Finite Group Databasebeta

← C23×#9455 #19157081

Subgroups — C23×#9455 #19157081

Order 5888.

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

normal subgroups only

This group has 554 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×#2542C211yesmaximal, characteristic
3C23×#1556C212yesmaximal, complemented
4C23×#1466C211yesmaximal, characteristic
5C23×#1285C212yesmaximal, complemented
6C2×C23×C4⋊Q8C2211yescharacteristic
7C2×C23×C22⋊Q8—21no—
8C2×C23×C22⋊Q8—21no—
9C2×C4×D4×C23C2211yescharacteristic
10C23×#435—24no—
11C23×#432C2212yes—
12C23×#432C2212yes—
13C23×#407—21no—
14C2×C23×C22⋊C8—21no—
15C23×#399—21no—
16C23×#399—21no—
17C23×#390C412yes—
18C23×#384C412yes—
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—
31C23×C22⋊Q8—44no—
32C23×C22⋊Q8—44no—
33C4×D4×C23—22no—
34C4×D4×C23—22no—
35C2×C4⋊C4×C23—21no—
36C2×C4⋊C4×C23—22no—
37C2×C4⋊C4×C23—22no—
38C2×C4⋊C4×C23—21no—
39C2×C4⋊C4×C23C2×C412yes—
40C2×C4⋊C4×C23C2×C411yescharacteristic
41C2×C22⋊C4×C23—22no—
42C2×C22⋊C4×C23—22no—
43C2×C22⋊C4×C23D411yescharacteristic
44C2×C22⋊C4×C23D411yescharacteristic
45C2×C42×C23C2×C412yes—
46C2×C42×C23C2311yescharacteristic
47C23×C4⋊C8—22no—
48C23×C4⋊C8D414yes—
49C23×C22⋊C8—22no—
50C23×C22⋊C8—22no—
51C23×C22⋊C8—22no—
52C23×C22⋊C8—22no—
53C23×#97—22no—
54C23×#97—22no—
55C23×#95—21no—
56C24×C23C22⋊C411yescharacteristic
57C24×C23C22⋊C411yescharacteristic
58C2×Q8×C23—42no—
59C2×Q8×C23—42no—
60C2×Q8×C23—41no—
61C2×Q8×C23—41no—
62C2×D4×C23—22no—
63C2×D4×C23—22no—
64C2×D4×C23—41no—
65C22×C4×C23—22no—
66C22×C4×C23—21no—
67C22×C4×C23—21no—
68C22×C4×C23—21no—
69C22×C4×C23—21no—
70C22×C4×C23—22no—
71C22×C4×C23—42no—
72C22×C4×C23—42no—
73C22×C4×C23C22⋊C411yescharacteristic
74C22×C4×C23C22⋊C411yescharacteristic
75C22×C4×C23C22×C411yescharacteristic
76C22×C4×C23C2×D411yescharacteristic
77C22×C4×C23C2×D411yescharacteristic
78M16×C23—82no—
79C2×C8×C23—24no—
80C2×C8×C23—42no—
81C2×C8×C23—82no—
82C4⋊C4×C23—24no—
83C4⋊C4×C23—22no—
84C4⋊C4×C23—24no—
85C4⋊C4×C23—44no—
86C4⋊C4×C23—44no—
87C4⋊C4×C23—42no—
88C22⋊C4×C23—22no—
89C22⋊C4×C23—22no—
90C22⋊C4×C23—44no—
91C22⋊C4×C23—44no—
92C42×C23—42no—
93C42×C23C2×D412yes—
94C42×C23#4112yes—
95#9455C2311yesmaximal, characteristic, direct product factor, Sylow
96C23×C23—21no—
97C23×C23—21no—
98C23×C23—21no—
99C23×C23—21no—
100C23×C23—21no—
101C23×C23—41no—
102C23×C23—41no—
103C23×C23—41no—
104C23×C23—41no—
105C23×C23#10112yes—
106C23×C23C2×C22⋊C411yescharacteristic
107Q8×C23—82no—
108Q8×C23—82no—
109D4×C23—44no—
110C2×C4×C23—21no—
111C2×C4×C23—21no—
112C2×C4×C23—22no—
113C2×C4×C23—24no—
114C2×C4×C23—21no—
115C2×C4×C23—21no—
116C2×C4×C23—41no—
117C2×C4×C23—42no—
118C2×C4×C23—42no—
119C2×C4×C23—42no—
120C2×C4×C23—42no—
121C2×C4×C23—41no—
122C2×C4×C23—41no—
123C2×C4×C23—42no—
124C2×C4×C23—42no—
125C2×C4×C23—42no—
126C2×C4×C23—81no—
127C2×C4×C23—81no—
128C2×C4×C23C4×D412yes—
129C2×C4×C23#12011yescharacteristic
130C2×C4×C23#12112yes—
131C2×C4×C23#12311yescharacteristic
132C8×C23—44no—
133C8×C23—82no—
134#2973C2×C2311yescharacteristic
135#2542C2×C2311yescharacteristic
136#1556C2×C2312yescomplemented
137#1466C2×C2311yescharacteristic
138#1285C2×C2312yescomplemented
139C22×C23—21no—
140C22×C23—21no—
141C22×C23—22no—
142C22×C23—42no—
143C22×C23—42no—
144C22×C23—41no—
145C22×C23—41no—
146C22×C23—41no—
147C22×C23—41no—
148C22×C23—41no—
149C22×C23—81no—
150C22×C23—81no—
151C22×C23#38611yescharacteristic
152C22×C23C2×#10111yescharacteristic
153C22×C23#41311yescharacteristic
154C4×C23—21no—
155C4×C23—22no—
156C4×C23—22no—
157C4×C23—44no—
158C4×C23—41no—
159C4×C23—41no—
160C4×C23—82no—
161C4×C23#45312yes—
162C2×C4⋊Q8C22×C2311yescharacteristic
163C2×C22⋊Q8—21no—
164C2×C22⋊Q8—21no—
165C2×C4×D4C22×C2311yescharacteristic
166#435—24no—
167#432C22×C2312yes—
168#432C22×C2312yes—
169#407—21no—
170C2×C22⋊C8—21no—
171#399—21no—
172#399—21no—
173#390C4×C2312yes—
174#384C4×C2312yes—
175C4×C4⋊C4C22×C2311yescharacteristic
176#332—24no—
177#323—21no—
178#323—21no—
179C2×C23—21no—
180C2×C23—21no—
181C2×C23—42no—
182C2×C23—81no—
183C2×C23#188711yescharacteristic
184C2×C23#190111yescharacteristic
185C2×C23#190911yescharacteristic
186C22×Q8—21no—
187C22×D4C2×C4×C2311yescharacteristic
188C23×C4D4×C2311yescharacteristic
189C23×C4D4×C2311yescharacteristic
190C2×M16—22no—
191C22×C8—22no—
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C22⋊Q8—44no—
195C22⋊Q8—44no—
196C4×D4—22no—
197C4×D4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4C2×C4×C2312yes—
203C2×C4⋊C4C2×C4×C2311yescharacteristic
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4D4×C2311yescharacteristic
207C2×C22⋊C4D4×C2311yescharacteristic
208C2×C42C2×C4×C2312yes—
209C2×C42C23×C2311yescharacteristic
210C4⋊C8—22no—
211C4⋊C8D4×C2314yes—
212C22⋊C8—22no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216#97—22no—
217#97—22no—
218#95—21no—
219C23#945511yescharacteristic, direct product factor, Sylow
220C24C22⋊C4×C2311yescharacteristic
221C24C22⋊C4×C2311yescharacteristic
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4C22⋊C4×C2311yescharacteristic
238C22×C4C22⋊C4×C2311yescharacteristic
239C22×C4C22×C4×C2311yescharacteristic
240C22×C4C2×D4×C2311yescharacteristic
241C22×C4C2×D4×C2311yescharacteristic
242M16—82no—
243C2×C8—24no—
244C2×C8—42no—
245C2×C8—82no—
246C4⋊C4—24no—
247C4⋊C4—22no—
248C4⋊C4—24no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—42no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C42—42no—
257C42C2×D4×C2312yes—
258C42#41×C2312yes—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23C23×#10112yes—
269C23C2×C22⋊C4×C2311yescharacteristic
270Q8—82no—
271Q8—82no—
272D4—44no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4C4×D4×C2312yes—
292C2×C4C23×#12011yescharacteristic
293C2×C4C23×#12112yes—
294C2×C4C23×#12311yescharacteristic
295C8—44no—
296C8—82no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22—81no—
309C22C23×#38611yescharacteristic
310C22C2×C23×#10111yescharacteristic
311C22C23×#41311yescharacteristic
312C4—21no—
313C4—22no—
314C4—22no—
315C4—44no—
316C4—41no—
317C4—41no—
318C4—82no—
319C4C23×#45312yes—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—81no—
324C2C23×#188711yescharacteristic
325C2C23×#190111yescharacteristic
326C2C23×#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×#1556C2semidirect
5. C23×#1285C2semidirect
95. #9455C23direct (N × H)
136. #1556C2×C23semidirect
138. #1285C2×C23semidirect
219. C23#9455direct (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×#25422
C23×#15562
C23×#14662
C23×#12852
#945523