Finite Group Databasebeta

← C23×#7496 #19155122

Subgroups — C23×#7496 #19155122

Order 5888.

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

normal subgroups only

This group has 274 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×#1884C211yesmaximal, characteristic
3C23×#1560C211yesmaximal, characteristic
4C23×#568C411yescharacteristic
5C23×#532C2211yescharacteristic
6C23×#423—21no—
7C23×#421—21no—
8C23×#413—21no—
9C23×#410C411yescharacteristic
10C23×#356—21no—
11C23×#355C412yescomplemented
12C23×#339—21no—
13C23×#333—21no—
14C23×#331C412yescomplemented
15C2×#41×C23C2×C411yescharacteristic
16C2×#41×C23D411yescharacteristic
17C23×#131—42no—
18C2×M16×C23—21no—
19C2×M16×C23—41no—
20C2×M16×C23—41no—
21C2×M16×C23—41no—
22C23×C4⋊Q8D411yescharacteristic
23C23×#127C2×C411yescharacteristic
24C23×#124C2×C411yescharacteristic
25C23×#124Q811yescharacteristic
26C23×#121—41no—
27C4×D4×C23—41no—
28C23×#117—21no—
29C2×C42×C23D411yescharacteristic
30C23×C4⋊C8—22no—
31C23×C4⋊C8—41no—
32C23×#104—42no—
33C23×#101—22no—
34C23×#100—41no—
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×C23C4211yescharacteristic
46C2×D4×C23C4⋊C411yescharacteristic
47C22×C4×C23—21no—
48C22×C4×C23—21no—
49C22×C4×C23—21no—
50C22×C4×C23C22⋊C411yescharacteristic
51M16×C23—22no—
52M16×C23—41no—
53M16×C23—42no—
54M16×C23—42no—
55M16×C23—82no—
56M16×C23—81no—
57C2×C8×C23—41no—
58C2×C8×C23—42no—
59C2×C8×C23—42no—
60C2×C8×C23—81no—
61C4⋊C4×C23—41no—
62C4⋊C4×C23—41no—
63C22⋊C4×C23—41no—
64C22⋊C4×C23—41no—
65C22⋊C4×C23—41no—
66C42×C23—21no—
67C42×C23—41no—
68C42×C23C22⋊C411yescharacteristic
69C42×C23C4⋊C411yescharacteristic
70#7496C2311yesmaximal, 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—42no—
79D4×C23—81no—
80D4×C23—81no—
81D4×C23—81no—
82C2×C4×C23—21no—
83C2×C4×C23—21no—
84C2×C4×C23—21no—
85C2×C4×C23—21no—
86C2×C4×C23—21no—
87C2×C4×C23—21no—
88C2×C4×C23—41no—
89C2×C4×C23—41no—
90C2×C4×C23—41no—
91C2×C4×C23—41no—
92C2×C4×C23—41no—
93C2×C4×C23—41no—
94C2×C4×C23—81no—
95C2×C4×C23#9511yescharacteristic
96C2×C4×C23#9911yescharacteristic
97C8×C23—42no—
98C8×C23—82no—
99C8×C23—82no—
100#2132C2×C2311yescharacteristic
101#1884C2×C2311yescharacteristic
102#1560C2×C2311yescharacteristic
103C22×C23—21no—
104C22×C23—41no—
105C22×C23—41no—
106C22×C23—41no—
107C22×C23—81no—
108C22×C23—81no—
109C22×C23#34211yescharacteristic
110C4×C23—21no—
111C4×C23—21no—
112C4×C23—21no—
113C4×C23—41no—
114C4×C23—41no—
115C4×C23—41no—
116C4×C23—161no—
117C4×C23AΣL(1,𝔽16)11yescharacteristic
118C4×C23#35211yescharacteristic
119#568C4×C2311yescharacteristic
120#532C22×C2311yescharacteristic
121#423—21no—
122#421—21no—
123#413—21no—
124#410C4×C2311yescharacteristic
125#356—21no—
126#355C4×C2312yescomplemented
127#339—21no—
128#333—21no—
129#331C4×C2312yescomplemented
130C2×C23—21no—
131C2×C23—41no—
132C2×C23—81no—
133C2×C23—81no—
134C2×C23#139011yescharacteristic
135C2×#41C2×C4×C2311yescharacteristic
136C2×#41D4×C2311yescharacteristic
137#131—42no—
138C2×M16—21no—
139C2×M16—41no—
140C2×M16—41no—
141C2×M16—41no—
142C4⋊Q8D4×C2311yescharacteristic
143#127C2×C4×C2311yescharacteristic
144#124C2×C4×C2311yescharacteristic
145#124Q8×C2311yescharacteristic
146#121—41no—
147C4×D4—41no—
148#117—21no—
149C2×C42D4×C2311yescharacteristic
150C4⋊C8—22no—
151C4⋊C8—41no—
152#104—42no—
153#101—22no—
154#100—41no—
155#99—41no—
156C23#749611yescharacteristic, direct product factor, Sylow
157#41—41no—
158#41—41no—
159#41—41no—
160#41—41no—
161C2×Q8C22⋊C4×C2311yescharacteristic
162C2×Q8C4⋊C4×C2311yescharacteristic
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—41no—
166C2×D4C42×C2311yescharacteristic
167C2×D4C4⋊C4×C2311yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4C22⋊C4×C2311yescharacteristic
172M16—22no—
173M16—41no—
174M16—42no—
175M16—42no—
176M16—82no—
177M16—81no—
178C2×C8—41no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—81no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C22⋊C4—41no—
185C22⋊C4—41no—
186C22⋊C4—41no—
187C42—21no—
188C42—41no—
189C42C22⋊C4×C2311yescharacteristic
190C42C4⋊C4×C2311yescharacteristic
191C23—21no—
192C23—21no—
193C23C23×#9911yescharacteristic
194Q8—41no—
195Q8—41no—
196D4—41no—
197D4—41no—
198D4—42no—
199D4—81no—
200D4—81no—
201D4—81no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—81no—
215C2×C4C23×#9511yescharacteristic
216C2×C4C23×#9911yescharacteristic
217C8—42no—
218C8—82no—
219C8—82no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—81no—
225C22—81no—
226C22C23×#34211yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—41no—
231C4—41no—
232C4—41no—
233C4—161no—
234C4C23×AΣL(1,𝔽16)11yescharacteristic
235C4C23×#35211yescharacteristic
236C2—21no—
237C2—41no—
238C2—81no—
239C2—81no—
240C2C23×#139011yescharacteristic
subgroupG/H
center, socleC2×C23#1390
commutatorC2×D4C42×C23
Frattini, cosocle#532C22×C23
prime radicalC23×#532C22
LCSG ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C2×C4×C23 ◁ C22×C4×C23 ◁ C23×#532 ◁ G

Complements

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

NH ≅ G/Ntype
11. C23×#355C4semidirect
14. C23×#331C4semidirect
70. #7496C23direct (N × H)
126. #355C4×C23semidirect
129. #331C4×C23semidirect
156. C23#7496direct (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×#18842
C23×#15602
#749623