Finite Group Databasebeta

← C23×#10502 #19158128

Subgroups — C23×#10502 #19158128

Order 5888.

412 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3227C211yesmaximal, characteristic
2C23×#1997C211yesmaximal, characteristic, complemented
3C23×#1987C211yesmaximal, characteristic, complemented
4C23×#1980C211yesmaximal, characteristic
5C23×#1565C211yesmaximal, characteristic, complemented
6C23×#1547C211yesmaximal, characteristic
7C23×#1303C211yesmaximal, characteristic, complemented
8C23×#532C2211yescharacteristic
9C2×C23×C4⋊Q8C2211yescharacteristic
10C23×#498—21no—
11C23×#486—21no—
12C23×#481—22no—
13C23×#476—21no—
14C23×#476—21no—
15C23×#431C2211yescharacteristic
16C23×#423C2211yescharacteristic
17C2×C23×C4⋊C8C2211yescharacteristic
18C2×C23×#104—21no—
19C2×C23×#104—21no—
20C4×M16×C23C2211yescharacteristic
21C4×C4⋊C4×C23C2211yescharacteristic
22C23×#343—21no—
23C23×#343—21no—
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×#106—22no—
52C23×#106—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×C22⋊C8—21no—
64C23×#97—21no—
65C4×C8×C23—21no—
66C4×C8×C23—21no—
67C23×#95—21no—
68#41×C23—41no—
69#41×C23—41no—
70#41×C23—41no—
71#41×C23—41no—
72C2×Q8×C23—21no—
73C2×Q8×C23—41no—
74C2×Q8×C23—41no—
75C2×Q8×C23—82no—
76C2×D4×C23—21no—
77C2×D4×C23—21no—
78C2×D4×C23—21no—
79C2×D4×C23—41no—
80C22×C4×C23—22no—
81C22×C4×C23—41no—
82C22×C4×C23—41no—
83C22×C4×C23C2×D411yescharacteristic
84C22×C4×C23C2×D411yescharacteristic
85C22×C4×C23#4111yescharacteristic
86M16×C23—21no—
87M16×C23—21no—
88M16×C23—21no—
89M16×C23—41no—
90M16×C23—41no—
91M16×C23SD1612yes—
92C2×C8×C23—21no—
93C2×C8×C23—21no—
94C2×C8×C23—21no—
95C2×C8×C23—21no—
96C2×C8×C23—42no—
97C2×C8×C23—41no—
98C2×C8×C23—41no—
99C4⋊C4×C23—22no—
100C4⋊C4×C23—22no—
101C4⋊C4×C23—21no—
102C4⋊C4×C23—22no—
103C4⋊C4×C23—21no—
104C4⋊C4×C23—41no—
105C4⋊C4×C23—41no—
106C4⋊C4×C23—41no—
107C22⋊C4×C23—41no—
108C22⋊C4×C23—41no—
109C42×C23—41no—
110C42×C23—41no—
111C42×C23C2×D411yescharacteristic
112C42×C23C2×D411yescharacteristic
113C42×C23#4111yescharacteristic
114C42×C23#4111yescharacteristic
115#10502C2311yesmaximal, characteristic, direct product factor, Sylow
116C23×C23—41no—
117C23×C23#12111yescharacteristic
118Q8×C23—41no—
119Q8×C23—41no—
120Q8×C23—82no—
121Q8×C23—82no—
122D4×C23—41no—
123D4×C23—41no—
124D4×C23—41no—
125D4×C23—41no—
126D4×C23—81no—
127D4×C23—81no—
128D4×C23—81no—
129C2×C4×C23—21no—
130C2×C4×C23—21no—
131C2×C4×C23—41no—
132C2×C4×C23—42no—
133C2×C4×C23—41no—
134C2×C4×C23—42no—
135C2×C4×C23—81no—
136C2×C4×C23—81no—
137C2×C4×C23—81no—
138C2×C4×C23—81no—
139C2×C4×C23—82no—
140C2×C4×C23#12011yescharacteristic
141C2×C4×C23#12111yescharacteristic
142C2×C4×C23#12111yescharacteristic
143C2×C4×C23#12311yescharacteristic
144C2×C4×C23#12311yescharacteristic
145C2×C4×C23#12411yescharacteristic
146C2×C4×C23C2×SD1611yescharacteristic
147C2×C4×C23#13511yescharacteristic
148C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C23#13711yescharacteristic
150C8×C23—22no—
151C8×C23—41no—
152C8×C23—41no—
153C8×C23—41no—
154C8×C23—81no—
155C8×C23—81no—
156#3227C2×C2311yescharacteristic
157#1997C2×C2311yescharacteristic, complemented
158#1987C2×C2311yescharacteristic, complemented
159#1980C2×C2311yescharacteristic
160#1565C2×C2311yescharacteristic, complemented
161#1547C2×C2311yescharacteristic
162#1303C2×C2311yescharacteristic, complemented
163C22×C23—21no—
164C22×C23—21no—
165C22×C23—81no—
166C22×C23—81no—
167C22×C23—81no—
168C22×C23#39411yescharacteristic
169C22×C23#46011yescharacteristic
170C22×C23#46411yescharacteristic
171C4×C23—21no—
172C4×C23—21no—
173C4×C23—21no—
174C4×C23—82no—
175C4×C23—82no—
176C4×C23—81no—
177C4×C23#45411yescharacteristic
178C4×C23#45611yescharacteristic
179C4×C23#46111yescharacteristic
180C4×C23#46311yescharacteristic
181C4×C23#46511yescharacteristic
182C4×C23#46811yescharacteristic
183#532C22×C2311yescharacteristic
184C2×C4⋊Q8C22×C2311yescharacteristic
185#498—21no—
186#486—21no—
187#481—22no—
188#476—21no—
189#476—21no—
190#431C22×C2311yescharacteristic
191#423C22×C2311yescharacteristic
192C2×C4⋊C8C22×C2311yescharacteristic
193C2×#104—21no—
194C2×#104—21no—
195C4×M16C22×C2311yescharacteristic
196C4×C4⋊C4C22×C2311yescharacteristic
197#343—21no—
198#343—21no—
199#340—22no—
200#335—21no—
201#334—21no—
202#332—21no—
203#331—21no—
204C2×C23—21no—
205C2×C23—21no—
206C2×C23—161no—
207C2×C23#203811yescharacteristic
208C2×C23#204811yescharacteristic
209C2×C23#205511yescharacteristic
210C2×#41—21no—
211C22×Q8—21no—
212C2×M16—21no—
213C2×M16D4×C2311yescharacteristic
214C2×M16D4×C2311yescharacteristic
215C22×C8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8D4×C2311yescharacteristic
221#127D4×C2311yescharacteristic
222#124—21no—
223#121—21no—
224#121—21no—
225C4×D4—21no—
226C4×D4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4D4×C2312yes—
230C2×C42—21no—
231C2×C42C23×C2311yescharacteristic
232#106—22no—
233#106—22no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8D4×C2311yescharacteristic
239C4⋊C8D4×C2311yescharacteristic
240#104—42no—
241#104—42no—
242#102—22no—
243#102—22no—
244C22⋊C8—21no—
245#97—21no—
246C4×C8—21no—
247C4×C8—21no—
248#95—21no—
249C23#1050211yescharacteristic, direct product factor, Sylow
250#41—41no—
251#41—41no—
252#41—41no—
253#41—41no—
254C2×Q8—21no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—82no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—41no—
262C22×C4—22no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4C2×D4×C2311yescharacteristic
266C22×C4C2×D4×C2311yescharacteristic
267C22×C4#41×C2311yescharacteristic
268M16—21no—
269M16—21no—
270M16—21no—
271M16—41no—
272M16—41no—
273M16SD16×C2312yes—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—42no—
279C2×C8—41no—
280C2×C8—41no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—21no—
284C4⋊C4—22no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C42—41no—
292C42—41no—
293C42C2×D4×C2311yescharacteristic
294C42C2×D4×C2311yescharacteristic
295C42#41×C2311yescharacteristic
296C42#41×C2311yescharacteristic
297C23—41no—
298C23C23×#12111yescharacteristic
299Q8—41no—
300Q8—41no—
301Q8—82no—
302Q8—82no—
303D4—41no—
304D4—41no—
305D4—41no—
306D4—41no—
307D4—81no—
308D4—81no—
309D4—81no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—82no—
321C2×C4C23×#12011yescharacteristic
322C2×C4C23×#12111yescharacteristic
323C2×C4C23×#12111yescharacteristic
324C2×C4C23×#12311yescharacteristic
325C2×C4C23×#12311yescharacteristic
326C2×C4C23×#12411yescharacteristic
327C2×C4C2×SD16×C2311yescharacteristic
328C2×C4C23×#13511yescharacteristic
329C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
330C2×C4C23×#13711yescharacteristic
331C8—22no—
332C8—41no—
333C8—41no—
334C8—41no—
335C8—81no—
336C8—81no—
337C22—21no—
338C22—21no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22C23×#39411yescharacteristic
343C22C23×#46011yescharacteristic
344C22C23×#46411yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—82no—
349C4—82no—
350C4—81no—
351C4C23×#45411yescharacteristic
352C4C23×#45611yescharacteristic
353C4C23×#46111yescharacteristic
354C4C23×#46311yescharacteristic
355C4C23×#46511yescharacteristic
356C4C23×#46811yescharacteristic
357C2—21no—
358C2—21no—
359C2—161no—
360C2C23×#203811yescharacteristic
361C2C23×#204811yescharacteristic
362C2C23×#205511yescharacteristic
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×#1997C2semidirect
3. C23×#1987C2semidirect
5. C23×#1565C2semidirect
7. C23×#1303C2semidirect
115. #10502C23direct (N × H)
157. #1997C2×C23semidirect
158. #1987C2×C23semidirect
160. #1565C2×C23semidirect
162. #1303C2×C23semidirect
249. C23#10502direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32272
C23×#19972
C23×#19872
C23×#19802
C23×#15652
C23×#15472
C23×#13032
#1050223