Finite Group Databasebeta

← C23×#9451 #19157077

Subgroups — C23×#9451 #19157077

Order 5888.

518 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2973C211yesmaximal, characteristic
2C23×#2547C211yesmaximal, characteristic
3C23×#1558C212yesmaximal, complemented
4C23×#1466C211yesmaximal, characteristic
5C23×#1285C212yesmaximal, complemented
6C2×C23×C4⋊Q8C2211yescharacteristic
7C2×C23×#124C412yes—
8C2×C4×D4×C23C2211yescharacteristic
9C23×#435—24no—
10C23×#432C2212yes—
11C23×#432C2212yes—
12C23×#407—21no—
13C2×C23×C22⋊C8—21no—
14C23×#397—21no—
15C23×#397—21no—
16C23×#391C412yes—
17C23×#380—21no—
18C23×#380—21no—
19C4×C4⋊C4×C23C2211yescharacteristic
20C23×#333—24no—
21C23×#323—21no—
22C23×#323—21no—
23C22×Q8×C23C2×C412yes—
24C22×D4×C23C2×C411yescharacteristic
25C23×C4×C23D411yescharacteristic
26C23×C4×C23D411yescharacteristic
27C2×M16×C23—22no—
28C22×C8×C23—22no—
29C23×C4⋊Q8—22no—
30C23×C4⋊Q8—22no—
31C23×#124—44no—
32C4×D4×C23—22no—
33C4×D4×C23—22no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23—21no—
36C2×C4⋊C4×C23—21no—
37C2×C4⋊C4×C23C2×C411yescharacteristic
38C2×C22⋊C4×C23—22no—
39C2×C22⋊C4×C23—22no—
40C2×C22⋊C4×C23D411yescharacteristic
41C2×C22⋊C4×C23D411yescharacteristic
42C2×C42×C23C2×C412yes—
43C2×C42×C23C2311yescharacteristic
44C23×C4⋊C8—22no—
45C23×C4⋊C8D414yes—
46C23×C22⋊C8—22no—
47C23×C22⋊C8—22no—
48C23×C22⋊C8—22no—
49C23×C22⋊C8—22no—
50C23×#97—22no—
51C23×#97—22no—
52C23×#95—22no—
53C23×#95—22no—
54C23×#95—21no—
55C24×C23C22⋊C411yescharacteristic
56C24×C23C22⋊C411yescharacteristic
57C2×Q8×C23—24no—
58C2×Q8×C23—42no—
59C2×Q8×C23—42no—
60C2×D4×C23—22no—
61C2×D4×C23—22no—
62C2×D4×C23—41no—
63C22×C4×C23—21no—
64C22×C4×C23—21no—
65C22×C4×C23—21no—
66C22×C4×C23—21no—
67C22×C4×C23—22no—
68C22×C4×C23—22no—
69C22×C4×C23—42no—
70C22×C4×C23—42no—
71C22×C4×C23C22⋊C411yescharacteristic
72C22×C4×C23C22⋊C411yescharacteristic
73C22×C4×C23C22×C411yescharacteristic
74C22×C4×C23C2×D411yescharacteristic
75C22×C4×C23C2×D411yescharacteristic
76M16×C23—82no—
77C2×C8×C23—24no—
78C2×C8×C23—42no—
79C2×C8×C23—82no—
80C4⋊C4×C23—22no—
81C4⋊C4×C23—42no—
82C4⋊C4×C23—41no—
83C4⋊C4×C23—42no—
84C4⋊C4×C23—41no—
85C22⋊C4×C23—22no—
86C22⋊C4×C23—22no—
87C22⋊C4×C23—44no—
88C22⋊C4×C23—44no—
89C42×C23—42no—
90C42×C23C2×D412yes—
91C42×C23#4112yes—
92#9451C2311yesmaximal, characteristic, direct product factor, Sylow
93C23×C23—21no—
94C23×C23—21no—
95C23×C23—21no—
96C23×C23—21no—
97C23×C23—21no—
98C23×C23—41no—
99C23×C23—41no—
100C23×C23—41no—
101C23×C23—41no—
102C23×C23#10012yes—
103C23×C23C2×C22⋊C411yescharacteristic
104Q8×C23—84no—
105D4×C23—44no—
106C2×C4×C23—21no—
107C2×C4×C23—21no—
108C2×C4×C23—22no—
109C2×C4×C23—21no—
110C2×C4×C23—21no—
111C2×C4×C23—41no—
112C2×C4×C23—42no—
113C2×C4×C23—42no—
114C2×C4×C23—41no—
115C2×C4×C23—41no—
116C2×C4×C23—42no—
117C2×C4×C23—42no—
118C2×C4×C23—42no—
119C2×C4×C23—42no—
120C2×C4×C23—42no—
121C2×C4×C23—42no—
122C2×C4×C23—81no—
123C2×C4×C23—81no—
124C2×C4×C23C4×D412yes—
125C2×C4×C23#12011yescharacteristic
126C2×C4×C23#12112yes—
127C2×C4×C23#12311yescharacteristic
128C8×C23—44no—
129C8×C23—82no—
130#2973C2×C2311yescharacteristic
131#2547C2×C2311yescharacteristic
132#1558C2×C2312yescomplemented
133#1466C2×C2311yescharacteristic
134#1285C2×C2312yescomplemented
135C22×C23—21no—
136C22×C23—21no—
137C22×C23—22no—
138C22×C23—42no—
139C22×C23—42no—
140C22×C23—41no—
141C22×C23—41no—
142C22×C23—41no—
143C22×C23—41no—
144C22×C23—41no—
145C22×C23—81no—
146C22×C23—81no—
147C22×C23#38611yescharacteristic
148C22×C23C2×#10011yescharacteristic
149C22×C23#41311yescharacteristic
150C4×C23—21no—
151C4×C23—22no—
152C4×C23—22no—
153C4×C23—41no—
154C4×C23—41no—
155C4×C23—82no—
156C4×C23—82no—
157C4×C23#45512yes—
158C2×C4⋊Q8C22×C2311yescharacteristic
159C2×#124C4×C2312yes—
160C2×C4×D4C22×C2311yescharacteristic
161#435—24no—
162#432C22×C2312yes—
163#432C22×C2312yes—
164#407—21no—
165C2×C22⋊C8—21no—
166#397—21no—
167#397—21no—
168#391C4×C2312yes—
169#380—21no—
170#380—21no—
171C4×C4⋊C4C22×C2311yescharacteristic
172#333—24no—
173#323—21no—
174#323—21no—
175C2×C23—21no—
176C2×C23—21no—
177C2×C23—42no—
178C2×C23—81no—
179C2×C23#188511yescharacteristic
180C2×C23#190011yescharacteristic
181C2×C23#190911yescharacteristic
182C22×Q8C2×C4×C2312yes—
183C22×D4C2×C4×C2311yescharacteristic
184C23×C4D4×C2311yescharacteristic
185C23×C4D4×C2311yescharacteristic
186C2×M16—22no—
187C22×C8—22no—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190#124—44no—
191C4×D4—22no—
192C4×D4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4C2×C4×C2311yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4D4×C2311yescharacteristic
200C2×C22⋊C4D4×C2311yescharacteristic
201C2×C42C2×C4×C2312yes—
202C2×C42C23×C2311yescharacteristic
203C4⋊C8—22no—
204C4⋊C8D4×C2314yes—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209#97—22no—
210#97—22no—
211#95—22no—
212#95—22no—
213#95—21no—
214C23#945111yescharacteristic, direct product factor, Sylow
215C24C22⋊C4×C2311yescharacteristic
216C24C22⋊C4×C2311yescharacteristic
217C2×Q8—24no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—41no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4C22⋊C4×C2311yescharacteristic
232C22×C4C22⋊C4×C2311yescharacteristic
233C22×C4C22×C4×C2311yescharacteristic
234C22×C4C2×D4×C2311yescharacteristic
235C22×C4C2×D4×C2311yescharacteristic
236M16—82no—
237C2×C8—24no—
238C2×C8—42no—
239C2×C8—82no—
240C4⋊C4—22no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C42—42no—
250C42C2×D4×C2312yes—
251C42#41×C2312yes—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23C23×#10012yes—
262C23C2×C22⋊C4×C2311yescharacteristic
263Q8—84no—
264D4—44no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4C4×D4×C2312yes—
284C2×C4C23×#12011yescharacteristic
285C2×C4C23×#12112yes—
286C2×C4C23×#12311yescharacteristic
287C8—44no—
288C8—82no—
289C22—21no—
290C22—21no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22—81no—
301C22C23×#38611yescharacteristic
302C22C2×C23×#10011yescharacteristic
303C22C23×#41311yescharacteristic
304C4—21no—
305C4—22no—
306C4—22no—
307C4—41no—
308C4—41no—
309C4—82no—
310C4—82no—
311C4C23×#45512yes—
312C2—21no—
313C2—21no—
314C2—42no—
315C2—81no—
316C2C23×#188511yescharacteristic
317C2C23×#190011yescharacteristic
318C2C23×#190911yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC22×D4×C23C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1558C2semidirect
5. C23×#1285C2semidirect
92. #9451C23direct (N × H)
132. #1558C2×C23semidirect
134. #1285C2×C23semidirect
214. C23#9451direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#29732
C23×#25472
C23×#15582
C23×#14662
C23×#12852
#945123