Finite Group Databasebeta

← C23×#12203 #19159829

Subgroups — C23×#12203 #19159829

Order 5888.

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

normal subgroups only

This group has 532 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×#2483C211yesmaximal, characteristic
2C23×#2029C211yesmaximal, characteristic
3C23×#1608C212yesmaximal, complemented
4C23×#1344C212yesmaximal, complemented
5C23×#1316C211yesmaximal, characteristic
6C23×#543C2211yescharacteristic
7C22×C22⋊C4×C23—21no—
8C23×#483—22no—
9C23×#397—21no—
10C23×#397C2212yes—
11C23×#385—22no—
12C23×#381—22no—
13C23×#379—21no—
14C23×#379C2211yescharacteristic
15C23×#342—21no—
16C23×#342C2211yescharacteristic
17C23×#328—22no—
18C23×#323C2212yes—
19C25×C23D411yescharacteristic
20C23×C4×C23—21no—
21C23×C4×C23—21no—
22C23×C22⋊Q8—22no—
23C23×C22⋊Q8—42no—
24C2×C4⋊C4×C23—21no—
25C2×C4⋊C4×C23D412yes—
26C2×C22⋊C4×C23—22no—
27C2×C22⋊C4×C23—21no—
28C2×C22⋊C4×C23—22no—
29C2×C22⋊C4×C23—22no—
30C2×C22⋊C4×C23—21no—
31C2×C22⋊C4×C23—42no—
32C2×C22⋊C4×C23—42no—
33C2×C22⋊C4×C23—42no—
34C2×C22⋊C4×C23D411yescharacteristic
35C2×C22⋊C4×C23C2311yescharacteristic
36C23×#106—42no—
37C23×#103—42no—
38C23×C22⋊C8—22no—
39C23×#95—21no—
40C23×#95—22no—
41C23×#95—22no—
42C23×#95D412yes—
43C24×C23—21no—
44C24×C23—21no—
45C24×C23—41no—
46C24×C23—44no—
47C24×C23—41no—
48C24×C23SD1612yes—
49C24×C23C2×D411yescharacteristic
50C2×Q8×C23—22no—
51C22×C4×C23—21no—
52C22×C4×C23—22no—
53C22×C4×C23—22no—
54C22×C4×C23—22no—
55C22×C4×C23—42no—
56C22×C4×C23—42no—
57C22×C4×C23—44no—
58C22×C4×C23—41no—
59C22×C4×C23—81no—
60C22×C4×C23—81no—
61C22×C4×C23C2×D412yes—
62C2×C8×C23—42no—
63C4⋊C4×C23—22no—
64C4⋊C4×C23—42no—
65C4⋊C4×C23—42no—
66C4⋊C4×C23—81no—
67C22⋊C4×C23—41no—
68C22⋊C4×C23—44no—
69C22⋊C4×C23—44no—
70C22⋊C4×C23—41no—
71C22⋊C4×C23—41no—
72C22⋊C4×C23—42no—
73C22⋊C4×C23—42no—
74C22⋊C4×C23—41no—
75C22⋊C4×C23—41no—
76C22⋊C4×C23—41no—
77C22⋊C4×C23—84no—
78#12203C2311yesmaximal, characteristic, direct product factor, Sylow
79C23×C23—24no—
80C23×C23—22no—
81C23×C23—21no—
82C23×C23—21no—
83C23×C23—42no—
84C23×C23—44no—
85C23×C23—42no—
86C23×C23—42no—
87C23×C23—44no—
88C23×C23—41no—
89C23×C23—41no—
90C23×C23—44no—
91C23×C23—81no—
92C23×C23—84no—
93C23×C23—82no—
94C23×C23#12011yescharacteristic
95C23×C23C2×SD1611yescharacteristic
96C23×C23AGL(1,ℤ/8)11yescharacteristic
97Q8×C23—82no—
98C2×C4×C23—22no—
99C2×C4×C23—42no—
100C2×C4×C23—42no—
101C2×C4×C23—44no—
102C2×C4×C23—42no—
103C2×C4×C23—42no—
104C2×C4×C23—41no—
105C2×C4×C23—42no—
106C2×C4×C23—42no—
107C2×C4×C23—42no—
108C2×C4×C23—84no—
109C2×C4×C23—84no—
110C2×C4×C23—82no—
111C2×C4×C23—81no—
112C2×C4×C23—82no—
113C8×C23—82no—
114#2483C2×C2311yescharacteristic
115#2029C2×C2311yescharacteristic
116#1608C2×C2312yescomplemented
117#1344C2×C2312yescomplemented
118#1316C2×C2311yescharacteristic
119C22×C23—22no—
120C22×C23—24no—
121C22×C23—21no—
122C22×C23—21no—
123C22×C23—41no—
124C22×C23—41no—
125C22×C23—42no—
126C22×C23—44no—
127C22×C23—44no—
128C22×C23—42no—
129C22×C23—42no—
130C22×C23—44no—
131C22×C23—81no—
132C22×C23—84no—
133C22×C23—82no—
134C22×C23#45011yescharacteristic
135C22×C23#45511yescharacteristic
136C22×C23B(4,𝔽2)11yescharacteristic
137C4×C23—42no—
138C4×C23—81no—
139C4×C23—82no—
140C4×C23—82no—
141C4×C23—82no—
142C4×C23—161no—
143#543C22×C2311yescharacteristic
144C22×C22⋊C4—21no—
145#483—22no—
146#397—21no—
147#397C22×C2312yes—
148#385—22no—
149#381—22no—
150#379—21no—
151#379C22×C2311yescharacteristic
152#342—21no—
153#342C22×C2311yescharacteristic
154#328—22no—
155#323C22×C2312yes—
156C2×C23—21no—
157C2×C23—21no—
158C2×C23—41no—
159C2×C23—44no—
160C2×C23—41no—
161C2×C23#160111yescharacteristic
162C2×C23#219712yes—
163C25D4×C2311yescharacteristic
164C23×C4—21no—
165C23×C4—21no—
166C22⋊Q8—22no—
167C22⋊Q8—42no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4D4×C2312yes—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4D4×C2311yescharacteristic
179C2×C22⋊C4C23×C2311yescharacteristic
180#106—42no—
181#103—42no—
182C22⋊C8—22no—
183#95—21no—
184#95—22no—
185#95—22no—
186#95D4×C2312yes—
187C23#1220311yescharacteristic, direct product factor, Sylow
188C24—21no—
189C24—21no—
190C24—41no—
191C24—44no—
192C24—41no—
193C24SD16×C2312yes—
194C24C2×D4×C2311yescharacteristic
195C2×Q8—22no—
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—44no—
203C22×C4—41no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4C2×D4×C2312yes—
207C2×C8—42no—
208C4⋊C4—22no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—81no—
212C22⋊C4—41no—
213C22⋊C4—44no—
214C22⋊C4—41no—
215C22⋊C4—44no—
216C22⋊C4—41no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—41no—
220C22⋊C4—41no—
221C22⋊C4—41no—
222C22⋊C4—84no—
223C23—24no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—42no—
228C23—44no—
229C23—42no—
230C23—42no—
231C23—44no—
232C23—41no—
233C23—41no—
234C23—44no—
235C23—81no—
236C23—84no—
237C23—82no—
238C23C23×#12011yescharacteristic
239C23C2×SD16×C2311yescharacteristic
240C23C23×AGL(1,ℤ/8)11yescharacteristic
241Q8—82no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—84no—
253C2×C4—84no—
254C2×C4—82no—
255C2×C4—81no—
256C2×C4—82no—
257C8—82no—
258C22—22no—
259C22—24no—
260C22—21no—
261C22—21no—
262C22—41no—
263C22—41no—
264C22—42no—
265C22—44no—
266C22—44no—
267C22—42no—
268C22—44no—
269C22—42no—
270C22—81no—
271C22—84no—
272C22—82no—
273C22C23×#45011yescharacteristic
274C22C23×#45511yescharacteristic
275C22C23×B(4,𝔽2)11yescharacteristic
276C4—42no—
277C4—82no—
278C4—81no—
279C4—82no—
280C4—82no—
281C4—161no—
282C2—21no—
283C2—21no—
284C2—41no—
285C2—44no—
286C2—41no—
287C2C23×#160111yescharacteristic
288C2C23×#219712yes—
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C23
prime radicalC25×C23D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C2×C22⋊C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1608C2semidirect
4. C23×#1344C2semidirect
78. #12203C23direct (N × H)
116. #1608C2×C23semidirect
117. #1344C2×C23semidirect
187. C23#12203direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#24832
C23×#20292
C23×#16082
C23×#13442
C23×#13162
#1220323