Finite Group Databasebeta

← C23×#7497 #19155123

Subgroups — C23×#7497 #19155123

Order 5888.

290 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 66 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×#2132C211yesmaximal, characteristic
2C23×#2126C211yesmaximal, characteristic
3C23×#1962C211yesmaximal, characteristic
4C23×#532C2211yescharacteristic
5C23×#423C411yescharacteristic, complemented
6C2×C23×#104—21no—
7C2×C23×#104—21no—
8C23×#413—21no—
9C23×#410—21no—
10C43×C23C411yescharacteristic, complemented
11C23×#356C412yescomplemented
12C23×#355—21no—
13C23×#354—21no—
14C23×#353C412yescomplemented
15C2×#41×C23D411yescharacteristic
16C2×#41×C23D411yescharacteristic
17C2×M16×C23—21no—
18C2×M16×C23—41no—
19C23×C4⋊Q8C2×C411yescharacteristic
20C23×#127C2×C411yescharacteristic
21C23×#124D411yescharacteristic
22C23×#124Q811yescharacteristic
23C23×#121—41no—
24C4×D4×C23—41no—
25C23×#117—41no—
26C2×C42×C23—21no—
27C2×C42×C23—41no—
28C2×C42×C23C2×C411yescharacteristic
29C23×C4⋊C8—41no—
30C23×#104—42no—
31C23×#104—42no—
32C23×#101—22no—
33C23×#100—41no—
34C23×#99—22no—
35C23×#99—41no—
36#41×C23—41no—
37#41×C23—41no—
38#41×C23—41no—
39#41×C23—41no—
40C2×Q8×C23C22⋊C411yescharacteristic
41C2×Q8×C23C4⋊C411yescharacteristic
42C2×D4×C23—21no—
43C2×D4×C23—21no—
44C2×D4×C23—41no—
45C2×D4×C23C22⋊C411yescharacteristic
46C2×D4×C23C4⋊C411yescharacteristic
47C22×C4×C23—21no—
48C22×C4×C23—21no—
49C22×C4×C23—21no—
50C22×C4×C23—41no—
51C22×C4×C23C22⋊C411yescharacteristic
52M16×C23—42no—
53M16×C23—41no—
54M16×C23—41no—
55M16×C23—81no—
56C2×C8×C23—41no—
57C2×C8×C23—81no—
58C4⋊C4×C23—41no—
59C4⋊C4×C23—81no—
60C22⋊C4×C23—41no—
61C22⋊C4×C23—41no—
62C22⋊C4×C23—42no—
63C42×C23—21no—
64C42×C23—22no—
65C42×C23—41no—
66C42×C23—44no—
67C42×C23—81no—
68C42×C23C4211yescharacteristic
69C42×C23C4⋊C411yescharacteristic
70#7497C2311yesmaximal, characteristic, direct product factor, Sylow
71C23×C23—21no—
72C23×C23—21no—
73C23×C23#9911yescharacteristic
74Q8×C23—41no—
75Q8×C23—41no—
76D4×C23—41no—
77D4×C23—41no—
78D4×C23—82no—
79D4×C23—81no—
80D4×C23—81no—
81C2×C4×C23—21no—
82C2×C4×C23—21no—
83C2×C4×C23—21no—
84C2×C4×C23—21no—
85C2×C4×C23—21no—
86C2×C4×C23—21no—
87C2×C4×C23—42no—
88C2×C4×C23—42no—
89C2×C4×C23—44no—
90C2×C4×C23—41no—
91C2×C4×C23—41no—
92C2×C4×C23—41no—
93C2×C4×C23—41no—
94C2×C4×C23—82no—
95C2×C4×C23#9511yescharacteristic
96C2×C4×C23#9911yescharacteristic
97C8×C23—82no—
98C8×C23—81no—
99#2132C2×C2311yescharacteristic
100#2126C2×C2311yescharacteristic
101#1962C2×C2311yescharacteristic
102C22×C23—21no—
103C22×C23—41no—
104C22×C23—41no—
105C22×C23—41no—
106C22×C23—81no—
107C22×C23—81no—
108C22×C23#34211yescharacteristic
109C4×C23—22no—
110C4×C23—21no—
111C4×C23—41no—
112C4×C23—44no—
113C4×C23—41no—
114C4×C23—41no—
115C4×C23—162no—
116C4×C23#35311yescharacteristic
117C4×C23#35411yescharacteristic
118#532C22×C2311yescharacteristic
119#423C4×C2311yescharacteristic, complemented
120C2×#104—21no—
121C2×#104—21no—
122#413—21no—
123#410—21no—
124C43C4×C2311yescharacteristic, complemented
125#356C4×C2312yescomplemented
126#355—21no—
127#354—21no—
128#353C4×C2312yescomplemented
129C2×C23—21no—
130C2×C23—41no—
131C2×C23—81no—
132C2×C23—81no—
133C2×C23#139311yescharacteristic
134C2×#41D4×C2311yescharacteristic
135C2×#41D4×C2311yescharacteristic
136C2×M16—21no—
137C2×M16—41no—
138C4⋊Q8C2×C4×C2311yescharacteristic
139#127C2×C4×C2311yescharacteristic
140#124D4×C2311yescharacteristic
141#124Q8×C2311yescharacteristic
142#121—41no—
143C4×D4—41no—
144#117—41no—
145C2×C42—21no—
146C2×C42—41no—
147C2×C42C2×C4×C2311yescharacteristic
148C4⋊C8—41no—
149#104—42no—
150#104—42no—
151#101—22no—
152#100—41no—
153#99—22no—
154#99—41no—
155C23#749711yescharacteristic, direct product factor, Sylow
156#41—41no—
157#41—41no—
158#41—41no—
159#41—41no—
160C2×Q8C22⋊C4×C2311yescharacteristic
161C2×Q8C4⋊C4×C2311yescharacteristic
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—41no—
165C2×D4C22⋊C4×C2311yescharacteristic
166C2×D4C4⋊C4×C2311yescharacteristic
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—41no—
171C22×C4C22⋊C4×C2311yescharacteristic
172M16—42no—
173M16—41no—
174M16—41no—
175M16—81no—
176C2×C8—41no—
177C2×C8—81no—
178C4⋊C4—41no—
179C4⋊C4—81no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—42no—
183C42—21no—
184C42—22no—
185C42—41no—
186C42—44no—
187C42—81no—
188C42C42×C2311yescharacteristic
189C42C4⋊C4×C2311yescharacteristic
190C23—21no—
191C23—21no—
192C23C23×#9911yescharacteristic
193Q8—41no—
194Q8—41no—
195D4—41no—
196D4—41no—
197D4—82no—
198D4—81no—
199D4—81no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—82no—
214C2×C4C23×#9511yescharacteristic
215C2×C4C23×#9911yescharacteristic
216C8—82no—
217C8—81no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—81no—
223C22—81no—
224C22C23×#34211yescharacteristic
225C4—22no—
226C4—21no—
227C4—41no—
228C4—44no—
229C4—41no—
230C4—41no—
231C4—162no—
232C4C23×#35311yescharacteristic
233C4C23×#35411yescharacteristic
234C2—21no—
235C2—41no—
236C2—81no—
237C2—81no—
238C2C23×#139311yescharacteristic
subgroupG/H
centerC4×C23#353
commutatorC42C42×C23
Frattini, cosocle#532C22×C23
socleC2×C23#1393
prime radicalC23×#532C22
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C23 ◁ C2×C4×C23 ◁ C22×C4×C23 ◁ C23×#532 ◁ G

Complements

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

NH ≅ G/Ntype
5. C23×#423C4semidirect
10. C43×C23C4semidirect
11. C23×#356C4semidirect
14. C23×#353C4semidirect
70. #7497C23direct (N × H)
119. #423C4×C23semidirect
124. C43C4×C23semidirect
125. #356C4×C23semidirect
128. #353C4×C23semidirect
155. C23#7497direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#21322
C23×#21262
C23×#19622
#749723