Finite Group Databasebeta

← C23×#12299 #19159925

Subgroups — C23×#12299 #19159925

Order 5888.

324 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3195C211yesmaximal, characteristic
2C23×#2239C212yesmaximal, complemented
3C23×#2091C211yesmaximal, characteristic
4C23×#1342C212yesmaximal, complemented
5C23×#1320C211yesmaximal, characteristic
6C2×C23×C22⋊Q8—21no—
7C23×#483—22no—
8C23×#470—21no—
9C23×#470C2212yes—
10C23×#451—22no—
11C23×#430—21no—
12C23×#430C2211yescharacteristic
13C23×#407—21no—
14C23×#407C2211yescharacteristic
15C23×#365—22no—
16C23×#362—22no—
17C23×#349C2212yes—
18C23×#338C2211yescharacteristic
19C22×Q8×C23—21no—
20C23×C4×C23D411yescharacteristic
21C2×Q16×C23—21no—
22C2×Q16×C23D412yes—
23C2×M16×C23—21no—
24C2×M16×C23—21no—
25C2×M16×C23D411yescharacteristic
26C2×M16×C23C2311yescharacteristic
27C23×C22⋊Q8—22no—
28C23×C22⋊Q8—22no—
29C23×C22⋊Q8—41no—
30C23×C22⋊Q8—41no—
31C2×C4⋊C4×C23—21no—
32C2×C4⋊C4×C23—41no—
33C2×C22⋊C4×C23—41no—
34C23×M32—22no—
35C23×#108—22no—
36C23×#108—42no—
37C23×#106—21no—
38C23×#106D412yes—
39C23×#103—22no—
40C23×#103—22no—
41C23×C22⋊C8—22no—
42C23×C22⋊C8—42no—
43C24×C23SD1611yescharacteristic
44C2×Q8×C23—22no—
45C2×Q8×C23—41no—
46C2×Q8×C23—81no—
47C2×Q8×C23—81no—
48C22×C4×C23—21no—
49C22×C4×C23—21no—
50C22×C4×C23—41no—
51C22×C4×C23—41no—
52C22×C4×C23—41no—
53C22×C4×C23—41no—
54C22×C4×C23SD1611yescharacteristic
55C22×C4×C23C2×D411yescharacteristic
56Q16×C23—42no—
57Q16×C23—81no—
58M16×C23—41no—
59M16×C23—41no—
60M16×C23—41no—
61M16×C23—41no—
62M16×C23—41no—
63M16×C23—41no—
64C2×C8×C23—22no—
65C2×C8×C23—21no—
66C2×C8×C23—41no—
67C2×C8×C23C2×D412yes—
68C4⋊C4×C23—22no—
69C4⋊C4×C23—41no—
70C4⋊C4×C23—42no—
71C4⋊C4×C23—42no—
72C22⋊C4×C23—42no—
73C22⋊C4×C23—42no—
74C16×C23—42no—
75#12299C2311yesmaximal, characteristic, direct product factor, Sylow
76C23×C23—21no—
77C23×C23—41no—
78C23×C23—42no—
79C23×C23C2×SD1611yescharacteristic
80Q8×C23—42no—
81Q8×C23—42no—
82Q8×C23—82no—
83C2×C4×C23—21no—
84C2×C4×C23—21no—
85C2×C4×C23—21no—
86C2×C4×C23—41no—
87C2×C4×C23—41no—
88C2×C4×C23—41no—
89C2×C4×C23—42no—
90C2×C4×C23—42no—
91C2×C4×C23—81no—
92C2×C4×C23—82no—
93C2×C4×C23—82no—
94C2×C4×C23#12011yescharacteristic
95C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
96C8×C23—22no—
97C8×C23—42no—
98C8×C23—41no—
99C8×C23—81no—
100#3195C2×C2311yescharacteristic
101#2239C2×C2312yescomplemented
102#2091C2×C2311yescharacteristic
103#1342C2×C2312yescomplemented
104#1320C2×C2311yescharacteristic
105C22×C23—22no—
106C22×C23—21no—
107C22×C23—41no—
108C22×C23—41no—
109C22×C23—42no—
110C22×C23—41no—
111C22×C23—81no—
112C22×C23#45011yescharacteristic
113C4×C23—21no—
114C4×C23—41no—
115C4×C23—82no—
116C4×C23—82no—
117C4×C23#45311yescharacteristic
118C4×C23#45811yescharacteristic
119C2×C22⋊Q8—21no—
120#483—22no—
121#470—21no—
122#470C22×C2312yes—
123#451—22no—
124#430—21no—
125#430C22×C2311yescharacteristic
126#407—21no—
127#407C22×C2311yescharacteristic
128#365—22no—
129#362—22no—
130#349C22×C2312yes—
131#338C22×C2311yescharacteristic
132C2×C23—21no—
133C2×C23—42no—
134C2×C23—41no—
135C2×C23#161311yescharacteristic
136C22×Q8—21no—
137C23×C4D4×C2311yescharacteristic
138C2×Q16—21no—
139C2×Q16D4×C2312yes—
140C2×M16—21no—
141C2×M16—21no—
142C2×M16D4×C2311yescharacteristic
143C2×M16C23×C2311yescharacteristic
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—41no—
150C2×C22⋊C4—41no—
151M32—22no—
152#108—22no—
153#108—42no—
154#106—21no—
155#106D4×C2312yes—
156#103—22no—
157#103—22no—
158C22⋊C8—22no—
159C22⋊C8—42no—
160C23#1229911yescharacteristic, direct product factor, Sylow
161C24SD16×C2311yescharacteristic
162C2×Q8—22no—
163C2×Q8—41no—
164C2×Q8—81no—
165C2×Q8—81no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4SD16×C2311yescharacteristic
173C22×C4C2×D4×C2311yescharacteristic
174Q16—42no—
175Q16—81no—
176M16—41no—
177M16—41no—
178M16—41no—
179M16—41no—
180M16—41no—
181M16—41no—
182C2×C8—22no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8C2×D4×C2312yes—
186C4⋊C4—22no—
187C4⋊C4—41no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C16—42no—
193C23—21no—
194C23—41no—
195C23—42no—
196C23C2×SD16×C2311yescharacteristic
197Q8—42no—
198Q8—42no—
199Q8—82no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—81no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4C23×#12011yescharacteristic
212C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
213C8—22no—
214C8—42no—
215C8—41no—
216C8—81no—
217C22—22no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—81no—
224C22C23×#45011yescharacteristic
225C4—21no—
226C4—41no—
227C4—82no—
228C4—82no—
229C4C23×#45311yescharacteristic
230C4C23×#45811yescharacteristic
231C2—21no—
232C2—42no—
233C2—41no—
234C2C23×#161311yescharacteristic
subgroupG/H
center, socleC2×C23#1613
commutator, Frattini, cosocleC2×M16C23×C23
prime radicalC24×C23SD16
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C2×C4×C23 ◁ C2×M16×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2239C2semidirect
4. C23×#1342C2semidirect
75. #12299C23direct (N × H)
101. #2239C2×C23semidirect
103. #1342C2×C23semidirect
160. C23#12299direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31952
C23×#22392
C23×#20912
C23×#13422
C23×#13202
#1229923