Finite Group Databasebeta

← C23×#13028 #19160654

Subgroups — C23×#13028 #19160654

Order 5888.

476 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#510C211yesmaximal, characteristic
2C2×C23×#443C211yesmaximal, characteristic
3C2×C23×#362C211yesmaximal, characteristic, complemented
4C23×#2139C211yesmaximal, characteristic
5C23×#1969C211yesmaximal, characteristic, complemented
6C23×#1387C211yesmaximal, characteristic, complemented
7C23×#1376C211yesmaximal, characteristic, complemented
8C2×C23×#135C2211yescharacteristic
9C22×Q16×C23C2211yescharacteristic
10C2×C23×#131—21no—
11C23×#510—22no—
12C23×#510C414yescomplemented
13C2×C23×Q32—21no—
14C2×C23×SD32—21no—
15C2×C23×M32C2211yescharacteristic
16C2×C23×M32C2211yescharacteristic
17C23×#443—22no—
18C23×#443—22no—
19C2×C23×#108C2211yescharacteristic
20C2×C23×#107C2211yescharacteristic
21C2×C23×#104—21no—
22C2×C23×#103—21no—
23C2×C4×C8×C23C2211yescharacteristic
24C23×#391—21no—
25C23×#362—22no—
26C23×#358—21no—
27C23×#357—21no—
28C2×#41×C23—21no—
29C22×Q8×C23—21no—
30C23×#135—21no—
31C23×#135—21no—
32C23×#135C2×C412yes—
33C23×#135D412yes—
34C2×Q16×C23—21no—
35C2×Q16×C23—21no—
36C2×Q16×C23—21no—
37C2×Q16×C23—21no—
38C2×Q16×C23—21no—
39C2×Q16×C23C2×C412yes—
40C2×Q16×C23D411yescharacteristic
41C2×SD16×C23—21no—
42C2×D8×C23D411yescharacteristic
43C23×#131—24no—
44C23×#131—42no—
45C2×M16×C23—21no—
46C2×M16×C23—41no—
47C22×C8×C23—41no—
48C22×C8×C23D411yescharacteristic
49C22×C8×C23C2311yescharacteristic
50C2×C4⋊C4×C23—21no—
51C2×C42×C23D411yescharacteristic
52C23×Q32—24no—
53C23×SD32—24no—
54C23×M32—21no—
55C23×M32—21no—
56C23×M32—21no—
57C23×M32C2×C412yes—
58C2×C16×C23—21no—
59C2×C16×C23—21no—
60C23×#108—21no—
61C23×#108D412yes—
62C23×#107—21no—
63C23×#107—21no—
64C23×#104—24no—
65C23×#103—41no—
66C23×#103—41no—
67C4×C8×C23—21no—
68C4×C8×C23—21no—
69C23×#95—42no—
70#41×C23—22no—
71#41×C23—22no—
72#41×C23—41no—
73#41×C23—41no—
74C2×Q8×C23—22no—
75C2×Q8×C23—21no—
76C2×Q8×C23—41no—
77C2×Q8×C23—41no—
78C2×Q8×C23—41no—
79C2×Q8×C23—41no—
80C2×Q8×C23—41no—
81C2×D4×C23—21no—
82C2×D4×C23—41no—
83C22×C4×C23—21no—
84C22×C4×C23—41no—
85C22×C4×C23—41no—
86C22×C4×C23—41no—
87C22×C4×C23C2×D411yescharacteristic
88Q16×C23—22no—
89Q16×C23—22no—
90Q16×C23—21no—
91Q16×C23—41no—
92Q16×C23—41no—
93Q16×C23C22⋊C412yes—
94SD16×C23—22no—
95SD16×C23—22no—
96D8×C23—21no—
97D8×C23C22⋊C412yes—
98M16×C23—22no—
99M16×C23—44no—
100M16×C23—41no—
101C2×C8×C23—21no—
102C2×C8×C23—21no—
103C2×C8×C23—44no—
104C2×C8×C23—41no—
105C2×C8×C23D811yescharacteristic
106C2×C8×C23D811yescharacteristic
107C2×C8×C23C22×C411yescharacteristic
108C2×C8×C23C2×D411yescharacteristic
109C2×C8×C23C2×D411yescharacteristic
110C2×C8×C23C2×D411yescharacteristic
111C2×C8×C23#4111yescharacteristic
112C2×C8×C23#4111yescharacteristic
113C4⋊C4×C23—41no—
114C4⋊C4×C23—41no—
115C42×C23—21no—
116C42×C23—21no—
117C16×C23—22no—
118C16×C23—41no—
119#13028C2311yesmaximal, characteristic, direct product factor, Sylow
120C23×C23—41no—
121C23×C23AGL(1,ℤ/8)11yescharacteristic
122Q8×C23—22no—
123Q8×C23—41no—
124Q8×C23—42no—
125Q8×C23—42no—
126Q8×C23—81no—
127Q8×C23—81no—
128D4×C23—22no—
129D4×C23—42no—
130D4×C23—42no—
131D4×C23—41no—
132C2×C4×C23—22no—
133C2×C4×C23—41no—
134C2×C4×C23—41no—
135C2×C4×C23—41no—
136C2×C4×C23—42no—
137C2×C4×C23—42no—
138C2×C4×C23—42no—
139C2×C4×C23—81no—
140C2×C4×C23—81no—
141C2×C4×C23—82no—
142C2×C4×C23—81no—
143C2×C4×C23C4×D411yescharacteristic
144C2×C4×C23#12011yescharacteristic
145C2×C4×C23#12111yescharacteristic
146C2×C4×C23C2×D811yescharacteristic
147C2×C4×C23#13511yescharacteristic
148C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
149C8×C23—21no—
150C8×C23—21no—
151C8×C23—42no—
152C8×C23C2×C22⋊C411yescharacteristic
153C8×C23C4×D411yescharacteristic
154C8×C23#12111yescharacteristic
155C8×C23#12311yescharacteristic
156C2×#510C2×C2311yescharacteristic
157C2×#443C2×C2311yescharacteristic
158C2×#362C2×C2311yescharacteristic, complemented
159#2139C2×C2311yescharacteristic
160#1969C2×C2311yescharacteristic, complemented
161#1387C2×C2311yescharacteristic, complemented
162#1376C2×C2311yescharacteristic, complemented
163C22×C23—21no—
164C22×C23—21no—
165C22×C23—42no—
166C22×C23—81no—
167C22×C23—81no—
168C22×C23#44211yescharacteristic
169C22×C23#44911yescharacteristic
170C22×C23#45911yescharacteristic
171C4×C23—42no—
172C4×C23—42no—
173C4×C23—82no—
174C4×C23—161no—
175C4×C23#38611yescharacteristic
176C4×C23C4×D811yescharacteristic
177C4×C23#44711yescharacteristic
178C4×C23#46311yescharacteristic
179C2×#135C22×C2311yescharacteristic
180C22×Q16C22×C2311yescharacteristic
181C2×#131—21no—
182#510—22no—
183#510C4×C2314yescomplemented
184C2×Q32—21no—
185C2×SD32—21no—
186C2×M32C22×C2311yescharacteristic
187C2×M32C22×C2311yescharacteristic
188#443—22no—
189#443—22no—
190C2×#108C22×C2311yescharacteristic
191C2×#107C22×C2311yescharacteristic
192C2×#104—21no—
193C2×#103—21no—
194C2×C4×C8C22×C2311yescharacteristic
195#391—21no—
196#362—22no—
197#358—21no—
198#357—21no—
199C2×C23—21no—
200C2×C23—21no—
201C2×C23—82no—
202C2×C23#187911yescharacteristic
203C2×C23#221211yescharacteristic
204C2×C23#221411yescharacteristic
205C2×#41—21no—
206C22×Q8—21no—
207#135—21no—
208#135—21no—
209#135C2×C4×C2312yes—
210#135D4×C2312yes—
211C2×Q16—21no—
212C2×Q16—21no—
213C2×Q16—21no—
214C2×Q16—21no—
215C2×Q16—21no—
216C2×Q16C2×C4×C2312yes—
217C2×Q16D4×C2311yescharacteristic
218C2×SD16—21no—
219C2×D8D4×C2311yescharacteristic
220#131—24no—
221#131—42no—
222C2×M16—21no—
223C2×M16—41no—
224C22×C8—41no—
225C22×C8D4×C2311yescharacteristic
226C22×C8C23×C2311yescharacteristic
227C2×C4⋊C4—21no—
228C2×C42D4×C2311yescharacteristic
229Q32—24no—
230SD32—24no—
231M32—21no—
232M32—21no—
233M32—21no—
234M32C2×C4×C2312yes—
235C2×C16—21no—
236C2×C16—21no—
237#108—21no—
238#108D4×C2312yes—
239#107—21no—
240#107—21no—
241#104—24no—
242#103—41no—
243#103—41no—
244C4×C8—21no—
245C4×C8—21no—
246#95—42no—
247C23#1302811yescharacteristic, direct product factor, Sylow
248#41—22no—
249#41—22no—
250#41—41no—
251#41—41no—
252C2×Q8—22no—
253C2×Q8—21no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×D4—21no—
260C2×D4—41no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4C2×D4×C2311yescharacteristic
266Q16—22no—
267Q16—22no—
268Q16—21no—
269Q16—41no—
270Q16—41no—
271Q16C22⋊C4×C2312yes—
272SD16—22no—
273SD16—22no—
274D8—21no—
275D8C22⋊C4×C2312yes—
276M16—22no—
277M16—44no—
278M16—41no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—44no—
282C2×C8—41no—
283C2×C8D8×C2311yescharacteristic
284C2×C8D8×C2311yescharacteristic
285C2×C8C22×C4×C2311yescharacteristic
286C2×C8C2×D4×C2311yescharacteristic
287C2×C8C2×D4×C2311yescharacteristic
288C2×C8C2×D4×C2311yescharacteristic
289C2×C8#41×C2311yescharacteristic
290C2×C8#41×C2311yescharacteristic
291C4⋊C4—41no—
292C4⋊C4—41no—
293C42—21no—
294C42—21no—
295C16—22no—
296C16—41no—
297C23—41no—
298C23C23×AGL(1,ℤ/8)11yescharacteristic
299Q8—22no—
300Q8—41no—
301Q8—42no—
302Q8—42no—
303Q8—81no—
304Q8—81no—
305D4—22no—
306D4—42no—
307D4—42no—
308D4—41no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—82no—
319C2×C4—81no—
320C2×C4C4×D4×C2311yescharacteristic
321C2×C4C23×#12011yescharacteristic
322C2×C4C23×#12111yescharacteristic
323C2×C4C2×D8×C2311yescharacteristic
324C2×C4C23×#13511yescharacteristic
325C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
326C8—21no—
327C8—21no—
328C8—42no—
329C8C2×C22⋊C4×C2311yescharacteristic
330C8C4×D4×C2311yescharacteristic
331C8C23×#12111yescharacteristic
332C8C23×#12311yescharacteristic
333C22—21no—
334C22—21no—
335C22—42no—
336C22—81no—
337C22—81no—
338C22C23×#44211yescharacteristic
339C22C23×#44911yescharacteristic
340C22C23×#45911yescharacteristic
341C4—42no—
342C4—42no—
343C4—82no—
344C4—161no—
345C4C23×#38611yescharacteristic
346C4C4×D8×C2311yescharacteristic
347C4C23×#44711yescharacteristic
348C4C23×#46311yescharacteristic
349C2—21no—
350C2—21no—
351C2—82no—
352C2C23×#187911yescharacteristic
353C2C23×#221211yescharacteristic
354C2C23×#221411yescharacteristic
subgroupG/H
center, socleC22×C23#459
commutatorC2×C8C22×C4×C23
Frattini, cosocleC22×C8C23×C23
prime radicalC2×C23×#135C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C4×C8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C23×#362C2semidirect
5. C23×#1969C2semidirect
6. C23×#1387C2semidirect
7. C23×#1376C2semidirect
12. C23×#510C4semidirect
119. #13028C23direct (N × H)
158. C2×#362C2×C23semidirect
160. #1969C2×C23semidirect
161. #1387C2×C23semidirect
162. #1376C2×C23semidirect
183. #510C4×C23semidirect
247. C23#13028direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5102
C2×C23×#4432
C2×C23×#3622
C23×#21392
C23×#19692
C23×#13872
C23×#13762
#1302823