Finite Group Databasebeta

← C23×#9487 #19157113

Subgroups — C23×#9487 #19157113

Order 5888.

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

normal subgroups only

This group has 518 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×#2975C211yesmaximal, characteristic
2C23×#2580C211yesmaximal, characteristic
3C23×#1800C211yesmaximal, characteristic
4C23×#1487C212yesmaximal, complemented
5C23×#1286C212yesmaximal, complemented
6C2×C4×Q8×C23C2211yescharacteristic
7C2×C4×D4×C23C2211yescharacteristic
8C23×#436—24no—
9C23×#431C2212yes—
10C23×#423C2212yes—
11C2×C23×#100—21no—
12C23×#407—22no—
13C23×#399C412yes—
14C23×#393C412yes—
15C23×#390—21no—
16C23×#389—21no—
17C23×#385—22no—
18C23×#382C2211yescharacteristic
19C23×#343—21no—
20C23×Q8⋊C8—24no—
21C22×Q8×C23C2×C411yescharacteristic
22C22×D4×C23D411yescharacteristic
23C23×C4×C23D412yes—
24C2×M16×C23—24no—
25C2×M16×C23—22no—
26C4×Q8×C23—22no—
27C4×Q8×C23—22no—
28C4×D4×C23—22no—
29C4×D4×C23—22no—
30C2×C4⋊C4×C23—21no—
31C2×C4⋊C4×C23—21no—
32C2×C4⋊C4×C23—22no—
33C2×C4⋊C4×C23C2×C411yescharacteristic
34C2×C4⋊C4×C23D411yescharacteristic
35C2×C22⋊C4×C23—22no—
36C2×C22⋊C4×C23—22no—
37C2×C22⋊C4×C23C2×C412yes—
38C2×C42×C23—21no—
39C2×C42×C23C2311yescharacteristic
40C23×C4⋊C8—22no—
41C23×C4⋊C8—22no—
42C23×C4⋊C8—22no—
43C23×#100—42no—
44C23×C22⋊C8—24no—
45C23×C22⋊C8—22no—
46C4×C8×C23D414yes—
47C23×#95—22no—
48C23×#95—21no—
49C23×#95C2×C412yes—
50C24×C23C22⋊C412yes—
51C2×Q8×C23—22no—
52C2×Q8×C23—42no—
53C2×D4×C23—22no—
54C2×D4×C23—21no—
55C2×D4×C23—22no—
56C2×D4×C23—21no—
57C22×C4×C23—21no—
58C22×C4×C23—22no—
59C22×C4×C23—22no—
60C22×C4×C23—21no—
61C22×C4×C23—22no—
62C22×C4×C23—82no—
63C22×C4×C23C22⋊C412yes—
64C22×C4×C23C22×C411yescharacteristic
65C22×C4×C23C2×D411yescharacteristic
66C22×C4×C23C2×D411yescharacteristic
67M16×C23—42no—
68M16×C23—42no—
69M16×C23—84no—
70C2×C8×C23—24no—
71C2×C8×C23—42no—
72C4⋊C4×C23—22no—
73C4⋊C4×C23—22no—
74C4⋊C4×C23—41no—
75C4⋊C4×C23—41no—
76C4⋊C4×C23—41no—
77C4⋊C4×C23—41no—
78C4⋊C4×C23—82no—
79C22⋊C4×C23—24no—
80C22⋊C4×C23—44no—
81C22⋊C4×C23—82no—
82C42×C23—42no—
83C42×C23C2×D412yes—
84C42×C23#4112yes—
85#9487C2311yesmaximal, characteristic, direct product factor, Sylow
86C23×C23—22no—
87C23×C23—22no—
88C23×C23—22no—
89C23×C23—42no—
90C23×C23—42no—
91C23×C23C2×C22⋊C411yescharacteristic
92Q8×C23—24no—
93Q8×C23—42no—
94D4×C23—42no—
95D4×C23—42no—
96C2×C4×C23—22no—
97C2×C4×C23—22no—
98C2×C4×C23—41no—
99C2×C4×C23—41no—
100C2×C4×C23—44no—
101C2×C4×C23—41no—
102C2×C4×C23—41no—
103C2×C4×C23—42no—
104C2×C4×C23—42no—
105C2×C4×C23—41no—
106C2×C4×C23—42no—
107C2×C4×C23—41no—
108C2×C4×C23—41no—
109C2×C4×C23—42no—
110C2×C4×C23—82no—
111C2×C4×C23C4×D412yes—
112C2×C4×C23#12011yescharacteristic
113C2×C4×C23#12112yes—
114C2×C4×C23#12311yescharacteristic
115C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
116C2×C4×C23#13712yes—
117C2×C4×C23#13711yescharacteristic
118C8×C23—44no—
119C8×C23—82no—
120#2975C2×C2311yescharacteristic
121#2580C2×C2311yescharacteristic
122#1800C2×C2311yescharacteristic
123#1487C2×C2312yescomplemented
124#1286C2×C2312yescomplemented
125C22×C23—21no—
126C22×C23—21no—
127C22×C23—44no—
128C22×C23—42no—
129C22×C23—42no—
130C22×C23—42no—
131C22×C23—82no—
132C22×C23#38611yescharacteristic
133C22×C23#41711yescharacteristic
134C22×C23#41911yescharacteristic
135C4×C23—22no—
136C4×C23—21no—
137C4×C23—21no—
138C4×C23—42no—
139C4×C23—42no—
140C4×C23—82no—
141C4×C23—81no—
142C4×C23#46812yes—
143C4×C23#47012yes—
144C2×C4×Q8C22×C2311yescharacteristic
145C2×C4×D4C22×C2311yescharacteristic
146#436—24no—
147#431C22×C2312yes—
148#423C22×C2312yes—
149C2×#100—21no—
150#407—22no—
151#399C4×C2312yes—
152#393C4×C2312yes—
153#390—21no—
154#389—21no—
155#385—22no—
156#382C22×C2311yescharacteristic
157#343—21no—
158Q8⋊C8—24no—
159C2×C23—21no—
160C2×C23—21no—
161C2×C23—82no—
162C2×C23#189111yescharacteristic
163C2×C23#189411yescharacteristic
164C2×C23#191111yescharacteristic
165C22×Q8C2×C4×C2311yescharacteristic
166C22×D4D4×C2311yescharacteristic
167C23×C4D4×C2312yes—
168C2×M16—24no—
169C2×M16—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×D4—22no—
173C4×D4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C2×C4×C2311yescharacteristic
178C2×C4⋊C4D4×C2311yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4C2×C4×C2312yes—
182C2×C42—21no—
183C2×C42C23×C2311yescharacteristic
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187#100—42no—
188C22⋊C8—24no—
189C22⋊C8—22no—
190C4×C8D4×C2314yes—
191#95—22no—
192#95—21no—
193#95C2×C4×C2312yes—
194C23#948711yescharacteristic, direct product factor, Sylow
195C24C22⋊C4×C2312yes—
196C2×Q8—22no—
197C2×Q8—42no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—82no—
208C22×C4C22⋊C4×C2312yes—
209C22×C4C22×C4×C2311yescharacteristic
210C22×C4C2×D4×C2311yescharacteristic
211C22×C4C2×D4×C2311yescharacteristic
212M16—42no—
213M16—42no—
214M16—84no—
215C2×C8—24no—
216C2×C8—42no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—82no—
224C22⋊C4—24no—
225C22⋊C4—44no—
226C22⋊C4—82no—
227C42—42no—
228C42C2×D4×C2312yes—
229C42#41×C2312yes—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—42no—
234C23—42no—
235C23C2×C22⋊C4×C2311yescharacteristic
236Q8—24no—
237Q8—42no—
238D4—42no—
239D4—42no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—44no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—82no—
255C2×C4C4×D4×C2312yes—
256C2×C4C23×#12011yescharacteristic
257C2×C4C23×#12112yes—
258C2×C4C23×#12311yescharacteristic
259C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
260C2×C4C23×#13712yes—
261C2×C4C23×#13711yescharacteristic
262C8—44no—
263C8—82no—
264C22—21no—
265C22—21no—
266C22—44no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—82no—
271C22C23×#38611yescharacteristic
272C22C23×#41711yescharacteristic
273C22C23×#41911yescharacteristic
274C4—22no—
275C4—21no—
276C4—21no—
277C4—42no—
278C4—42no—
279C4—82no—
280C4—81no—
281C4C23×#46812yes—
282C4C23×#47012yes—
283C2—21no—
284C2—21no—
285C2—82no—
286C2C23×#189111yescharacteristic
287C2C23×#189411yescharacteristic
288C2C23×#191111yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC22×D4×C23D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×#1487C2semidirect
5. C23×#1286C2semidirect
85. #9487C23direct (N × H)
123. #1487C2×C23semidirect
124. #1286C2×C23semidirect
194. C23#9487direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#29752
C23×#25802
C23×#18002
C23×#14872
C23×#12862
#948723