Finite Group Databasebeta

← #19212307

Subgroups — #19212307

Order 5888.

514 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15673377C212yesmaximal
2#15671220C214yesmaximal, complemented
3C23×#2296C211yesmaximal, characteristic
4C2×C4⋊(C23⋊C8)—22no—
5C2×C22⋊(C23⋊C8)—22no—
6#13092172C2214yes—
7C2×C23×#117C411yescharacteristic
8C2×C23×#117C411yescharacteristic
9C22×C42×C23C2211yescharacteristic
10C23×#389C412yes—
11C23×#389C2212yes—
12C23×#385—22no—
13C4×C4⋊C4×C23—22no—
14C23×C4×C23C2×C411yescharacteristic
15C23×C4×C23D412yes—
16C23×#117—22no—
17C23×#117—42no—
18C23×#117C814yescomplemented
19C2×C4⋊C4×C23—22no—
20C2×C4⋊C4×C23—22no—
21C2×C4⋊C4×C23C2×C412yes—
22C2×C4⋊C4×C23C2×C412yes—
23C2×C22⋊C4×C23—21no—
24C2×C22⋊C4×C23—21no—
25C2×C42×C23—21no—
26C2×C42×C23—21no—
27C2×C42×C23—42no—
28C2×C42×C23C2×C411yescharacteristic
29C2×C42×C23D412yes—
30C2×C42×C23C2311yescharacteristic
31C22×C23⋊C8—24no—
32C23×#95—22no—
33C23×#95—22no—
34C22⋊(C23⋊C8)—44no—
35C4⋊(C23⋊C8)—44no—
36C24×C23C22⋊C411yescharacteristic
37C22×C4×C23—22no—
38C22×C4×C23—22no—
39C22×C4×C23—22no—
40C22×C4×C23—22no—
41C22×C4×C23—22no—
42C22×C4×C23—22no—
43C22×C4×C23—42no—
44C22×C4×C23—42no—
45C22×C4×C23C22⋊C412yes—
46C22×C4×C23C22⋊C411yescharacteristic
47C22×C4×C23C2×C812yes—
48C22×C4×C23M1612yes—
49C22×C4×C23C22×C411yescharacteristic
50C22×C4×C23C2×D412yes—
51C4⋊C4×C23—22no—
52C4⋊C4×C23—42no—
53C4⋊C4×C23—44no—
54C4⋊C4×C23—42no—
55C4⋊C4×C23—42no—
56C4⋊C4×C23C2×C814yes—
57C22⋊C4×C23—24no—
58C22⋊C4×C23—42no—
59C42×C23—24no—
60C42×C23—24no—
61C42×C23—41no—
62C42×C23—41no—
63C42×C23—42no—
64C2×C23⋊C8—44no—
65C2×C23⋊C8—44no—
66C2×C23⋊C8—44no—
67#7598—231nomaximal, Sylow
68C23×C23—21no—
69C23×C23—21no—
70C23×C23—22no—
71C23×C23—22no—
72C23×C23C22⋊C812yes—
73C23×C23C2×C22⋊C411yescharacteristic
74C2×C4×C23—22no—
75C2×C4×C23—21no—
76C2×C4×C23—21no—
77C2×C4×C23—22no—
78C2×C4×C23—24no—
79C2×C4×C23—24no—
80C2×C4×C23—24no—
81C2×C4×C23—22no—
82C2×C4×C23—22no—
83C2×C4×C23—22no—
84C2×C4×C23—44no—
85C2×C4×C23—44no—
86C2×C4×C23—42no—
87C2×C4×C23—42no—
88C2×C4×C23—42no—
89C2×C4×C23—42no—
90C2×C4×C23—42no—
91C2×C4×C23C22⋊C812yes—
92C2×C4×C23#9912yes—
93C2×C4×C23#10112yes—
94C2×C4×C23C22×C811yescharacteristic
95C2×C4×C23C2×M1611yescharacteristic
96C2×C4×C23AGL(1,ℤ/8)12yes—
97C2×C4×C23#13712yes—
98C23⋊C8—84no—
99#2296D2311yescharacteristic
100#1797—232no—
101#1271—234no—
102C22×C23—22no—
103C22×C23—24no—
104C22×C23—41no—
105C22×C23—41no—
106C22×C23—42no—
107C22×C23C2×C22⋊C811yescharacteristic
108C22×C23C2×#9911yescharacteristic
109C22×C23C2×#10111yescharacteristic
110C22×C23#41712yes—
111C22×C23#42112yes—
112C4×C23—21no—
113C4×C23—21no—
114C4×C23—22no—
115C4×C23—22no—
116C4×C23—44no—
117C4×C23—42no—
118C4×C23—82no—
119C4×C23#32414yes—
120C2×#117Dic2311yescharacteristic
121C2×#117Dic2311yescharacteristic
122C22×C42C2×D2311yescharacteristic
123C2×C4⋊C8—462no—
124C2×C22⋊C8—462no—
125#389Dic2312yes—
126#389C2×D2312yes—
127#385—22no—
128C4×C4⋊C4—22no—
129#336—234no—
130C2×C23—42no—
131C2×C23C2×#32411yescharacteristic
132C2×C23#147712yes—
133C2×C23#151312yes—
134C2×C23#153711yescharacteristic
135C2×C23#156111yescharacteristic
136C23×C4C2×Dic2311yescharacteristic
137C23×C4C22⋊D2312yes—
138C22×C8—464no—
139#117—22no—
140#117—42no—
141#117C23⋊C814yescomplemented
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4C2×Dic2312yes—
145C2×C4⋊C4C2×Dic2312yes—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—42no—
151C2×C42C2×Dic2311yescharacteristic
152C2×C42C22⋊D2312yes—
153C2×C42C22×D2311yescharacteristic
154C4⋊C8—924no—
155C22⋊C8—924no—
156#95—22no—
157#95—22no—
158C23#759811yescharacteristic, complemented, Sylow
159C24C22⋊Dic2311yescharacteristic
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4C2×C23⋊C812yes—
169C22×C4#6726712yes—
170C22×C4C22⋊Dic2312yes—
171C22×C4C22⋊Dic2311yescharacteristic
172C22×C4C22×Dic2311yescharacteristic
173C22×C4C2×C22⋊D2312yes—
174C2×C8—924no—
175C2×C8—924no—
176C2×C8—924no—
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—44no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4C2×C23⋊C814yes—
183C22⋊C4—24no—
184C22⋊C4—42no—
185C42—24no—
186C42—24no—
187C42—41no—
188C42—41no—
189C42—42no—
190C23—21no—
191C23—21no—
192C23—22no—
193C23—22no—
194C23C22⋊(C23⋊C8)12yes—
195C23C2×C22⋊Dic2311yescharacteristic
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4C22⋊(C23⋊C8)12yes—
214C2×C4#1062501612yes—
215C2×C4#1062501812yes—
216C2×C4C22×C23⋊C811yescharacteristic
217C2×C4C2×#6726711yescharacteristic
218C2×C4#1062512012yes—
219C2×C4#1062512212yes—
220C8—1844no—
221C22—22no—
222C22—24no—
223C22—41no—
224C22—42no—
225C22—41no—
226C22C2×C22⋊(C23⋊C8)11yescharacteristic
227C22C2×#1062501611yescharacteristic
228C22C2×#1062501811yescharacteristic
229C22#1309241812yes—
230C22#1309243412yes—
231C4—22no—
232C4—22no—
233C4—21no—
234C4—21no—
235C4—42no—
236C4—44no—
237C4—82no—
238C4#1309214314yes—
239C2—42no—
240C2C2×#1309214311yescharacteristic
241C2#1567178412yes—
242C2#1567194012yes—
243C2#1567203511yescharacteristic
244C2#1567213511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic23
commutatorC2×C4×C23C22×C8
FrattiniC2×C42C22×D23
FittingC23×#2296C2
socleC23×C23C2×C22⋊C4
hypercenter#2296D23
hypocenterC23#7598
LCSG ▷ C2×C4×C23 ▷ C22×C23 ▷ C23
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2296

Complements

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

NH ≅ G/Ntype
2. #15671220C2semidirect
18. C23×#117C8semidirect
141. #117C23⋊C8semidirect
158. C23#7598semidirect