Finite Group Databasebeta

← C23×#7712 #19155338

Subgroups — C23×#7712 #19155338

Order 5888.

426 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2860C211yesmaximal, characteristic
2C23×#1839C211yesmaximal, characteristic, complemented
3C23×#1767C211yesmaximal, characteristic, complemented
4C23×#1271C212yesmaximal, complemented
5C23×#1270C212yesmaximal
6C2×C23×#126—21no—
7C2×C23×C4⋊C8—21no—
8C2×C23×#97—21no—
9C23×#390C411yescharacteristic
10C23×#390C411yescharacteristic
11C23×#390C2211yescharacteristic
12C23×#389C411yescharacteristic
13C23×#389C411yescharacteristic
14C23×#389C2211yescharacteristic
15C23×#388—21no—
16C23×#388—21no—
17C23×#388—21no—
18C43×C23C2211yescharacteristic
19C23×#336—21no—
20C23×#336—21no—
21C23×#336C2212yes—
22C23×#336C2212yes—
23C22×D4×C23C2×C411yescharacteristic
24C22×C8×C23—22no—
25C22×C8×C23—22no—
26C23×#126—44no—
27C2×C4⋊C4×C23—21no—
28C2×C4⋊C4×C23—21no—
29C2×C4⋊C4×C23—21no—
30C2×C4⋊C4×C23C2×C411yescharacteristic
31C2×C4⋊C4×C23C2×C411yescharacteristic
32C2×C4⋊C4×C23C2×C411yescharacteristic
33C2×C22⋊C4×C23—21no—
34C2×C22⋊C4×C23—21no—
35C2×C22⋊C4×C23—21no—
36C2×C22⋊C4×C23—21no—
37C2×C22⋊C4×C23—21no—
38C2×C22⋊C4×C23—21no—
39C2×C42×C23C2×C411yescharacteristic
40C2×C42×C23C2×C411yescharacteristic
41C2×C42×C23D411yescharacteristic
42C2×C42×C23D411yescharacteristic
43C2×C42×C23D411yescharacteristic
44C2×C42×C23D411yescharacteristic
45C2×C42×C23C2311yescharacteristic
46C23×C4⋊C8—24no—
47C23×#97—42no—
48C23×#95—21no—
49C23×#95—21no—
50C23×#95—21no—
51C24×C23—21no—
52C2×D4×C23—22no—
53C2×D4×C23—41no—
54C2×D4×C23—41no—
55C22×C4×C23—21no—
56C22×C4×C23—21no—
57C22×C4×C23—21no—
58C22×C4×C23C22⋊C411yescharacteristic
59C22×C4×C23C22⋊C411yescharacteristic
60C22×C4×C23C22⋊C411yescharacteristic
61C22×C4×C23C22⋊C411yescharacteristic
62C22×C4×C23C22×C411yescharacteristic
63C22×C4×C23C2×D411yescharacteristic
64C22×C4×C23C2×D411yescharacteristic
65C2×C8×C23—24no—
66C2×C8×C23—42no—
67C2×C8×C23—42no—
68C2×C8×C23—42no—
69C2×C8×C23—42no—
70C2×C8×C23—42no—
71C4⋊C4×C23—22no—
72C4⋊C4×C23—41no—
73C4⋊C4×C23—41no—
74C4⋊C4×C23—42no—
75C4⋊C4×C23—42no—
76C4⋊C4×C23—42no—
77C22⋊C4×C23—42no—
78C22⋊C4×C23—42no—
79C22⋊C4×C23—42no—
80C22⋊C4×C23—42no—
81C22⋊C4×C23—41no—
82C22⋊C4×C23—42no—
83C22⋊C4×C23—41no—
84C42×C23—22no—
85C42×C23—22no—
86C42×C23—22no—
87C42×C23—21no—
88C42×C23—21no—
89C42×C23—41no—
90C42×C23—41no—
91C42×C23—41no—
92#7712C2311yesmaximal, characteristic, direct product factor, Sylow
93C23×C23—41no—
94C23×C23—41no—
95C23×C23—41no—
96C23×C23—41no—
97C23×C23—41no—
98C23×C23—41no—
99C23×C23—41no—
100C23×C23C2×C22⋊C411yescharacteristic
101D4×C23—82no—
102C2×C4×C23—21no—
103C2×C4×C23—21no—
104C2×C4×C23—21no—
105C2×C4×C23—21no—
106C2×C4×C23—21no—
107C2×C4×C23—21no—
108C2×C4×C23—21no—
109C2×C4×C23—21no—
110C2×C4×C23—21no—
111C2×C4×C23—21no—
112C2×C4×C23—21no—
113C2×C4×C23—21no—
114C2×C4×C23—21no—
115C2×C4×C23—21no—
116C2×C4×C23—21no—
117C2×C4×C23—21no—
118C2×C4×C23—21no—
119C2×C4×C23—41no—
120C2×C4×C23—41no—
121C2×C4×C23—41no—
122C2×C4×C23—41no—
123C2×C4×C23—41no—
124C2×C4×C23—41no—
125C2×C4×C23—41no—
126C2×C4×C23—41no—
127C2×C4×C23—41no—
128C2×C4×C23#10412yes—
129C2×C4×C23#13112yes—
130C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C23AGL(1,ℤ/8)12yes—
132C2×C4×C23#13711yescharacteristic
133C8×C23—44no—
134C8×C23—82no—
135#2860C2×C2311yescharacteristic
136#1839C2×C2311yescharacteristic, complemented
137#1767C2×C2311yescharacteristic, complemented
138#1271C2×C2312yescomplemented
139#1270C2×C2312yes—
140C22×C23—81no—
141C22×C23—81no—
142C22×C23—81no—
143C22×C23—81no—
144C22×C23—81no—
145C22×C23—81no—
146C22×C23—81no—
147C22×C23#40811yescharacteristic
148C22×C23#41011yescharacteristic
149C22×C23#41311yescharacteristic
150C22×C23#41711yescharacteristic
151C22×C23#41811yescharacteristic
152C22×C23C2×#10411yescharacteristic
153C22×C23#42111yescharacteristic
154C4×C23—22no—
155C4×C23—21no—
156C4×C23—21no—
157C4×C23—22no—
158C4×C23—22no—
159C4×C23—41no—
160C4×C23—41no—
161C4×C23—41no—
162C4×C23—81no—
163C4×C23—81no—
164C4×C23—81no—
165C2×#126—21no—
166C2×C4⋊C8—21no—
167C2×#97—21no—
168#390C4×C2311yescharacteristic
169#390C4×C2311yescharacteristic
170#390C22×C2311yescharacteristic
171#389C4×C2311yescharacteristic
172#389C4×C2311yescharacteristic
173#389C22×C2311yescharacteristic
174#388—21no—
175#388—21no—
176#388—21no—
177C43C22×C2311yescharacteristic
178#336—21no—
179#336—21no—
180#336C22×C2312yes—
181#336C22×C2312yes—
182C2×C23—161no—
183C2×C23#147311yescharacteristic
184C2×C23#148411yescharacteristic
185C2×C23#149511yescharacteristic
186C2×C23#150911yescharacteristic
187C2×C23#151411yescharacteristic
188C2×C23#153211yescharacteristic
189C2×C23#155711yescharacteristic
190C22×D4C2×C4×C2311yescharacteristic
191C22×C8—22no—
192C22×C8—22no—
193#126—44no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2×C4×C2311yescharacteristic
198C2×C4⋊C4C2×C4×C2311yescharacteristic
199C2×C4⋊C4C2×C4×C2311yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C42C2×C4×C2311yescharacteristic
207C2×C42C2×C4×C2311yescharacteristic
208C2×C42D4×C2311yescharacteristic
209C2×C42D4×C2311yescharacteristic
210C2×C42D4×C2311yescharacteristic
211C2×C42D4×C2311yescharacteristic
212C2×C42C23×C2311yescharacteristic
213C4⋊C8—24no—
214#97—42no—
215#95—21no—
216#95—21no—
217#95—21no—
218C23#771211yescharacteristic, direct product factor, Sylow
219C24—21no—
220C2×D4—22no—
221C2×D4—41no—
222C2×D4—41no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4C22⋊C4×C2311yescharacteristic
227C22×C4C22⋊C4×C2311yescharacteristic
228C22×C4C22⋊C4×C2311yescharacteristic
229C22×C4C22⋊C4×C2311yescharacteristic
230C22×C4C22×C4×C2311yescharacteristic
231C22×C4C2×D4×C2311yescharacteristic
232C22×C4C2×D4×C2311yescharacteristic
233C2×C8—24no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C4⋊C4—22no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C42—22no—
253C42—22no—
254C42—21no—
255C42—22no—
256C42—21no—
257C42—41no—
258C42—41no—
259C42—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23C2×C22⋊C4×C2311yescharacteristic
268D4—82no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4C23×#10412yes—
296C2×C4C23×#13112yes—
297C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
298C2×C4C23×AGL(1,ℤ/8)12yes—
299C2×C4C23×#13711yescharacteristic
300C8—44no—
301C8—82no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22C23×#40811yescharacteristic
310C22C23×#41011yescharacteristic
311C22C23×#41311yescharacteristic
312C22C23×#41711yescharacteristic
313C22C23×#41811yescharacteristic
314C22C2×C23×#10411yescharacteristic
315C22C23×#42111yescharacteristic
316C4—22no—
317C4—21no—
318C4—21no—
319C4—22no—
320C4—22no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—81no—
325C4—81no—
326C4—81no—
327C2—161no—
328C2C23×#147311yescharacteristic
329C2C23×#148411yescharacteristic
330C2C23×#149511yescharacteristic
331C2C23×#150911yescharacteristic
332C2C23×#151411yescharacteristic
333C2C23×#153211yescharacteristic
334C2C23×#155711yescharacteristic
subgroupG/H
center, socleC23×C23C2×C22⋊C4
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC22×D4×C23C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C43×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1839C2semidirect
3. C23×#1767C2semidirect
4. C23×#1271C2semidirect
92. #7712C23direct (N × H)
136. #1839C2×C23semidirect
137. #1767C2×C23semidirect
138. #1271C2×C23semidirect
218. C23#7712direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#28602
C23×#18392
C23×#17672
C23×#12712
C23×#12702
#771223