Finite Group Databasebeta

← C23×#9757 #19157383

Subgroups — C23×#9757 #19157383

Order 5888.

400 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 134 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×#3131C211yesmaximal, characteristic
2C23×#1835C211yesmaximal, characteristic, complemented
3C23×#1518C212yesmaximal, complemented
4C23×#1517C212yesmaximal
5C23×#1450C211yesmaximal, characteristic, complemented
6C23×#533C2211yescharacteristic
7C23×#532C2211yescharacteristic
8C23×#488—21no—
9C23×#488—21no—
10C23×#487C412yescomplemented
11C23×#486C412yescomplemented
12C23×#431C2212yes—
13C23×#416—22no—
14C2×C4×C8×C23C2211yescharacteristic
15C23×#382C2212yes—
16C23×#340—22no—
17C23×#328—24no—
18C23×#327—24no—
19C23×#322—21no—
20C82×C23—21no—
21C2×#41×C23—21no—
22C22×C8×C23D412yes—
23C23×C4⋊Q8C2×C412yes—
24C23×C4⋊Q8C2×C411yescharacteristic
25C23×#127C2×C411yescharacteristic
26C23×#125—21no—
27C23×#124—21no—
28C23×C22⋊Q8—22no—
29C23×#121—22no—
30C4×Q8×C23—22no—
31C4×D4×C23—22no—
32C23×#117—21no—
33C2×C4⋊C4×C23—22no—
34C2×C42×C23C2311yescharacteristic
35C23×C4⋊C8—22no—
36C23×#103—22no—
37C23×#103—22no—
38C23×#102—22no—
39C23×#102—22no—
40C23×C22⋊C8D414yes—
41C4×C8×C23—22no—
42C4×C8×C23C2×C412yes—
43C4×C8×C23D411yescharacteristic
44C4×C8×C23D411yescharacteristic
45C23×#95—22no—
46C23×#95—22no—
47#41×C23—44no—
48C2×Q8×C23—21no—
49C2×Q8×C23—22no—
50C2×D4×C23—21no—
51C2×D4×C23—22no—
52C2×D4×C23—41no—
53C22×C4×C23—41no—
54C22×C4×C23—42no—
55C22×C4×C23C2×D412yes—
56C22×C4×C23C2×D411yescharacteristic
57C2×C8×C23—24no—
58C2×C8×C23—24no—
59C2×C8×C23C22⋊C412yes—
60C2×C8×C23C22⋊C412yes—
61C4⋊C4×C23—22no—
62C4⋊C4×C23—22no—
63C4⋊C4×C23—22no—
64C4⋊C4×C23—42no—
65C4⋊C4×C23—44no—
66C22⋊C4×C23—41no—
67C22⋊C4×C23—42no—
68C42×C23—41no—
69C42×C23C22×C411yescharacteristic
70C42×C23C2×D411yescharacteristic
71C42×C23#4112yes—
72#9757C2311yesmaximal, characteristic, direct product factor, Sylow
73C23×C23—41no—
74C23×C23#12711yescharacteristic
75Q8×C23—42no—
76Q8×C23—82no—
77D4×C23—42no—
78D4×C23—82no—
79D4×C23—82no—
80C2×C4×C23—42no—
81C2×C4×C23—42no—
82C2×C4×C23—41no—
83C2×C4×C23—82no—
84C2×C4×C23—84no—
85C2×C4×C23—82no—
86C2×C4×C23C2×C22⋊C411yescharacteristic
87C2×C4×C23C4×D412yes—
88C2×C4×C23#12112yes—
89C2×C4×C23#12411yescharacteristic
90C2×C4×C23#13514yes—
91C8×C23—22no—
92C8×C23—22no—
93C8×C23—44no—
94#3131C2×C2311yescharacteristic
95#1835C2×C2311yescharacteristic, complemented
96#1518C2×C2312yescomplemented
97#1517C2×C2312yes—
98#1450C2×C2311yescharacteristic, complemented
99C22×C23—22no—
100C22×C23—81no—
101C22×C23—82no—
102C22×C23#39011yescharacteristic
103C22×C23#49512yes—
104C4×C23—22no—
105C4×C23—22no—
106C4×C23—81no—
107C4×C23—82no—
108C4×C23—162no—
109C4×C23#41612yes—
110C4×C23#48812yes—
111#533C22×C2311yescharacteristic
112#532C22×C2311yescharacteristic
113#488—21no—
114#488—21no—
115#487C4×C2312yescomplemented
116#486C4×C2312yescomplemented
117#431C22×C2312yes—
118#416—22no—
119C2×C4×C8C22×C2311yescharacteristic
120#382C22×C2312yes—
121#340—22no—
122#328—24no—
123#327—24no—
124#322—21no—
125C82—21no—
126C2×C23—22no—
127C2×C23—161no—
128C2×C23#195811yescharacteristic
129C2×C23#197612yes—
130C2×#41—21no—
131C22×C8D4×C2312yes—
132C4⋊Q8C2×C4×C2311yescharacteristic
133C4⋊Q8C2×C4×C2312yes—
134#127C2×C4×C2311yescharacteristic
135#125—21no—
136#124—21no—
137C22⋊Q8—22no—
138#121—22no—
139C4×Q8—22no—
140C4×D4—22no—
141#117—21no—
142C2×C4⋊C4—22no—
143C2×C42C23×C2311yescharacteristic
144C4⋊C8—22no—
145#103—22no—
146#103—22no—
147#102—22no—
148#102—22no—
149C22⋊C8D4×C2314yes—
150C4×C8—22no—
151C4×C8C2×C4×C2312yes—
152C4×C8D4×C2311yescharacteristic
153C4×C8D4×C2311yescharacteristic
154#95—22no—
155#95—22no—
156C23#975711yescharacteristic, direct product factor, Sylow
157#41—44no—
158C2×Q8—21no—
159C2×Q8—22no—
160C2×D4—21no—
161C2×D4—22no—
162C2×D4—41no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4C2×D4×C2312yes—
166C22×C4C2×D4×C2311yescharacteristic
167C2×C8—24no—
168C2×C8—24no—
169C2×C8C22⋊C4×C2312yes—
170C2×C8C22⋊C4×C2312yes—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C4⋊C4—44no—
176C22⋊C4—41no—
177C22⋊C4—42no—
178C42—41no—
179C42C22×C4×C2311yescharacteristic
180C42C2×D4×C2311yescharacteristic
181C42#41×C2312yes—
182C23—41no—
183C23C23×#12711yescharacteristic
184Q8—42no—
185Q8—82no—
186D4—42no—
187D4—82no—
188D4—82no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—82no—
193C2×C4—84no—
194C2×C4—82no—
195C2×C4C2×C22⋊C4×C2311yescharacteristic
196C2×C4C4×D4×C2312yes—
197C2×C4C23×#12112yes—
198C2×C4C23×#12411yescharacteristic
199C2×C4C23×#13514yes—
200C8—22no—
201C8—22no—
202C8—44no—
203C22—22no—
204C22—81no—
205C22—82no—
206C22C23×#39011yescharacteristic
207C22C23×#49512yes—
208C4—22no—
209C4—22no—
210C4—81no—
211C4—82no—
212C4—162no—
213C4C23×#41612yes—
214C4C23×#48812yes—
215C2—22no—
216C2—161no—
217C2C23×#195811yescharacteristic
218C2C23×#197612yes—
subgroupG/H
center, socleC22×C23#390
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1835C2semidirect
3. C23×#1518C2semidirect
5. C23×#1450C2semidirect
10. C23×#487C4semidirect
11. C23×#486C4semidirect
72. #9757C23direct (N × H)
95. #1835C2×C23semidirect
96. #1518C2×C23semidirect
98. #1450C2×C23semidirect
115. #487C4×C23semidirect
116. #486C4×C23semidirect
156. C23#9757direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31312
C23×#18352
C23×#15182
C23×#15172
C23×#14502
#975723