Finite Group Databasebeta

← C23×#12152 #19159778

Subgroups — C23×#12152 #19159778

Order 5888.

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

normal subgroups only

This group has 588 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×#2537C211yesmaximal, characteristic, complemented
2C23×#2039C211yesmaximal, characteristic, complemented
3C23×#1598C212yesmaximal, complemented
4C23×#1345C212yesmaximal, complemented
5C23×#1325C211yesmaximal, characteristic, complemented
6C23×#537C2211yescharacteristic
7C2×C23×#121—22no—
8C2×C4×D4×C23—21no—
9C23×#447—22no—
10C2×C23×#104—21no—
11C23×#417—21no—
12C23×#394—21no—
13C23×#394C2212yes—
14C23×#384C2211yescharacteristic
15C23×#380—22no—
16C23×#343C2211yescharacteristic
17C23×#327—22no—
18C23×#324C2212yes—
19C2×#41×C23—21no—
20C22×D4×C23—21no—
21C22×D4×C23—21no—
22C22×D4×C23D412yes—
23C23×C4×C23—22no—
24C2×D8×C23—42no—
25C2×M16×C23—21no—
26C23×C22⋊Q8—41no—
27C23×#121—22no—
28C23×#121—41no—
29C23×#121—48no—
30C4×D4×C23—42no—
31C4×D4×C23—42no—
32C2×C4⋊C4×C23—21no—
33C2×C4⋊C4×C23—22no—
34C2×C4⋊C4×C23D411yescharacteristic
35C2×C4⋊C4×C23C2311yescharacteristic
36C2×C22⋊C4×C23—22no—
37C2×C22⋊C4×C23—22no—
38C2×C22⋊C4×C23—22no—
39C2×C42×C23D411yescharacteristic
40C23×#104—42no—
41C23×#103—41no—
42C23×#102—42no—
43C23×#102—41no—
44C23×C22⋊C8—22no—
45C23×#95—21no—
46C23×#95D412yes—
47C24×C23—22no—
48#41×C23—41no—
49#41×C23—41no—
50#41×C23—41no—
51#41×C23—41no—
52C2×Q8×C23—21no—
53C2×D4×C23—24no—
54C2×D4×C23—21no—
55C2×D4×C23—22no—
56C2×D4×C23—42no—
57C2×D4×C23—42no—
58C2×D4×C23—42no—
59C2×D4×C23—44no—
60C2×D4×C23—44no—
61C2×D4×C23—41no—
62C2×D4×C23—81no—
63C22×C4×C23—21no—
64C22×C4×C23—22no—
65C22×C4×C23—21no—
66C22×C4×C23—44no—
67C22×C4×C23—44no—
68C22×C4×C23—41no—
69C22×C4×C23—44no—
70C22×C4×C23C2×D411yescharacteristic
71C22×C4×C23C2×D412yes—
72D8×C23—84no—
73M16×C23—41no—
74M16×C23—41no—
75C2×C8×C23—41no—
76C2×C8×C23—42no—
77C4⋊C4×C23—22no—
78C4⋊C4×C23—41no—
79C4⋊C4×C23—41no—
80C4⋊C4×C23—42no—
81C4⋊C4×C23—44no—
82C22⋊C4×C23—42no—
83C22⋊C4×C23—44no—
84C22⋊C4×C23—44no—
85C22⋊C4×C23—42no—
86C22⋊C4×C23—42no—
87C42×C23—21no—
88C42×C23D812yescomplemented
89#12152C2311yesmaximal, characteristic, direct product factor, Sylow
90C23×C23—24no—
91C23×C23—42no—
92C23×C23—42no—
93C23×C23—42no—
94C23×C23—42no—
95C23×C23—42no—
96C23×C23—42no—
97C23×C23—41no—
98C23×C23#12011yescharacteristic
99Q8×C23—41no—
100Q8×C23—41no—
101D4×C23—48no—
102D4×C23—41no—
103D4×C23—41no—
104D4×C23—44no—
105D4×C23—82no—
106D4×C23—82no—
107D4×C23—84no—
108D4×C23—81no—
109D4×C23—81no—
110C2×C4×C23—22no—
111C2×C4×C23—22no—
112C2×C4×C23—22no—
113C2×C4×C23—21no—
114C2×C4×C23—21no—
115C2×C4×C23—22no—
116C2×C4×C23—42no—
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—88no—
124C2×C4×C23—81no—
125C2×C4×C23—81no—
126C2×C4×C23C2×D811yescharacteristic
127C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
128C8×C23—81no—
129C8×C23—82no—
130#2537C2×C2311yescharacteristic, complemented
131#2039C2×C2311yescharacteristic, complemented
132#1598C2×C2312yescomplemented
133#1345C2×C2312yescomplemented
134#1325C2×C2311yescharacteristic, complemented
135C22×C23—21no—
136C22×C23—21no—
137C22×C23—44no—
138C22×C23—44no—
139C22×C23—44no—
140C22×C23—82no—
141C22×C23—82no—
142C22×C23—82no—
143C22×C23—82no—
144C22×C23—81no—
145C22×C23—81no—
146C22×C23—81no—
147C22×C23#44711yescharacteristic
148C22×C23#45411yescharacteristic
149C22×C23B(4,𝔽2)11yescharacteristic
150C4×C23—24no—
151C4×C23—21no—
152C4×C23—21no—
153C4×C23—42no—
154C4×C23—82no—
155C4×C23—81no—
156C4×C23—81no—
157#537C22×C2311yescharacteristic
158C2×#121—22no—
159C2×C4×D4—21no—
160#447—22no—
161C2×#104—21no—
162#417—21no—
163#394—21no—
164#394C22×C2312yes—
165#384C22×C2311yescharacteristic
166#380—22no—
167#343C22×C2311yescharacteristic
168#327—22no—
169#324C22×C2312yes—
170C2×C23—21no—
171C2×C23—21no—
172C2×C23—84no—
173C2×C23—161no—
174C2×C23#159811yescharacteristic
175C2×C23#219611yescharacteristic
176C2×C23#220411yescharacteristic
177C2×#41—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4D4×C2312yes—
181C23×C4—22no—
182C2×D8—42no—
183C2×M16—21no—
184C22⋊Q8—41no—
185#121—22no—
186#121—41no—
187#121—48no—
188C4×D4—42no—
189C4×D4—42no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4D4×C2311yescharacteristic
193C2×C4⋊C4C23×C2311yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C42D4×C2311yescharacteristic
198#104—42no—
199#103—41no—
200#102—42no—
201#102—41no—
202C22⋊C8—22no—
203#95—21no—
204#95D4×C2312yes—
205C23#1215211yescharacteristic, direct product factor, Sylow
206C24—22no—
207#41—41no—
208#41—41no—
209#41—41no—
210#41—41no—
211C2×Q8—21no—
212C2×D4—24no—
213C2×D4—21no—
214C2×D4—22no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—44no—
219C2×D4—44no—
220C2×D4—41no—
221C2×D4—81no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—41no—
228C22×C4—44no—
229C22×C4C2×D4×C2311yescharacteristic
230C22×C4C2×D4×C2312yes—
231D8—84no—
232M16—41no—
233M16—41no—
234C2×C8—42no—
235C2×C8—41no—
236C4⋊C4—22no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—44no—
241C22⋊C4—42no—
242C22⋊C4—44no—
243C22⋊C4—44no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C42—21no—
247C42D8×C2312yescomplemented
248C23—24no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—41no—
256C23C23×#12011yescharacteristic
257Q8—41no—
258Q8—41no—
259D4—48no—
260D4—41no—
261D4—41no—
262D4—44no—
263D4—82no—
264D4—82no—
265D4—84no—
266D4—81no—
267D4—81no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—88no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4C2×D8×C2311yescharacteristic
285C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
286C8—81no—
287C8—82no—
288C22—21no—
289C22—21no—
290C22—44no—
291C22—44no—
292C22—44no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22C23×#44711yescharacteristic
301C22C23×#45411yescharacteristic
302C22C23×B(4,𝔽2)11yescharacteristic
303C4—24no—
304C4—21no—
305C4—21no—
306C4—42no—
307C4—82no—
308C4—81no—
309C4—81no—
310C2—21no—
311C2—21no—
312C2—84no—
313C2—161no—
314C2C23×#159811yescharacteristic
315C2C23×#219611yescharacteristic
316C2C23×#220411yescharacteristic
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×#2537C2semidirect
2. C23×#2039C2semidirect
3. C23×#1598C2semidirect
4. C23×#1345C2semidirect
5. C23×#1325C2semidirect
88. C42×C23D8semidirect
89. #12152C23direct (N × H)
130. #2537C2×C23semidirect
131. #2039C2×C23semidirect
132. #1598C2×C23semidirect
133. #1345C2×C23semidirect
134. #1325C2×C23semidirect
205. C23#12152direct (N × H)
247. C42D8×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#25372
C23×#20392
C23×#15982
C23×#13452
C23×#13252
#1215223