Finite Group Databasebeta

← C23×#8357 #19155983

Subgroups — C23×#8357 #19155983

Order 5888.

710 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3507C211yesmaximal, characteristic
2C23×#1984C211yesmaximal, characteristic, complemented
3C23×#1801C211yesmaximal, characteristic, complemented
4C23×#1550C211yesmaximal, characteristic, complemented
5C23×#1547C211yesmaximal, characteristic
6C23×#1493C211yesmaximal, characteristic, complemented
7C23×#1492C211yesmaximal, characteristic
8C23×#559—21no—
9C23×#557—21no—
10C23×#556—21no—
11C23×#556—22no—
12C23×#554—21no—
13C23×#552—22no—
14C23×#539C414yes—
15C23×#536—21no—
16C23×#533C2211yescharacteristic
17C23×#532C2211yescharacteristic
18C23×#529—22no—
19C23×#520—21no—
20C2×C4×Q8×C23C2211yescharacteristic
21C23×#423C2211yescharacteristic
22C23×#423C2211yescharacteristic
23C2×C23×C4⋊C8C2211yescharacteristic
24C2×C23×#101—21no—
25C2×C4×C8×C23C2211yescharacteristic
26C23×#343—21no—
27C23×#336—22no—
28C23×#333—21no—
29C23×#332—21no—
30C23×#332—21no—
31C23×#331—21no—
32C23×Q8⋊C8—22no—
33C23×D4⋊C8—22no—
34C2×#41×C23—21no—
35C22×Q8×C23D411yescharacteristic
36C2×M16×C23—21no—
37C2×M16×C23—21no—
38C22×C8×C23—21no—
39C22×C8×C23—21no—
40C23×C4⋊Q8—22no—
41C23×C4⋊Q8D411yescharacteristic
42C23×C4⋊Q8D411yescharacteristic
43C23×C4⋊Q8D411yescharacteristic
44C23×#127D411yescharacteristic
45C23×#126—24no—
46C23×#126—24no—
47C23×#125—22no—
48C23×#125—24no—
49C23×#125—21no—
50C23×#124—22no—
51C23×#124—22no—
52C23×#124—21no—
53C23×#123—24no—
54C23×#123—24no—
55C23×C22⋊Q8—22no—
56C23×C22⋊Q8—22no—
57C23×C22⋊Q8—24no—
58C23×C22⋊Q8—21no—
59C23×C22⋊Q8C2×C412yes—
60C23×#121—21no—
61C23×#121C2×C412yes—
62C4×Q8×C23—22no—
63C4×Q8×C23—22no—
64C4×Q8×C23—21no—
65C4×Q8×C23—21no—
66C4×Q8×C23—21no—
67C4×Q8×C23—21no—
68C4×Q8×C23—21no—
69C4×Q8×C23D412yes—
70C4×D4×C23—22no—
71C4×D4×C23—22no—
72C4×D4×C23—24no—
73C4×D4×C23—21no—
74C4×D4×C23D412yes—
75C23×#117—22no—
76C23×#117—24no—
77C23×#117—21no—
78C2×C4⋊C4×C23—22no—
79C2×C4⋊C4×C23C2×C412yes—
80C2×C4⋊C4×C23D411yescharacteristic
81C2×C42×C23D412yes—
82C2×C42×C23C2311yescharacteristic
83C23×C4⋊C8—21no—
84C23×C4⋊C8—21no—
85C23×C4⋊C8—21no—
86C23×C4⋊C8—21no—
87C23×C4⋊C8—21no—
88C23×C4⋊C8—21no—
89C23×#101—42no—
90C4×C8×C23—21no—
91C4×C8×C23—21no—
92#41×C23—41no—
93#41×C23—41no—
94#41×C23—41no—
95#41×C23—41no—
96C2×Q8×C23—21no—
97C2×Q8×C23—21no—
98C2×Q8×C23—21no—
99C2×Q8×C23—21no—
100C2×Q8×C23—21no—
101C2×Q8×C23—21no—
102C2×Q8×C23—21no—
103C2×Q8×C23—21no—
104C2×Q8×C23C22⋊C412yes—
105C2×D4×C23—22no—
106C2×D4×C23—21no—
107C2×D4×C23—21no—
108C2×D4×C23C22⋊C412yes—
109C22×C4×C23—24no—
110C22×C4×C23—22no—
111C22×C4×C23C22⋊C412yes—
112C22×C4×C23C22⋊C412yes—
113C22×C4×C23C22×C411yescharacteristic
114C22×C4×C23C2×D411yescharacteristic
115C22×C4×C23C2×D411yescharacteristic
116M16×C23—41no—
117M16×C23—41no—
118M16×C23—41no—
119M16×C23—41no—
120C2×C8×C23—22no—
121C2×C8×C23—41no—
122C2×C8×C23—41no—
123C2×C8×C23—42no—
124C2×C8×C23—41no—
125C2×C8×C23—41no—
126C2×C8×C23—41no—
127C4⋊C4×C23—24no—
128C4⋊C4×C23—24no—
129C4⋊C4×C23—24no—
130C4⋊C4×C23—22no—
131C4⋊C4×C23—22no—
132C4⋊C4×C23—22no—
133C4⋊C4×C23—21no—
134C4⋊C4×C23—21no—
135C4⋊C4×C23—21no—
136C4⋊C4×C23—21no—
137C4⋊C4×C23—22no—
138C4⋊C4×C23—21no—
139C4⋊C4×C23—21no—
140C4⋊C4×C23C22⋊C412yes—
141C4⋊C4×C23C22⋊C412yes—
142C22⋊C4×C23—24no—
143C22⋊C4×C23—24no—
144C22⋊C4×C23—24no—
145C22⋊C4×C23—22no—
146C22⋊C4×C23—22no—
147C22⋊C4×C23—22no—
148C42×C23—24no—
149C42×C23—22no—
150C42×C23—22no—
151C42×C23—22no—
152C42×C23C2×D411yescharacteristic
153C42×C23C2×D411yescharacteristic
154C42×C23C2×D411yescharacteristic
155C42×C23C2×D411yescharacteristic
156#8357C2311yesmaximal, characteristic, direct product factor, Sylow
157C23×C23—22no—
158C23×C23C2×C22⋊C411yescharacteristic
159Q8×C23—42no—
160Q8×C23—41no—
161Q8×C23—41no—
162Q8×C23—41no—
163Q8×C23—41no—
164Q8×C23—41no—
165Q8×C23—41no—
166Q8×C23—41no—
167Q8×C23—41no—
168D4×C23—42no—
169D4×C23—42no—
170D4×C23—42no—
171D4×C23—41no—
172D4×C23—41no—
173D4×C23—41no—
174D4×C23—41no—
175C2×C4×C23—21no—
176C2×C4×C23—21no—
177C2×C4×C23—22no—
178C2×C4×C23—22no—
179C2×C4×C23—22no—
180C2×C4×C23—22no—
181C2×C4×C23—22no—
182C2×C4×C23—22no—
183C2×C4×C23—22no—
184C2×C4×C23—41no—
185C2×C4×C23—42no—
186C2×C4×C23—42no—
187C2×C4×C23—42no—
188C2×C4×C23—84no—
189C2×C4×C23C2×C22⋊C411yescharacteristic
190C2×C4×C23C2×C22⋊C411yescharacteristic
191C2×C4×C23#12011yescharacteristic
192C2×C4×C23#12011yescharacteristic
193C2×C4×C23#12011yescharacteristic
194C2×C4×C23#12011yescharacteristic
195C2×C4×C23#13511yescharacteristic
196C2×C4×C23#13511yescharacteristic
197C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
198C2×C4×C23#13711yescharacteristic
199C8×C23—42no—
200C8×C23—81no—
201C8×C23—81no—
202C8×C23—81no—
203#3507C2×C2311yescharacteristic
204#1984C2×C2311yescharacteristic, complemented
205#1801C2×C2311yescharacteristic, complemented
206#1550C2×C2311yescharacteristic, complemented
207#1547C2×C2311yescharacteristic
208#1493C2×C2311yescharacteristic, complemented
209#1492C2×C2311yescharacteristic
210C22×C23—21no—
211C22×C23—21no—
212C22×C23—42no—
213C22×C23—42no—
214C22×C23—42no—
215C22×C23#37911yescharacteristic
216C22×C23#41611yescharacteristic
217C22×C23#41711yescharacteristic
218C4×C23—21no—
219C4×C23—21no—
220C4×C23—21no—
221C4×C23—42no—
222C4×C23—42no—
223C4×C23—42no—
224C4×C23—42no—
225C4×C23—42no—
226C4×C23#44911yescharacteristic
227C4×C23#44911yescharacteristic
228C4×C23#45211yescharacteristic
229C4×C23#45211yescharacteristic
230C4×C23#45411yescharacteristic
231C4×C23#45611yescharacteristic
232#559—21no—
233#557—21no—
234#556—21no—
235#556—22no—
236#554—21no—
237#552—22no—
238#539C4×C2314yes—
239#536—21no—
240#533C22×C2311yescharacteristic
241#532C22×C2311yescharacteristic
242#529—22no—
243#520—21no—
244C2×C4×Q8C22×C2311yescharacteristic
245#423C22×C2311yescharacteristic
246#423C22×C2311yescharacteristic
247C2×C4⋊C8C22×C2311yescharacteristic
248C2×#101—21no—
249C2×C4×C8C22×C2311yescharacteristic
250#343—21no—
251#336—22no—
252#333—21no—
253#332—21no—
254#332—21no—
255#331—21no—
256Q8⋊C8—22no—
257D4⋊C8—22no—
258C2×C23—21no—
259C2×C23—21no—
260C2×C23—82no—
261C2×C23#178811yescharacteristic
262C2×C23#179111yescharacteristic
263C2×C23#179311yescharacteristic
264C2×#41—21no—
265C22×Q8D4×C2311yescharacteristic
266C2×M16—21no—
267C2×M16—21no—
268C22×C8—21no—
269C22×C8—21no—
270C4⋊Q8—22no—
271C4⋊Q8D4×C2311yescharacteristic
272C4⋊Q8D4×C2311yescharacteristic
273C4⋊Q8D4×C2311yescharacteristic
274#127D4×C2311yescharacteristic
275#126—24no—
276#126—24no—
277#125—22no—
278#125—24no—
279#125—21no—
280#124—22no—
281#124—22no—
282#124—21no—
283#123—24no—
284#123—24no—
285C22⋊Q8—22no—
286C22⋊Q8—22no—
287C22⋊Q8—24no—
288C22⋊Q8—21no—
289C22⋊Q8C2×C4×C2312yes—
290#121—21no—
291#121C2×C4×C2312yes—
292C4×Q8—22no—
293C4×Q8—22no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×Q8—21no—
297C4×Q8—21no—
298C4×Q8—21no—
299C4×Q8D4×C2312yes—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—24no—
303C4×D4—21no—
304C4×D4D4×C2312yes—
305#117—22no—
306#117—24no—
307#117—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4C2×C4×C2312yes—
310C2×C4⋊C4D4×C2311yescharacteristic
311C2×C42D4×C2312yes—
312C2×C42C23×C2311yescharacteristic
313C4⋊C8—21no—
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319#101—42no—
320C4×C8—21no—
321C4×C8—21no—
322C23#835711yescharacteristic, direct product factor, Sylow
323#41—41no—
324#41—41no—
325#41—41no—
326#41—41no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8C22⋊C4×C2312yes—
336C2×D4—22no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4C22⋊C4×C2312yes—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4C22⋊C4×C2312yes—
343C22×C4C22⋊C4×C2312yes—
344C22×C4C22×C4×C2311yescharacteristic
345C22×C4C2×D4×C2311yescharacteristic
346C22×C4C2×D4×C2311yescharacteristic
347M16—41no—
348M16—41no—
349M16—41no—
350M16—41no—
351C2×C8—22no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—42no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4C22⋊C4×C2312yes—
372C4⋊C4C22⋊C4×C2312yes—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C42—24no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42C2×D4×C2311yescharacteristic
384C42C2×D4×C2311yescharacteristic
385C42C2×D4×C2311yescharacteristic
386C42C2×D4×C2311yescharacteristic
387C23—22no—
388C23C2×C22⋊C4×C2311yescharacteristic
389Q8—42no—
390Q8—41no—
391Q8—41no—
392Q8—41no—
393Q8—41no—
394Q8—41no—
395Q8—41no—
396Q8—41no—
397Q8—41no—
398D4—42no—
399D4—42no—
400D4—42no—
401D4—41no—
402D4—41no—
403D4—41no—
404D4—41no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—84no—
419C2×C4C2×C22⋊C4×C2311yescharacteristic
420C2×C4C2×C22⋊C4×C2311yescharacteristic
421C2×C4C23×#12011yescharacteristic
422C2×C4C23×#12011yescharacteristic
423C2×C4C23×#12011yescharacteristic
424C2×C4C23×#12011yescharacteristic
425C2×C4C23×#13511yescharacteristic
426C2×C4C23×#13511yescharacteristic
427C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
428C2×C4C23×#13711yescharacteristic
429C8—42no—
430C8—81no—
431C8—81no—
432C8—81no—
433C22—21no—
434C22—21no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22C23×#37911yescharacteristic
439C22C23×#41611yescharacteristic
440C22C23×#41711yescharacteristic
441C4—21no—
442C4—21no—
443C4—21no—
444C4—42no—
445C4—42no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4C23×#44911yescharacteristic
450C4C23×#44911yescharacteristic
451C4C23×#45211yescharacteristic
452C4C23×#45211yescharacteristic
453C4C23×#45411yescharacteristic
454C4C23×#45611yescharacteristic
455C2—21no—
456C2—21no—
457C2—82no—
458C2C23×#178811yescharacteristic
459C2C23×#179111yescharacteristic
460C2C23×#179311yescharacteristic
subgroupG/H
center, socleC22×C23#379
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1984C2semidirect
3. C23×#1801C2semidirect
4. C23×#1550C2semidirect
6. C23×#1493C2semidirect
156. #8357C23direct (N × H)
204. #1984C2×C23semidirect
205. #1801C2×C23semidirect
206. #1550C2×C23semidirect
208. #1493C2×C23semidirect
322. C23#8357direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35072
C23×#19842
C23×#18012
C23×#15502
C23×#15472
C23×#14932
C23×#14922
#835723