Finite Group Databasebeta

← C23×#12921 #19160547

Subgroups — C23×#12921 #19160547

Order 5888.

442 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#427C211yesmaximal, characteristic
2C2×C23×#367C211yesmaximal, characteristic, complemented
3C2×C23×#366C211yesmaximal, characteristic, complemented
4C2×C23×C22⋊C16C211yesmaximal, characteristic
5C23×#1808C211yesmaximal, characteristic
6C23×#1380C211yesmaximal, characteristic, complemented
7C23×#1380C211yesmaximal, characteristic, complemented
8C23×C8×C23C2211yescharacteristic
9C2×C23×#117—21no—
10C22×C16×C23C2211yescharacteristic
11C22×C16×C23C2211yescharacteristic
12C23×#427—22no—
13C23×#427—22no—
14C2×C23×#107C2211yescharacteristic
15C2×C23×#107C2211yescharacteristic
16C2×C23×#107C2211yescharacteristic
17C2×C23×#107C2211yescharacteristic
18C2×C23×#106—21no—
19C23×#380—21no—
20C23×#367—22no—
21C23×#366—22no—
22C23×C22⋊C16C414yescomplemented
23C23×#340—21no—
24C23×C4×C23D411yescharacteristic
25C22×C8×C23—21no—
26C22×C8×C23—21no—
27C22×C8×C23—21no—
28C22×C8×C23—21no—
29C22×C8×C23C2×C412yes—
30C22×C8×C23D411yescharacteristic
31C22×C8×C23Q812yes—
32C22×C8×C23C2311yescharacteristic
33C23×#117—24no—
34C23×#117—42no—
35C2×C4⋊C4×C23—21no—
36C2×C4⋊C4×C23—21no—
37C2×C4⋊C4×C23—21no—
38C2×C4⋊C4×C23—21no—
39C2×C22⋊C4×C23—41no—
40C2×C22⋊C4×C23—41no—
41C2×C42×C23—41no—
42C2×C16×C23—21no—
43C2×C16×C23—21no—
44C2×C16×C23—21no—
45C2×C16×C23—21no—
46C2×C16×C23C2×C412yes—
47C2×C16×C23C2×C412yes—
48C23×#107—21no—
49C23×#107—21no—
50C23×#107—21no—
51C23×#107—21no—
52C23×#107—21no—
53C23×#107—21no—
54C23×#107D412yes—
55C23×#107D412yes—
56C23×#106—24no—
57C23×#95—41no—
58C24×C23D811yescharacteristic
59C22×C4×C23—21no—
60C22×C4×C23—21no—
61C22×C4×C23—21no—
62C22×C4×C23—21no—
63C22×C4×C23—41no—
64C22×C4×C23—41no—
65C22×C4×C23—41no—
66C22×C4×C23—41no—
67C22×C4×C23C4⋊C412yes—
68C22×C4×C23D811yescharacteristic
69C22×C4×C23Q1612yes—
70C22×C4×C23C2×D411yescharacteristic
71C2×C8×C23—21no—
72C2×C8×C23—21no—
73C2×C8×C23—22no—
74C2×C8×C23—22no—
75C2×C8×C23—22no—
76C2×C8×C23—22no—
77C2×C8×C23C4⋊C412yes—
78C2×C8×C23C22×C411yescharacteristic
79C2×C8×C23C2×D411yescharacteristic
80C2×C8×C23C2×D411yescharacteristic
81C2×C8×C23C2×Q811yescharacteristic
82C2×C8×C23#4111yescharacteristic
83C2×C8×C23#4111yescharacteristic
84C4⋊C4×C23—22no—
85C4⋊C4×C23—22no—
86C4⋊C4×C23—41no—
87C4⋊C4×C23—41no—
88C4⋊C4×C23—41no—
89C4⋊C4×C23—41no—
90C4⋊C4×C23—41no—
91C4⋊C4×C23—41no—
92C22⋊C4×C23—44no—
93C22⋊C4×C23—82no—
94C42×C23—44no—
95C16×C23—22no—
96C16×C23—22no—
97#12921C2311yesmaximal, characteristic, direct product factor, Sylow
98C23×C23—21no—
99C23×C23—21no—
100C23×C23—21no—
101C23×C23—21no—
102C23×C23—21no—
103C23×C23—21no—
104C23×C23#10712yes—
105C23×C23C2×D811yescharacteristic
106C2×C4×C23—21no—
107C2×C4×C23—21no—
108C2×C4×C23—22no—
109C2×C4×C23—22no—
110C2×C4×C23—22no—
111C2×C4×C23—22no—
112C2×C4×C23—42no—
113C2×C4×C23—42no—
114C2×C4×C23—81no—
115C2×C4×C23—81no—
116C2×C4×C23—81no—
117C2×C4×C23—81no—
118C2×C4×C23—81no—
119C2×C4×C23—81no—
120C2×C4×C23—81no—
121C2×C4×C23—81no—
122C2×C4×C23—81no—
123C2×C4×C23—81no—
124C2×C4×C23#10712yes—
125C2×C4×C23C2×C4⋊C411yescharacteristic
126C2×C4×C23#12011yescharacteristic
127C2×C4×C23#12311yescharacteristic
128C2×C4×C23C2×Q1611yescharacteristic
129C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C23#13711yescharacteristic
131C8×C23—22no—
132C8×C23C4×D411yescharacteristic
133C8×C23C4×D411yescharacteristic
134C8×C23C22⋊Q811yescharacteristic
135C8×C23C22⋊Q811yescharacteristic
136C2×#427C2×C2311yescharacteristic
137C2×#367C2×C2311yescharacteristic, complemented
138C2×#366C2×C2311yescharacteristic, complemented
139C2×C22⋊C16C2×C2311yescharacteristic
140#1808C2×C2311yescharacteristic
141#1380C2×C2311yescharacteristic, complemented
142#1380C2×C2311yescharacteristic, complemented
143C22×C23—22no—
144C22×C23—22no—
145C22×C23—22no—
146C22×C23—41no—
147C22×C23—41no—
148C22×C23—41no—
149C22×C23—41no—
150C22×C23C2×#10711yescharacteristic
151C22×C23#44711yescharacteristic
152C22×C23#48011yescharacteristic
153C22×C23#50811yescharacteristic
154C22×C23#50811yescharacteristic
155C22×C23#50911yescharacteristic
156C22×C23#51011yescharacteristic
157C4×C23—22no—
158C4×C23—82no—
159C4×C23—82no—
160C4×C23—161no—
161C4×C23—161no—
162C4×C23#38511yescharacteristic
163C4×C23#42811yescharacteristic
164C4×C23#45111yescharacteristic
165C4×C23#48411yescharacteristic
166C23×C8C22×C2311yescharacteristic
167C2×#117—21no—
168C22×C16C22×C2311yescharacteristic
169C22×C16C22×C2311yescharacteristic
170#427—22no—
171#427—22no—
172C2×#107C22×C2311yescharacteristic
173C2×#107C22×C2311yescharacteristic
174C2×#107C22×C2311yescharacteristic
175C2×#107C22×C2311yescharacteristic
176C2×#106—21no—
177#380—21no—
178#367—22no—
179#366—22no—
180C22⋊C16C4×C2314yescomplemented
181#340—21no—
182C2×C23—42no—
183C2×C23#185211yescharacteristic
184C2×C23#215811yescharacteristic
185C2×C23#215911yescharacteristic
186C2×C23#218611yescharacteristic
187C2×C23#218911yescharacteristic
188C2×C23#223411yescharacteristic
189C2×C23#223711yescharacteristic
190C23×C4D4×C2311yescharacteristic
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8C2×C4×C2312yes—
196C22×C8D4×C2311yescharacteristic
197C22×C8Q8×C2312yes—
198C22×C8C23×C2311yescharacteristic
199#117—24no—
200#117—42no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C42—41no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16C2×C4×C2312yes—
213C2×C16C2×C4×C2312yes—
214#107—21no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107D4×C2312yes—
221#107D4×C2312yes—
222#106—24no—
223#95—41no—
224C23#1292111yescharacteristic, direct product factor, Sylow
225C24D8×C2311yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4C4⋊C4×C2312yes—
235C22×C4D8×C2311yescharacteristic
236C22×C4Q16×C2312yes—
237C22×C4C2×D4×C2311yescharacteristic
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8C4⋊C4×C2312yes—
245C2×C8C22×C4×C2311yescharacteristic
246C2×C8C2×D4×C2311yescharacteristic
247C2×C8C2×D4×C2311yescharacteristic
248C2×C8C2×Q8×C2311yescharacteristic
249C2×C8#41×C2311yescharacteristic
250C2×C8#41×C2311yescharacteristic
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C22⋊C4—44no—
260C22⋊C4—82no—
261C42—44no—
262C16—22no—
263C16—22no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23C23×#10712yes—
271C23C2×D8×C2311yescharacteristic
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4C23×#10712yes—
291C2×C4C2×C4⋊C4×C2311yescharacteristic
292C2×C4C23×#12011yescharacteristic
293C2×C4C23×#12311yescharacteristic
294C2×C4C2×Q16×C2311yescharacteristic
295C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
296C2×C4C23×#13711yescharacteristic
297C8—22no—
298C8C4×D4×C2311yescharacteristic
299C8C4×D4×C2311yescharacteristic
300C8C23×C22⋊Q811yescharacteristic
301C8C23×C22⋊Q811yescharacteristic
302C22—22no—
303C22—22no—
304C22—22no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22C2×C23×#10711yescharacteristic
310C22C23×#44711yescharacteristic
311C22C23×#48011yescharacteristic
312C22C23×#50811yescharacteristic
313C22C23×#50811yescharacteristic
314C22C23×#50911yescharacteristic
315C22C23×#51011yescharacteristic
316C4—22no—
317C4—82no—
318C4—82no—
319C4—161no—
320C4—161no—
321C4C23×#38511yescharacteristic
322C4C23×#42811yescharacteristic
323C4C23×#45111yescharacteristic
324C4C23×#48411yescharacteristic
325C2—42no—
326C2C23×#185211yescharacteristic
327C2C23×#215811yescharacteristic
328C2C23×#215911yescharacteristic
329C2C23×#218611yescharacteristic
330C2C23×#218911yescharacteristic
331C2C23×#223411yescharacteristic
332C2C23×#223711yescharacteristic
subgroupG/H
center, socleC23×C23C2×D8
commutatorC2×C8C22×C4×C23
Frattini, cosocleC22×C8C23×C23
prime radicalC24×C23D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C23 ◁ C23×C4×C23 ◁ C23×C8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#367C2semidirect
3. C2×C23×#366C2semidirect
6. C23×#1380C2semidirect
7. C23×#1380C2semidirect
22. C23×C22⋊C16C4semidirect
97. #12921C23direct (N × H)
137. C2×#367C2×C23semidirect
138. C2×#366C2×C23semidirect
141. #1380C2×C23semidirect
142. #1380C2×C23semidirect
180. C22⋊C16C4×C23semidirect
224. C23#12921direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#4272
C2×C23×#3672
C2×C23×#3662
C2×C23×C22⋊C162
C23×#18082
C23×#13802
#1292123