Finite Group Databasebeta

← C23×#10427 #19158053

Subgroups — C23×#10427 #19158053

Order 5888.

460 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2820C211yesmaximal, characteristic
2C23×#1992C211yesmaximal, characteristic, complemented
3C23×#1959C211yesmaximal, characteristic
4C23×#1847C211yesmaximal, characteristic, complemented
5C2×C23×#330C211yesmaximal, characteristic, complemented
6C2×C23×#329C211yesmaximal, characteristic
7C23×#1295C211yesmaximal, characteristic, complemented
8C2×C23×#125C2211yescharacteristic
9C2×C23×#124C2211yescharacteristic
10C2×C23×#107—21no—
11C2×C23×#106—21no—
12C2×C23×#103—21no—
13C2×C23×#102—21no—
14C2×C23×#97C2211yescharacteristic
15C2×C23×#97C2211yescharacteristic
16C2×C23×#97C2211yescharacteristic
17C23×#399—21no—
18C23×#397—21no—
19C23×#396—21no—
20C23×#396—21no—
21C23×#395—21no—
22C23×#394—21no—
23C23×#384C2211yescharacteristic
24C23×#382C2211yescharacteristic
25C23×#340—21no—
26C23×#340—21no—
27C23×#330—22no—
28C23×#329—22no—
29C22×Q8×C23D411yescharacteristic
30C22×D4×C23D411yescharacteristic
31C22×C8×C23—21no—
32C22×C8×C23—21no—
33C22×C8×C23—21no—
34C23×#125—21no—
35C23×#125—21no—
36C23×#125—21no—
37C23×#125—21no—
38C23×#124—21no—
39C23×#124—21no—
40C23×#124—21no—
41C23×#124—21no—
42C2×C4⋊C4×C23—21no—
43C2×C4⋊C4×C23—21no—
44C2×C4⋊C4×C23—21no—
45C2×C4⋊C4×C23—21no—
46C2×C4⋊C4×C23—21no—
47C2×C4⋊C4×C23—21no—
48C2×C4⋊C4×C23—21no—
49C2×C4⋊C4×C23D411yescharacteristic
50C2×C4⋊C4×C23D411yescharacteristic
51C2×C4⋊C4×C23D411yescharacteristic
52C2×C4⋊C4×C23D411yescharacteristic
53C2×C22⋊C4×C23—21no—
54C2×C22⋊C4×C23—21no—
55C2×C22⋊C4×C23—21no—
56C2×C22⋊C4×C23—21no—
57C2×C22⋊C4×C23—21no—
58C2×C22⋊C4×C23—21no—
59C2×C42×C23C2311yescharacteristic
60C23×#107—42no—
61C23×#106—42no—
62C23×#103—42no—
63C23×#102—42no—
64C23×#97—21no—
65C23×#97—21no—
66C23×#97—21no—
67C23×#97—21no—
68C23×#97—21no—
69C23×#97D412yes—
70C23×#95—21no—
71C23×#95—21no—
72C23×#95—21no—
73C23×#95—21no—
74C24×C23—21no—
75C2×Q8×C23—21no—
76C2×Q8×C23—21no—
77C2×Q8×C23—41no—
78C2×Q8×C23—41no—
79C2×D4×C23—21no—
80C2×D4×C23—21no—
81C2×D4×C23—41no—
82C2×D4×C23—41no—
83C22×C4×C23—21no—
84C22×C4×C23—21no—
85C22×C4×C23—21no—
86C22×C4×C23—21no—
87C22×C4×C23—21no—
88C22×C4×C23—41no—
89C22×C4×C23C2×D411yescharacteristic
90C22×C4×C23C2×D411yescharacteristic
91C22×C4×C23C2×D411yescharacteristic
92C2×C8×C23—22no—
93C2×C8×C23—22no—
94C2×C8×C23—22no—
95C2×C8×C23—22no—
96C2×C8×C23—22no—
97C2×C8×C23—41no—
98C2×C8×C23—41no—
99C2×C8×C23—41no—
100C2×C8×C23—41no—
101C4⋊C4×C23—21no—
102C4⋊C4×C23—21no—
103C4⋊C4×C23—21no—
104C4⋊C4×C23—21no—
105C4⋊C4×C23—41no—
106C4⋊C4×C23—41no—
107C4⋊C4×C23—41no—
108C4⋊C4×C23—41no—
109C4⋊C4×C23—41no—
110C4⋊C4×C23—41no—
111C4⋊C4×C23—42no—
112C4⋊C4×C23—41no—
113C4⋊C4×C23—41no—
114C4⋊C4×C23—41no—
115C4⋊C4×C23—41no—
116C4⋊C4×C23—81no—
117C4⋊C4×C23—81no—
118C22⋊C4×C23—41no—
119C22⋊C4×C23—41no—
120C22⋊C4×C23—41no—
121C22⋊C4×C23—41no—
122C22⋊C4×C23—81no—
123C22⋊C4×C23—81no—
124C22⋊C4×C23—81no—
125C22⋊C4×C23—81no—
126C42×C23C2×D411yescharacteristic
127C42×C23#4111yescharacteristic
128C42×C23#4111yescharacteristic
129C42×C23#4111yescharacteristic
130#10427C2311yesmaximal, characteristic, direct product factor, Sylow
131C23×C23—41no—
132C23×C23—41no—
133C23×C23—41no—
134C23×C23—41no—
135C23×C23—41no—
136C23×C23—41no—
137C23×C23—41no—
138C23×C23#12011yescharacteristic
139Q8×C23—42no—
140Q8×C23—81no—
141D4×C23—42no—
142D4×C23—81no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—21no—
146C2×C4×C23—21no—
147C2×C4×C23—21no—
148C2×C4×C23—21no—
149C2×C4×C23—41no—
150C2×C4×C23—41no—
151C2×C4×C23—41no—
152C2×C4×C23—41no—
153C2×C4×C23—41no—
154C2×C4×C23—41no—
155C2×C4×C23—41no—
156C2×C4×C23—41no—
157C2×C4×C23—41no—
158C2×C4×C23—41no—
159C2×C4×C23—41no—
160C2×C4×C23—41no—
161C2×C4×C23—41no—
162C2×C4×C23—41no—
163C2×C4×C23—41no—
164C2×C4×C23—81no—
165C2×C4×C23—81no—
166C2×C4×C23—81no—
167C2×C4×C23#12111yescharacteristic
168C2×C4×C23#12111yescharacteristic
169C2×C4×C23#12111yescharacteristic
170C2×C4×C23#12311yescharacteristic
171C2×C4×C23#12311yescharacteristic
172C2×C4×C23#12411yescharacteristic
173C8×C23—24no—
174C8×C23—42no—
175C8×C23—42no—
176#2820C2×C2311yescharacteristic
177#1992C2×C2311yescharacteristic, complemented
178#1959C2×C2311yescharacteristic
179#1847C2×C2311yescharacteristic, complemented
180C2×#330C2×C2311yescharacteristic, complemented
181C2×#329C2×C2311yescharacteristic
182#1295C2×C2311yescharacteristic, complemented
183C22×C23—81no—
184C22×C23—81no—
185C22×C23—81no—
186C22×C23—81no—
187C22×C23—81no—
188C22×C23—81no—
189C22×C23—81no—
190C22×C23#39411yescharacteristic
191C22×C23#45311yescharacteristic
192C22×C23#45311yescharacteristic
193C22×C23#45411yescharacteristic
194C22×C23#45511yescharacteristic
195C22×C23#45611yescharacteristic
196C22×C23#45611yescharacteristic
197C4×C23—21no—
198C4×C23—21no—
199C4×C23—21no—
200C4×C23—21no—
201C4×C23—21no—
202C4×C23—21no—
203C4×C23—81no—
204C4×C23—81no—
205C4×C23—81no—
206C4×C23—81no—
207C4×C23—81no—
208C4×C23—161no—
209C2×#125C22×C2311yescharacteristic
210C2×#124C22×C2311yescharacteristic
211C2×#107—21no—
212C2×#106—21no—
213C2×#103—21no—
214C2×#102—21no—
215C2×#97C22×C2311yescharacteristic
216C2×#97C22×C2311yescharacteristic
217C2×#97C22×C2311yescharacteristic
218#399—21no—
219#397—21no—
220#396—21no—
221#396—21no—
222#395—21no—
223#394—21no—
224#384C22×C2311yescharacteristic
225#382C22×C2311yescharacteristic
226#340—21no—
227#340—21no—
228#330—22no—
229#329—22no—
230C2×C23—161no—
231C2×C23#165611yescharacteristic
232C2×C23#166011yescharacteristic
233C2×C23#166111yescharacteristic
234C2×C23#166411yescharacteristic
235C2×C23#203711yescharacteristic
236C2×C23#203811yescharacteristic
237C2×C23#203911yescharacteristic
238C22×Q8D4×C2311yescharacteristic
239C22×D4D4×C2311yescharacteristic
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243#125—21no—
244#125—21no—
245#125—21no—
246#125—21no—
247#124—21no—
248#124—21no—
249#124—21no—
250#124—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4D4×C2311yescharacteristic
259C2×C4⋊C4D4×C2311yescharacteristic
260C2×C4⋊C4D4×C2311yescharacteristic
261C2×C4⋊C4D4×C2311yescharacteristic
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C42C23×C2311yescharacteristic
269#107—42no—
270#106—42no—
271#103—42no—
272#102—42no—
273#97—21no—
274#97—21no—
275#97—21no—
276#97—21no—
277#97—21no—
278#97D4×C2312yes—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283C23#1042711yescharacteristic, direct product factor, Sylow
284C24—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—41no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—41no—
299C22×C4C2×D4×C2311yescharacteristic
300C22×C4C2×D4×C2311yescharacteristic
301C22×C4C2×D4×C2311yescharacteristic
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—81no—
327C4⋊C4—81no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—81no—
333C22⋊C4—81no—
334C22⋊C4—81no—
335C22⋊C4—81no—
336C42C2×D4×C2311yescharacteristic
337C42#41×C2311yescharacteristic
338C42#41×C2311yescharacteristic
339C42#41×C2311yescharacteristic
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23C23×#12011yescharacteristic
348Q8—42no—
349Q8—81no—
350D4—42no—
351D4—81no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4C23×#12111yescharacteristic
377C2×C4C23×#12111yescharacteristic
378C2×C4C23×#12111yescharacteristic
379C2×C4C23×#12311yescharacteristic
380C2×C4C23×#12311yescharacteristic
381C2×C4C23×#12411yescharacteristic
382C8—24no—
383C8—42no—
384C8—42no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22C23×#39411yescharacteristic
393C22C23×#45311yescharacteristic
394C22C23×#45311yescharacteristic
395C22C23×#45411yescharacteristic
396C22C23×#45511yescharacteristic
397C22C23×#45611yescharacteristic
398C22C23×#45611yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—81no—
406C4—81no—
407C4—81no—
408C4—81no—
409C4—81no—
410C4—161no—
411C2—161no—
412C2C23×#165611yescharacteristic
413C2C23×#166011yescharacteristic
414C2C23×#166111yescharacteristic
415C2C23×#166411yescharacteristic
416C2C23×#203711yescharacteristic
417C2C23×#203811yescharacteristic
418C2C23×#203911yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC22×D4×C23D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1992C2semidirect
4. C23×#1847C2semidirect
5. C2×C23×#330C2semidirect
7. C23×#1295C2semidirect
130. #10427C23direct (N × H)
177. #1992C2×C23semidirect
179. #1847C2×C23semidirect
180. C2×#330C2×C23semidirect
182. #1295C2×C23semidirect
283. C23#10427direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#28202
C23×#19922
C23×#19592
C23×#18472
C2×C23×#3302
C2×C23×#3292
C23×#12952
#1042723