Finite Group Databasebeta

← C23×#7426 #19155052

Subgroups — C23×#7426 #19155052

Order 5888.

332 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 42 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×#2029C211yesmaximal, characteristic
2C23×#1316C211yesmaximal, characteristic
3C23×#1304C211yesmaximal, characteristic
4C23×#543C411yescharacteristic, complemented
5C23×#379—21no—
6C23×#379—21no—
7C23×#379C2211yescharacteristic
8C23×#342—21no—
9C23×#342C411yescharacteristic, complemented
10C23×#323—21no—
11C25×C23D411yescharacteristic
12C23×C22⋊Q8—42no—
13C23×C22⋊Q8—41no—
14C2×C22⋊C4×C23—21no—
15C2×C22⋊C4×C23—21no—
16C2×C22⋊C4×C23—22no—
17C2×C22⋊C4×C23—41no—
18C2×C22⋊C4×C23—42no—
19C2×C22⋊C4×C23—41no—
20C2×C22⋊C4×C23—41no—
21C2×C22⋊C4×C23C2×C411yescharacteristic
22C2×C22⋊C4×C23D411yescharacteristic
23C23×C22⋊C8—41no—
24C24×C23—21no—
25C24×C23—21no—
26C24×C23—41no—
27C24×C23—41no—
28C24×C23—82no—
29C24×C23C22⋊C411yescharacteristic
30C24×C23SD1611yescharacteristic
31C24×C23Q1611yescharacteristic
32C2×Q8×C23—41no—
33C22×C4×C23—22no—
34C22×C4×C23—21no—
35C22×C4×C23—21no—
36C22×C4×C23—41no—
37C22×C4×C23—42no—
38C2×C8×C23—81no—
39C4⋊C4×C23—42no—
40C4⋊C4×C23—41no—
41C22⋊C4×C23—41no—
42C22⋊C4×C23—41no—
43C22⋊C4×C23—41no—
44C22⋊C4×C23—41no—
45C22⋊C4×C23—41no—
46C22⋊C4×C23—41no—
47C22⋊C4×C23—82no—
48C22⋊C4×C23—84no—
49C22⋊C4×C23—82no—
50C22⋊C4×C23—81no—
51C22⋊C4×C23—81no—
52C22⋊C4×C23—82no—
53#7426C2311yesmaximal, characteristic, direct product factor, Sylow
54C23×C23—21no—
55C23×C23—21no—
56C23×C23—21no—
57C23×C23—21no—
58C23×C23—42no—
59C23×C23—41no—
60C23×C23—41no—
61C23×C23—42no—
62C23×C23—42no—
63C23×C23—44no—
64C23×C23—41no—
65C23×C23—41no—
66C23×C23—81no—
67C23×C23—81no—
68C23×C23—81no—
69C23×C23—81no—
70C23×C23—81no—
71C23×C23—82no—
72C23×C23—82no—
73C23×C23—82no—
74C23×C23#9911yescharacteristic
75C23×C23#10311yescharacteristic
76C23×C23#10411yescharacteristic
77Q8×C23—161no—
78C2×C4×C23—42no—
79C2×C4×C23—41no—
80C2×C4×C23—41no—
81C2×C4×C23—81no—
82C2×C4×C23—81no—
83C2×C4×C23—82no—
84C2×C4×C23—81no—
85C2×C4×C23—82no—
86C2×C4×C23—82no—
87C2×C4×C23—82no—
88C2×C4×C23—82no—
89C8×C23—161no—
90#2029C2×C2311yescharacteristic
91#1316C2×C2311yescharacteristic
92#1304C2×C2311yescharacteristic
93C22×C23—22no—
94C22×C23—21no—
95C22×C23—21no—
96C22×C23—41no—
97C22×C23—41no—
98C22×C23—41no—
99C22×C23—42no—
100C22×C23—41no—
101C22×C23—41no—
102C22×C23—41no—
103C22×C23—42no—
104C22×C23—41no—
105C22×C23—42no—
106C22×C23—41no—
107C22×C23—81no—
108C22×C23—82no—
109C22×C23—82no—
110C22×C23—81no—
111C22×C23—84no—
112C22×C23—81no—
113C22×C23#32811yescharacteristic
114C22×C23AΣL(1,𝔽16)11yescharacteristic
115C22×C23#35411yescharacteristic
116C4×C23—81no—
117C4×C23—82no—
118C4×C23—81no—
119C4×C23—161no—
120C4×C23—162no—
121#543C4×C2311yescharacteristic, complemented
122#379—21no—
123#379—21no—
124#379C22×C2311yescharacteristic
125#342—21no—
126#342C4×C2311yescharacteristic, complemented
127#323—21no—
128C2×C23—21no—
129C2×C23—21no—
130C2×C23—21no—
131C2×C23—42no—
132C2×C23—41no—
133C2×C23—41no—
134C2×C23—81no—
135C2×C23#135111yescharacteristic
136C25D4×C2311yescharacteristic
137C22⋊Q8—42no—
138C22⋊Q8—41no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4C2×C4×C2311yescharacteristic
147C2×C22⋊C4D4×C2311yescharacteristic
148C22⋊C8—41no—
149C23#742611yescharacteristic, direct product factor, Sylow
150C24—21no—
151C24—21no—
152C24—41no—
153C24—41no—
154C24—82no—
155C24C22⋊C4×C2311yescharacteristic
156C24SD16×C2311yescharacteristic
157C24Q16×C2311yescharacteristic
158C2×Q8—41no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—41no—
163C22×C4—42no—
164C2×C8—81no—
165C4⋊C4—42no—
166C4⋊C4—41no—
167C22⋊C4—41no—
168C22⋊C4—41no—
169C22⋊C4—41no—
170C22⋊C4—41no—
171C22⋊C4—41no—
172C22⋊C4—41no—
173C22⋊C4—82no—
174C22⋊C4—84no—
175C22⋊C4—82no—
176C22⋊C4—81no—
177C22⋊C4—81no—
178C22⋊C4—82no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—42no—
184C23—41no—
185C23—41no—
186C23—42no—
187C23—42no—
188C23—44no—
189C23—41no—
190C23—41no—
191C23—81no—
192C23—81no—
193C23—81no—
194C23—81no—
195C23—81no—
196C23—82no—
197C23—82no—
198C23—82no—
199C23C23×#9911yescharacteristic
200C23C23×#10311yescharacteristic
201C23C23×#10411yescharacteristic
202Q8—161no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—82no—
209C2×C4—81no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C8—161no—
215C22—22no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22—41no—
227C22—42no—
228C22—41no—
229C22—81no—
230C22—82no—
231C22—82no—
232C22—81no—
233C22—84no—
234C22—81no—
235C22C23×#32811yescharacteristic
236C22C23×AΣL(1,𝔽16)11yescharacteristic
237C22C23×#35411yescharacteristic
238C4—81no—
239C4—82no—
240C4—81no—
241C4—161no—
242C4—162no—
243C2—21no—
244C2—21no—
245C2—21no—
246C2—42no—
247C2—41no—
248C2—41no—
249C2—81no—
250C2C23×#135111yescharacteristic
subgroupG/H
center, socleC2×C23#1351
commutatorC2×C22⋊C4C2×C4×C23
Frattini, cosocle#379C22×C23
prime radicalC25×C23D4
LCSG ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C23×C23 ◁ C24×C23 ◁ C23×#379 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×#543C4semidirect
9. C23×#342C4semidirect
53. #7426C23direct (N × H)
121. #543C4×C23semidirect
126. #342C4×C23semidirect
149. C23#7426direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#20292
C23×#13162
C23×#13042
#742623