Finite Group Databasebeta

← C23×#13417 #19161043

Subgroups — C23×#13417 #19161043

Order 5888.

454 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3155C211yesmaximal, characteristic
2C23×#2148C211yesmaximal, characteristic, complemented
3C2×C23×#361C211yesmaximal, characteristic, complemented
4C2×C23×#361C211yesmaximal, characteristic, complemented
5C23×#2143C211yesmaximal, characteristic, complemented
6C23×#1953C211yesmaximal, characteristic, complemented
7C23×#1378C211yesmaximal, characteristic, complemented
8C2×C23×AGL(1,ℤ/8)—21no—
9C2×C23×#135C2211yescharacteristic
10C22×D8×C23C2211yescharacteristic
11C23×#535—21no—
12C2×C23×M32C2211yescharacteristic
13C2×C23×M32C2211yescharacteristic
14C23×#496—22no—
15C23×#495—21no—
16C23×#493—21no—
17C23×#430—21no—
18C2×C23×#108C2211yescharacteristic
19C2×C23×#108C2211yescharacteristic
20C23×#405C2211yescharacteristic
21C23×#362—21no—
22C23×#361—22no—
23C23×#361—21no—
24C23×#361—22no—
25C23×#359—22no—
26C23×#344—21no—
27C2×#41×C23—21no—
28C22×D4×C23—21no—
29C23×AGL(1,ℤ/8)—42no—
30C23×AGL(1,ℤ/8)—42no—
31C23×#135—21no—
32C23×#135—21no—
33C23×#135—21no—
34C23×#135—21no—
35C2×Q16×C23D411yescharacteristic
36C2×SD16×C23—22no—
37C2×SD16×C23—21no—
38C2×D8×C23—22no—
39C2×D8×C23—21no—
40C2×D8×C23—21no—
41C2×D8×C23—21no—
42C2×D8×C23—21no—
43C2×D8×C23—21no—
44C2×D8×C23D412yes—
45C2×D8×C23D411yescharacteristic
46C2×M16×C23—21no—
47C2×M16×C23—21no—
48C2×M16×C23D411yescharacteristic
49C22×C8×C23C2311yescharacteristic
50C23×#127—22no—
51C23×#124—22no—
52C23×#121—42no—
53C23×#120—42no—
54C23×#117D411yescharacteristic
55C23×M32—21no—
56C23×M32—21no—
57C23×M32—21no—
58C23×M32—21no—
59C2×C16×C23—21no—
60C2×C16×C23—21no—
61C23×#108—21no—
62C23×#108—21no—
63C23×#108—21no—
64C23×#108—42no—
65C23×#108D412yescomplemented
66C23×#97—21no—
67C4×C8×C23—21no—
68C24×C23—41no—
69#41×C23—41no—
70#41×C23—41no—
71#41×C23—41no—
72#41×C23—41no—
73C2×Q8×C23—21no—
74C2×D4×C23—22no—
75C2×D4×C23—21no—
76C2×D4×C23—44no—
77C2×D4×C23—41no—
78C2×D4×C23—41no—
79C2×D4×C23—41no—
80C2×D4×C23—41no—
81C2×D4×C23—41no—
82C2×D4×C23—41no—
83C22×C4×C23—41no—
84C22×C4×C23C2×D411yescharacteristic
85Q16×C23—21no—
86Q16×C23—21no—
87SD16×C23—44no—
88SD16×C23—41no—
89SD16×C23—41no—
90D8×C23—22no—
91D8×C23—21no—
92D8×C23—22no—
93D8×C23—21no—
94D8×C23—44no—
95D8×C23—41no—
96D8×C23—41no—
97M16×C23—21no—
98M16×C23—22no—
99M16×C23—41no—
100M16×C23—41no—
101M16×C23—41no—
102M16×C23D812yes—
103C2×C8×C23—21no—
104C2×C8×C23—41no—
105C2×C8×C23—41no—
106C2×C8×C23C2×D411yescharacteristic
107C2×C8×C23C2×D411yescharacteristic
108C2×C8×C23C2×D411yescharacteristic
109C2×C8×C23#4111yescharacteristic
110C2×C8×C23#4111yescharacteristic
111C2×C8×C23#4111yescharacteristic
112C4⋊C4×C23—21no—
113C22⋊C4×C23—44no—
114C22⋊C4×C23SD1612yes—
115C42×C23—21no—
116C16×C23—41no—
117C16×C23—41no—
118#13417C2311yesmaximal, characteristic, direct product factor, Sylow
119C23×C23—42no—
120C23×C23—41no—
121C23×C23—81no—
122C23×C23—81no—
123C23×C23—162no—
124C23×C23C2×SD1611yescharacteristic
125Q8×C23—41no—
126Q8×C23—41no—
127D4×C23—42no—
128D4×C23—41no—
129D4×C23—41no—
130D4×C23—42no—
131D4×C23—81no—
132D4×C23—81no—
133D4×C23—81no—
134D4×C23—81no—
135D4×C23—164no—
136C2×C4×C23—22no—
137C2×C4×C23—41no—
138C2×C4×C23—81no—
139C2×C4×C23—81no—
140C2×C4×C23#12011yescharacteristic
141C2×C4×C23#12111yescharacteristic
142C2×C4×C23#12411yescharacteristic
143C2×C4×C23C2×D811yescharacteristic
144C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
146C8×C23—22no—
147C8×C23—42no—
148C8×C23—81no—
149C8×C23#12111yescharacteristic
150C8×C23#12111yescharacteristic
151C8×C23#12311yescharacteristic
152C8×C23#12311yescharacteristic
153#3155C2×C2311yescharacteristic
154#2148C2×C2311yescharacteristic, complemented
155C2×#361C2×C2311yescharacteristic, complemented
156C2×#361C2×C2311yescharacteristic, complemented
157#2143C2×C2311yescharacteristic, complemented
158#1953C2×C2311yescharacteristic, complemented
159#1378C2×C2311yescharacteristic, complemented
160C22×C23—41no—
161C22×C23—82no—
162C22×C23—84no—
163C22×C23—81no—
164C22×C23—82no—
165C22×C23—162no—
166C22×C23—162no—
167C22×C23#45011yescharacteristic
168C22×C23#46011yescharacteristic
169C22×C23#48611yescharacteristic
170C4×C23—42no—
171C4×C23—81no—
172C4×C23#39411yescharacteristic
173C4×C23#44711yescharacteristic
174C4×C23#45911yescharacteristic
175C4×C23#49011yescharacteristic
176C2×AGL(1,ℤ/8)—21no—
177C2×#135C22×C2311yescharacteristic
178C22×D8C22×C2311yescharacteristic
179#535—21no—
180C2×M32C22×C2311yescharacteristic
181C2×M32C22×C2311yescharacteristic
182#496—22no—
183#495—21no—
184#493—21no—
185#430—21no—
186C2×#108C22×C2311yescharacteristic
187C2×#108C22×C2311yescharacteristic
188#405C22×C2311yescharacteristic
189#362—21no—
190#361—22no—
191#361—21no—
192#361—22no—
193#359—22no—
194#344—21no—
195C2×C23—21no—
196C2×C23—21no—
197C2×C23—21no—
198C2×C23—161no—
199C2×C23—162no—
200C2×C23#204111yescharacteristic
201C2×#41—21no—
202C22×D4—21no—
203AGL(1,ℤ/8)—42no—
204AGL(1,ℤ/8)—42no—
205#135—21no—
206#135—21no—
207#135—21no—
208#135—21no—
209C2×Q16D4×C2311yescharacteristic
210C2×SD16—22no—
211C2×SD16—21no—
212C2×D8—22no—
213C2×D8—21no—
214C2×D8—21no—
215C2×D8—21no—
216C2×D8—21no—
217C2×D8—21no—
218C2×D8D4×C2312yes—
219C2×D8D4×C2311yescharacteristic
220C2×M16—21no—
221C2×M16—21no—
222C2×M16D4×C2311yescharacteristic
223C22×C8C23×C2311yescharacteristic
224#127—22no—
225#124—22no—
226#121—42no—
227#120—42no—
228#117D4×C2311yescharacteristic
229M32—21no—
230M32—21no—
231M32—21no—
232M32—21no—
233C2×C16—21no—
234C2×C16—21no—
235#108—21no—
236#108—21no—
237#108—21no—
238#108—42no—
239#108D4×C2312yescomplemented
240#97—21no—
241C4×C8—21no—
242C23#1341711yescharacteristic, direct product factor, Sylow
243C24—41no—
244#41—41no—
245#41—41no—
246#41—41no—
247#41—41no—
248C2×Q8—21no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—44no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C22×C4—41no—
259C22×C4C2×D4×C2311yescharacteristic
260Q16—21no—
261Q16—21no—
262SD16—44no—
263SD16—41no—
264SD16—41no—
265D8—22no—
266D8—21no—
267D8—22no—
268D8—21no—
269D8—44no—
270D8—41no—
271D8—41no—
272M16—21no—
273M16—22no—
274M16—41no—
275M16—41no—
276M16—41no—
277M16D8×C2312yes—
278C2×C8—21no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8C2×D4×C2311yescharacteristic
282C2×C8C2×D4×C2311yescharacteristic
283C2×C8C2×D4×C2311yescharacteristic
284C2×C8#41×C2311yescharacteristic
285C2×C8#41×C2311yescharacteristic
286C2×C8#41×C2311yescharacteristic
287C4⋊C4—21no—
288C22⋊C4—44no—
289C22⋊C4SD16×C2312yes—
290C42—21no—
291C16—41no—
292C16—41no—
293C23—42no—
294C23—41no—
295C23—81no—
296C23—81no—
297C23—162no—
298C23C2×SD16×C2311yescharacteristic
299Q8—41no—
300Q8—41no—
301D4—42no—
302D4—41no—
303D4—41no—
304D4—42no—
305D4—81no—
306D4—81no—
307D4—81no—
308D4—81no—
309D4—164no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4C23×#12011yescharacteristic
315C2×C4C23×#12111yescharacteristic
316C2×C4C23×#12411yescharacteristic
317C2×C4C2×D8×C2311yescharacteristic
318C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
319C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
320C8—22no—
321C8—42no—
322C8—81no—
323C8C23×#12111yescharacteristic
324C8C23×#12111yescharacteristic
325C8C23×#12311yescharacteristic
326C8C23×#12311yescharacteristic
327C22—41no—
328C22—82no—
329C22—84no—
330C22—81no—
331C22—82no—
332C22—162no—
333C22—162no—
334C22C23×#45011yescharacteristic
335C22C23×#46011yescharacteristic
336C22C23×#48611yescharacteristic
337C4—42no—
338C4—81no—
339C4C23×#39411yescharacteristic
340C4C23×#44711yescharacteristic
341C4C23×#45911yescharacteristic
342C4C23×#49011yescharacteristic
343C2—21no—
344C2—21no—
345C2—21no—
346C2—162no—
347C2—161no—
348C2C23×#204111yescharacteristic
subgroupG/H
center, socleC2×C23#2041
commutator, Frattini, cosocleC22×C8C23×C23
prime radicalC23×#3155C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C22×C4×C23 ◁ C23×#405 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2148C2semidirect
3. C2×C23×#361C2semidirect
4. C2×C23×#361C2semidirect
5. C23×#2143C2semidirect
6. C23×#1953C2semidirect
7. C23×#1378C2semidirect
65. C23×#108D4semidirect
118. #13417C23direct (N × H)
154. #2148C2×C23semidirect
155. C2×#361C2×C23semidirect
156. C2×#361C2×C23semidirect
157. #2143C2×C23semidirect
158. #1953C2×C23semidirect
159. #1378C2×C23semidirect
239. #108D4×C23semidirect
242. C23#13417direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31552
C23×#21482
C2×C23×#3612
C23×#21432
C23×#19532
C23×#13782
#1341723