Finite Group Databasebeta

← C23×#13321 #19160947

Subgroups — C23×#13321 #19160947

Order 5888.

552 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C23×SD32C211yesmaximal, characteristic, complemented
2C2×C23×#473C211yesmaximal, characteristic, complemented
3C23×#3058C211yesmaximal, characteristic
4C2×C23×#361C211yesmaximal, characteristic, complemented
5C23×#2141C211yesmaximal, characteristic, complemented
6C2×C23×#350C211yesmaximal, characteristic, complemented
7C23×#1382C211yesmaximal, characteristic, complemented
8C2×C23×#137—21no—
9C22×Q16×C23C2211yescharacteristic
10C22×D8×C23C2211yescharacteristic
11C2×C23×#131C2211yescharacteristic
12C23×#535—21no—
13C2×C23×SD32—22no—
14C2×C23×SD32—22no—
15C2×C23×SD32—22no—
16C2×C23×M32C2211yescharacteristic
17C22×C16×C23C2211yescharacteristic
18C23×#473—22no—
19C23×#473—22no—
20C23×#469—22no—
21C23×#466—22no—
22C2×C23×#108C2211yescharacteristic
23C23×#427C2211yescharacteristic
24C23×#418—21no—
25C2×C23×#101—21no—
26C23×#361—22no—
27C23×#358—22no—
28C23×#350—22no—
29C2×#41×C23D411yescharacteristic
30C22×Q8×C23—21no—
31C22×D4×C23—21no—
32C23×#137—24no—
33C23×#137—42no—
34C2×Q16×C23—21no—
35C2×Q16×C23—21no—
36C2×Q16×C23—21no—
37C2×Q16×C23—21no—
38C2×Q16×C23—21no—
39C2×Q16×C23—41no—
40C2×Q16×C23D412yescomplemented
41C2×SD16×C23—41no—
42C2×D8×C23—21no—
43C2×D8×C23—21no—
44C2×D8×C23—21no—
45C2×D8×C23—21no—
46C2×D8×C23—21no—
47C2×D8×C23D412yes—
48C23×#131—21no—
49C23×#131—21no—
50C23×#131—21no—
51C23×#131—21no—
52C2×M16×C23—21no—
53C2×M16×C23—21no—
54C2×M16×C23D411yescharacteristic
55C22×C8×C23—21no—
56C22×C8×C23C2311yescharacteristic
57C23×#127—41no—
58C23×#124—41no—
59C23×#121—24no—
60C23×#120—42no—
61C23×#117—21no—
62C23×SD32—28no—
63C23×M32—21no—
64C23×M32—21no—
65C2×C16×C23—21no—
66C2×C16×C23—21no—
67C2×C16×C23—21no—
68C2×C16×C23D412yescomplemented
69C23×#108—21no—
70C23×#108D412yescomplemented
71C23×#107D412yes—
72C23×#106—21no—
73C23×#102—24no—
74C23×#101—24no—
75C24×C23—41no—
76#41×C23—21no—
77#41×C23—21no—
78#41×C23—21no—
79#41×C23—21no—
80C2×Q8×C23—22no—
81C2×Q8×C23—41no—
82C2×Q8×C23—41no—
83C2×Q8×C23—41no—
84C2×Q8×C23—41no—
85C2×Q8×C23—41no—
86C2×Q8×C23D811yescharacteristic
87C2×D4×C23—21no—
88C2×D4×C23—22no—
89C2×D4×C23—44no—
90C2×D4×C23—41no—
91C2×D4×C23—41no—
92C2×D4×C23—41no—
93C2×D4×C23—41no—
94C2×D4×C23—41no—
95C2×D4×C23D811yescharacteristic
96C22×C4×C23—21no—
97C22×C4×C23—41no—
98C22×C4×C23C2×D411yescharacteristic
99Q16×C23—22no—
100Q16×C23—22no—
101Q16×C23—44no—
102Q16×C23—41no—
103Q16×C23—41no—
104SD16×C23—44no—
105D8×C23—22no—
106D8×C23—22no—
107D8×C23—41no—
108D8×C23—41no—
109M16×C23—21no—
110M16×C23—22no—
111M16×C23—21no—
112M16×C23—41no—
113M16×C23—41no—
114M16×C23—41no—
115C2×C8×C23—21no—
116C2×C8×C23—41no—
117C2×C8×C23—41no—
118C2×C8×C23—41no—
119C2×C8×C23C2×D411yescharacteristic
120C2×C8×C23C2×D411yescharacteristic
121C2×C8×C23C2×D411yescharacteristic
122C2×C8×C23C2×D411yescharacteristic
123C2×C8×C23C2×D411yescharacteristic
124C2×C8×C23#4111yescharacteristic
125C4⋊C4×C23—22no—
126C22⋊C4×C23—44no—
127C22⋊C4×C23—41no—
128C42×C23—41no—
129C16×C23—22no—
130C16×C23—41no—
131#13321C2311yesmaximal, characteristic, direct product factor, Sylow
132C23×C23—21no—
133C23×C23—42no—
134C23×C23—81no—
135C23×C23—81no—
136C23×C23—162no—
137C23×C23AGL(1,ℤ/8)11yescharacteristic
138Q8×C23—21no—
139Q8×C23—21no—
140Q8×C23—42no—
141Q8×C23—42no—
142Q8×C23—81no—
143Q8×C23—81no—
144D4×C23—21no—
145D4×C23—21no—
146D4×C23—41no—
147D4×C23—41no—
148D4×C23—42no—
149D4×C23—42no—
150D4×C23—81no—
151D4×C23—81no—
152D4×C23—164no—
153C2×C4×C23—21no—
154C2×C4×C23—41no—
155C2×C4×C23—41no—
156C2×C4×C23—42no—
157C2×C4×C23—42no—
158C2×C4×C23—81no—
159C2×C4×C23—81no—
160C2×C4×C23#12011yescharacteristic
161C2×C4×C23#12011yescharacteristic
162C2×C4×C23#12111yescharacteristic
163C2×C4×C23C2×D811yescharacteristic
164C2×C4×C23#13511yescharacteristic
165C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
166C8×C23—41no—
167C8×C23—42no—
168C8×C23#12011yescharacteristic
169C8×C23#12111yescharacteristic
170C8×C23#12111yescharacteristic
171C8×C23#12711yescharacteristic
172C22×SD32C2×C2311yescharacteristic, complemented
173C2×#473C2×C2311yescharacteristic, complemented
174#3058C2×C2311yescharacteristic
175C2×#361C2×C2311yescharacteristic, complemented
176#2141C2×C2311yescharacteristic, complemented
177C2×#350C2×C2311yescharacteristic, complemented
178#1382C2×C2311yescharacteristic, complemented
179C22×C23—22no—
180C22×C23—42no—
181C22×C23—41no—
182C22×C23—84no—
183C22×C23—82no—
184C22×C23—162no—
185C22×C23—162no—
186C22×C23#44711yescharacteristic
187C22×C23#44911yescharacteristic
188C22×C23#46911yescharacteristic
189C4×C23—41no—
190C4×C23—82no—
191C4×C23—82no—
192C4×C23#39211yescharacteristic
193C4×C23#44711yescharacteristic
194C4×C23#44911yescharacteristic
195C4×C23#46611yescharacteristic
196C2×#137—21no—
197C22×Q16C22×C2311yescharacteristic
198C22×D8C22×C2311yescharacteristic
199C2×#131C22×C2311yescharacteristic
200#535—21no—
201C2×SD32—22no—
202C2×SD32—22no—
203C2×SD32—22no—
204C2×M32C22×C2311yescharacteristic
205C22×C16C22×C2311yescharacteristic
206#473—22no—
207#473—22no—
208#469—22no—
209#466—22no—
210C2×#108C22×C2311yescharacteristic
211#427C22×C2311yescharacteristic
212#418—21no—
213C2×#101—21no—
214#361—22no—
215#358—22no—
216#350—22no—
217C2×C23—21no—
218C2×C23—21no—
219C2×C23—81no—
220C2×C23—162no—
221C2×C23#199911yescharacteristic
222C2×C23#219412yes—
223C2×#41D4×C2311yescharacteristic
224C22×Q8—21no—
225C22×D4—21no—
226#137—24no—
227#137—42no—
228C2×Q16—21no—
229C2×Q16—21no—
230C2×Q16—21no—
231C2×Q16—21no—
232C2×Q16—21no—
233C2×Q16—41no—
234C2×Q16D4×C2312yescomplemented
235C2×SD16—41no—
236C2×D8—21no—
237C2×D8—21no—
238C2×D8—21no—
239C2×D8—21no—
240C2×D8—21no—
241C2×D8D4×C2312yes—
242#131—21no—
243#131—21no—
244#131—21no—
245#131—21no—
246C2×M16—21no—
247C2×M16—21no—
248C2×M16D4×C2311yescharacteristic
249C22×C8—21no—
250C22×C8C23×C2311yescharacteristic
251#127—41no—
252#124—41no—
253#121—24no—
254#120—42no—
255#117—21no—
256SD32—28no—
257M32—21no—
258M32—21no—
259C2×C16—21no—
260C2×C16—21no—
261C2×C16—21no—
262C2×C16D4×C2312yescomplemented
263#108—21no—
264#108D4×C2312yescomplemented
265#107D4×C2312yes—
266#106—21no—
267#102—24no—
268#101—24no—
269C23#1332111yescharacteristic, direct product factor, Sylow
270C24—41no—
271#41—21no—
272#41—21no—
273#41—21no—
274#41—21no—
275C2×Q8—22no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8D8×C2311yescharacteristic
282C2×D4—21no—
283C2×D4—22no—
284C2×D4—44no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4D8×C2311yescharacteristic
291C22×C4—21no—
292C22×C4—41no—
293C22×C4C2×D4×C2311yescharacteristic
294Q16—22no—
295Q16—22no—
296Q16—44no—
297Q16—41no—
298Q16—41no—
299SD16—44no—
300D8—22no—
301D8—22no—
302D8—41no—
303D8—41no—
304M16—21no—
305M16—22no—
306M16—21no—
307M16—41no—
308M16—41no—
309M16—41no—
310C2×C8—21no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8C2×D4×C2311yescharacteristic
315C2×C8C2×D4×C2311yescharacteristic
316C2×C8C2×D4×C2311yescharacteristic
317C2×C8C2×D4×C2311yescharacteristic
318C2×C8C2×D4×C2311yescharacteristic
319C2×C8#41×C2311yescharacteristic
320C4⋊C4—22no—
321C22⋊C4—44no—
322C22⋊C4—41no—
323C42—41no—
324C16—22no—
325C16—41no—
326C23—21no—
327C23—42no—
328C23—81no—
329C23—81no—
330C23—162no—
331C23C23×AGL(1,ℤ/8)11yescharacteristic
332Q8—21no—
333Q8—21no—
334Q8—42no—
335Q8—42no—
336Q8—81no—
337Q8—81no—
338D4—21no—
339D4—21no—
340D4—41no—
341D4—41no—
342D4—42no—
343D4—42no—
344D4—81no—
345D4—81no—
346D4—164no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4C23×#12011yescharacteristic
355C2×C4C23×#12011yescharacteristic
356C2×C4C23×#12111yescharacteristic
357C2×C4C2×D8×C2311yescharacteristic
358C2×C4C23×#13511yescharacteristic
359C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
360C8—41no—
361C8—42no—
362C8C23×#12011yescharacteristic
363C8C23×#12111yescharacteristic
364C8C23×#12111yescharacteristic
365C8C23×#12711yescharacteristic
366C22—22no—
367C22—42no—
368C22—41no—
369C22—84no—
370C22—82no—
371C22—162no—
372C22—162no—
373C22C23×#44711yescharacteristic
374C22C23×#44911yescharacteristic
375C22C23×#46911yescharacteristic
376C4—41no—
377C4—82no—
378C4—82no—
379C4C23×#39211yescharacteristic
380C4C23×#44711yescharacteristic
381C4C23×#44911yescharacteristic
382C4C23×#46611yescharacteristic
383C2—21no—
384C2—21no—
385C2—81no—
386C2—162no—
387C2C23×#199911yescharacteristic
388C2C23×#219412yes—
subgroupG/H
center, socleC22×C23#447
commutator, Frattini, cosocleC22×C8C23×C23
prime radicalC23×#3058C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C23×#131 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C23×SD32C2semidirect
2. C2×C23×#473C2semidirect
4. C2×C23×#361C2semidirect
5. C23×#2141C2semidirect
6. C2×C23×#350C2semidirect
7. C23×#1382C2semidirect
40. C2×Q16×C23D4semidirect
68. C2×C16×C23D4semidirect
70. C23×#108D4semidirect
131. #13321C23direct (N × H)
172. C22×SD32C2×C23semidirect
173. C2×#473C2×C23semidirect
175. C2×#361C2×C23semidirect
176. #2141C2×C23semidirect
177. C2×#350C2×C23semidirect
178. #1382C2×C23semidirect
234. C2×Q16D4×C23semidirect
262. C2×C16D4×C23semidirect
264. #108D4×C23semidirect
269. C23#13321direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C23×SD322
C2×C23×#4732
C23×#30582
C2×C23×#3612
C23×#21412
C2×C23×#3502
C23×#13822
#1332123