄	C₂×D₆₁	1	1	yes	characteristic
103	#368	—	61	1	no	—
104	#367	—	61	1	no	—
105	#359	—	122	1	no	—
106	#357	—	122	1	no	—
107	#346	—	61	1	no	—
108	C₆₁	#13658	1	1	yes	characteristic, complemented, Sylow
109	#135	—	2	1	no	—
110	C₂×D₈	—	2	1	no	—
111	#131	—	244	1	no	—
112	C₄×D₄	—	244	1	no	—
113	Q₃₂	C₂×Dic₆₁	1	1	yes	characteristic, complemented
114	SD₃₂	—	2	1	no	—
115	D₁₆	—	2	1	no	—
116	D₁₆	C₂×Dic₆₁	1	2	yes	—
117	D₁₆	C₂×Dic₆₁	1	1	yes	characteristic, complemented
118	C₂×C₁₆	—	61	1	no	—
119	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
120	#108	—	122	1	no	—
121	#107	—	122	1	no	—
122	#104	—	244	1	no	—
123	#102	—	244	1	no	—
124	C₄×C₈	—	61	1	no	—
125	C₃₂	—	122	1	no	—
126	C₃₂	C₂×Dic₆₁	1	2	yes	complemented
127	#41	—	4	1	no	—
128	C₂×D₄	—	4	1	no	—
129	C₂²×C₄	—	488	1	no	—
130	Q₁₆	—	2	1	no	—
131	SD₁₆	—	4	1	no	—
132	D₈	—	2	2	no	—
133	D₈	—	2	1	no	—
134	D₈	—	4	1	no	—
135	M₁₆	—	488	1	no	—
136	M₁₆	—	244	1	no	—
137	C₂×C₈	—	61	1	no	—
138	C₂×C₈	—	488	1	no	—
139	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
140	C₄⋊C₄	—	244	1	no	—
141	C₂²⋊C₄	—	488	1	no	—
142	C₄²	—	61	1	no	—
143	C₁₆	—	122	1	no	—
144	C₁₆	#11758224	1	1	yes	characteristic
145	C₁₆	C₂²×Dic₆₁	1	1	yes	characteristic
146	C₂³	—	8	1	no	—
147	Q₈	—	4	1	no	—
148	D₄	—	4	2	no	—
149	D₄	—	4	1	no	—
150	D₄	—	8	1	no	—
151	D₄	—	8	1	no	—
152	C₂×C₄	—	8	1	no	—
153	C₂×C₄	—	122	1	no	—
154	C₂×C₄	—	976	2	no	—
155	C₂×C₄	—	488	1	no	—
156	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
157	C₈	—	122	1	no	—
158	C₈	—	488	1	no	—
159	C₈	D₄×Dic₆₁	1	1	yes	characteristic
160	C₈	#14519796	1	1	yes	characteristic
161	C₂²	—	8	1	no	—
162	C₂²	—	8	2	no	—
163	C₂²	—	16	1	no	—
164	C₂²	—	16	1	no	—
165	C₂²	#16840196	1	1	yes	characteristic
166	C₄	—	8	1	no	—
167	C₄	—	976	1	no	—
168	C₄	—	244	1	no	—
169	C₄	D₈×Dic₆₁	1	1	yes	characteristic
170	C₄	#16840196	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	16	2	no	—
173	C₂	—	16	1	no	—
174	C₂	#21447695	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21447695
commutator	C₁₆×C₆₁	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₆₁
Fitting, prime radical	C₆₁×#2263	C₂
socle	C₂×C₆₁	#2177
cosocle	C₂×C₁₆×C₆₁	C₂³
hypercenter	#2263	D₆₁
hypocenter	C₆₁	#13658
LCS	G ▷ C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #2263

Complements

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

N	H ≅ G/N	type
1. #21447705	C₂	semidirect
2. #21447695	C₂	semidirect
3. #21444025	C₂	semidirect
4. #21444009	C₂	semidirect
5. #21444006	C₂	semidirect
6. #21443954	C₂	semidirect
18. C₆₁×SD₆₄	C₄	semidirect
19. C₆₁×D₃₂	C₄	semidirect
32. Q₃₂×C₆₁	C₂×C₄	semidirect
36. D₁₆×C₆₁	C₂×C₄	semidirect
38. C₃₂×C₆₁	C₂×C₄	semidirect
99. SD₆₄	Dic₆₁	semidirect
100. D₃₂	Dic₆₁	semidirect
108. C₆₁	#13658	semidirect
113. Q₃₂	C₂×Dic₆₁	semidirect
117. D₁₆	C₂×Dic₆₁	semidirect
126. C₃₂	C₂×Dic₆₁	semidirect

Maximal subgroups

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

subgroup	index
#21447705	2
#21447695	2
#21444025	2
#21444009	2
#21444006	2
#21443954	2
C₆₁×#2263	2
#13658	61