Finite Group Databasebeta

← C23×#10551 #19158177

Subgroups — C23×#10551 #19158177

Order 5888.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3141C211yesmaximal, characteristic
2C23×#1950C211yesmaximal, characteristic, complemented
3C23×#1549C211yesmaximal, characteristic, complemented
4C23×#1547C211yesmaximal, characteristic
5C2×C23×#333C211yesmaximal, 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—82no—
72C2×D4×C23—21no—
73C2×D4×C23—21no—
74C2×D4×C23—21no—
75C2×D4×C23—41no—
76C22×C4×C23—41no—
77C22×C4×C23—41no—
78C22×C4×C23C2×D411yescharacteristic
79C22×C4×C23#4111yescharacteristic
80C22×C4×C23#4111yescharacteristic
81M16×C23—21no—
82M16×C23—21no—
83M16×C23—21no—
84M16×C23—21no—
85M16×C23—41no—
86M16×C23—41no—
87C2×C8×C23—21no—
88C2×C8×C23—21no—
89C2×C8×C23—22no—
90C2×C8×C23—41no—
91C2×C8×C23—41no—
92C2×C8×C23—41no—
93C2×C8×C23—41no—
94C2×C8×C23—41no—
95C2×C8×C23—81no—
96C4⋊C4×C23—22no—
97C4⋊C4×C23—21no—
98C4⋊C4×C23—22no—
99C4⋊C4×C23—21no—
100C4⋊C4×C23—41no—
101C4⋊C4×C23—41no—
102C22⋊C4×C23—41no—
103C22⋊C4×C23—41no—
104C42×C23C2×D411yescharacteristic
105C42×C23C2×D411yescharacteristic
106C42×C23C2×D411yescharacteristic
107C42×C23#4111yescharacteristic
108#10551C2311yesmaximal, characteristic, direct product factor, Sylow
109C23×C23—41no—
110C23×C23#12411yescharacteristic
111Q8×C23—41no—
112Q8×C23—41no—
113Q8×C23—82no—
114Q8×C23—82no—
115D4×C23—41no—
116D4×C23—41no—
117D4×C23—41no—
118D4×C23—41no—
119D4×C23—81no—
120D4×C23—81no—
121D4×C23—81no—
122C2×C4×C23—21no—
123C2×C4×C23—21no—
124C2×C4×C23—41no—
125C2×C4×C23—41no—
126C2×C4×C23—42no—
127C2×C4×C23—82no—
128C2×C4×C23—81no—
129C2×C4×C23—81no—
130C2×C4×C23#12011yescharacteristic
131C2×C4×C23#12111yescharacteristic
132C2×C4×C23#12111yescharacteristic
133C2×C4×C23#12111yescharacteristic
134C2×C4×C23#12311yescharacteristic
135C2×C4×C23#12311yescharacteristic
136C2×C4×C23#13511yescharacteristic
137C2×C4×C23#13511yescharacteristic
138C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C23#13711yescharacteristic
140C8×C23—41no—
141C8×C23—41no—
142C8×C23—42no—
143C8×C23—81no—
144C8×C23—81no—
145C8×C23—81no—
146#3141C2×C2311yescharacteristic
147#1950C2×C2311yescharacteristic, complemented
148#1549C2×C2311yescharacteristic, complemented
149#1547C2×C2311yescharacteristic
150C2×#333C2×C2311yescharacteristic, complemented
151#1480C2×C2311yescharacteristic
152#1283C2×C2311yescharacteristic, complemented
153C22×C23—21no—
154C22×C23—21no—
155C22×C23—81no—
156C22×C23—81no—
157C22×C23—81no—
158C22×C23#39411yescharacteristic
159C22×C23#48811yescharacteristic
160C22×C23#48911yescharacteristic
161C4×C23—21no—
162C4×C23—21no—
163C4×C23—21no—
164C4×C23—82no—
165C4×C23—81no—
166C4×C23#44911yescharacteristic
167C4×C23#45211yescharacteristic
168C4×C23#46311yescharacteristic
169C4×C23#46411yescharacteristic
170C4×C23#47111yescharacteristic
171C4×C23#47311yescharacteristic
172#532C22×C2311yescharacteristic
173C2×C4⋊Q8C22×C2311yescharacteristic
174#489—22no—
175#487—21no—
176#486—21no—
177#431C22×C2311yescharacteristic
178C2×#108—21no—
179#423C22×C2311yescharacteristic
180C2×C4⋊C8C22×C2311yescharacteristic
181C2×C4⋊C8C22×C2311yescharacteristic
182#417—21no—
183#417—21no—
184C4×M16C22×C2311yescharacteristic
185#341—21no—
186#333—22no—
187#332—21no—
188#332—22no—
189#331—21no—
190D4⋊C8—22no—
191C2×C23—21no—
192C2×C23—21no—
193C2×C23—161no—
194C2×C23#204211yescharacteristic
195C2×C23#204811yescharacteristic
196C2×C23#205211yescharacteristic
197C2×#41—21no—
198C22×Q8—21no—
199C2×M16—21no—
200C2×M16D4×C2311yescharacteristic
201C2×M16D4×C2311yescharacteristic
202C22×C8—21no—
203C22×C8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8D4×C2312yes—
208C4⋊Q8D4×C2311yescharacteristic
209#127D4×C2311yescharacteristic
210#124—21no—
211#121—21no—
212#121—21no—
213C4×D4—21no—
214C4×D4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C42C23×C2311yescharacteristic
218#108—42no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8D4×C2312yes—
226#103—22no—
227#103—22no—
228#102—22no—
229#102—22no—
230C22⋊C8—21no—
231#97—21no—
232C4×C8—21no—
233C4×C8—21no—
234C23#1055111yescharacteristic, direct product factor, Sylow
235#41—41no—
236#41—41no—
237#41—41no—
238#41—41no—
239C2×Q8—21no—
240C2×Q8—22no—
241C2×Q8—41no—
242C2×Q8—82no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4C2×D4×C2311yescharacteristic
250C22×C4#41×C2311yescharacteristic
251C22×C4#41×C2311yescharacteristic
252M16—21no—
253M16—21no—
254M16—21no—
255M16—21no—
256M16—41no—
257M16—41no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—22no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—81no—
267C4⋊C4—22no—
268C4⋊C4—21no—
269C4⋊C4—22no—
270C4⋊C4—21no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C42C2×D4×C2311yescharacteristic
276C42C2×D4×C2311yescharacteristic
277C42C2×D4×C2311yescharacteristic
278C42#41×C2311yescharacteristic
279C23—41no—
280C23C23×#12411yescharacteristic
281Q8—41no—
282Q8—41no—
283Q8—82no—
284Q8—82no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4—81no—
290D4—81no—
291D4—81no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4C23×#12011yescharacteristic
301C2×C4C23×#12111yescharacteristic
302C2×C4C23×#12111yescharacteristic
303C2×C4C23×#12111yescharacteristic
304C2×C4C23×#12311yescharacteristic
305C2×C4C23×#12311yescharacteristic
306C2×C4C23×#13511yescharacteristic
307C2×C4C23×#13511yescharacteristic
308C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
309C2×C4C23×#13711yescharacteristic
310C8—41no—
311C8—41no—
312C8—42no—
313C8—81no—
314C8—81no—
315C8—81no—
316C22—21no—
317C22—21no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22C23×#39411yescharacteristic
322C22C23×#48811yescharacteristic
323C22C23×#48911yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—81no—
328C4—82no—
329C4C23×#44911yescharacteristic
330C4C23×#45211yescharacteristic
331C4C23×#46311yescharacteristic
332C4C23×#46411yescharacteristic
333C4C23×#47111yescharacteristic
334C4C23×#47311yescharacteristic
335C2—21no—
336C2—21no—
337C2—161no—
338C2C23×#204211yescharacteristic
339C2C23×#204811yescharacteristic
340C2C23×#205211yescharacteristic
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×#1549C2semidirect
5. C2×C23×#333C2semidirect
7. C23×#1283C2semidirect
108. #10551C23direct (N × H)
147. #1950C2×C23semidirect
148. #1549C2×C23semidirect
150. C2×#333C2×C23semidirect
152. #1283C2×C23semidirect
234. C23#10551direct (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×#15492
C23×#15472
C2×C23×#3332
C23×#14802
C23×#12832
#1055123