Finite Group Databasebeta

Subgroups — #29600635

Order 15616.

194 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 59 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	#21448131	C₂	1	1	yes	maximal, characteristic
2	#21447705	C₂	1	1	yes	maximal, characteristic, complemented
3	#21447694	C₂	1	1	yes	maximal, characteristic, complemented
4	#21444024	C₂	1	1	yes	maximal, characteristic, complemented
5	#21444010	C₂	1	1	yes	maximal, characteristic, complemented
6	#21443954	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1418	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₉₇₆	C₂²	1	1	yes	characteristic
9	#16839879	—	2	1	no	—
10	#16839844	—	2	1	no	—
11	#16839599	C₄	1	2	yes	complemented
12	#16839597	C₄	1	2	yes	complemented
13	#16839596	C₂²	1	1	yes	characteristic
14	#16839592	C₂²	1	1	yes	characteristic
15	#16839569	—	2	1	no	—
16	#16839562	—	2	1	no	—
17	#16839537	C₂²	1	1	yes	characteristic
18	C₆₁×#508	C₂²	1	1	yes	characteristic
19	C₆₁×M₆₄	C₂²	1	1	yes	characteristic
20	C₆₁×#367	C₂²	1	1	yes	characteristic
21	C₂×D₄₈₈	—	2	1	no	—
22	#14519784	—	4	1	no	—
23	#14519739	—	4	1	no	—
24	D₉₇₆	—	2	1	no	—
25	D₉₇₆	C₂×C₄	1	2	yes	—
26	C₂×#11758185	D₄	1	1	yes	characteristic
27	#14519713	—	2	1	no	—
28	#14519704	—	4	1	no	—
29	#14519697	—	4	1	no	—
30	C₈×Dic₆₁	D₄	1	1	yes	characteristic
31	#14519683	C₂×C₄	1	2	yes	complemented
32	#135×C₆₁	—	2	1	no	—
33	Q₃₂×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
34	SD₃₂×C₆₁	—	2	1	no	—
35	D₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
36	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
37	#107×C₆₁	—	2	1	no	—
38	C₃₂×C₆₁	—	2	1	no	—
39	#41×C₆₁	—	4	1	no	—
40	C₂×D₂₄₄	—	4	1	no	—
41	C₂×C₄×D₆₁	—	8	1	no	—
42	Q₁₆×C₆₁	—	2	1	no	—
43	SD₁₆×C₆₁	—	4	1	no	—
44	D₈×C₆₁	—	2	1	no	—
45	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₄⋊C₄×C₆₁	—	4	1	no	—
47	#11758198	—	8	1	no	—
48	C₄×Dic₆₁	D₈	1	1	yes	characteristic
49	#11758194	—	4	1	no	—
50	#11758194	—	8	1	no	—
51	C₂×#92418	—	8	1	no	—
52	C₂×#92418	D₈	1	1	yes	characteristic
53	D₄₈₈	—	2	2	no	—
54	D₄₈₈	—	4	1	no	—
55	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
56	C₁₆×C₆₁	#41	1	1	yes	characteristic
57	#11758185	—	2	1	no	—
58	C₂²×D₆₁	—	8	1	no	—
59	Q₈×C₆₁	—	4	1	no	—
60	D₄×C₆₁	—	4	1	no	—
61	D₄×C₆₁	—	8	1	no	—
62	C₂×C₄×C₆₁	—	8	1	no	—
63	C₂×C₄×C₆₁	—	8	1	no	—
64	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
65	C₂×Dic₆₁	—	2	1	no	—
66	D₂₄₄	—	4	2	no	—
67	D₂₄₄	—	8	1	no	—
68	C₄×D₆₁	—	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	#509	1	1	yes	characteristic
76	C₂×D₆₁	—	8	2	no	—
77	C₂×D₆₁	—	16	1	no	—
78	C₂×D₆₁	—	16	1	no	—
79	C₄×C₆₁	—	8	1	no	—
80	C₄×C₆₁	—	16	1	no	—
81	C₄×C₆₁	C₄×D₈	1	1	yes	characteristic
82	C₄×C₆₁	#509	1	1	yes	characteristic
83	Dic₆₁	—	4	1	no	—
84	#2263	—	61	1	no	—
85	#2179	—	61	1	no	—
86	#2177	—	61	1	no	—
87	#1426	—	61	1	no	—
88	#1419	—	61	1	no	—
89	#1418	D₆₁	1	1	yes	characteristic, complemented
90	#1398	—	61	1	no	—
91	C₂×C₆₁	—	2	1	no	—
92	C₂×C₆₁	—	16	1	no	—
93	C₂×C₆₁	#2177	1	1	yes	characteristic
94	D₆₁	—	16	2	no	—
95	#508	C₂×D₆₁	1	1	yes	characteristic
96	C₂×D₁₆	—	61	1	no	—
97	#443	—	122	1	no	—
98	C₄×D₈	—	122	1	no	—
99	SD₆₄	—	61	2	no	—
100	D₃₂	—	61	2	no	—
101	M₆₄	—	61	1	no	—
102	M₆₄	C₂×D₆₁	1	1	yes	characteristic
103	#368	—	61	1	no	—
104	#367	C₂×D₆₁	1	1	yes	characteristic
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	—	122	1	no	—
111	#131	—	244	1	no	—
112	C₄×D₄	—	244	1	no	—
113	Q₃₂	C₄×D₆₁	1	1	yes	characteristic, complemented
114	SD₃₂	—	2	1	no	—
115	D₁₆	—	61	2	no	—
116	D₁₆	—	122	1	no	—
117	D₁₆	C₄×D₆₁	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	—	2	1	no	—
122	#104	—	244	1	no	—
123	#102	—	244	1	no	—
124	C₄×C₈	—	61	1	no	—
125	C₃₂	—	2	1	no	—
126	C₃₂	—	61	2	no	—
127	#41	—	4	1	no	—
128	C₂×D₄	—	244	1	no	—
129	C₂²×C₄	—	488	1	no	—
130	Q₁₆	—	2	1	no	—
131	SD₁₆	—	4	1	no	—
132	D₈	—	2	1	no	—
133	D₈	—	122	2	no	—
134	D₈	—	244	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	—	4	1	no	—
141	C₂²⋊C₄	—	488	1	no	—
142	C₄²	—	61	1	no	—
143	C₁₆	—	122	1	no	—
144	C₁₆	C₂×C₄×D₆₁	1	1	yes	characteristic
145	C₁₆	#11758224	1	1	yes	characteristic
146	C₂³	—	488	1	no	—
147	Q₈	—	4	1	no	—
148	D₄	—	4	1	no	—
149	D₄	—	8	1	no	—
150	D₄	—	488	1	no	—
151	D₄	—	244	2	no	—
152	C₂×C₄	—	8	1	no	—
153	C₂×C₄	—	8	1	no	—
154	C₂×C₄	—	122	1	no	—
155	C₂×C₄	—	976	2	no	—
156	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
157	C₈	—	122	1	no	—
158	C₈	—	488	1	no	—
159	C₈	#14519737	1	1	yes	characteristic
160	C₈	#14519798	1	1	yes	characteristic
161	C₂²	—	8	1	no	—
162	C₂²	—	976	1	no	—
163	C₂²	—	976	1	no	—
164	C₂²	—	488	2	no	—
165	C₂²	#16840196	1	1	yes	characteristic
166	C₄	—	8	1	no	—
167	C₄	—	16	1	no	—
168	C₄	—	244	1	no	—
169	C₄	#16839842	1	1	yes	characteristic
170	C₄	#16840199	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	16	1	no	—
173	C₂	—	976	2	no	—
174	C₂	#21447692	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21447692
commutator	C₁₆×C₆₁	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₆₁
Fitting	C₆₁×#1418	C₂
socle	C₂×C₆₁	#2177
prime radical	#21448131	C₂
cosocle	C₂×C₁₆×C₆₁	C₂³
hypercenter	#1418	D₆₁
hypocenter	C₆₁	#13658
LCS	G ▷ C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1418

Complements

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

N	H ≅ G/N	type
2. #21447705	C₂	semidirect
3. #21447694	C₂	semidirect
4. #21444024	C₂	semidirect
5. #21444010	C₂	semidirect
6. #21443954	C₂	semidirect
7. C₆₁×#1418	C₂	semidirect
11. #16839599	C₄	semidirect
12. #16839597	C₄	semidirect
31. #14519683	C₂×C₄	semidirect
33. Q₃₂×C₆₁	C₂×C₄	semidirect
35. D₁₆×C₆₁	C₂×C₄	semidirect
89. #1418	D₆₁	semidirect
108. C₆₁	#13658	semidirect
113. Q₃₂	C₄×D₆₁	semidirect
117. D₁₆	C₄×D₆₁	semidirect

Maximal subgroups

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

subgroup	index
#21448131	2
#21447705	2
#21447694	2
#21444024	2
#21444010	2
#21443954	2
C₆₁×#1418	2
#13658	61