Finite Group Databasebeta

← C23×#10552 #19158178

Subgroups — C23×#10552 #19158178

Order 5888.

386 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3141C211yesmaximal, characteristic
2C23×#1950C211yesmaximal, characteristic, complemented
3C23×#1548C211yesmaximal, characteristic, complemented
4C23×#1547C211yesmaximal, characteristic
5C2×C23×#332C211yesmaximal, characteristic, complemented
6C23×#1480C211yesmaximal, characteristic
7C23×#1283C211yesmaximal, characteristic, complemented
8C23×#532C2211yescharacteristic
9C2×C23×C4⋊Q8C2211yescharacteristic
10C23×#489—22no—
11C23×#487—21no—
12C23×#486—21no—
13C23×#431C2211yescharacteristic
14C2×C23×#108—21no—
15C23×#423C2211yescharacteristic
16C2×C23×C4⋊C8C2211yescharacteristic
17C2×C23×C4⋊C8C2211yescharacteristic
18C23×#417—21no—
19C23×#417—21no—
20C4×M16×C23C2211yescharacteristic
21C23×#341—21no—
22C23×#333—22no—
23C23×#332—21no—
24C23×#332—22no—
25C23×#331—21no—
26C23×D4⋊C8—22no—
27C2×#41×C23—21no—
28C22×Q8×C23—21no—
29C2×M16×C23—21no—
30C2×M16×C23D411yescharacteristic
31C2×M16×C23D411yescharacteristic
32C22×C8×C23—21no—
33C22×C8×C23—21no—
34C23×C4⋊Q8—21no—
35C23×C4⋊Q8—21no—
36C23×C4⋊Q8—21no—
37C23×C4⋊Q8D411yescharacteristic
38C23×C4⋊Q8D412yes—
39C23×#127D411yescharacteristic
40C23×#124—21no—
41C23×#121—21no—
42C23×#121—21no—
43C4×D4×C23—21no—
44C4×D4×C23—21no—
45C2×C4⋊C4×C23—21no—
46C2×C4⋊C4×C23—21no—
47C2×C42×C23C2311yescharacteristic
48C23×#108—42no—
49C23×C4⋊C8—21no—
50C23×C4⋊C8—21no—
51C23×C4⋊C8—21no—
52C23×C4⋊C8—21no—
53C23×C4⋊C8—21no—
54C23×C4⋊C8—21no—
55C23×C4⋊C8D412yes—
56C23×#103—22no—
57C23×#103—22no—
58C23×#102—22no—
59C23×#102—22no—
60C23×C22⋊C8—21no—
61C23×#97—21no—
62C4×C8×C23—21no—
63C4×C8×C23—21no—
64#41×C23—41no—
65#41×C23—41no—
66#41×C23—41no—
67#41×C23—41no—
68C2×Q8×C23—21no—
69C2×Q8×C23—22no—
70C2×Q8×C23—41no—
71C2×Q8×C23—81no—
72C2×Q8×C23—81no—
73C2×D4×C23—21no—
74C2×D4×C23—21no—
75C2×D4×C23—21no—
76C2×D4×C23—41no—
77C22×C4×C23—41no—
78C22×C4×C23—41no—
79C22×C4×C23C2×D411yescharacteristic
80C22×C4×C23#4111yescharacteristic
81C22×C4×C23#4111yescharacteristic
82M16×C23—21no—
83M16×C23—21no—
84M16×C23—21no—
85M16×C23—21no—
86M16×C23—41no—
87M16×C23—41no—
88C2×C8×C23—21no—
89C2×C8×C23—21no—
90C2×C8×C23—22no—
91C2×C8×C23—41no—
92C2×C8×C23—41no—
93C2×C8×C23—41no—
94C2×C8×C23—41no—
95C2×C8×C23—41no—
96C2×C8×C23—81no—
97C4⋊C4×C23—22no—
98C4⋊C4×C23—22no—
99C4⋊C4×C23—21no—
100C4⋊C4×C23—21no—
101C4⋊C4×C23—41no—
102C4⋊C4×C23—41no—
103C22⋊C4×C23—41no—
104C22⋊C4×C23—41no—
105C42×C23C2×D411yescharacteristic
106C42×C23C2×D411yescharacteristic
107C42×C23C2×D411yescharacteristic
108C42×C23#4111yescharacteristic
109#10552C2311yesmaximal, characteristic, direct product factor, Sylow
110C23×C23—41no—
111C23×C23#12411yescharacteristic
112Q8×C23—41no—
113Q8×C23—41no—
114Q8×C23—82no—
115Q8×C23—81no—
116Q8×C23—81no—
117D4×C23—41no—
118D4×C23—41no—
119D4×C23—41no—
120D4×C23—41no—
121D4×C23—81no—
122D4×C23—81no—
123D4×C23—81no—
124C2×C4×C23—21no—
125C2×C4×C23—21no—
126C2×C4×C23—41no—
127C2×C4×C23—41no—
128C2×C4×C23—42no—
129C2×C4×C23—82no—
130C2×C4×C23—81no—
131C2×C4×C23—81no—
132C2×C4×C23#12011yescharacteristic
133C2×C4×C23#12111yescharacteristic
134C2×C4×C23#12111yescharacteristic
135C2×C4×C23#12111yescharacteristic
136C2×C4×C23#12311yescharacteristic
137C2×C4×C23#12311yescharacteristic
138C2×C4×C23#13511yescharacteristic
139C2×C4×C23#13511yescharacteristic
140C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C23#13711yescharacteristic
142C8×C23—41no—
143C8×C23—41no—
144C8×C23—42no—
145C8×C23—81no—
146C8×C23—81no—
147C8×C23—81no—
148#3141C2×C2311yescharacteristic
149#1950C2×C2311yescharacteristic, complemented
150#1548C2×C2311yescharacteristic, complemented
151#1547C2×C2311yescharacteristic
152C2×#332C2×C2311yescharacteristic, complemented
153#1480C2×C2311yescharacteristic
154#1283C2×C2311yescharacteristic, complemented
155C22×C23—21no—
156C22×C23—21no—
157C22×C23—81no—
158C22×C23—81no—
159C22×C23—81no—
160C22×C23#39411yescharacteristic
161C22×C23#48811yescharacteristic
162C22×C23#48911yescharacteristic
163C4×C23—21no—
164C4×C23—21no—
165C4×C23—21no—
166C4×C23—82no—
167C4×C23—81no—
168C4×C23#44911yescharacteristic
169C4×C23#45211yescharacteristic
170C4×C23#46311yescharacteristic
171C4×C23#46411yescharacteristic
172C4×C23#47111yescharacteristic
173C4×C23#47211yescharacteristic
174#532C22×C2311yescharacteristic
175C2×C4⋊Q8C22×C2311yescharacteristic
176#489—22no—
177#487—21no—
178#486—21no—
179#431C22×C2311yescharacteristic
180C2×#108—21no—
181#423C22×C2311yescharacteristic
182C2×C4⋊C8C22×C2311yescharacteristic
183C2×C4⋊C8C22×C2311yescharacteristic
184#417—21no—
185#417—21no—
186C4×M16C22×C2311yescharacteristic
187#341—21no—
188#333—22no—
189#332—21no—
190#332—22no—
191#331—21no—
192D4⋊C8—22no—
193C2×C23—21no—
194C2×C23—21no—
195C2×C23—161no—
196C2×C23#204211yescharacteristic
197C2×C23#204911yescharacteristic
198C2×C23#205111yescharacteristic
199C2×#41—21no—
200C22×Q8—21no—
201C2×M16—21no—
202C2×M16D4×C2311yescharacteristic
203C2×M16D4×C2311yescharacteristic
204C22×C8—21no—
205C22×C8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8D4×C2312yes—
210C4⋊Q8D4×C2311yescharacteristic
211#127D4×C2311yescharacteristic
212#124—21no—
213#121—21no—
214#121—21no—
215C4×D4—21no—
216C4×D4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C42C23×C2311yescharacteristic
220#108—42no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8D4×C2312yes—
228#103—22no—
229#103—22no—
230#102—22no—
231#102—22no—
232C22⋊C8—21no—
233#97—21no—
234C4×C8—21no—
235C4×C8—21no—
236C23#1055211yescharacteristic, direct product factor, Sylow
237#41—41no—
238#41—41no—
239#41—41no—
240#41—41no—
241C2×Q8—21no—
242C2×Q8—22no—
243C2×Q8—41no—
244C2×Q8—81no—
245C2×Q8—81no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4C2×D4×C2311yescharacteristic
253C22×C4#41×C2311yescharacteristic
254C22×C4#41×C2311yescharacteristic
255M16—21no—
256M16—21no—
257M16—21no—
258M16—21no—
259M16—41no—
260M16—41no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—22no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—81no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C42C2×D4×C2311yescharacteristic
279C42C2×D4×C2311yescharacteristic
280C42C2×D4×C2311yescharacteristic
281C42#41×C2311yescharacteristic
282C23—41no—
283C23C23×#12411yescharacteristic
284Q8—41no—
285Q8—41no—
286Q8—82no—
287Q8—81no—
288Q8—81no—
289D4—41no—
290D4—41no—
291D4—41no—
292D4—41no—
293D4—81no—
294D4—81no—
295D4—81no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—82no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4C23×#12011yescharacteristic
305C2×C4C23×#12111yescharacteristic
306C2×C4C23×#12111yescharacteristic
307C2×C4C23×#12111yescharacteristic
308C2×C4C23×#12311yescharacteristic
309C2×C4C23×#12311yescharacteristic
310C2×C4C23×#13511yescharacteristic
311C2×C4C23×#13511yescharacteristic
312C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
313C2×C4C23×#13711yescharacteristic
314C8—41no—
315C8—41no—
316C8—42no—
317C8—81no—
318C8—81no—
319C8—81no—
320C22—21no—
321C22—21no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22C23×#39411yescharacteristic
326C22C23×#48811yescharacteristic
327C22C23×#48911yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—81no—
332C4—82no—
333C4C23×#44911yescharacteristic
334C4C23×#45211yescharacteristic
335C4C23×#46311yescharacteristic
336C4C23×#46411yescharacteristic
337C4C23×#47111yescharacteristic
338C4C23×#47211yescharacteristic
339C2—21no—
340C2—21no—
341C2—161no—
342C2C23×#204211yescharacteristic
343C2C23×#204911yescharacteristic
344C2C23×#205111yescharacteristic
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×#1950C2semidirect
3. C23×#1548C2semidirect
5. C2×C23×#332C2semidirect
7. C23×#1283C2semidirect
109. #10552C23direct (N × H)
149. #1950C2×C23semidirect
150. #1548C2×C23semidirect
152. C2×#332C2×C23semidirect
154. #1283C2×C23semidirect
236. C23#10552direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31412
C23×#19502
C23×#15482
C23×#15472
C2×C23×#3322
C23×#14802
C23×#12832
#1055223