Finite Group Databasebeta

← C23×#9465 #19157091

Subgroups — C23×#9465 #19157091

Order 5888.

554 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C23×#2504C211yesmaximal, characteristic
2C23×#2313C211yesmaximal, characteristic, complemented
3C23×#1821C211yesmaximal, characteristic, complemented
4C23×#1510C212yesmaximal
5C23×#1284C212yesmaximal, complemented
6C2×C23×#121C2211yescharacteristic
7C22×C4⋊C4×C23C2211yescharacteristic
8C23×#407C2212yes—
9C23×#397—22no—
10C23×#390C412yescomplemented
11C23×#387—24no—
12C23×#386—21no—
13C23×#386—21no—
14C23×#382C412yescomplemented
15C23×#381C2212yes—
16C23×#380—21no—
17C4×C22⋊C4×C23—21no—
18C4×C22⋊C4×C23C2211yescharacteristic
19C23×#343—22no—
20C23×#327—24no—
21C22×D4×C23—21no—
22C22×D4×C23C2×C411yescharacteristic
23C23×C4×C23D411yescharacteristic
24C23×C4×C23D411yescharacteristic
25C23×C4×C23C2311yescharacteristic
26C2×M16×C23—22no—
27C23×#121—22no—
28C23×#121—22no—
29C2×C4⋊C4×C23—22no—
30C2×C4⋊C4×C23—22no—
31C2×C4⋊C4×C23—22no—
32C2×C4⋊C4×C23C2×C411yescharacteristic
33C2×C4⋊C4×C23C2×C412yes—
34C2×C4⋊C4×C23D412yes—
35C2×C22⋊C4×C23—22no—
36C2×C22⋊C4×C23—21no—
37C2×C22⋊C4×C23—21no—
38C2×C22⋊C4×C23—22no—
39C2×C22⋊C4×C23—22no—
40C2×C42×C23—22no—
41C2×C42×C23C2×C412yes—
42C23×C22⋊C8—22no—
43C23×C22⋊C8—22no—
44C23×#95—22no—
45C23×#95—22no—
46C23×#95—22no—
47C23×#95—22no—
48C23×#95—21no—
49C23×#95—22no—
50C23×#95D414yes—
51C24×C23—21no—
52C24×C23C2×D411yescharacteristic
53C2×D4×C23—22no—
54C2×D4×C23—42no—
55C2×D4×C23—44no—
56C2×D4×C23—41no—
57C2×D4×C23—41no—
58C22×C4×C23—21no—
59C22×C4×C23—21no—
60C22×C4×C23—21no—
61C22×C4×C23—21no—
62C22×C4×C23—22no—
63C22×C4×C23—21no—
64C22×C4×C23—24no—
65C22×C4×C23—42no—
66C22×C4×C23—81no—
67C22×C4×C23—81no—
68C22×C4×C23C22⋊C412yes—
69C22×C4×C23C22⋊C412yes—
70C22×C4×C23C22×C411yescharacteristic
71C22×C4×C23C2×D411yescharacteristic
72C22×C4×C23C2×D412yes—
73C22×C4×C23#4112yes—
74M16×C23—44no—
75C2×C8×C23—42no—
76C4⋊C4×C23—24no—
77C4⋊C4×C23—24no—
78C4⋊C4×C23—22no—
79C4⋊C4×C23—82no—
80C22⋊C4×C23—44no—
81C22⋊C4×C23—42no—
82C22⋊C4×C23—42no—
83C22⋊C4×C23—82no—
84C22⋊C4×C23—82no—
85C42×C23—42no—
86C42×C23—82no—
87#9465C2311yesmaximal, characteristic, direct product factor, Sylow
88C23×C23—22no—
89C23×C23—22no—
90C23×C23—41no—
91C23×C23—42no—
92C23×C23—42no—
93C23×C23—41no—
94C23×C23—42no—
95C23×C23C2×C22⋊C411yescharacteristic
96C23×C23#12011yescharacteristic
97C23×C23#12311yescharacteristic
98D4×C23—82no—
99D4×C23—82no—
100D4×C23—82no—
101C2×C4×C23—22no—
102C2×C4×C23—22no—
103C2×C4×C23—22no—
104C2×C4×C23—22no—
105C2×C4×C23—42no—
106C2×C4×C23—42no—
107C2×C4×C23—42no—
108C2×C4×C23—42no—
109C2×C4×C23—42no—
110C2×C4×C23—42no—
111C2×C4×C23—44no—
112C2×C4×C23—42no—
113C2×C4×C23—44no—
114C2×C4×C23—42no—
115C2×C4×C23—42no—
116C2×C4×C23—41no—
117C2×C4×C23—42no—
118C2×C4×C23—44no—
119C2×C4×C23—81no—
120C2×C4×C23—81no—
121C2×C4×C23C4×D412yes—
122C2×C4×C23#12112yes—
123C8×C23—82no—
124#2504C2×C2311yescharacteristic
125#2313C2×C2311yescharacteristic, complemented
126#1821C2×C2311yescharacteristic, complemented
127#1510C2×C2312yes—
128#1284C2×C2312yescomplemented
129C22×C23—21no—
130C22×C23—21no—
131C22×C23—21no—
132C22×C23—21no—
133C22×C23—21no—
134C22×C23—21no—
135C22×C23—21no—
136C22×C23—44no—
137C22×C23—84no—
138C22×C23—81no—
139C22×C23—81no—
140C22×C23—81no—
141C22×C23#38611yescharacteristic
142C22×C23#41012yes—
143C22×C23#45412yes—
144C4×C23—22no—
145C4×C23—22no—
146C4×C23—42no—
147C4×C23—42no—
148C4×C23—84no—
149C4×C23—82no—
150C4×C23—81no—
151C2×#121C22×C2311yescharacteristic
152C22×C4⋊C4C22×C2311yescharacteristic
153#407C22×C2312yes—
154#397—22no—
155#390C4×C2312yescomplemented
156#387—24no—
157#386—21no—
158#386—21no—
159#382C4×C2312yescomplemented
160#381C22×C2312yes—
161#380—21no—
162C4×C22⋊C4—21no—
163C4×C22⋊C4C22×C2311yescharacteristic
164#343—22no—
165#327—24no—
166C2×C23—22no—
167C2×C23—22no—
168C2×C23—41no—
169C2×C23—161no—
170C2×C23#188811yescharacteristic
171C2×C23#189911yescharacteristic
172C2×C23#191111yescharacteristic
173C22×D4—21no—
174C22×D4C2×C4×C2311yescharacteristic
175C23×C4D4×C2311yescharacteristic
176C23×C4D4×C2311yescharacteristic
177C23×C4C23×C2311yescharacteristic
178C2×M16—22no—
179#121—22no—
180#121—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4C2×C4×C2311yescharacteristic
185C2×C4⋊C4C2×C4×C2312yes—
186C2×C4⋊C4D4×C2312yes—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C42—22no—
193C2×C42C2×C4×C2312yes—
194C22⋊C8—22no—
195C22⋊C8—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—21no—
201#95—22no—
202#95D4×C2314yes—
203C23#946511yescharacteristic, direct product factor, Sylow
204C24—21no—
205C24C2×D4×C2311yescharacteristic
206C2×D4—22no—
207C2×D4—42no—
208C2×D4—44no—
209C2×D4—41no—
210C2×D4—41no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—42no—
219C22×C4—81no—
220C22×C4—81no—
221C22×C4C22⋊C4×C2312yes—
222C22×C4C22⋊C4×C2312yes—
223C22×C4C22×C4×C2311yescharacteristic
224C22×C4C2×D4×C2311yescharacteristic
225C22×C4C2×D4×C2312yes—
226C22×C4#41×C2312yes—
227M16—44no—
228C2×C8—42no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—22no—
232C4⋊C4—82no—
233C22⋊C4—44no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—82no—
237C22⋊C4—82no—
238C42—42no—
239C42—82no—
240C23—22no—
241C23—22no—
242C23—41no—
243C23—42no—
244C23—42no—
245C23—41no—
246C23—42no—
247C23C2×C22⋊C4×C2311yescharacteristic
248C23C23×#12011yescharacteristic
249C23C23×#12311yescharacteristic
250D4—82no—
251D4—82no—
252D4—82no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4C4×D4×C2312yes—
274C2×C4C23×#12112yes—
275C8—82no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—44no—
284C22—84no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22C23×#38611yescharacteristic
289C22C23×#41012yes—
290C22C23×#45412yes—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—84no—
296C4—82no—
297C4—81no—
298C2—22no—
299C2—22no—
300C2—41no—
301C2—161no—
302C2C23×#188811yescharacteristic
303C2C23×#189911yescharacteristic
304C2C23×#191111yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC2×C23×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2313C2semidirect
3. C23×#1821C2semidirect
5. C23×#1284C2semidirect
10. C23×#390C4semidirect
14. C23×#382C4semidirect
87. #9465C23direct (N × H)
125. #2313C2×C23semidirect
126. #1821C2×C23semidirect
128. #1284C2×C23semidirect
155. #390C4×C23semidirect
159. #382C4×C23semidirect
203. C23#9465direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25042
C23×#23132
C23×#18212
C23×#15102
C23×#12842
#946523