Finite Group Databasebeta

← C23×#13021 #19160647

Subgroups — C23×#13021 #19160647

Order 5888.

462 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3414C211yesmaximal, characteristic
2C23×#2957C211yesmaximal, characteristic, complemented
3C2×C23×#359C211yesmaximal, characteristic, complemented
4C23×#2138C211yesmaximal, characteristic, complemented
5C23×#1976C211yesmaximal, characteristic, complemented
6C23×#1389C211yesmaximal, characteristic, complemented
7C23×#1379C211yesmaximal, characteristic, complemented
8C2×C23×#135C2211yescharacteristic
9C2×C23×#135C2211yescharacteristic
10C23×#568—21no—
11C23×#510—21no—
12C23×#510C412yescomplemented
13C23×#509—21no—
14C23×#509C412yescomplemented
15C23×#508—22no—
16C2×C23×M32C2211yescharacteristic
17C22×C16×C23C2211yescharacteristic
18C23×#444—22no—
19C23×#443—22no—
20C2×C23×#108C2211yescharacteristic
21C23×#427C2211yescharacteristic
22C23×#421—21no—
23C23×#421—21no—
24C23×#410—21no—
25C23×#405C2211yescharacteristic
26C23×#359—22no—
27C23×#358—21no—
28C23×#357—21no—
29C2×#41×C23—21no—
30C2×#41×C23—21no—
31C23×#135—21no—
32C23×#135—21no—
33C23×#135—21no—
34C23×#135—21no—
35C23×#135—21no—
36C23×#135—21no—
37C23×#135C2×C412yes—
38C23×#135D412yes—
39C2×Q16×C23C2×C411yescharacteristic
40C2×Q16×C23D411yescharacteristic
41C2×SD16×C23—21no—
42C2×SD16×C23—21no—
43C2×D8×C23C2×C411yescharacteristic
44C2×D8×C23D411yescharacteristic
45C23×#131—24no—
46C23×#131—42no—
47C2×M16×C23—21no—
48C2×M16×C23—42no—
49C2×M16×C23D411yescharacteristic
50C22×C8×C23C2311yescharacteristic
51C23×#117—21no—
52C23×#117D411yescharacteristic
53C23×Q32—22no—
54C23×SD32—22no—
55C23×SD32—22no—
56C23×D16—22no—
57C23×M32—21no—
58C23×M32C2×C412yes—
59C2×C16×C23—21no—
60C2×C16×C23—21no—
61C2×C16×C23—21no—
62C2×C16×C23—21no—
63C23×#108—21no—
64C23×#108D412yes—
65C23×#107—21no—
66C23×#106—21no—
67C23×#104—24no—
68C23×#104—42no—
69C23×#99—42no—
70C23×#97—21no—
71C4×C8×C23—21no—
72#41×C23—22no—
73#41×C23—22no—
74#41×C23—41no—
75#41×C23—41no—
76#41×C23—41no—
77#41×C23—41no—
78#41×C23—41no—
79#41×C23—41no—
80C2×Q8×C23—21no—
81C2×Q8×C23—21no—
82C2×D4×C23—21no—
83C2×D4×C23—21no—
84C2×D4×C23—41no—
85C2×D4×C23—41no—
86C22×C4×C23—41no—
87C22×C4×C23—41no—
88C22×C4×C23C2×D411yescharacteristic
89Q16×C23—21no—
90Q16×C23—21no—
91Q16×C23—21no—
92Q16×C23C22⋊C412yes—
93SD16×C23—22no—
94SD16×C23—22no—
95SD16×C23—41no—
96SD16×C23—41no—
97D8×C23—21no—
98D8×C23—21no—
99D8×C23—21no—
100D8×C23C22⋊C412yes—
101M16×C23—21no—
102M16×C23—22no—
103M16×C23—44no—
104M16×C23—41no—
105M16×C23D812yes—
106C2×C8×C23—21no—
107C2×C8×C23—44no—
108C2×C8×C23—41no—
109C2×C8×C23C22×C411yescharacteristic
110C2×C8×C23C2×D411yescharacteristic
111C2×C8×C23C2×D411yescharacteristic
112C2×C8×C23C2×D411yescharacteristic
113C2×C8×C23#4111yescharacteristic
114C2×C8×C23#4111yescharacteristic
115C4⋊C4×C23—21no—
116C4⋊C4×C23—41no—
117C22⋊C4×C23—21no—
118C22⋊C4×C23—41no—
119C42×C23—21no—
120C42×C23—41no—
121C16×C23—22no—
122C16×C23—41no—
123#13021C2311yesmaximal, characteristic, direct product factor, Sylow
124C23×C23—41no—
125C23×C23—41no—
126C23×C23AGL(1,ℤ/8)11yescharacteristic
127Q8×C23—22no—
128Q8×C23—41no—
129Q8×C23—41no—
130Q8×C23—41no—
131D4×C23—22no—
132D4×C23—42no—
133D4×C23—42no—
134D4×C23—41no—
135D4×C23—41no—
136D4×C23—41no—
137D4×C23—81no—
138D4×C23—81no—
139C2×C4×C23—41no—
140C2×C4×C23—42no—
141C2×C4×C23—42no—
142C2×C4×C23—41no—
143C2×C4×C23—41no—
144C2×C4×C23—81no—
145C2×C4×C23—81no—
146C2×C4×C23—81no—
147C2×C4×C23C4×D411yescharacteristic
148C2×C4×C23#12011yescharacteristic
149C2×C4×C23#12111yescharacteristic
150C2×C4×C23C2×D811yescharacteristic
151C2×C4×C23#13511yescharacteristic
152C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
153C8×C23—22no—
154C8×C23—42no—
155C8×C23C2×C22⋊C411yescharacteristic
156C8×C23C4×D411yescharacteristic
157C8×C23#12111yescharacteristic
158C8×C23#12311yescharacteristic
159#3414C2×C2311yescharacteristic
160#2957C2×C2311yescharacteristic, complemented
161C2×#359C2×C2311yescharacteristic, complemented
162#2138C2×C2311yescharacteristic, complemented
163#1976C2×C2311yescharacteristic, complemented
164#1389C2×C2311yescharacteristic, complemented
165#1379C2×C2311yescharacteristic, complemented
166C22×C23—41no—
167C22×C23—42no—
168C22×C23—81no—
169C22×C23—161no—
170C22×C23—161no—
171C22×C23#44211yescharacteristic
172C22×C23#44911yescharacteristic
173C22×C23#45911yescharacteristic
174C4×C23—42no—
175C4×C23—81no—
176C4×C23—81no—
177C4×C23—161no—
178C4×C23#38611yescharacteristic
179C4×C23C4×D811yescharacteristic
180C4×C23#44711yescharacteristic
181C4×C23#46311yescharacteristic
182C2×#135C22×C2311yescharacteristic
183C2×#135C22×C2311yescharacteristic
184#568—21no—
185#510—21no—
186#510C4×C2312yescomplemented
187#509—21no—
188#509C4×C2312yescomplemented
189#508—22no—
190C2×M32C22×C2311yescharacteristic
191C22×C16C22×C2311yescharacteristic
192#444—22no—
193#443—22no—
194C2×#108C22×C2311yescharacteristic
195#427C22×C2311yescharacteristic
196#421—21no—
197#421—21no—
198#410—21no—
199#405C22×C2311yescharacteristic
200#359—22no—
201#358—21no—
202#357—21no—
203C2×C23—21no—
204C2×C23—21no—
205C2×C23—21no—
206C2×C23—82no—
207C2×C23—161no—
208C2×C23#187911yescharacteristic
209C2×#41—21no—
210C2×#41—21no—
211#135—21no—
212#135—21no—
213#135—21no—
214#135—21no—
215#135—21no—
216#135—21no—
217#135C2×C4×C2312yes—
218#135D4×C2312yes—
219C2×Q16C2×C4×C2311yescharacteristic
220C2×Q16D4×C2311yescharacteristic
221C2×SD16—21no—
222C2×SD16—21no—
223C2×D8C2×C4×C2311yescharacteristic
224C2×D8D4×C2311yescharacteristic
225#131—24no—
226#131—42no—
227C2×M16—21no—
228C2×M16—42no—
229C2×M16D4×C2311yescharacteristic
230C22×C8C23×C2311yescharacteristic
231#117—21no—
232#117D4×C2311yescharacteristic
233Q32—22no—
234SD32—22no—
235SD32—22no—
236D16—22no—
237M32—21no—
238M32C2×C4×C2312yes—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243#108—21no—
244#108D4×C2312yes—
245#107—21no—
246#106—21no—
247#104—24no—
248#104—42no—
249#99—42no—
250#97—21no—
251C4×C8—21no—
252C23#1302111yescharacteristic, direct product factor, Sylow
253#41—22no—
254#41—22no—
255#41—41no—
256#41—41no—
257#41—41no—
258#41—41no—
259#41—41no—
260#41—41no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—41no—
266C2×D4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4C2×D4×C2311yescharacteristic
270Q16—21no—
271Q16—21no—
272Q16—21no—
273Q16C22⋊C4×C2312yes—
274SD16—22no—
275SD16—22no—
276SD16—41no—
277SD16—41no—
278D8—21no—
279D8—21no—
280D8—21no—
281D8C22⋊C4×C2312yes—
282M16—21no—
283M16—22no—
284M16—44no—
285M16—41no—
286M16D8×C2312yes—
287C2×C8—21no—
288C2×C8—44no—
289C2×C8—41no—
290C2×C8C22×C4×C2311yescharacteristic
291C2×C8C2×D4×C2311yescharacteristic
292C2×C8C2×D4×C2311yescharacteristic
293C2×C8C2×D4×C2311yescharacteristic
294C2×C8#41×C2311yescharacteristic
295C2×C8#41×C2311yescharacteristic
296C4⋊C4—21no—
297C4⋊C4—41no—
298C22⋊C4—21no—
299C22⋊C4—41no—
300C42—21no—
301C42—41no—
302C16—22no—
303C16—41no—
304C23—41no—
305C23—41no—
306C23C23×AGL(1,ℤ/8)11yescharacteristic
307Q8—22no—
308Q8—41no—
309Q8—41no—
310Q8—41no—
311D4—22no—
312D4—42no—
313D4—42no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—81no—
318D4—81no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4C4×D4×C2311yescharacteristic
328C2×C4C23×#12011yescharacteristic
329C2×C4C23×#12111yescharacteristic
330C2×C4C2×D8×C2311yescharacteristic
331C2×C4C23×#13511yescharacteristic
332C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
333C8—22no—
334C8—42no—
335C8C2×C22⋊C4×C2311yescharacteristic
336C8C4×D4×C2311yescharacteristic
337C8C23×#12111yescharacteristic
338C8C23×#12311yescharacteristic
339C22—41no—
340C22—42no—
341C22—81no—
342C22—161no—
343C22—161no—
344C22C23×#44211yescharacteristic
345C22C23×#44911yescharacteristic
346C22C23×#45911yescharacteristic
347C4—42no—
348C4—81no—
349C4—81no—
350C4—161no—
351C4C23×#38611yescharacteristic
352C4C4×D8×C2311yescharacteristic
353C4C23×#44711yescharacteristic
354C4C23×#46311yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—82no—
359C2—161no—
360C2C23×#187911yescharacteristic
subgroupG/H
centerC4×C23#447
commutatorC2×C8C22×C4×C23
Frattini, cosocleC22×C8C23×C23
socleC2×C23#1879
prime radicalC23×#3414C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C23 ◁ C22×C4×C23 ◁ C23×#405 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2957C2semidirect
3. C2×C23×#359C2semidirect
4. C23×#2138C2semidirect
5. C23×#1976C2semidirect
6. C23×#1389C2semidirect
7. C23×#1379C2semidirect
12. C23×#510C4semidirect
14. C23×#509C4semidirect
123. #13021C23direct (N × H)
160. #2957C2×C23semidirect
161. C2×#359C2×C23semidirect
162. #2138C2×C23semidirect
163. #1976C2×C23semidirect
164. #1389C2×C23semidirect
165. #1379C2×C23semidirect
186. #510C4×C23semidirect
188. #509C4×C23semidirect
252. C23#13021direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34142
C23×#29572
C2×C23×#3592
C23×#21382
C23×#19762
C23×#13892
C23×#13792
#1302123