Finite Group Databasebeta

Subgroups — #36475369

Order 22784.

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

normal subgroups only

This group has 176 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	#26022340	C₂	1	1	yes	maximal, characteristic
2	#366×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#26018454	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018453	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018433	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018389	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1416	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
9	#19151836	—	2	1	no	—
10	#107×D₈₉	—	2	1	no	—
11	C₂×#15667918	C₂²	1	1	yes	characteristic
12	#19151424	C₂²	1	1	yes	characteristic
13	#19151424	C₂²	1	1	yes	characteristic
14	#19151423	—	2	1	no	—
15	#19151402	—	2	1	no	—
16	C₂×Q₃₂×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
18	#366×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
20	#15667988	—	4	1	no	—
21	C₄⋊C₄×D₈₉	—	4	1	no	—
22	C₂×#13083549	D₄	1	1	yes	characteristic
23	C₁₆×D₈₉	—	2	1	no	—
24	C₁₆×D₈₉	—	2	1	no	—
25	#15667946	—	2	1	no	—
26	#15667946	—	2	1	no	—
27	#15667945	—	4	1	no	—
28	#15667936	—	4	1	no	—
29	#15667918	—	2	1	no	—
30	C₂×Q₁₆×C₈₉	—	2	1	no	—
31	Q₃₂×C₈₉	—	2	1	no	—
32	Q₃₂×C₈₉	—	2	1	no	—
33	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
34	#107×C₈₉	—	2	1	no	—
35	C₃₂×C₈₉	—	2	1	no	—
36	C₂×C₄×D₈₉	—	8	1	no	—
37	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
38	C₂×#10623494	D₈	1	1	yes	characteristic
39	C₈×D₈₉	—	2	1	no	—
40	C₈×D₈₉	—	2	1	no	—
41	C₄⋊Dic₈₉	—	4	1	no	—
42	C₄⋊Dic₈₉	—	4	1	no	—
43	#13083553	—	8	1	no	—
44	#13083553	—	8	1	no	—
45	#13083551	—	8	1	no	—
46	#13083549	—	2	1	no	—
47	C₂×Q₈×C₈₉	—	4	1	no	—
48	Q₁₆×C₈₉	—	4	1	no	—
49	Q₁₆×C₈₉	—	4	1	no	—
50	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₄⋊C₄×C₈₉	—	4	1	no	—
52	C₁₆×C₈₉	#41	1	1	yes	characteristic
53	C₁₆×C₈₉	#41	1	1	yes	characteristic
54	C₂²×D₈₉	D₁₆	1	1	yes	characteristic
55	Q₈×C₈₉	—	8	1	no	—
56	Q₈×C₈₉	—	8	1	no	—
57	C₂×C₄×C₈₉	—	8	1	no	—
58	C₂×C₄×C₈₉	—	8	1	no	—
59	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
60	C₂×Dic₈₉	—	8	1	no	—
61	C₂×Dic₈₉	—	8	1	no	—
62	C₂×Dic₈₉	D₁₆	1	1	yes	characteristic
63	C₄×D₈₉	—	2	1	no	—
64	C₄×D₈₉	—	2	1	no	—
65	C₄×D₈₉	—	16	2	no	—
66	C₈×C₈₉	#123	1	1	yes	characteristic
67	C₈×C₈₉	#137	1	1	yes	characteristic
68	#10623494	—	2	1	no	—
69	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
70	C₂×D₈₉	—	2	1	no	—
71	C₂×D₈₉	—	2	1	no	—
72	C₂×D₈₉	SD₆₄	1	2	yes	—
73	C₄×C₈₉	—	16	1	no	—
74	C₄×C₈₉	—	16	1	no	—
75	C₄×C₈₉	#480	1	1	yes	characteristic
76	C₄×C₈₉	#510	1	1	yes	characteristic
77	Dic₈₉	—	2	1	no	—
78	Dic₈₉	—	16	1	no	—
79	Dic₈₉	—	16	1	no	—
80	#13714	—	89	1	no	maximal, Sylow
81	C₂×C₈₉	#2232	1	1	yes	characteristic
82	C₂×C₈₉	C₂×SD₆₄	1	1	yes	characteristic
83	C₂×C₈₉	#2264	1	1	yes	characteristic
84	D₈₉	—	2	2	no	—
85	#2216	—	89	1	no	—
86	C₂×#366	—	89	1	no	—
87	#1424	—	89	1	no	—
88	#1424	—	89	1	no	—
89	#1416	—	89	1	no	—
90	#1416	D₈₉	1	1	yes	characteristic, complemented
91	C₂²⋊C₃₂	—	89	1	no	—
92	C₈₉	#13714	1	1	yes	characteristic, complemented, Sylow
93	C₂×Q₃₂	C₂×D₈₉	1	1	yes	characteristic
94	C₂²×C₁₆	—	89	1	no	—
95	#467	—	178	1	no	—
96	C₂×#107	—	178	1	no	—
97	C₂×C₃₂	—	89	1	no	—
98	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
99	#366	—	89	1	no	—
100	#366	—	89	1	no	—
101	#366	—	178	1	no	—
102	#366	C₂×D₈₉	1	1	yes	characteristic
103	#358	—	178	1	no	—
104	C₂×Q₁₆	—	2	1	no	—
105	C₂²×C₈	—	89	1	no	—
106	C₂²⋊Q₈	—	356	1	no	—
107	C₂×C₄⋊C₄	—	356	1	no	—
108	Q₃₂	—	2	1	no	—
109	Q₃₂	—	2	1	no	—
110	C₂×C₁₆	—	89	1	no	—
111	C₂×C₁₆	—	178	1	no	—
112	C₂×C₁₆	—	178	1	no	—
113	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
114	#107	—	2	1	no	—
115	#107	—	356	1	no	—
116	#107	—	178	1	no	—
117	#107	—	178	1	no	—
118	#103	—	356	1	no	—
119	C₃₂	—	2	1	no	—
120	C₃₂	—	178	1	no	—
121	C₂×Q₈	—	4	1	no	—
122	C₂²×C₄	—	89	1	no	—
123	C₂²×C₄	—	712	1	no	—
124	Q₁₆	—	4	1	no	—
125	Q₁₆	—	4	1	no	—
126	C₂×C₈	—	89	1	no	—
127	C₂×C₈	—	178	1	no	—
128	C₂×C₈	—	178	1	no	—
129	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
130	C₄⋊C₄	—	4	1	no	—
131	C₄⋊C₄	—	356	1	no	—
132	C₄⋊C₄	—	356	1	no	—
133	C₄⋊C₄	—	712	1	no	—
134	C₄⋊C₄	—	712	1	no	—
135	C₂²⋊C₄	—	712	1	no	—
136	C₁₆	—	178	1	no	—
137	C₁₆	#13083573	1	1	yes	characteristic
138	C₁₆	#13083574	1	1	yes	characteristic
139	C₂³	—	89	1	no	—
140	Q₈	—	8	1	no	—
141	Q₈	—	8	1	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	8	1	no	—
144	C₂×C₄	—	89	1	no	—
145	C₂×C₄	—	1424	2	no	—
146	C₂×C₄	—	178	1	no	—
147	C₂×C₄	—	178	1	no	—
148	C₂×C₄	—	712	1	no	—
149	C₂×C₄	—	712	1	no	—
150	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
151	C₈	—	178	1	no	—
152	C₈	#15667994	1	1	yes	characteristic
153	C₈	#15668041	1	1	yes	characteristic
154	C₂²	—	89	2	no	—
155	C₂²	—	178	1	no	—
156	C₂²	—	178	1	no	—
157	C₂²	D₁₆×D₈₉	1	1	yes	characteristic
158	C₄	—	16	1	no	—
159	C₄	—	16	1	no	—
160	C₄	—	1424	1	no	—
161	C₄	—	1424	1	no	—
162	C₄	—	178	1	no	—
163	C₄	#19151910	1	1	yes	characteristic
164	C₄	#19152037	1	1	yes	characteristic
165	C₂	—	178	2	no	—
166	C₂	#26022431	1	1	yes	characteristic
167	C₂	SD₆₄×D₈₉	1	1	yes	characteristic
168	C₂	#26022571	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₈₉
commutator, cosocle	C₂×C₁₆×C₈₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₈₉×#1416	C₂
socle	C₂²×C₈₉	C₂×D₁₆
prime radical	C₂²×D₈₉	D₁₆
hypercenter	#1416	D₈₉
hypocenter	C₈₉	#13714
LCS	G ▷ C₂×C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

Complements

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

N	H ≅ G/N	type
3. #26018454	C₂	semidirect
4. #26018453	C₂	semidirect
5. #26018433	C₂	semidirect
7. C₈₉×#1416	C₂	semidirect
90. #1416	D₈₉	semidirect
92. C₈₉	#13714	semidirect

Maximal subgroups

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

subgroup	index
#26022340	2
#366×D₈₉	2
#26018454	2
#26018453	2
#26018433	2
#26018389	2
C₈₉×#1416	2
#13714	89