Finite Group Databasebeta

← C23×#12106 #19159732

Subgroups — C23×#12106 #19159732

Order 5888.

424 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 82 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×#3189C211yesmaximal, characteristic
2C23×#2188C212yesmaximal, complemented
3C23×#2001C211yesmaximal, characteristic
4C23×#1348C212yesmaximal, complemented
5C23×#1314C211yesmaximal, characteristic
6C22×Q16×C23C2211yescharacteristic
7C2×C23×C22⋊Q8—21no—
8C2×C23×C22⋊Q8—21no—
9C2×C23×Q32—22no—
10C23×#484—22no—
11C23×#467—21no—
12C23×#467C2212yes—
13C23×#451—22no—
14C2×C23×#103—21no—
15C2×C23×C22⋊C8C2211yescharacteristic
16C23×#358—22no—
17C23×C22⋊C16C2212yes—
18C23×#340C2211yescharacteristic
19C22×Q8×C23—21no—
20C22×Q8×C23—21no—
21C23×C4×C23D411yescharacteristic
22C2×Q16×C23—21no—
23C2×Q16×C23—22no—
24C2×Q16×C23—42no—
25C2×Q16×C23D412yes—
26C22×C8×C23D411yescharacteristic
27C22×C8×C23C2311yescharacteristic
28C23×C22⋊Q8—22no—
29C23×C22⋊Q8—22no—
30C23×C22⋊Q8—42no—
31C23×C22⋊Q8—41no—
32C23×C22⋊Q8—42no—
33C23×C22⋊Q8—41no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23—21no—
36C2×C4⋊C4×C23—41no—
37C2×C4⋊C4×C23—41no—
38C2×C22⋊C4×C23—41no—
39C2×C22⋊C4×C23—41no—
40C23×Q32—44no—
41C2×C16×C23—22no—
42C23×#107—21no—
43C23×#107D412yes—
44C23×#103—22no—
45C23×#103—22no—
46C23×#103—42no—
47C23×C22⋊C8—22no—
48C24×C23D811yescharacteristic
49C2×Q8×C23—22no—
50C2×Q8×C23—22no—
51C2×Q8×C23—41no—
52C2×Q8×C23—41no—
53C2×Q8×C23—41no—
54C2×Q8×C23—41no—
55C2×Q8×C23—41no—
56C2×Q8×C23—41no—
57C2×Q8×C23—81no—
58C2×Q8×C23—81no—
59C22×C4×C23—21no—
60C22×C4×C23—21no—
61C22×C4×C23—41no—
62C22×C4×C23—41no—
63C22×C4×C23—41no—
64C22×C4×C23—41no—
65C22×C4×C23—41no—
66C22×C4×C23—41no—
67C22×C4×C23D811yescharacteristic
68C22×C4×C23C2×D411yescharacteristic
69Q16×C23—24no—
70Q16×C23—42no—
71Q16×C23—42no—
72Q16×C23—81no—
73C2×C8×C23—21no—
74C2×C8×C23—41no—
75C2×C8×C23—41no—
76C2×C8×C23C2×D412yes—
77C4⋊C4×C23—22no—
78C4⋊C4×C23—22no—
79C4⋊C4×C23—41no—
80C4⋊C4×C23—41no—
81C4⋊C4×C23—42no—
82C4⋊C4×C23—44no—
83C4⋊C4×C23—42no—
84C22⋊C4×C23—42no—
85C22⋊C4×C23—44no—
86C22⋊C4×C23—42no—
87C16×C23—42no—
88#12106C2311yesmaximal, characteristic, direct product factor, Sylow
89C23×C23—21no—
90C23×C23—21no—
91C23×C23—41no—
92C23×C23—41no—
93C23×C23Q3212yescomplemented
94C23×C23C2×D811yescharacteristic
95Q8×C23—42no—
96Q8×C23—42no—
97Q8×C23—42no—
98Q8×C23—42no—
99Q8×C23—81no—
100Q8×C23—82no—
101Q8×C23—81no—
102C2×C4×C23—21no—
103C2×C4×C23—21no—
104C2×C4×C23—21no—
105C2×C4×C23—41no—
106C2×C4×C23—41no—
107C2×C4×C23—41no—
108C2×C4×C23—42no—
109C2×C4×C23—42no—
110C2×C4×C23—42no—
111C2×C4×C23—42no—
112C2×C4×C23—81no—
113C2×C4×C23—81no—
114C2×C4×C23—81no—
115C2×C4×C23—82no—
116C2×C4×C23—82no—
117C2×C4×C23—81no—
118C2×C4×C23—81no—
119C2×C4×C23#12011yescharacteristic
120C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
121C8×C23—22no—
122C8×C23—41no—
123#3189C2×C2311yescharacteristic
124#2188C2×C2312yescomplemented
125#2001C2×C2311yescharacteristic
126#1348C2×C2312yescomplemented
127#1314C2×C2311yescharacteristic
128C22×C23—21no—
129C22×C23—21no—
130C22×C23—22no—
131C22×C23—42no—
132C22×C23—41no—
133C22×C23—41no—
134C22×C23—81no—
135C22×C23#44711yescharacteristic
136C22×C23C2×Q3211yescharacteristic
137C22×C23#51011yescharacteristic
138C4×C23—21no—
139C4×C23—41no—
140C4×C23—82no—
141C4×C23—82no—
142C4×C23—82no—
143C4×C23—82no—
144C4×C23#45311yescharacteristic
145C4×C23B(4,𝔽2)11yescharacteristic
146C22×Q16C22×C2311yescharacteristic
147C2×C22⋊Q8—21no—
148C2×C22⋊Q8—21no—
149C2×Q32—22no—
150#484—22no—
151#467—21no—
152#467C22×C2312yes—
153#451—22no—
154C2×#103—21no—
155C2×C22⋊C8C22×C2311yescharacteristic
156#358—22no—
157C22⋊C16C22×C2312yes—
158#340C22×C2311yescharacteristic
159C2×C23—21no—
160C2×C23—21no—
161C2×C23—42no—
162C2×C23#159511yescharacteristic
163C2×C23#218811yescharacteristic
164C2×C23#219111yescharacteristic
165C22×Q8—21no—
166C22×Q8—21no—
167C23×C4D4×C2311yescharacteristic
168C2×Q16—21no—
169C2×Q16—22no—
170C2×Q16—42no—
171C2×Q16D4×C2312yes—
172C22×C8D4×C2311yescharacteristic
173C22×C8C23×C2311yescharacteristic
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C22⋊Q8—42no—
177C22⋊Q8—41no—
178C22⋊Q8—42no—
179C22⋊Q8—41no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186Q32—44no—
187C2×C16—22no—
188#107—21no—
189#107D4×C2312yes—
190#103—22no—
191#103—22no—
192#103—42no—
193C22⋊C8—22no—
194C23#1210611yescharacteristic, direct product factor, Sylow
195C24D8×C2311yescharacteristic
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—81no—
205C2×Q8—81no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4D8×C2311yescharacteristic
215C22×C4C2×D4×C2311yescharacteristic
216Q16—24no—
217Q16—42no—
218Q16—42no—
219Q16—81no—
220C2×C8—21no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8C2×D4×C2312yes—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C4⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—44no—
233C22⋊C4—42no—
234C16—42no—
235C23—21no—
236C23—21no—
237C23—41no—
238C23—41no—
239C23C23×Q3212yescomplemented
240C23C2×D8×C2311yescharacteristic
241Q8—42no—
242Q8—42no—
243Q8—42no—
244Q8—42no—
245Q8—81no—
246Q8—82no—
247Q8—81no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4C23×#12011yescharacteristic
266C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
267C8—22no—
268C8—41no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—81no—
276C22C23×#44711yescharacteristic
277C22C2×C23×Q3211yescharacteristic
278C22C23×#51011yescharacteristic
279C4—21no—
280C4—41no—
281C4—82no—
282C4—82no—
283C4—82no—
284C4—82no—
285C4C23×#45311yescharacteristic
286C4C23×B(4,𝔽2)11yescharacteristic
287C2—21no—
288C2—21no—
289C2—42no—
290C2C23×#159511yescharacteristic
291C2C23×#218811yescharacteristic
292C2C23×#219111yescharacteristic
subgroupG/H
center, socleC22×C23#447
commutator, Frattini, cosocleC22×C8C23×C23
prime radicalC24×C23D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C23×C22⋊C8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2188C2semidirect
4. C23×#1348C2semidirect
88. #12106C23direct (N × H)
93. C23×C23Q32semidirect
124. #2188C2×C23semidirect
126. #1348C2×C23semidirect
194. C23#12106direct (N × H)
239. C23C23×Q32semidirect

Maximal subgroups

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

subgroupindex
C23×#31892
C23×#21882
C23×#20012
C23×#13482
C23×#13142
#1210623