Finite Group Databasebeta

← C23×#7215 #19154841

Subgroups — C23×#7215 #19154841

Order 5888.

422 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#1821C211yesmaximal, characteristic
2C23×#1463C211yesmaximal, characteristic
3C23×#1436C211yesmaximal, characteristic
4C22×C4⋊C4×C23C2211yescharacteristic
5C22×C42×C23C411yescharacteristic
6C23×#407—21no—
7C23×#407—21no—
8C2×C23×#95—21no—
9C2×C23×#95—21no—
10C2×C23×#95C411yescharacteristic
11C23×#343—21no—
12C23×#343C412yescomplemented
13C23×#324—21no—
14C23×#324C412yescomplemented
15C23×C4×C23—21no—
16C23×C4×C23—21no—
17C23×C4×C23C2×C411yescharacteristic
18C23×C4×C23D411yescharacteristic
19C23×C4×C23D411yescharacteristic
20C2×M16×C23—22no—
21C2×M16×C23—41no—
22C2×C4⋊C4×C23—41no—
23C2×C4⋊C4×C23—41no—
24C2×C4⋊C4×C23C2×C411yescharacteristic
25C2×C4⋊C4×C23C2×C411yescharacteristic
26C2×C4⋊C4×C23D411yescharacteristic
27C2×C4⋊C4×C23Q811yescharacteristic
28C2×C42×C23—41no—
29C2×C42×C23—42no—
30C23×C22⋊C8—22no—
31C23×C22⋊C8—41no—
32C23×C22⋊C8—41no—
33C23×C22⋊C8—41no—
34C23×#95—24no—
35C23×#95—41no—
36C23×#95—42no—
37C24×C23C22⋊C411yescharacteristic
38C22×C4×C23—22no—
39C22×C4×C23—22no—
40C22×C4×C23—22no—
41C22×C4×C23—21no—
42C22×C4×C23—21no—
43C22×C4×C23—21no—
44C22×C4×C23—21no—
45C22×C4×C23—21no—
46C22×C4×C23—21no—
47C22×C4×C23—22no—
48C22×C4×C23—22no—
49C22×C4×C23—41no—
50C22×C4×C23—41no—
51C22×C4×C23—41no—
52C22×C4×C23—41no—
53C22×C4×C23—41no—
54C22×C4×C23—41no—
55C22×C4×C23—41no—
56C22×C4×C23—41no—
57C22×C4×C23—41no—
58C22×C4×C23—41no—
59C22×C4×C23—41no—
60C22×C4×C23—41no—
61C22×C4×C23—41no—
62C22×C4×C23—41no—
63C22×C4×C23—41no—
64C22×C4×C23C4211yescharacteristic
65C22×C4×C23C22⋊C411yescharacteristic
66C22×C4×C23C22⋊C411yescharacteristic
67C22×C4×C23C4⋊C411yescharacteristic
68C22×C4×C23C4⋊C411yescharacteristic
69C22×C4×C23C4⋊C411yescharacteristic
70M16×C23—42no—
71M16×C23—42no—
72M16×C23—81no—
73M16×C23—81no—
74C2×C8×C23—42no—
75C2×C8×C23—42no—
76C4⋊C4×C23—41no—
77C4⋊C4×C23—41no—
78C4⋊C4×C23—41no—
79C4⋊C4×C23—41no—
80C42×C23—42no—
81C42×C23—42no—
82#7215C2311yesmaximal, characteristic, direct product factor, Sylow
83C23×C23—21no—
84C23×C23—21no—
85C23×C23—21no—
86C23×C23—21no—
87C23×C23—21no—
88C23×C23—21no—
89C23×C23#9511yescharacteristic
90C23×C23#9911yescharacteristic
91C23×C23#9911yescharacteristic
92C2×C4×C23—21no—
93C2×C4×C23—21no—
94C2×C4×C23—24no—
95C2×C4×C23—21no—
96C2×C4×C23—21no—
97C2×C4×C23—21no—
98C2×C4×C23—21no—
99C2×C4×C23—21no—
100C2×C4×C23—21no—
101C2×C4×C23—21no—
102C2×C4×C23—42no—
103C2×C4×C23—42no—
104C2×C4×C23—42no—
105C2×C4×C23—42no—
106C2×C4×C23—42no—
107C2×C4×C23—42no—
108C2×C4×C23—42no—
109C2×C4×C23—42no—
110C2×C4×C23—41no—
111C2×C4×C23—42no—
112C2×C4×C23—41no—
113C2×C4×C23—41no—
114C2×C4×C23—41no—
115C2×C4×C23—41no—
116C2×C4×C23—41no—
117C2×C4×C23—41no—
118C2×C4×C23—41no—
119C2×C4×C23—42no—
120C2×C4×C23—42no—
121C2×C4×C23—42no—
122C2×C4×C23—81no—
123C2×C4×C23—81no—
124C2×C4×C23—81no—
125C2×C4×C23—81no—
126C2×C4×C23—81no—
127C2×C4×C23#10412yes—
128C8×C23—82no—
129C8×C23—82no—
130#1821C2×C2311yescharacteristic
131#1463C2×C2311yescharacteristic
132#1436C2×C2311yescharacteristic
133C22×C23—21no—
134C22×C23—21no—
135C22×C23—21no—
136C22×C23—21no—
137C22×C23—21no—
138C22×C23—21no—
139C22×C23—21no—
140C22×C23—21no—
141C22×C23—21no—
142C22×C23—21no—
143C22×C23—21no—
144C22×C23—21no—
145C22×C23—21no—
146C22×C23—21no—
147C22×C23H(3,ℤ/4)11yescharacteristic
148C22×C23#33911yescharacteristic
149C22×C23#34211yescharacteristic
150C22×C23AΣL(1,𝔽16)11yescharacteristic
151C22×C23#35211yescharacteristic
152C22×C23#35511yescharacteristic
153C22×C23#35611yescharacteristic
154C4×C23—22no—
155C4×C23—44no—
156C4×C23—41no—
157C4×C23—41no—
158C4×C23—41no—
159C4×C23—41no—
160C4×C23—41no—
161C4×C23—82no—
162C22×C4⋊C4C22×C2311yescharacteristic
163C22×C42C4×C2311yescharacteristic
164#407—21no—
165#407—21no—
166C2×#95—21no—
167C2×#95—21no—
168C2×#95C4×C2311yescharacteristic
169#343—21no—
170#343C4×C2312yescomplemented
171#324—21no—
172#324C4×C2312yescomplemented
173C2×C23—21no—
174C2×C23—21no—
175C2×C23—21no—
176C2×C23—21no—
177C2×C23—21no—
178C2×C23—21no—
179C2×C23#128411yescharacteristic
180C2×C23#139011yescharacteristic
181C2×C23#139411yescharacteristic
182C23×C4—21no—
183C23×C4—21no—
184C23×C4C2×C4×C2311yescharacteristic
185C23×C4D4×C2311yescharacteristic
186C23×C4D4×C2311yescharacteristic
187C2×M16—22no—
188C2×M16—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4C2×C4×C2311yescharacteristic
192C2×C4⋊C4C2×C4×C2311yescharacteristic
193C2×C4⋊C4D4×C2311yescharacteristic
194C2×C4⋊C4Q8×C2311yescharacteristic
195C2×C42—41no—
196C2×C42—42no—
197C22⋊C8—22no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200C22⋊C8—41no—
201#95—24no—
202#95—41no—
203#95—42no—
204C23#721511yescharacteristic, direct product factor, Sylow
205C24C22⋊C4×C2311yescharacteristic
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4C42×C2311yescharacteristic
233C22×C4C22⋊C4×C2311yescharacteristic
234C22×C4C22⋊C4×C2311yescharacteristic
235C22×C4C4⋊C4×C2311yescharacteristic
236C22×C4C4⋊C4×C2311yescharacteristic
237C22×C4C4⋊C4×C2311yescharacteristic
238M16—42no—
239M16—42no—
240M16—81no—
241M16—81no—
242C2×C8—42no—
243C2×C8—42no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C42—42no—
249C42—42no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23C23×#9511yescharacteristic
257C23C23×#9911yescharacteristic
258C23C23×#9911yescharacteristic
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—24no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4C23×#10412yes—
295C8—82no—
296C8—82no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22C23×H(3,ℤ/4)11yescharacteristic
312C22C23×#33911yescharacteristic
313C22C23×#34211yescharacteristic
314C22C23×AΣL(1,𝔽16)11yescharacteristic
315C22C23×#35211yescharacteristic
316C22C23×#35511yescharacteristic
317C22C23×#35611yescharacteristic
318C4—22no—
319C4—44no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—82no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2C23×#128411yescharacteristic
333C2C23×#139011yescharacteristic
334C2C23×#139411yescharacteristic
subgroupG/H
center, socleC22×C23#342
commutatorC22×C4C42×C23
Frattini, cosocleC22×C4⋊C4C22×C23
prime radicalC24×C23C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C24×C23 ◁ C22×C4⋊C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
12. C23×#343C4semidirect
14. C23×#324C4semidirect
82. #7215C23direct (N × H)
170. #343C4×C23semidirect
172. #324C4×C23semidirect
204. C23#7215direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#18212
C23×#14632
C23×#14362
#721523