Finite Group Databasebeta

← C23×#10503 #19158129

Subgroups — C23×#10503 #19158129

Order 5888.

428 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3226C211yesmaximal, characteristic
2C23×#1988C211yesmaximal, characteristic, complemented
3C23×#1980C211yesmaximal, characteristic
4C23×#1966C211yesmaximal, characteristic, complemented
5C23×#1565C211yesmaximal, characteristic, complemented
6C23×#1547C211yesmaximal, characteristic
7C23×#1303C211yesmaximal, characteristic, complemented
8C23×#532C2211yescharacteristic
9C2×C23×C4⋊Q8C2211yescharacteristic
10C23×#500—21no—
11C23×#486—21no—
12C23×#480—22no—
13C23×#474—21no—
14C23×#474—21no—
15C23×#431C2211yescharacteristic
16C23×#423C2211yescharacteristic
17C2×C23×C4⋊C8C2211yescharacteristic
18C2×C23×#104—21no—
19C2×C23×#104—21no—
20C2×C23×#101—21no—
21C2×C23×#101—21no—
22C4×M16×C23C2211yescharacteristic
23C4×C4⋊C4×C23C2211yescharacteristic
24C23×#340—22no—
25C23×#335—21no—
26C23×#334—21no—
27C23×#332—21no—
28C23×#331—21no—
29C2×#41×C23—21no—
30C22×Q8×C23—21no—
31C2×M16×C23—21no—
32C2×M16×C23D411yescharacteristic
33C2×M16×C23D411yescharacteristic
34C22×C8×C23—21no—
35C23×C4⋊Q8—21no—
36C23×C4⋊Q8—21no—
37C23×C4⋊Q8—21no—
38C23×C4⋊Q8—21no—
39C23×C4⋊Q8D411yescharacteristic
40C23×#127D411yescharacteristic
41C23×#124—21no—
42C23×#121—21no—
43C23×#121—21no—
44C4×D4×C23—21no—
45C4×D4×C23—21no—
46C2×C4⋊C4×C23—21no—
47C2×C4⋊C4×C23—21no—
48C2×C4⋊C4×C23D412yes—
49C2×C42×C23—21no—
50C2×C42×C23C2311yescharacteristic
51C23×#107—22no—
52C23×#107—22no—
53C23×C4⋊C8—21no—
54C23×C4⋊C8—21no—
55C23×C4⋊C8—21no—
56C23×C4⋊C8—21no—
57C23×C4⋊C8D411yescharacteristic
58C23×C4⋊C8D411yescharacteristic
59C23×#104—42no—
60C23×#104—42no—
61C23×#102—22no—
62C23×#102—22no—
63C23×#101—42no—
64C23×#101—42no—
65C23×C22⋊C8—21no—
66C23×#97—21no—
67C4×C8×C23—21no—
68C4×C8×C23—21no—
69C23×#95—21no—
70#41×C23—41no—
71#41×C23—41no—
72#41×C23—41no—
73#41×C23—41no—
74C2×Q8×C23—21no—
75C2×Q8×C23—41no—
76C2×Q8×C23—41no—
77C2×Q8×C23—41no—
78C2×Q8×C23—41no—
79C2×Q8×C23—41no—
80C2×Q8×C23—41no—
81C2×D4×C23—21no—
82C2×D4×C23—21no—
83C2×D4×C23—21no—
84C2×D4×C23—41no—
85C22×C4×C23—22no—
86C22×C4×C23—41no—
87C22×C4×C23—41no—
88C22×C4×C23C2×D411yescharacteristic
89C22×C4×C23C2×D411yescharacteristic
90C22×C4×C23#4111yescharacteristic
91M16×C23—21no—
92M16×C23—21no—
93M16×C23—21no—
94M16×C23—41no—
95M16×C23—41no—
96M16×C23D812yes—
97C2×C8×C23—21no—
98C2×C8×C23—21no—
99C2×C8×C23—21no—
100C2×C8×C23—21no—
101C2×C8×C23—42no—
102C2×C8×C23—41no—
103C2×C8×C23—41no—
104C4⋊C4×C23—22no—
105C4⋊C4×C23—22no—
106C4⋊C4×C23—21no—
107C4⋊C4×C23—22no—
108C4⋊C4×C23—21no—
109C4⋊C4×C23—41no—
110C4⋊C4×C23—41no—
111C4⋊C4×C23—41no—
112C22⋊C4×C23—41no—
113C22⋊C4×C23—41no—
114C42×C23—41no—
115C42×C23—41no—
116C42×C23C2×D411yescharacteristic
117C42×C23C2×D411yescharacteristic
118C42×C23#4111yescharacteristic
119C42×C23#4111yescharacteristic
120#10503C2311yesmaximal, characteristic, direct product factor, Sylow
121C23×C23—41no—
122C23×C23#12111yescharacteristic
123Q8×C23—41no—
124Q8×C23—41no—
125Q8×C23—81no—
126Q8×C23—81no—
127Q8×C23—81no—
128Q8×C23—81no—
129D4×C23—41no—
130D4×C23—41no—
131D4×C23—41no—
132D4×C23—41no—
133D4×C23—81no—
134D4×C23—81no—
135D4×C23—81no—
136C2×C4×C23—21no—
137C2×C4×C23—21no—
138C2×C4×C23—41no—
139C2×C4×C23—42no—
140C2×C4×C23—41no—
141C2×C4×C23—42no—
142C2×C4×C23—42no—
143C2×C4×C23—42no—
144C2×C4×C23—81no—
145C2×C4×C23—81no—
146C2×C4×C23—81no—
147C2×C4×C23—81no—
148C2×C4×C23#12011yescharacteristic
149C2×C4×C23#12111yescharacteristic
150C2×C4×C23#12111yescharacteristic
151C2×C4×C23#12311yescharacteristic
152C2×C4×C23#12311yescharacteristic
153C2×C4×C23#12411yescharacteristic
154C2×C4×C23C2×D811yescharacteristic
155C2×C4×C23#13511yescharacteristic
156C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
158C8×C23—22no—
159C8×C23—41no—
160C8×C23—41no—
161C8×C23—41no—
162C8×C23—81no—
163C8×C23—81no—
164#3226C2×C2311yescharacteristic
165#1988C2×C2311yescharacteristic, complemented
166#1980C2×C2311yescharacteristic
167#1966C2×C2311yescharacteristic, complemented
168#1565C2×C2311yescharacteristic, complemented
169#1547C2×C2311yescharacteristic
170#1303C2×C2311yescharacteristic, complemented
171C22×C23—21no—
172C22×C23—21no—
173C22×C23—81no—
174C22×C23—81no—
175C22×C23—81no—
176C22×C23#39411yescharacteristic
177C22×C23#45911yescharacteristic
178C22×C23#46311yescharacteristic
179C4×C23—21no—
180C4×C23—21no—
181C4×C23—21no—
182C4×C23—82no—
183C4×C23—82no—
184C4×C23—81no—
185C4×C23#45411yescharacteristic
186C4×C23#45611yescharacteristic
187C4×C23#45911yescharacteristic
188C4×C23#46311yescharacteristic
189C4×C23#46611yescharacteristic
190C4×C23#46911yescharacteristic
191#532C22×C2311yescharacteristic
192C2×C4⋊Q8C22×C2311yescharacteristic
193#500—21no—
194#486—21no—
195#480—22no—
196#474—21no—
197#474—21no—
198#431C22×C2311yescharacteristic
199#423C22×C2311yescharacteristic
200C2×C4⋊C8C22×C2311yescharacteristic
201C2×#104—21no—
202C2×#104—21no—
203C2×#101—21no—
204C2×#101—21no—
205C4×M16C22×C2311yescharacteristic
206C4×C4⋊C4C22×C2311yescharacteristic
207#340—22no—
208#335—21no—
209#334—21no—
210#332—21no—
211#331—21no—
212C2×C23—21no—
213C2×C23—21no—
214C2×C23—161no—
215C2×C23#203811yescharacteristic
216C2×C23#204911yescharacteristic
217C2×C23#205411yescharacteristic
218C2×#41—21no—
219C22×Q8—21no—
220C2×M16—21no—
221C2×M16D4×C2311yescharacteristic
222C2×M16D4×C2311yescharacteristic
223C22×C8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8D4×C2311yescharacteristic
229#127D4×C2311yescharacteristic
230#124—21no—
231#121—21no—
232#121—21no—
233C4×D4—21no—
234C4×D4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4D4×C2312yes—
238C2×C42—21no—
239C2×C42C23×C2311yescharacteristic
240#107—22no—
241#107—22no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8D4×C2311yescharacteristic
247C4⋊C8D4×C2311yescharacteristic
248#104—42no—
249#104—42no—
250#102—22no—
251#102—22no—
252#101—42no—
253#101—42no—
254C22⋊C8—21no—
255#97—21no—
256C4×C8—21no—
257C4×C8—21no—
258#95—21no—
259C23#1050311yescharacteristic, direct product factor, Sylow
260#41—41no—
261#41—41no—
262#41—41no—
263#41—41no—
264C2×Q8—21no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4C2×D4×C2311yescharacteristic
279C22×C4C2×D4×C2311yescharacteristic
280C22×C4#41×C2311yescharacteristic
281M16—21no—
282M16—21no—
283M16—21no—
284M16—41no—
285M16—41no—
286M16D8×C2312yes—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—42no—
292C2×C8—41no—
293C2×C8—41no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—21no—
297C4⋊C4—22no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C42—41no—
305C42—41no—
306C42C2×D4×C2311yescharacteristic
307C42C2×D4×C2311yescharacteristic
308C42#41×C2311yescharacteristic
309C42#41×C2311yescharacteristic
310C23—41no—
311C23C23×#12111yescharacteristic
312Q8—41no—
313Q8—41no—
314Q8—81no—
315Q8—81no—
316Q8—81no—
317Q8—81no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—81no—
323D4—81no—
324D4—81no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4C23×#12011yescharacteristic
338C2×C4C23×#12111yescharacteristic
339C2×C4C23×#12111yescharacteristic
340C2×C4C23×#12311yescharacteristic
341C2×C4C23×#12311yescharacteristic
342C2×C4C23×#12411yescharacteristic
343C2×C4C2×D8×C2311yescharacteristic
344C2×C4C23×#13511yescharacteristic
345C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
346C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
347C8—22no—
348C8—41no—
349C8—41no—
350C8—41no—
351C8—81no—
352C8—81no—
353C22—21no—
354C22—21no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22C23×#39411yescharacteristic
359C22C23×#45911yescharacteristic
360C22C23×#46311yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—82no—
365C4—82no—
366C4—81no—
367C4C23×#45411yescharacteristic
368C4C23×#45611yescharacteristic
369C4C23×#45911yescharacteristic
370C4C23×#46311yescharacteristic
371C4C23×#46611yescharacteristic
372C4C23×#46911yescharacteristic
373C2—21no—
374C2—21no—
375C2—161no—
376C2C23×#203811yescharacteristic
377C2C23×#204911yescharacteristic
378C2C23×#205411yescharacteristic
subgroupG/H
center, socleC22×C23#394
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC23×#532C22
LCSG ▷ C2×C42 ▷ 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×#1988C2semidirect
4. C23×#1966C2semidirect
5. C23×#1565C2semidirect
7. C23×#1303C2semidirect
120. #10503C23direct (N × H)
165. #1988C2×C23semidirect
167. #1966C2×C23semidirect
168. #1565C2×C23semidirect
170. #1303C2×C23semidirect
259. C23#10503direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32262
C23×#19882
C23×#19802
C23×#19662
C23×#15652
C23×#15472
C23×#13032
#1050323