Finite Group Databasebeta

← C23×#9453 #19157079

Subgroups — C23×#9453 #19157079

Order 5888.

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

normal subgroups only

This group has 498 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×#2563C211yesmaximal, characteristic
3C23×#1556C212yesmaximal, complemented
4C23×#1466C211yesmaximal, characteristic
5C23×#1285C212yesmaximal, complemented
6C2×C23×C4⋊Q8C2211yescharacteristic
7C2×C4×D4×C23C2211yescharacteristic
8C23×#435—24no—
9C23×#432C2212yes—
10C23×#432C2212yes—
11C23×#407—21no—
12C2×C23×C22⋊C8—21no—
13C23×#393—21no—
14C23×#393—21no—
15C23×#390C412yes—
16C23×#382C412yes—
17C23×#380—21no—
18C23×#380—21no—
19C4×C4⋊C4×C23C2211yescharacteristic
20C23×#332—24no—
21C23×#323—21no—
22C23×#323—21no—
23C22×Q8×C23—21no—
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—
31C4×D4×C23—22no—
32C4×D4×C23—22no—
33C2×C4⋊C4×C23—21no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23C2×C412yes—
36C2×C4⋊C4×C23C2×C411yescharacteristic
37C2×C22⋊C4×C23—22no—
38C2×C22⋊C4×C23—22no—
39C2×C22⋊C4×C23D411yescharacteristic
40C2×C22⋊C4×C23D411yescharacteristic
41C2×C42×C23C2×C412yes—
42C2×C42×C23C2311yescharacteristic
43C23×C4⋊C8—22no—
44C23×C4⋊C8D414yes—
45C23×C22⋊C8—22no—
46C23×C22⋊C8—22no—
47C23×C22⋊C8—22no—
48C23×C22⋊C8—22no—
49C23×#97—22no—
50C23×#97—22no—
51C23×#95—22no—
52C23×#95—22no—
53C23×#95—21no—
54C24×C23C22⋊C411yescharacteristic
55C24×C23C22⋊C411yescharacteristic
56C2×Q8×C23—42no—
57C2×Q8×C23—82no—
58C2×D4×C23—22no—
59C2×D4×C23—22no—
60C2×D4×C23—41no—
61C22×C4×C23—21no—
62C22×C4×C23—21no—
63C22×C4×C23—21no—
64C22×C4×C23—21no—
65C22×C4×C23—22no—
66C22×C4×C23—22no—
67C22×C4×C23—42no—
68C22×C4×C23—42no—
69C22×C4×C23C22⋊C411yescharacteristic
70C22×C4×C23C22⋊C411yescharacteristic
71C22×C4×C23C22×C411yescharacteristic
72C22×C4×C23C2×D411yescharacteristic
73C22×C4×C23C2×D411yescharacteristic
74M16×C23—82no—
75C2×C8×C23—24no—
76C2×C8×C23—42no—
77C2×C8×C23—82no—
78C4⋊C4×C23—22no—
79C4⋊C4×C23—24no—
80C4⋊C4×C23—42no—
81C4⋊C4×C23—41no—
82C4⋊C4×C23—41no—
83C22⋊C4×C23—22no—
84C22⋊C4×C23—22no—
85C22⋊C4×C23—82no—
86C22⋊C4×C23—82no—
87C42×C23—42no—
88C42×C23C2×D412yes—
89C42×C23#4112yes—
90#9453C2311yesmaximal, characteristic, direct product factor, Sylow
91C23×C23—21no—
92C23×C23—21no—
93C23×C23—21no—
94C23×C23—21no—
95C23×C23—21no—
96C23×C23—41no—
97C23×C23—41no—
98C23×C23—41no—
99C23×C23—41no—
100C23×C23#10112yes—
101C23×C23C2×C22⋊C411yescharacteristic
102Q8×C23—84no—
103D4×C23—44no—
104C2×C4×C23—21no—
105C2×C4×C23—21no—
106C2×C4×C23—22no—
107C2×C4×C23—21no—
108C2×C4×C23—21no—
109C2×C4×C23—41no—
110C2×C4×C23—42no—
111C2×C4×C23—42no—
112C2×C4×C23—41no—
113C2×C4×C23—41no—
114C2×C4×C23—42no—
115C2×C4×C23—42no—
116C2×C4×C23—42no—
117C2×C4×C23—42no—
118C2×C4×C23—42no—
119C2×C4×C23—42no—
120C2×C4×C23—81no—
121C2×C4×C23—81no—
122C2×C4×C23C4×D412yes—
123C2×C4×C23#12011yescharacteristic
124C2×C4×C23#12112yes—
125C2×C4×C23#12311yescharacteristic
126C8×C23—44no—
127C8×C23—82no—
128#2973C2×C2311yescharacteristic
129#2563C2×C2311yescharacteristic
130#1556C2×C2312yescomplemented
131#1466C2×C2311yescharacteristic
132#1285C2×C2312yescomplemented
133C22×C23—21no—
134C22×C23—21no—
135C22×C23—22no—
136C22×C23—42no—
137C22×C23—42no—
138C22×C23—41no—
139C22×C23—41no—
140C22×C23—41no—
141C22×C23—41no—
142C22×C23—41no—
143C22×C23—81no—
144C22×C23—81no—
145C22×C23#38611yescharacteristic
146C22×C23C2×#10111yescharacteristic
147C22×C23#41311yescharacteristic
148C4×C23—21no—
149C4×C23—22no—
150C4×C23—22no—
151C4×C23—41no—
152C4×C23—41no—
153C4×C23—82no—
154C4×C23—82no—
155C4×C23#45512yes—
156C2×C4⋊Q8C22×C2311yescharacteristic
157C2×C4×D4C22×C2311yescharacteristic
158#435—24no—
159#432C22×C2312yes—
160#432C22×C2312yes—
161#407—21no—
162C2×C22⋊C8—21no—
163#393—21no—
164#393—21no—
165#390C4×C2312yes—
166#382C4×C2312yes—
167#380—21no—
168#380—21no—
169C4×C4⋊C4C22×C2311yescharacteristic
170#332—24no—
171#323—21no—
172#323—21no—
173C2×C23—21no—
174C2×C23—21no—
175C2×C23—42no—
176C2×C23—81no—
177C2×C23#188511yescharacteristic
178C2×C23#190111yescharacteristic
179C2×C23#191011yescharacteristic
180C22×Q8—21no—
181C22×D4C2×C4×C2311yescharacteristic
182C23×C4D4×C2311yescharacteristic
183C23×C4D4×C2311yescharacteristic
184C2×M16—22no—
185C22×C8—22no—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4×D4—22no—
189C4×D4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C2×C4×C2312yes—
193C2×C4⋊C4C2×C4×C2311yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4D4×C2311yescharacteristic
197C2×C22⋊C4D4×C2311yescharacteristic
198C2×C42C2×C4×C2312yes—
199C2×C42C23×C2311yescharacteristic
200C4⋊C8—22no—
201C4⋊C8D4×C2314yes—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206#97—22no—
207#97—22no—
208#95—22no—
209#95—22no—
210#95—21no—
211C23#945311yescharacteristic, direct product factor, Sylow
212C24C22⋊C4×C2311yescharacteristic
213C24C22⋊C4×C2311yescharacteristic
214C2×Q8—42no—
215C2×Q8—82no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—41no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4C22⋊C4×C2311yescharacteristic
228C22×C4C22⋊C4×C2311yescharacteristic
229C22×C4C22×C4×C2311yescharacteristic
230C22×C4C2×D4×C2311yescharacteristic
231C22×C4C2×D4×C2311yescharacteristic
232M16—82no—
233C2×C8—24no—
234C2×C8—42no—
235C2×C8—82no—
236C4⋊C4—22no—
237C4⋊C4—24no—
238C4⋊C4—42no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—82no—
244C22⋊C4—82no—
245C42—42no—
246C42C2×D4×C2312yes—
247C42#41×C2312yes—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23C23×#10112yes—
258C23C2×C22⋊C4×C2311yescharacteristic
259Q8—84no—
260D4—44no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4C4×D4×C2312yes—
280C2×C4C23×#12011yescharacteristic
281C2×C4C23×#12112yes—
282C2×C4C23×#12311yescharacteristic
283C8—44no—
284C8—82no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22C23×#38611yescharacteristic
298C22C2×C23×#10111yescharacteristic
299C22C23×#41311yescharacteristic
300C4—21no—
301C4—22no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—82no—
306C4—82no—
307C4C23×#45512yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—81no—
312C2C23×#188511yescharacteristic
313C2C23×#190111yescharacteristic
314C2C23×#191011yescharacteristic
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×#1556C2semidirect
5. C23×#1285C2semidirect
90. #9453C23direct (N × H)
130. #1556C2×C23semidirect
132. #1285C2×C23semidirect
211. C23#9453direct (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×#25632
C23×#15562
C23×#14662
C23×#12852
#945323