Finite Group Databasebeta

← C23×#13310 #19160936

Subgroups — C23×#13310 #19160936

Order 5888.

552 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#509C211yesmaximal, characteristic, complemented
2C23×#3067C211yesmaximal, characteristic, complemented
3C23×#3058C211yesmaximal, characteristic, complemented
4C2×C23×#361C211yesmaximal, characteristic, complemented
5C23×#2145C211yesmaximal, characteristic, complemented
6C23×#2117C211yesmaximal, characteristic, complemented
7C23×#1389C211yesmaximal, characteristic, complemented
8C23×#575—21no—
9C2×C23×#135C2211yescharacteristic
10C22×D8×C23C2211yescharacteristic
11C2×C23×#131C2211yescharacteristic
12C23×#535—21no—
13C23×#509—22no—
14C23×#509—22no—
15C2×C23×SD32—21no—
16C2×C23×D16—21no—
17C2×C23×M32C2211yescharacteristic
18C2×C23×M32C2211yescharacteristic
19C23×#473—21no—
20C23×#472—21no—
21C23×#471—22no—
22C23×#469—22no—
23C23×#466—22no—
24C2×C23×#108C2211yescharacteristic
25C23×#427C2211yescharacteristic
26C23×#418—21no—
27C23×#413—21no—
28C23×#361—22no—
29C23×#360—22no—
30C23×#350—22no—
31C2×#41×C23—21no—
32C2×#41×C23D411yescharacteristic
33C22×D4×C23—21no—
34C23×#137—22no—
35C23×#137—41no—
36C23×AGL(1,ℤ/8)—22no—
37C23×AGL(1,ℤ/8)—41no—
38C23×#135—21no—
39C23×#135—21no—
40C23×#135—21no—
41C23×#135—21no—
42C23×#135—42no—
43C2×Q16×C23D411yescharacteristic, complemented
44C2×SD16×C23—21no—
45C2×D8×C23—21no—
46C2×D8×C23—21no—
47C2×D8×C23—21no—
48C2×D8×C23—21no—
49C2×D8×C23—21no—
50C2×D8×C23D412yes—
51C2×D8×C23D411yescharacteristic, complemented
52C23×#131—21no—
53C23×#131—21no—
54C23×#131—21no—
55C23×#131—21no—
56C2×M16×C23—21no—
57C2×M16×C23—21no—
58C2×M16×C23D411yescharacteristic
59C22×C8×C23—21no—
60C22×C8×C23C2311yescharacteristic
61C23×#127—41no—
62C23×#124—41no—
63C23×#121—24no—
64C23×#120—42no—
65C23×#117—21no—
66C23×SD32—24no—
67C23×D16—24no—
68C23×M32—21no—
69C23×M32—21no—
70C23×M32—21no—
71C23×M32D412yescomplemented
72C2×C16×C23—21no—
73C2×C16×C23—21no—
74C23×#108—21no—
75C23×#108D412yescomplemented
76C23×#107—21no—
77C23×#106D412yes—
78C23×#102—24no—
79C23×#101—22no—
80C23×#100—22no—
81C24×C23—41no—
82#41×C23—21no—
83#41×C23—21no—
84#41×C23—21no—
85#41×C23—21no—
86#41×C23—41no—
87#41×C23—41no—
88#41×C23—41no—
89#41×C23—41no—
90C2×Q8×C23—21no—
91C2×Q8×C23D811yescharacteristic
92C2×D4×C23—21no—
93C2×D4×C23—21no—
94C2×D4×C23—22no—
95C2×D4×C23—41no—
96C2×D4×C23—44no—
97C2×D4×C23—41no—
98C2×D4×C23—41no—
99C2×D4×C23—41no—
100C2×D4×C23—41no—
101C2×D4×C23—41no—
102C2×D4×C23D811yescharacteristic
103C22×C4×C23—21no—
104C22×C4×C23—41no—
105C22×C4×C23C2×D411yescharacteristic
106Q16×C23—21no—
107Q16×C23—21no—
108Q16×C23—42no—
109SD16×C23—42no—
110SD16×C23—42no—
111SD16×C23—41no—
112SD16×C23—41no—
113D8×C23—21no—
114D8×C23—22no—
115D8×C23—22no—
116D8×C23—21no—
117D8×C23—42no—
118D8×C23—41no—
119D8×C23—41no—
120M16×C23—21no—
121M16×C23—22no—
122M16×C23—21no—
123M16×C23—41no—
124M16×C23—41no—
125M16×C23—41no—
126C2×C8×C23—21no—
127C2×C8×C23—41no—
128C2×C8×C23—41no—
129C2×C8×C23—41no—
130C2×C8×C23C2×D411yescharacteristic
131C2×C8×C23C2×D411yescharacteristic
132C2×C8×C23C2×D411yescharacteristic
133C2×C8×C23C2×D411yescharacteristic
134C2×C8×C23C2×D411yescharacteristic
135C2×C8×C23#4111yescharacteristic
136C4⋊C4×C23—22no—
137C22⋊C4×C23—44no—
138C22⋊C4×C23—41no—
139C42×C23—41no—
140C16×C23—22no—
141C16×C23—41no—
142#13310C2311yesmaximal, characteristic, direct product factor, Sylow
143C23×C23—21no—
144C23×C23—42no—
145C23×C23—41no—
146C23×C23—81no—
147C23×C23—81no—
148C23×C23—162no—
149C23×C23AGL(1,ℤ/8)11yescharacteristic
150Q8×C23—21no—
151Q8×C23—21no—
152Q8×C23—41no—
153Q8×C23—41no—
154D4×C23—21no—
155D4×C23—21no—
156D4×C23—41no—
157D4×C23—41no—
158D4×C23—42no—
159D4×C23—41no—
160D4×C23—41no—
161D4×C23—42no—
162D4×C23—81no—
163D4×C23—81no—
164D4×C23—81no—
165D4×C23—81no—
166D4×C23—164no—
167C2×C4×C23—21no—
168C2×C4×C23—41no—
169C2×C4×C23—41no—
170C2×C4×C23—42no—
171C2×C4×C23—41no—
172C2×C4×C23—81no—
173C2×C4×C23—81no—
174C2×C4×C23#12011yescharacteristic
175C2×C4×C23#12011yescharacteristic
176C2×C4×C23#12111yescharacteristic
177C2×C4×C23C2×D811yescharacteristic
178C2×C4×C23#13511yescharacteristic
179C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
180C8×C23—41no—
181C8×C23—42no—
182C8×C23#12011yescharacteristic
183C8×C23#12111yescharacteristic
184C8×C23#12111yescharacteristic
185C8×C23#12711yescharacteristic
186C2×#509C2×C2311yescharacteristic, complemented
187#3067C2×C2311yescharacteristic, complemented
188#3058C2×C2311yescharacteristic, complemented
189C2×#361C2×C2311yescharacteristic, complemented
190#2145C2×C2311yescharacteristic, complemented
191#2117C2×C2311yescharacteristic, complemented
192#1389C2×C2311yescharacteristic, complemented
193C22×C23—41no—
194C22×C23—42no—
195C22×C23—41no—
196C22×C23—84no—
197C22×C23—81no—
198C22×C23—82no—
199C22×C23—161no—
200C22×C23—162no—
201C22×C23—162no—
202C22×C23#44711yescharacteristic
203C22×C23#44911yescharacteristic
204C22×C23#46911yescharacteristic
205C4×C23—41no—
206C4×C23—82no—
207C4×C23—81no—
208C4×C23#39211yescharacteristic
209C4×C23#44711yescharacteristic
210C4×C23#44911yescharacteristic
211C4×C23#46611yescharacteristic
212#575—21no—
213C2×#135C22×C2311yescharacteristic
214C22×D8C22×C2311yescharacteristic
215C2×#131C22×C2311yescharacteristic
216#535—21no—
217#509—22no—
218#509—22no—
219C2×SD32—21no—
220C2×D16—21no—
221C2×M32C22×C2311yescharacteristic
222C2×M32C22×C2311yescharacteristic
223#473—21no—
224#472—21no—
225#471—22no—
226#469—22no—
227#466—22no—
228C2×#108C22×C2311yescharacteristic
229#427C22×C2311yescharacteristic
230#418—21no—
231#413—21no—
232#361—22no—
233#360—22no—
234#350—22no—
235C2×C23—21no—
236C2×C23—21no—
237C2×C23—21no—
238C2×C23—81no—
239C2×C23—162no—
240C2×C23—161no—
241C2×C23#199911yescharacteristic
242C2×#41—21no—
243C2×#41D4×C2311yescharacteristic
244C22×D4—21no—
245#137—22no—
246#137—41no—
247AGL(1,ℤ/8)—22no—
248AGL(1,ℤ/8)—41no—
249#135—21no—
250#135—21no—
251#135—21no—
252#135—21no—
253#135—42no—
254C2×Q16D4×C2311yescharacteristic, complemented
255C2×SD16—21no—
256C2×D8—21no—
257C2×D8—21no—
258C2×D8—21no—
259C2×D8—21no—
260C2×D8—21no—
261C2×D8D4×C2312yes—
262C2×D8D4×C2311yescharacteristic, complemented
263#131—21no—
264#131—21no—
265#131—21no—
266#131—21no—
267C2×M16—21no—
268C2×M16—21no—
269C2×M16D4×C2311yescharacteristic
270C22×C8—21no—
271C22×C8C23×C2311yescharacteristic
272#127—41no—
273#124—41no—
274#121—24no—
275#120—42no—
276#117—21no—
277SD32—24no—
278D16—24no—
279M32—21no—
280M32—21no—
281M32—21no—
282M32D4×C2312yescomplemented
283C2×C16—21no—
284C2×C16—21no—
285#108—21no—
286#108D4×C2312yescomplemented
287#107—21no—
288#106D4×C2312yes—
289#102—24no—
290#101—22no—
291#100—22no—
292C23#1331011yescharacteristic, direct product factor, Sylow
293C24—41no—
294#41—21no—
295#41—21no—
296#41—21no—
297#41—21no—
298#41—41no—
299#41—41no—
300#41—41no—
301#41—41no—
302C2×Q8—21no—
303C2×Q8D8×C2311yescharacteristic
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—22no—
307C2×D4—41no—
308C2×D4—44no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4D8×C2311yescharacteristic
315C22×C4—21no—
316C22×C4—41no—
317C22×C4C2×D4×C2311yescharacteristic
318Q16—21no—
319Q16—21no—
320Q16—42no—
321SD16—42no—
322SD16—42no—
323SD16—41no—
324SD16—41no—
325D8—21no—
326D8—22no—
327D8—22no—
328D8—21no—
329D8—42no—
330D8—41no—
331D8—41no—
332M16—21no—
333M16—22no—
334M16—21no—
335M16—41no—
336M16—41no—
337M16—41no—
338C2×C8—21no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8C2×D4×C2311yescharacteristic
343C2×C8C2×D4×C2311yescharacteristic
344C2×C8C2×D4×C2311yescharacteristic
345C2×C8C2×D4×C2311yescharacteristic
346C2×C8C2×D4×C2311yescharacteristic
347C2×C8#41×C2311yescharacteristic
348C4⋊C4—22no—
349C22⋊C4—44no—
350C22⋊C4—41no—
351C42—41no—
352C16—22no—
353C16—41no—
354C23—21no—
355C23—42no—
356C23—41no—
357C23—81no—
358C23—81no—
359C23—162no—
360C23C23×AGL(1,ℤ/8)11yescharacteristic
361Q8—21no—
362Q8—21no—
363Q8—41no—
364Q8—41no—
365D4—21no—
366D4—21no—
367D4—41no—
368D4—41no—
369D4—42no—
370D4—41no—
371D4—41no—
372D4—42no—
373D4—81no—
374D4—81no—
375D4—81no—
376D4—81no—
377D4—164no—
378C2×C4—21no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4C23×#12011yescharacteristic
386C2×C4C23×#12011yescharacteristic
387C2×C4C23×#12111yescharacteristic
388C2×C4C2×D8×C2311yescharacteristic
389C2×C4C23×#13511yescharacteristic
390C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
391C8—41no—
392C8—42no—
393C8C23×#12011yescharacteristic
394C8C23×#12111yescharacteristic
395C8C23×#12111yescharacteristic
396C8C23×#12711yescharacteristic
397C22—41no—
398C22—42no—
399C22—41no—
400C22—84no—
401C22—81no—
402C22—82no—
403C22—161no—
404C22—162no—
405C22—162no—
406C22C23×#44711yescharacteristic
407C22C23×#44911yescharacteristic
408C22C23×#46911yescharacteristic
409C4—41no—
410C4—82no—
411C4—81no—
412C4C23×#39211yescharacteristic
413C4C23×#44711yescharacteristic
414C4C23×#44911yescharacteristic
415C4C23×#46611yescharacteristic
416C2—21no—
417C2—21no—
418C2—21no—
419C2—81no—
420C2—162no—
421C2—161no—
422C2C23×#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. C2×C23×#509C2semidirect
2. C23×#3067C2semidirect
3. C23×#3058C2semidirect
4. C2×C23×#361C2semidirect
5. C23×#2145C2semidirect
6. C23×#2117C2semidirect
7. C23×#1389C2semidirect
43. C2×Q16×C23D4semidirect
51. C2×D8×C23D4semidirect
71. C23×M32D4semidirect
75. C23×#108D4semidirect
142. #13310C23direct (N × H)
186. C2×#509C2×C23semidirect
187. #3067C2×C23semidirect
188. #3058C2×C23semidirect
189. C2×#361C2×C23semidirect
190. #2145C2×C23semidirect
191. #2117C2×C23semidirect
192. #1389C2×C23semidirect
254. C2×Q16D4×C23semidirect
262. C2×D8D4×C23semidirect
282. M32D4×C23semidirect
286. #108D4×C23semidirect
292. C23#13310direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5092
C23×#30672
C23×#30582
C2×C23×#3612
C23×#21452
C23×#21172
C23×#13892
#1331023