Finite Group Databasebeta

← C23×#13318 #19160944

Subgroups — C23×#13318 #19160944

Order 5888.

532 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3414C211yesmaximal, characteristic, complemented
2C23×#3067C211yesmaximal, characteristic, complemented
3C23×#3060C211yesmaximal, characteristic, complemented
4C2×C23×#360C211yesmaximal, characteristic, complemented
5C23×#2143C211yesmaximal, characteristic, complemented
6C23×#2117C211yesmaximal, characteristic, complemented
7C23×#1383C211yesmaximal, characteristic, complemented
8C23×#575—21no—
9C2×C23×#135C2211yescharacteristic
10C2×C23×#135C2211yescharacteristic
11C2×C23×#131C2211yescharacteristic
12C23×#537—21no—
13C23×#510—21no—
14C23×#510—21no—
15C23×#509—21no—
16C23×#509—21no—
17C23×#508—22no—
18C2×C23×M32C2211yescharacteristic
19C2×C23×M32C2211yescharacteristic
20C23×#473—21no—
21C23×#472—21no—
22C23×#471—22no—
23C23×#470—21no—
24C23×#469—21no—
25C23×#465—22no—
26C2×C23×#108C2211yescharacteristic
27C2×C23×#106C2211yescharacteristic
28C23×#417—21no—
29C23×#413—21no—
30C23×#360—22no—
31C23×#359—22no—
32C23×#350—22no—
33C2×#41×C23—21no—
34C2×#41×C23—21no—
35C2×#41×C23D411yescharacteristic
36C23×#137—22no—
37C23×#137—41no—
38C23×AGL(1,ℤ/8)—22no—
39C23×AGL(1,ℤ/8)—41no—
40C23×#135—21no—
41C23×#135—21no—
42C23×#135—21no—
43C23×#135—21no—
44C23×#135—21no—
45C23×#135—21no—
46C23×#135—21no—
47C23×#135—21no—
48C23×#135—42no—
49C2×Q16×C23D411yescharacteristic
50C2×Q16×C23D411yescharacteristic, complemented
51C2×SD16×C23—21no—
52C2×SD16×C23—21no—
53C2×D8×C23D411yescharacteristic
54C2×D8×C23D411yescharacteristic, complemented
55C23×#131—21no—
56C23×#131—21no—
57C23×#131—21no—
58C23×#131—21no—
59C2×M16×C23—21no—
60C2×M16×C23—21no—
61C2×M16×C23D411yescharacteristic
62C22×C8×C23—21no—
63C22×C8×C23C2311yescharacteristic
64C23×C22⋊Q8—22no—
65C23×C22⋊Q8—41no—
66C23×#121—22no—
67C23×#121—42no—
68C23×#121—41no—
69C2×C4⋊C4×C23—21no—
70C23×Q32—22no—
71C23×SD32—22no—
72C23×SD32—22no—
73C23×D16—22no—
74C23×M32—21no—
75C23×M32—21no—
76C23×M32—21no—
77C23×M32—21no—
78C2×C16×C23—21no—
79C2×C16×C23D412yescomplemented
80C23×#108—21no—
81C23×#108D412yescomplemented
82C23×#106—21no—
83C23×#106D412yes—
84C23×#103—22no—
85C23×#102—22no—
86C23×#101—22no—
87C23×#100—22no—
88#41×C23—21no—
89#41×C23—21no—
90#41×C23—21no—
91#41×C23—21no—
92#41×C23—41no—
93#41×C23—41no—
94#41×C23—41no—
95#41×C23—41no—
96#41×C23—41no—
97#41×C23—41no—
98#41×C23—41no—
99#41×C23—41no—
100C2×Q8×C23—21no—
101C2×Q8×C23—21no—
102C2×Q8×C23D811yescharacteristic
103C2×D4×C23—21no—
104C2×D4×C23—21no—
105C2×D4×C23—21no—
106C2×D4×C23—41no—
107C2×D4×C23—41no—
108C2×D4×C23—42no—
109C2×D4×C23D811yescharacteristic
110C22×C4×C23—21no—
111C22×C4×C23—41no—
112C22×C4×C23—41no—
113C22×C4×C23—41no—
114C22×C4×C23C2×D411yescharacteristic
115Q16×C23—21no—
116Q16×C23—21no—
117Q16×C23—21no—
118Q16×C23—21no—
119Q16×C23—42no—
120SD16×C23—42no—
121SD16×C23—42no—
122SD16×C23—41no—
123SD16×C23—41no—
124SD16×C23—41no—
125SD16×C23—41no—
126D8×C23—21no—
127D8×C23—21no—
128D8×C23—21no—
129D8×C23—21no—
130D8×C23—42no—
131M16×C23—21no—
132M16×C23—22no—
133M16×C23—21no—
134M16×C23—41no—
135M16×C23—41no—
136M16×C23—41no—
137C2×C8×C23—21no—
138C2×C8×C23—41no—
139C2×C8×C23—41no—
140C2×C8×C23—41no—
141C2×C8×C23C2×D411yescharacteristic
142C2×C8×C23C2×D411yescharacteristic
143C2×C8×C23C2×D411yescharacteristic
144C2×C8×C23C2×D411yescharacteristic
145C2×C8×C23C2×D411yescharacteristic
146C2×C8×C23#4111yescharacteristic
147C4⋊C4×C23—22no—
148C4⋊C4×C23—42no—
149C4⋊C4×C23—41no—
150C22⋊C4×C23—42no—
151C22⋊C4×C23—42no—
152C16×C23—22no—
153C16×C23—41no—
154#13318C2311yesmaximal, characteristic, direct product factor, Sylow
155C23×C23—21no—
156C23×C23—41no—
157C23×C23—41no—
158C23×C23AGL(1,ℤ/8)11yescharacteristic
159Q8×C23—21no—
160Q8×C23—21no—
161Q8×C23—41no—
162Q8×C23—41no—
163Q8×C23—41no—
164Q8×C23—41no—
165D4×C23—21no—
166D4×C23—21no—
167D4×C23—41no—
168D4×C23—41no—
169D4×C23—41no—
170D4×C23—41no—
171D4×C23—41no—
172D4×C23—41no—
173D4×C23—81no—
174D4×C23—81no—
175D4×C23—81no—
176D4×C23—81no—
177D4×C23—162no—
178C2×C4×C23—21no—
179C2×C4×C23—41no—
180C2×C4×C23—41no—
181C2×C4×C23—42no—
182C2×C4×C23—41no—
183C2×C4×C23—41no—
184C2×C4×C23—81no—
185C2×C4×C23—81no—
186C2×C4×C23—81no—
187C2×C4×C23—81no—
188C2×C4×C23—161no—
189C2×C4×C23#12011yescharacteristic
190C2×C4×C23#12011yescharacteristic
191C2×C4×C23#12111yescharacteristic
192C2×C4×C23C2×D811yescharacteristic
193C2×C4×C23#13511yescharacteristic
194C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
195C8×C23—41no—
196C8×C23—42no—
197C8×C23#12011yescharacteristic
198C8×C23#12111yescharacteristic
199C8×C23#12111yescharacteristic
200C8×C23#12711yescharacteristic
201#3414C2×C2311yescharacteristic, complemented
202#3067C2×C2311yescharacteristic, complemented
203#3060C2×C2311yescharacteristic, complemented
204C2×#360C2×C2311yescharacteristic, complemented
205#2143C2×C2311yescharacteristic, complemented
206#2117C2×C2311yescharacteristic, complemented
207#1383C2×C2311yescharacteristic, complemented
208C22×C23—41no—
209C22×C23—42no—
210C22×C23—41no—
211C22×C23—81no—
212C22×C23—81no—
213C22×C23—81no—
214C22×C23—81no—
215C22×C23—161no—
216C22×C23#44711yescharacteristic
217C22×C23#44911yescharacteristic
218C22×C23#46911yescharacteristic
219C4×C23—41no—
220C4×C23—82no—
221C4×C23—81no—
222C4×C23—81no—
223C4×C23#39211yescharacteristic
224C4×C23#44711yescharacteristic
225C4×C23#44911yescharacteristic
226C4×C23#46611yescharacteristic
227#575—21no—
228C2×#135C22×C2311yescharacteristic
229C2×#135C22×C2311yescharacteristic
230C2×#131C22×C2311yescharacteristic
231#537—21no—
232#510—21no—
233#510—21no—
234#509—21no—
235#509—21no—
236#508—22no—
237C2×M32C22×C2311yescharacteristic
238C2×M32C22×C2311yescharacteristic
239#473—21no—
240#472—21no—
241#471—22no—
242#470—21no—
243#469—21no—
244#465—22no—
245C2×#108C22×C2311yescharacteristic
246C2×#106C22×C2311yescharacteristic
247#417—21no—
248#413—21no—
249#360—22no—
250#359—22no—
251#350—22no—
252C2×C23—21no—
253C2×C23—21no—
254C2×C23—21no—
255C2×C23—81no—
256C2×C23—161no—
257C2×C23—161no—
258C2×C23#199911yescharacteristic
259C2×#41—21no—
260C2×#41—21no—
261C2×#41D4×C2311yescharacteristic
262#137—22no—
263#137—41no—
264AGL(1,ℤ/8)—22no—
265AGL(1,ℤ/8)—41no—
266#135—21no—
267#135—21no—
268#135—21no—
269#135—21no—
270#135—21no—
271#135—21no—
272#135—21no—
273#135—21no—
274#135—42no—
275C2×Q16D4×C2311yescharacteristic
276C2×Q16D4×C2311yescharacteristic, complemented
277C2×SD16—21no—
278C2×SD16—21no—
279C2×D8D4×C2311yescharacteristic
280C2×D8D4×C2311yescharacteristic, complemented
281#131—21no—
282#131—21no—
283#131—21no—
284#131—21no—
285C2×M16—21no—
286C2×M16—21no—
287C2×M16D4×C2311yescharacteristic
288C22×C8—21no—
289C22×C8C23×C2311yescharacteristic
290C22⋊Q8—22no—
291C22⋊Q8—41no—
292#121—22no—
293#121—42no—
294#121—41no—
295C2×C4⋊C4—21no—
296Q32—22no—
297SD32—22no—
298SD32—22no—
299D16—22no—
300M32—21no—
301M32—21no—
302M32—21no—
303M32—21no—
304C2×C16—21no—
305C2×C16D4×C2312yescomplemented
306#108—21no—
307#108D4×C2312yescomplemented
308#106—21no—
309#106D4×C2312yes—
310#103—22no—
311#102—22no—
312#101—22no—
313#100—22no—
314C23#1331811yescharacteristic, direct product factor, Sylow
315#41—21no—
316#41—21no—
317#41—21no—
318#41—21no—
319#41—41no—
320#41—41no—
321#41—41no—
322#41—41no—
323#41—41no—
324#41—41no—
325#41—41no—
326#41—41no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8D8×C2311yescharacteristic
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—42no—
336C2×D4D8×C2311yescharacteristic
337C22×C4—21no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4C2×D4×C2311yescharacteristic
342Q16—21no—
343Q16—21no—
344Q16—21no—
345Q16—21no—
346Q16—42no—
347SD16—42no—
348SD16—42no—
349SD16—41no—
350SD16—41no—
351SD16—41no—
352SD16—41no—
353D8—21no—
354D8—21no—
355D8—21no—
356D8—21no—
357D8—42no—
358M16—21no—
359M16—22no—
360M16—21no—
361M16—41no—
362M16—41no—
363M16—41no—
364C2×C8—21no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8C2×D4×C2311yescharacteristic
369C2×C8C2×D4×C2311yescharacteristic
370C2×C8C2×D4×C2311yescharacteristic
371C2×C8C2×D4×C2311yescharacteristic
372C2×C8C2×D4×C2311yescharacteristic
373C2×C8#41×C2311yescharacteristic
374C4⋊C4—22no—
375C4⋊C4—42no—
376C4⋊C4—41no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C16—22no—
380C16—41no—
381C23—21no—
382C23—41no—
383C23—41no—
384C23C23×AGL(1,ℤ/8)11yescharacteristic
385Q8—21no—
386Q8—21no—
387Q8—41no—
388Q8—41no—
389Q8—41no—
390Q8—41no—
391D4—21no—
392D4—21no—
393D4—41no—
394D4—41no—
395D4—41no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—81no—
400D4—81no—
401D4—81no—
402D4—81no—
403D4—162no—
404C2×C4—21no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—161no—
415C2×C4C23×#12011yescharacteristic
416C2×C4C23×#12011yescharacteristic
417C2×C4C23×#12111yescharacteristic
418C2×C4C2×D8×C2311yescharacteristic
419C2×C4C23×#13511yescharacteristic
420C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
421C8—41no—
422C8—42no—
423C8C23×#12011yescharacteristic
424C8C23×#12111yescharacteristic
425C8C23×#12111yescharacteristic
426C8C23×#12711yescharacteristic
427C22—41no—
428C22—42no—
429C22—41no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—161no—
435C22C23×#44711yescharacteristic
436C22C23×#44911yescharacteristic
437C22C23×#46911yescharacteristic
438C4—41no—
439C4—82no—
440C4—81no—
441C4—81no—
442C4C23×#39211yescharacteristic
443C4C23×#44711yescharacteristic
444C4C23×#44911yescharacteristic
445C4C23×#46611yescharacteristic
446C2—21no—
447C2—21no—
448C2—21no—
449C2—81no—
450C2—161no—
451C2—161no—
452C2C23×#199911yescharacteristic
subgroupG/H
center, socleC2×C23#1999
commutator, Frattini, cosocleC22×C8C23×C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C22×C4×C23 ◁ C2×C23×#131 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3414C2semidirect
2. C23×#3067C2semidirect
3. C23×#3060C2semidirect
4. C2×C23×#360C2semidirect
5. C23×#2143C2semidirect
6. C23×#2117C2semidirect
7. C23×#1383C2semidirect
50. C2×Q16×C23D4semidirect
54. C2×D8×C23D4semidirect
79. C2×C16×C23D4semidirect
81. C23×#108D4semidirect
154. #13318C23direct (N × H)
201. #3414C2×C23semidirect
202. #3067C2×C23semidirect
203. #3060C2×C23semidirect
204. C2×#360C2×C23semidirect
205. #2143C2×C23semidirect
206. #2117C2×C23semidirect
207. #1383C2×C23semidirect
276. C2×Q16D4×C23semidirect
280. C2×D8D4×C23semidirect
305. C2×C16D4×C23semidirect
307. #108D4×C23semidirect
314. C23#13318direct (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×#30672
C23×#30602
C2×C23×#3602
C23×#21432
C23×#21172
C23×#13832
#1331823