Finite Group Databasebeta

← C23×#12212 #19159838

Subgroups — C23×#12212 #19159838

Order 5888.

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

normal subgroups only

This group has 460 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×#2733C211yesmaximal, characteristic
2C23×#2028C211yesmaximal, characteristic, complemented
3C23×#1599C212yesmaximal, complemented
4C23×#1345C212yesmaximal, complemented
5C23×#1319C211yesmaximal, characteristic, complemented
6C23×#544C2211yescharacteristic
7C2×C23×#124—21no—
8C23×#450—22no—
9C23×#419—21no—
10C23×#394C2212yes—
11C23×#392—22no—
12C23×#391C2211yescharacteristic
13C23×#390—21no—
14C23×#388—22no—
15C23×#343C2211yescharacteristic
16C23×#339—21no—
17C23×#328—22no—
18C23×#324C2212yes—
19C22×Q8×C23D411yescharacteristic
20C22×D4×C23—21no—
21C22×D4×C23—21no—
22C22×D4×C23D412yes—
23C2×SD16×C23—42no—
24C2×M16×C23—21no—
25C23×C4⋊Q8—41no—
26C23×#125—41no—
27C23×#124—42no—
28C23×#124—42no—
29C23×C22⋊Q8—22no—
30C23×#117—21no—
31C2×C4⋊C4×C23—21no—
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×C23—21no—
38C2×C42×C23D411yescharacteristic
39C23×#103—41no—
40C23×#103—41no—
41C23×#102—42no—
42C23×C22⋊C8—22no—
43C23×#95—22no—
44C23×#95D412yes—
45C24×C23—22no—
46C2×Q8×C23—21no—
47C2×Q8×C23—22no—
48C2×Q8×C23—41no—
49C2×Q8×C23—41no—
50C2×Q8×C23SD1612yescomplemented
51C2×D4×C23—42no—
52C2×D4×C23—42no—
53C2×D4×C23—42no—
54C2×D4×C23—82no—
55C2×D4×C23—81no—
56C2×D4×C23—81no—
57C2×D4×C23—81no—
58C2×D4×C23—81no—
59C22×C4×C23—21no—
60C22×C4×C23—21no—
61C22×C4×C23—22no—
62C22×C4×C23C2×D411yescharacteristic
63C22×C4×C23C2×D412yes—
64SD16×C23—84no—
65M16×C23—41no—
66M16×C23—41no—
67C2×C8×C23—42no—
68C2×C8×C23—41no—
69C4⋊C4×C23—22no—
70C4⋊C4×C23—21no—
71C4⋊C4×C23—21no—
72C4⋊C4×C23—41no—
73C4⋊C4×C23—42no—
74C4⋊C4×C23—81no—
75C22⋊C4×C23—44no—
76C22⋊C4×C23—44no—
77C22⋊C4×C23—41no—
78C22⋊C4×C23—42no—
79C22⋊C4×C23—42no—
80C22⋊C4×C23—82no—
81C42×C23—41no—
82C42×C23—42no—
83C42×C23—41no—
84#12212C2311yesmaximal, characteristic, direct product factor, Sylow
85C23×C23—42no—
86C23×C23—42no—
87C23×C23—42no—
88C23×C23—42no—
89C23×C23—42no—
90C23×C23—42no—
91C23×C23—42no—
92C23×C23#12011yescharacteristic
93Q8×C23—42no—
94Q8×C23—81no—
95Q8×C23—82no—
96D4×C23—44no—
97D4×C23—82no—
98D4×C23—84no—
99D4×C23—84no—
100C2×C4×C23—21no—
101C2×C4×C23—21no—
102C2×C4×C23—22no—
103C2×C4×C23—22no—
104C2×C4×C23—22no—
105C2×C4×C23—41no—
106C2×C4×C23—41no—
107C2×C4×C23—41no—
108C2×C4×C23—41no—
109C2×C4×C23—42no—
110C2×C4×C23—42no—
111C2×C4×C23—42no—
112C2×C4×C23—42no—
113C2×C4×C23—41no—
114C2×C4×C23—41no—
115C2×C4×C23C2×SD1611yescharacteristic
116C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
117C8×C23—81no—
118C8×C23—82no—
119#2733C2×C2311yescharacteristic
120#2028C2×C2311yescharacteristic, complemented
121#1599C2×C2312yescomplemented
122#1345C2×C2312yescomplemented
123#1319C2×C2311yescharacteristic, complemented
124C22×C23—21no—
125C22×C23—21no—
126C22×C23—82no—
127C22×C23—82no—
128C22×C23—82no—
129C22×C23—82no—
130C22×C23—82no—
131C22×C23—82no—
132C22×C23—82no—
133C22×C23#45011yescharacteristic
134C22×C23#45511yescharacteristic
135C22×C23B(4,𝔽2)11yescharacteristic
136C4×C23—21no—
137C4×C23—21no—
138C4×C23—42no—
139C4×C23—42no—
140C4×C23—42no—
141C4×C23—82no—
142C4×C23—81no—
143C4×C23—81no—
144#544C22×C2311yescharacteristic
145C2×#124—21no—
146#450—22no—
147#419—21no—
148#394C22×C2312yes—
149#392—22no—
150#391C22×C2311yescharacteristic
151#390—21no—
152#388—22no—
153#343C22×C2311yescharacteristic
154#339—21no—
155#328—22no—
156#324C22×C2312yes—
157C2×C23—21no—
158C2×C23—21no—
159C2×C23—162no—
160C2×C23#160111yescharacteristic
161C2×C23#219911yescharacteristic
162C2×C23#220411yescharacteristic
163C22×Q8D4×C2311yescharacteristic
164C22×D4—21no—
165C22×D4—21no—
166C22×D4D4×C2312yes—
167C2×SD16—42no—
168C2×M16—21no—
169C4⋊Q8—41no—
170#125—41no—
171#124—42no—
172#124—42no—
173C22⋊Q8—22no—
174#117—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4C23×C2311yescharacteristic
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C42—21no—
182C2×C42D4×C2311yescharacteristic
183#103—41no—
184#103—41no—
185#102—42no—
186C22⋊C8—22no—
187#95—22no—
188#95D4×C2312yes—
189C23#1221211yescharacteristic, direct product factor, Sylow
190C24—22no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8SD16×C2312yescomplemented
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—82no—
200C2×D4—81no—
201C2×D4—81no—
202C2×D4—81no—
203C2×D4—81no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4C2×D4×C2311yescharacteristic
208C22×C4C2×D4×C2312yes—
209SD16—84no—
210M16—41no—
211M16—41no—
212C2×C8—42no—
213C2×C8—41no—
214C4⋊C4—22no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—41no—
218C4⋊C4—42no—
219C4⋊C4—81no—
220C22⋊C4—44no—
221C22⋊C4—44no—
222C22⋊C4—41no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—82no—
226C42—41no—
227C42—42no—
228C42—41no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23C23×#12011yescharacteristic
237Q8—42no—
238Q8—81no—
239Q8—82no—
240D4—44no—
241D4—82no—
242D4—84no—
243D4—84no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4C2×SD16×C2311yescharacteristic
260C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
261C8—81no—
262C8—82no—
263C22—21no—
264C22—21no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22—82no—
269C22—82no—
270C22—82no—
271C22—82no—
272C22C23×#45011yescharacteristic
273C22C23×#45511yescharacteristic
274C22C23×B(4,𝔽2)11yescharacteristic
275C4—21no—
276C4—21no—
277C4—42no—
278C4—42no—
279C4—42no—
280C4—82no—
281C4—81no—
282C4—81no—
283C2—21no—
284C2—21no—
285C2—162no—
286C2C23×#160111yescharacteristic
287C2C23×#219911yescharacteristic
288C2C23×#220411yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC23×#2733C2
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×#2028C2semidirect
3. C23×#1599C2semidirect
4. C23×#1345C2semidirect
5. C23×#1319C2semidirect
50. C2×Q8×C23SD16semidirect
84. #12212C23direct (N × H)
120. #2028C2×C23semidirect
121. #1599C2×C23semidirect
122. #1345C2×C23semidirect
123. #1319C2×C23semidirect
189. C23#12212direct (N × H)
195. C2×Q8SD16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#27332
C23×#20282
C23×#15992
C23×#13452
C23×#13192
#1221223