Finite Group Databasebeta

← C23×#10030 #19157656

Subgroups — C23×#10030 #19157656

Order 5888.

520 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 102 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×#3249C211yesmaximal, characteristic, complemented
2C23×#3110C212yesmaximal
3C23×#1955C211yesmaximal, characteristic
4C23×#1531C212yesmaximal, complemented
5C23×#1290C211yesmaximal, characteristic, complemented
6C23×#533C2212yes—
7C2×C4×D4×C23C2211yescharacteristic
8C23×#484—22no—
9C23×#477—22no—
10C23×#474—22no—
11C23×#467—22no—
12C23×#462—22no—
13C23×#461—22no—
14C23×#450—22no—
15C23×#440—24no—
16C23×#432C2212yes—
17C2×C23×C4⋊C8C2211yescharacteristic
18C2×C23×C22⋊C8—22no—
19C2×C23×#97C2211yescharacteristic
20C23×#330—24no—
21C22×D4×C23D411yescharacteristic
22C23×C4×C23—21no—
23C2×Q16×C23—22no—
24C2×SD16×C23—24no—
25C22×C8×C23—21no—
26C22×C8×C23D412yes—
27C23×C4⋊Q8D412yes—
28C23×C4⋊Q8D412yes—
29C23×#125—22no—
30C23×C22⋊Q8—22no—
31C23×C22⋊Q8—22no—
32C4×Q8×C23—22no—
33C4×Q8×C23—22no—
34C4×D4×C23—22no—
35C4×D4×C23—22no—
36C23×#117—22no—
37C2×C4⋊C4×C23D411yescharacteristic
38C2×C22⋊C4×C23—21no—
39C2×C42×C23C2311yescharacteristic
40C23×#107—24no—
41C23×#107—22no—
42C23×C4⋊C8—22no—
43C23×C4⋊C8—22no—
44C23×#103—24no—
45C23×#103—24no—
46C23×#102—24no—
47C23×C22⋊C8—22no—
48C23×C22⋊C8—44no—
49C23×#97—22no—
50C23×#97D414yes—
51C24×C23—21no—
52C2×Q8×C23—22no—
53C2×Q8×C23—22no—
54C2×D4×C23—22no—
55C2×D4×C23—42no—
56C22×C4×C23—21no—
57C22×C4×C23—42no—
58C22×C4×C23—42no—
59C22×C4×C23—41no—
60C22×C4×C23—41no—
61C22×C4×C23C2×D411yescharacteristic
62C22×C4×C23C2×D411yescharacteristic
63C22×C4×C23#4111yescharacteristic
64Q16×C23—82no—
65SD16×C23—48no—
66C2×C8×C23—22no—
67C2×C8×C23—24no—
68C2×C8×C23—22no—
69C2×C8×C23—41no—
70C2×C8×C23—41no—
71C2×C8×C23—42no—
72C4⋊C4×C23—22no—
73C4⋊C4×C23—22no—
74C4⋊C4×C23—22no—
75C4⋊C4×C23—22no—
76C4⋊C4×C23—22no—
77C4⋊C4×C23—42no—
78C4⋊C4×C23—42no—
79C22⋊C4×C23—42no—
80C22⋊C4×C23—42no—
81C42×C23—42no—
82C42×C23C2×D412yes—
83C42×C23C2×D412yes—
84#10030C2311yesmaximal, characteristic, direct product factor, Sylow
85C23×C23—22no—
86C23×C23—42no—
87C23×C23—41no—
88C23×C23—41no—
89C23×C23—42no—
90C23×C23#12111yescharacteristic
91Q8×C23—44no—
92Q8×C23—42no—
93Q8×C23—42no—
94D4×C23—24no—
95D4×C23—42no—
96C2×C4×C23—22no—
97C2×C4×C23—21no—
98C2×C4×C23—21no—
99C2×C4×C23—41no—
100C2×C4×C23—42no—
101C2×C4×C23—42no—
102C2×C4×C23—42no—
103C2×C4×C23—41no—
104C2×C4×C23—82no—
105C2×C4×C23—81no—
106C2×C4×C23—81no—
107C2×C4×C23#12012yes—
108C2×C4×C23#12011yescharacteristic
109C2×C4×C23#12112yes—
110C2×C4×C23#12711yescharacteristic
111C8×C23—42no—
112C8×C23—44no—
113C8×C23—42no—
114#3249C2×C2311yescharacteristic, complemented
115#3110C2×C2312yes—
116#1955C2×C2311yescharacteristic
117#1531C2×C2312yescomplemented
118#1290C2×C2311yescharacteristic, complemented
119C22×C23—21no—
120C22×C23—21no—
121C22×C23—44no—
122C22×C23—42no—
123C22×C23—82no—
124C22×C23—81no—
125C22×C23—81no—
126C22×C23#39211yescharacteristic
127C22×C23#47312yes—
128C4×C23—21no—
129C4×C23—22no—
130C4×C23—22no—
131C4×C23—81no—
132C4×C23—82no—
133C4×C23—82no—
134C4×C23#45311yescharacteristic
135C4×C23#45411yescharacteristic
136#533C22×C2312yes—
137C2×C4×D4C22×C2311yescharacteristic
138#484—22no—
139#477—22no—
140#474—22no—
141#467—22no—
142#462—22no—
143#461—22no—
144#450—22no—
145#440—24no—
146#432C22×C2312yes—
147C2×C4⋊C8C22×C2311yescharacteristic
148C2×C22⋊C8—22no—
149C2×#97C22×C2311yescharacteristic
150#330—24no—
151C2×C23—22no—
152C2×C23—82no—
153C2×C23#200611yescharacteristic
154C2×C23#201011yescharacteristic
155C2×C23#202011yescharacteristic
156C22×D4D4×C2311yescharacteristic
157C23×C4—21no—
158C2×Q16—22no—
159C2×SD16—24no—
160C22×C8—21no—
161C22×C8D4×C2312yes—
162C4⋊Q8D4×C2312yes—
163C4⋊Q8D4×C2312yes—
164#125—22no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×D4—22no—
170C4×D4—22no—
171#117—22no—
172C2×C4⋊C4D4×C2311yescharacteristic
173C2×C22⋊C4—21no—
174C2×C42C23×C2311yescharacteristic
175#107—24no—
176#107—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179#103—24no—
180#103—24no—
181#102—24no—
182C22⋊C8—22no—
183C22⋊C8—44no—
184#97—22no—
185#97D4×C2314yes—
186C23#1003011yescharacteristic, direct product factor, Sylow
187C24—21no—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×D4—22no—
191C2×D4—42no—
192C22×C4—21no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4C2×D4×C2311yescharacteristic
198C22×C4C2×D4×C2311yescharacteristic
199C22×C4#41×C2311yescharacteristic
200Q16—82no—
201SD16—48no—
202C2×C8—22no—
203C2×C8—24no—
204C2×C8—22no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—42no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C42—42no—
218C42C2×D4×C2312yes—
219C42C2×D4×C2312yes—
220C23—22no—
221C23—42no—
222C23—41no—
223C23—41no—
224C23—42no—
225C23C23×#12111yescharacteristic
226Q8—44no—
227Q8—42no—
228Q8—42no—
229D4—24no—
230D4—42no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—82no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4C23×#12012yes—
243C2×C4C23×#12011yescharacteristic
244C2×C4C23×#12112yes—
245C2×C4C23×#12711yescharacteristic
246C8—42no—
247C8—44no—
248C8—42no—
249C22—21no—
250C22—21no—
251C22—44no—
252C22—42no—
253C22—82no—
254C22—81no—
255C22—81no—
256C22C23×#39211yescharacteristic
257C22C23×#47312yes—
258C4—21no—
259C4—22no—
260C4—22no—
261C4—81no—
262C4—82no—
263C4—82no—
264C4C23×#45311yescharacteristic
265C4C23×#45411yescharacteristic
266C2—22no—
267C2—82no—
268C2C23×#200611yescharacteristic
269C2C23×#201011yescharacteristic
270C2C23×#202011yescharacteristic
subgroupG/H
center, socleC22×C23#392
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC22×D4×C23D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3249C2semidirect
4. C23×#1531C2semidirect
5. C23×#1290C2semidirect
84. #10030C23direct (N × H)
114. #3249C2×C23semidirect
117. #1531C2×C23semidirect
118. #1290C2×C23semidirect
186. C23#10030direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32492
C23×#31102
C23×#19552
C23×#15312
C23×#12902
#1003023