Finite Group Databasebeta

← C23×#12156 #19159782

Subgroups — C23×#12156 #19159782

Order 5888.

484 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2819C211yesmaximal, characteristic, complemented
2C23×#2040C211yesmaximal, characteristic, complemented
3C23×#1598C212yesmaximal, complemented
4C23×#1345C212yesmaximal, complemented
5C23×#1324C211yesmaximal, characteristic, complemented
6C23×#537C2211yescharacteristic
7C2×C23×#124—21no—
8C23×#447—22no—
9C2×C23×#104—21no—
10C23×#417—21no—
11C23×#394—21no—
12C23×#394C2212yes—
13C23×#393—22no—
14C23×#392—22no—
15C23×#382C2211yescharacteristic
16C23×#343C2211yescharacteristic
17C23×#327—22no—
18C23×#324C2212yes—
19C2×#41×C23—21no—
20C22×Q8×C23—21no—
21C22×D4×C23—21no—
22C22×D4×C23—21no—
23C22×D4×C23D412yes—
24C2×D8×C23—42no—
25C2×M16×C23—21no—
26C23×#124—42no—
27C23×#124—42no—
28C23×C22⋊Q8—41no—
29C23×#121—22no—
30C23×#121—41no—
31C2×C4⋊C4×C23D411yescharacteristic
32C2×C4⋊C4×C23C2311yescharacteristic
33C2×C22⋊C4×C23—22no—
34C2×C22⋊C4×C23—22no—
35C2×C22⋊C4×C23—22no—
36C2×C22⋊C4×C23—22no—
37C2×C42×C23D411yescharacteristic
38C23×#104—42no—
39C23×#103—41no—
40C23×#102—42no—
41C23×#102—41no—
42C23×C22⋊C8—22no—
43C23×#95—21no—
44C23×#95—22no—
45C23×#95D412yes—
46C24×C23—22no—
47#41×C23—41no—
48#41×C23—41no—
49#41×C23—41no—
50#41×C23—41no—
51C2×Q8×C23—21no—
52C2×Q8×C23—42no—
53C2×Q8×C23—81no—
54C2×Q8×C23—81no—
55C2×D4×C23—21no—
56C2×D4×C23—22no—
57C2×D4×C23—42no—
58C2×D4×C23—42no—
59C2×D4×C23—41no—
60C2×D4×C23—82no—
61C2×D4×C23—81no—
62C2×D4×C23—81no—
63C2×D4×C23—82no—
64C2×D4×C23—81no—
65C22×C4×C23—21no—
66C22×C4×C23—22no—
67C22×C4×C23—21no—
68C22×C4×C23—41no—
69C22×C4×C23C2×D411yescharacteristic
70C22×C4×C23C2×D412yes—
71D8×C23—84no—
72M16×C23—41no—
73M16×C23—41no—
74C2×C8×C23—41no—
75C2×C8×C23—42no—
76C4⋊C4×C23—22no—
77C4⋊C4×C23—41no—
78C4⋊C4×C23—41no—
79C22⋊C4×C23—44no—
80C22⋊C4×C23—42no—
81C22⋊C4×C23—42no—
82C22⋊C4×C23—42no—
83C22⋊C4×C23—82no—
84C22⋊C4×C23—82no—
85C42×C23—21no—
86C42×C23D812yescomplemented
87#12156C2311yesmaximal, characteristic, direct product factor, Sylow
88C23×C23—42no—
89C23×C23—42no—
90C23×C23—42no—
91C23×C23—42no—
92C23×C23—42no—
93C23×C23—42no—
94C23×C23—41no—
95C23×C23—42no—
96C23×C23#12011yescharacteristic
97Q8×C23—41no—
98Q8×C23—41no—
99Q8×C23—84no—
100D4×C23—41no—
101D4×C23—41no—
102D4×C23—44no—
103D4×C23—82no—
104D4×C23—82no—
105D4×C23—84no—
106D4×C23—84no—
107D4×C23—81no—
108D4×C23—81no—
109C2×C4×C23—21no—
110C2×C4×C23—21no—
111C2×C4×C23—22no—
112C2×C4×C23—22no—
113C2×C4×C23—41no—
114C2×C4×C23—41no—
115C2×C4×C23—42no—
116C2×C4×C23—42no—
117C2×C4×C23—42no—
118C2×C4×C23—42no—
119C2×C4×C23—42no—
120C2×C4×C23—41no—
121C2×C4×C23—41no—
122C2×C4×C23—81no—
123C2×C4×C23—81no—
124C2×C4×C23C2×D811yescharacteristic
125C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
126C8×C23—81no—
127C8×C23—82no—
128#2819C2×C2311yescharacteristic, complemented
129#2040C2×C2311yescharacteristic, complemented
130#1598C2×C2312yescomplemented
131#1345C2×C2312yescomplemented
132#1324C2×C2311yescharacteristic, complemented
133C22×C23—21no—
134C22×C23—21no—
135C22×C23—82no—
136C22×C23—82no—
137C22×C23—82no—
138C22×C23—82no—
139C22×C23—81no—
140C22×C23—82no—
141C22×C23—82no—
142C22×C23—81no—
143C22×C23—82no—
144C22×C23—81no—
145C22×C23#44711yescharacteristic
146C22×C23#45411yescharacteristic
147C22×C23B(4,𝔽2)11yescharacteristic
148C4×C23—21no—
149C4×C23—21no—
150C4×C23—42no—
151C4×C23—42no—
152C4×C23—82no—
153C4×C23—81no—
154C4×C23—81no—
155#537C22×C2311yescharacteristic
156C2×#124—21no—
157#447—22no—
158C2×#104—21no—
159#417—21no—
160#394—21no—
161#394C22×C2312yes—
162#393—22no—
163#392—22no—
164#382C22×C2311yescharacteristic
165#343C22×C2311yescharacteristic
166#327—22no—
167#324C22×C2312yes—
168C2×C23—21no—
169C2×C23—21no—
170C2×C23—161no—
171C2×C23—162no—
172C2×C23#159811yescharacteristic
173C2×C23#219911yescharacteristic
174C2×C23#220211yescharacteristic
175C2×#41—21no—
176C22×Q8—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4D4×C2312yes—
180C2×D8—42no—
181C2×M16—21no—
182#124—42no—
183#124—42no—
184C22⋊Q8—41no—
185#121—22no—
186#121—41no—
187C2×C4⋊C4D4×C2311yescharacteristic
188C2×C4⋊C4C23×C2311yescharacteristic
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C42D4×C2311yescharacteristic
194#104—42no—
195#103—41no—
196#102—42no—
197#102—41no—
198C22⋊C8—22no—
199#95—21no—
200#95—22no—
201#95D4×C2312yes—
202C23#1215611yescharacteristic, direct product factor, Sylow
203C24—22no—
204#41—41no—
205#41—41no—
206#41—41no—
207#41—41no—
208C2×Q8—21no—
209C2×Q8—42no—
210C2×Q8—81no—
211C2×Q8—81no—
212C2×D4—21no—
213C2×D4—22no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—41no—
217C2×D4—82no—
218C2×D4—81no—
219C2×D4—81no—
220C2×D4—82no—
221C2×D4—81no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—41no—
226C22×C4C2×D4×C2311yescharacteristic
227C22×C4C2×D4×C2312yes—
228D8—84no—
229M16—41no—
230M16—41no—
231C2×C8—42no—
232C2×C8—41no—
233C4⋊C4—22no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C22⋊C4—44no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—82no—
241C22⋊C4—82no—
242C42—21no—
243C42D8×C2312yescomplemented
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—41no—
251C23—42no—
252C23C23×#12011yescharacteristic
253Q8—41no—
254Q8—41no—
255Q8—84no—
256D4—41no—
257D4—41no—
258D4—44no—
259D4—82no—
260D4—82no—
261D4—84no—
262D4—84no—
263D4—81no—
264D4—81no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4C2×D8×C2311yescharacteristic
281C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
282C8—81no—
283C8—82no—
284C22—21no—
285C22—21no—
286C22—82no—
287C22—82no—
288C22—82no—
289C22—82no—
290C22—82no—
291C22—81no—
292C22—82no—
293C22—81no—
294C22—81no—
295C22—82no—
296C22C23×#44711yescharacteristic
297C22C23×#45411yescharacteristic
298C22C23×B(4,𝔽2)11yescharacteristic
299C4—21no—
300C4—21no—
301C4—42no—
302C4—42no—
303C4—82no—
304C4—81no—
305C4—81no—
306C2—21no—
307C2—21no—
308C2—162no—
309C2—161no—
310C2C23×#159811yescharacteristic
311C2C23×#219911yescharacteristic
312C2C23×#220211yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C23
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C2×C4⋊C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2819C2semidirect
2. C23×#2040C2semidirect
3. C23×#1598C2semidirect
4. C23×#1345C2semidirect
5. C23×#1324C2semidirect
86. C42×C23D8semidirect
87. #12156C23direct (N × H)
128. #2819C2×C23semidirect
129. #2040C2×C23semidirect
130. #1598C2×C23semidirect
131. #1345C2×C23semidirect
132. #1324C2×C23semidirect
202. C23#12156direct (N × H)
243. C42D8×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#28192
C23×#20402
C23×#15982
C23×#13452
C23×#13242
#1215623