Finite Group Databasebeta

Subgroups — #29600623

Order 15616.

184 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 61 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	#21448126	C₂	1	1	yes	maximal, characteristic
2	#21447700	C₂	1	1	yes	maximal, characteristic, complemented
3	#21447699	C₂	1	1	yes	maximal, characteristic, complemented
4	#21444022	C₂	1	1	yes	maximal, characteristic, complemented
5	#21444004	C₂	1	1	yes	maximal, characteristic, complemented
6	#21443952	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1417	C₂	1	1	yes	maximal, characteristic, complemented
8	#16840194	C₂²	1	1	yes	characteristic
9	#16839879	—	2	1	no	—
10	#16839878	—	2	1	no	—
11	#16839602	C₄	1	1	yes	characteristic, complemented
12	#16839600	C₄	1	1	yes	characteristic, complemented
13	#16839599	C₄	1	1	yes	characteristic, complemented
14	#16839597	C₄	1	1	yes	characteristic, complemented
15	C₂×#14519683	C₂²	1	1	yes	characteristic
16	#16839592	C₂²	1	1	yes	characteristic
17	#16839569	—	2	1	no	—
18	#16839568	—	2	1	no	—
19	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
20	C₆₁×#508	C₂²	1	1	yes	characteristic
21	C₂×C₃₂×C₆₁	C₂²	1	1	yes	characteristic
22	C₆₁×#368	C₂²	1	1	yes	characteristic
23	#14519799	—	2	1	no	—
24	#14519785	—	4	1	no	—
25	#14519784	—	4	1	no	—
26	Dic₄₈₈	C₂×C₄	1	1	yes	characteristic
27	#14519722	—	2	1	no	—
28	D₉₇₆	C₂×C₄	1	1	yes	characteristic
29	C₂×#11758185	D₄	1	1	yes	characteristic
30	#14519713	—	2	1	no	—
31	#14519704	—	4	1	no	—
32	#14519703	—	4	1	no	—
33	C₈×Dic₆₁	D₄	1	1	yes	characteristic
34	#14519683	C₂×C₄	1	1	yes	characteristic
35	#14519683	C₂×C₄	1	1	yes	characteristic
36	#135×C₆₁	—	2	1	no	—
37	Q₃₂×C₆₁	C₂×C₄	1	1	yes	characteristic
38	SD₃₂×C₆₁	—	2	1	no	—
39	D₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
41	#108×C₆₁	—	2	1	no	—
42	C₃₂×C₆₁	—	2	1	no	—
43	#41×C₆₁	—	4	1	no	—
44	#11758222	—	4	1	no	—
45	Q₁₆×C₆₁	—	2	1	no	—
46	SD₁₆×C₆₁	—	4	1	no	—
47	D₈×C₆₁	—	2	1	no	—
48	M₁₆×C₆₁	—	4	1	no	—
49	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
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	Dic₂₄₄	—	2	1	no	—
56	D₄₈₈	—	2	1	no	—
57	#11758190	—	4	1	no	—
58	#11758189	—	8	1	no	—
59	C₈×D₆₁	—	8	1	no	—
60	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
61	C₁₆×C₆₁	#41	1	1	yes	characteristic
62	#11758185	—	2	1	no	—
63	Q₈×C₆₁	—	4	1	no	—
64	D₄×C₆₁	—	4	1	no	—
65	D₄×C₆₁	—	8	1	no	—
66	C₂×C₄×C₆₁	—	8	1	no	—
67	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
68	C₂²⋊D₆₁	—	8	1	no	—
69	C₂×Dic₆₁	—	2	1	no	—
70	D₂₄₄	—	4	1	no	—
71	C₄×D₆₁	—	8	1	no	—
72	Dic₁₂₂	—	4	1	no	—
73	C₈×C₆₁	—	8	1	no	—
74	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
75	C₈×C₆₁	#135	1	1	yes	characteristic
76	#92418	—	8	1	no	—
77	#92418	D₁₆	1	1	yes	characteristic
78	#92418	D₁₆	1	1	yes	characteristic
79	#13656	—	61	1	no	maximal, Sylow
80	C₂²×C₆₁	—	8	1	no	—
81	C₂²×C₆₁	#508	1	1	yes	characteristic
82	C₂×D₆₁	—	8	1	no	—
83	C₄×C₆₁	—	8	1	no	—
84	C₄×C₆₁	C₄×D₈	1	1	yes	characteristic
85	C₄×C₆₁	C₂×D₁₆	1	1	yes	characteristic
86	Dic₆₁	—	4	1	no	—
87	Dic₆₁	—	8	1	no	—
88	#2262	—	61	1	no	—
89	#2178	—	61	1	no	—
90	#2178	—	61	1	no	—
91	#1425	—	61	1	no	—
92	#1417	—	61	1	no	—
93	#1417	D₆₁	1	1	yes	characteristic, complemented
94	#1397	—	61	1	no	—
95	C₂×C₆₁	—	2	1	no	—
96	C₂×C₆₁	—	16	1	no	—
97	C₂×C₆₁	C₄×D₁₆	1	1	yes	characteristic
98	D₆₁	—	16	1	no	—
99	#508	—	61	1	no	—
100	#508	C₂×D₆₁	1	1	yes	characteristic
101	#443	—	122	1	no	—
102	#443	—	122	1	no	—
103	Q₆₄	—	61	1	no	—
104	SD₆₄	—	61	1	no	—
105	SD₆₄	—	61	1	no	—
106	D₃₂	—	61	1	no	—
107	C₂×C₃₂	—	61	1	no	—
108	C₂×C₃₂	C₂×D₆₁	1	1	yes	characteristic
109	#368	—	61	1	no	—
110	#368	C₂×D₆₁	1	1	yes	characteristic
111	#359	—	122	1	no	—
112	#359	—	122	1	no	—
113	C₄×C₁₆	—	61	1	no	—
114	C₆₁	#13656	1	1	yes	characteristic, complemented, Sylow
115	#135	—	2	1	no	—
116	#135	—	122	1	no	—
117	#131	—	244	1	no	—
118	#131	—	244	1	no	—
119	Q₃₂	—	61	1	no	—
120	Q₃₂	C₄×D₆₁	1	1	yes	characteristic
121	SD₃₂	—	2	1	no	—
122	SD₃₂	—	122	1	no	—
123	D₁₆	—	61	1	no	—
124	D₁₆	C₄×D₆₁	1	1	yes	characteristic
125	C₂×C₁₆	—	61	1	no	—
126	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
127	#108	—	2	1	no	—
128	#108	—	122	1	no	—
129	#104	—	244	1	no	—
130	#104	—	244	1	no	—
131	C₄×C₈	—	61	1	no	—
132	C₃₂	—	2	1	no	—
133	C₃₂	—	61	1	no	—
134	C₃₂	—	61	1	no	—
135	#41	—	4	1	no	—
136	#41	—	244	1	no	—
137	Q₁₆	—	2	1	no	—
138	Q₁₆	—	122	1	no	—
139	SD₁₆	—	4	1	no	—
140	SD₁₆	—	244	1	no	—
141	D₈	—	2	1	no	—
142	D₈	—	122	1	no	—
143	M₁₆	—	4	1	no	—
144	M₁₆	—	488	1	no	—
145	M₁₆	—	488	1	no	—
146	M₁₆	—	244	1	no	—
147	C₂×C₈	—	61	1	no	—
148	C₂×C₈	—	488	1	no	—
149	C₂×C₈	—	488	1	no	—
150	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
151	C₄²	—	61	1	no	—
152	C₁₆	—	122	1	no	—
153	C₁₆	C₂×C₄×D₆₁	1	1	yes	characteristic
154	C₁₆	#11758224	1	1	yes	characteristic
155	Q₈	—	4	1	no	—
156	Q₈	—	244	1	no	—
157	D₄	—	4	1	no	—
158	D₄	—	8	1	no	—
159	D₄	—	488	1	no	—
160	D₄	—	244	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	122	1	no	—
163	C₂×C₄	—	488	1	no	—
164	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
165	C₈	—	8	1	no	—
166	C₈	—	61	1	no	—
167	C₈	—	61	1	no	—
168	C₈	—	488	1	no	—
169	C₈	#14519737	1	1	yes	characteristic
170	C₈	#14519798	1	1	yes	characteristic
171	C₂²	—	8	1	no	—
172	C₂²	—	488	1	no	—
173	C₂²	#16840193	1	1	yes	characteristic
174	C₄	—	8	1	no	—
175	C₄	—	488	1	no	—
176	C₄	—	244	1	no	—
177	C₄	#16839842	1	1	yes	characteristic
178	C₄	D₁₆×D₆₁	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	16	1	no	—
181	C₂	—	976	1	no	—
182	C₂	#21447663	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₆×D₆₁
commutator	C₁₆×C₆₁	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₆₁
Fitting	C₆₁×#1417	C₂
socle	C₂×C₆₁	C₄×D₁₆
prime radical	#21448126	C₂
cosocle	C₂×C₁₆×C₆₁	C₂³
hypercenter	#1417	D₆₁
hypocenter	C₆₁	#13656
LCS	G ▷ C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1417

Complements

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

N	H ≅ G/N	type
2. #21447700	C₂	semidirect
3. #21447699	C₂	semidirect
4. #21444022	C₂	semidirect
5. #21444004	C₂	semidirect
6. #21443952	C₂	semidirect
7. C₆₁×#1417	C₂	semidirect
11. #16839602	C₄	semidirect
12. #16839600	C₄	semidirect
13. #16839599	C₄	semidirect
14. #16839597	C₄	semidirect
93. #1417	D₆₁	semidirect
114. C₆₁	#13656	semidirect

Maximal subgroups

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

subgroup	index
#21448126	2
#21447700	2
#21447699	2
#21444022	2
#21444004	2
#21443952	2
C₆₁×#1417	2
#13656	61