Finite Group Databasebeta

← C23×#12208 #19159834

Subgroups — C23×#12208 #19159834

Order 5888.

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

normal subgroups only

This group has 664 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×#2679C211yesmaximal, characteristic
2C23×#2027C211yesmaximal, characteristic, complemented
3C23×#1599C212yesmaximal, complemented
4C23×#1345C212yesmaximal, complemented
5C23×#1318C211yesmaximal, characteristic, complemented
6C23×#544C2211yescharacteristic, complemented
7C2×C23×#120—22no—
8C2×C23×#120—21no—
9C23×#450—22no—
10C23×#418—21no—
11C23×#394—22no—
12C23×#394C2212yes—
13C23×#390—21no—
14C23×#390C2211yescharacteristic
15C23×#343C2211yescharacteristic, complemented
16C23×#339—21no—
17C23×#328—22no—
18C23×#324C2212yescomplemented
19C25×C23—21no—
20C22×D4×C23—21no—
21C22×D4×C23—22no—
22C22×D4×C23D411yescharacteristic
23C22×D4×C23D412yes—
24C2×SD16×C23—42no—
25C2×M16×C23—21no—
26C23×C4⋊Q8—41no—
27C23×#125—41no—
28C23×C22⋊Q8—22no—
29C23×#120—48no—
30C23×#120—42no—
31C23×#120—42no—
32C23×#117—21no—
33C2×C4⋊C4×C23—21no—
34C2×C4⋊C4×C23C2311yescharacteristic
35C2×C22⋊C4×C23—22no—
36C2×C22⋊C4×C23—21no—
37C2×C22⋊C4×C23—22no—
38C2×C22⋊C4×C23—22no—
39C2×C22⋊C4×C23—22no—
40C2×C42×C23D411yescharacteristic
41C23×#102—41no—
42C23×#102—41no—
43C23×#102—42no—
44C23×C22⋊C8—22no—
45C23×#95D412yes—
46C24×C23—22no—
47C24×C23—21no—
48C24×C23—44no—
49C24×C23—42no—
50C24×C23—82no—
51C24×C23—81no—
52C24×C23—81no—
53C2×Q8×C23—22no—
54C2×D4×C23—24no—
55C2×D4×C23—21no—
56C2×D4×C23—41no—
57C2×D4×C23—42no—
58C2×D4×C23—41no—
59C2×D4×C23—44no—
60C2×D4×C23—44no—
61C2×D4×C23—44no—
62C2×D4×C23—81no—
63C2×D4×C23—81no—
64C2×D4×C23—82no—
65C2×D4×C23SD1612yescomplemented
66C22×C4×C23—21no—
67C22×C4×C23—22no—
68C22×C4×C23C2×D411yescharacteristic
69C22×C4×C23C2×D412yes—
70SD16×C23—84no—
71M16×C23—41no—
72M16×C23—41no—
73C2×C8×C23—41no—
74C2×C8×C23—42no—
75C4⋊C4×C23—21no—
76C4⋊C4×C23—21no—
77C4⋊C4×C23—22no—
78C4⋊C4×C23—41no—
79C4⋊C4×C23—42no—
80C4⋊C4×C23—81no—
81C22⋊C4×C23—44no—
82C22⋊C4×C23—44no—
83C22⋊C4×C23—44no—
84C22⋊C4×C23—42no—
85C22⋊C4×C23—41no—
86C22⋊C4×C23—42no—
87C22⋊C4×C23—42no—
88C42×C23—41no—
89C42×C23—41no—
90#12208C2311yesmaximal, characteristic, direct product factor, Sylow
91C23×C23—24no—
92C23×C23—22no—
93C23×C23—42no—
94C23×C23—42no—
95C23×C23—42no—
96C23×C23—42no—
97C23×C23—42no—
98C23×C23—42no—
99C23×C23—41no—
100C23×C23—42no—
101C23×C23—41no—
102C23×C23—42no—
103C23×C23—88no—
104C23×C23—84no—
105C23×C23—88no—
106C23×C23—84no—
107C23×C23—84no—
108C23×C23#12011yescharacteristic
109Q8×C23—82no—
110D4×C23—42no—
111D4×C23—44no—
112D4×C23—81no—
113D4×C23—84no—
114D4×C23—82no—
115D4×C23—88no—
116C2×C4×C23—21no—
117C2×C4×C23—21no—
118C2×C4×C23—22no—
119C2×C4×C23—22no—
120C2×C4×C23—41no—
121C2×C4×C23—41no—
122C2×C4×C23—42no—
123C2×C4×C23—42no—
124C2×C4×C23—42no—
125C2×C4×C23—42no—
126C2×C4×C23—41no—
127C2×C4×C23—41no—
128C2×C4×C23C2×SD1611yescharacteristic
129C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
130C8×C23—81no—
131C8×C23—82no—
132#2679C2×C2311yescharacteristic
133#2027C2×C2311yescharacteristic, complemented
134#1599C2×C2312yescomplemented
135#1345C2×C2312yescomplemented
136#1318C2×C2311yescharacteristic, complemented
137C22×C23—21no—
138C22×C23—21no—
139C22×C23—44no—
140C22×C23—44no—
141C22×C23—44no—
142C22×C23—42no—
143C22×C23—44no—
144C22×C23—82no—
145C22×C23—82no—
146C22×C23—82no—
147C22×C23—816no—
148C22×C23—82no—
149C22×C23—81no—
150C22×C23—81no—
151C22×C23—82no—
152C22×C23#45011yescharacteristic
153C22×C23#45511yescharacteristic
154C22×C23B(4,𝔽2)11yescharacteristic
155C4×C23—21no—
156C4×C23—21no—
157C4×C23—42no—
158C4×C23—42no—
159C4×C23—82no—
160C4×C23—81no—
161C4×C23—81no—
162#544C22×C2311yescharacteristic, complemented
163C2×#120—22no—
164C2×#120—21no—
165#450—22no—
166#418—21no—
167#394—22no—
168#394C22×C2312yes—
169#390—21no—
170#390C22×C2311yescharacteristic
171#343C22×C2311yescharacteristic, complemented
172#339—21no—
173#328—22no—
174#324C22×C2312yescomplemented
175C2×C23—21no—
176C2×C23—21no—
177C2×C23—82no—
178C2×C23—84no—
179C2×C23#160111yescharacteristic
180C2×C23#219611yescharacteristic
181C2×C23#220211yescharacteristic
182C25—21no—
183C22×D4—21no—
184C22×D4—22no—
185C22×D4D4×C2311yescharacteristic
186C22×D4D4×C2312yes—
187C2×SD16—42no—
188C2×M16—21no—
189C4⋊Q8—41no—
190#125—41no—
191C22⋊Q8—22no—
192#120—48no—
193#120—42no—
194#120—42no—
195#117—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C23×C2311yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C42D4×C2311yescharacteristic
204#102—41no—
205#102—41no—
206#102—42no—
207C22⋊C8—22no—
208#95D4×C2312yes—
209C23#1220811yescharacteristic, direct product factor, Sylow
210C24—22no—
211C24—21no—
212C24—44no—
213C24—42no—
214C24—82no—
215C24—81no—
216C24—81no—
217C2×Q8—22no—
218C2×D4—24no—
219C2×D4—21no—
220C2×D4—41no—
221C2×D4—42no—
222C2×D4—41no—
223C2×D4—44no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—81no—
227C2×D4—81no—
228C2×D4—82no—
229C2×D4SD16×C2312yescomplemented
230C22×C4—21no—
231C22×C4—22no—
232C22×C4C2×D4×C2311yescharacteristic
233C22×C4C2×D4×C2312yes—
234SD16—84no—
235M16—41no—
236M16—41no—
237C2×C8—42no—
238C2×C8—41no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—22no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—81no—
245C22⋊C4—44no—
246C22⋊C4—44no—
247C22⋊C4—44no—
248C22⋊C4—42no—
249C22⋊C4—41no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C42—41no—
253C42—41no—
254C23—24no—
255C23—22no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—41no—
263C23—42no—
264C23—42no—
265C23—41no—
266C23—88no—
267C23—84no—
268C23—88no—
269C23—84no—
270C23—84no—
271C23C23×#12011yescharacteristic
272Q8—82no—
273D4—42no—
274D4—44no—
275D4—81no—
276D4—84no—
277D4—82no—
278D4—88no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4C2×SD16×C2311yescharacteristic
292C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
293C8—81no—
294C8—82no—
295C22—21no—
296C22—21no—
297C22—44no—
298C22—44no—
299C22—44no—
300C22—44no—
301C22—42no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22—816no—
306C22—81no—
307C22—82no—
308C22—81no—
309C22—82no—
310C22C23×#45011yescharacteristic
311C22C23×#45511yescharacteristic
312C22C23×B(4,𝔽2)11yescharacteristic
313C4—21no—
314C4—21no—
315C4—42no—
316C4—42no—
317C4—82no—
318C4—81no—
319C4—81no—
320C2—21no—
321C2—21no—
322C2—84no—
323C2—82no—
324C2C23×#160111yescharacteristic
325C2C23×#219611yescharacteristic
326C2C23×#220211yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC23×#2679C2
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
2. C23×#2027C2semidirect
3. C23×#1599C2semidirect
4. C23×#1345C2semidirect
5. C23×#1318C2semidirect
6. C23×#544C22semidirect
15. C23×#343C22semidirect
18. C23×#324C22semidirect
65. C2×D4×C23SD16semidirect
90. #12208C23direct (N × H)
133. #2027C2×C23semidirect
134. #1599C2×C23semidirect
135. #1345C2×C23semidirect
136. #1318C2×C23semidirect
162. #544C22×C23semidirect
171. #343C22×C23semidirect
174. #324C22×C23semidirect
209. C23#12208direct (N × H)
229. C2×D4SD16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#26792
C23×#20272
C23×#15992
C23×#13452
C23×#13182
#1220823