Finite Group Databasebeta

← C23×#17419 #19165045

Subgroups — C23×#17419 #19165045

Order 5888.

1342 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#550C211yesmaximal, characteristic
2C23×#2880C212yesmaximal, complemented
3C23×#1471C216yesmaximal, complemented
4C23×#1465C213yesmaximal, complemented
5C23×#1459C213yesmaximal, complemented
6C22×#41×C23C411yescharacteristic
7C22×#41×C23C2211yescharacteristic
8C2×C23×#131—22no—
9C23×#550—22no—
10C23×#550—26no—
11C2×C23×#127C413yes—
12C2×C23×#121C413yes—
13C2×C23×#121C2213yes—
14C2×C4×Q8×C23C411yescharacteristic
15C2×C4×D4×C23C2213yes—
16C23×#408—26no—
17C23×#407C2216yes—
18C2×C23×C22⋊C8C2216yes—
19C23×#324C2211yescharacteristic
20C23×#324C2213yes—
21C23×#323C2216yes—
22C23×#323C2216yes—
23C2×#41×C23—23no—
24C2×#41×C23—21no—
25C2×#41×C23—44no—
26C2×#41×C23—44no—
27C2×#41×C23D412yes—
28C2×#41×C23D416yes—
29C22×Q8×C23C2×C411yescharacteristic
30C22×D4×C23C2×C413yes—
31C22×D4×C23C2×C413yes—
32C22×D4×C23C2×C416yes—
33C22×D4×C23C2×C413yes—
34C23×C4×C23C2×C413yes—
35C23×C4×C23C2313yes—
36C23×#131—48no—
37C2×M16×C23—26no—
38C22×C8×C23—26no—
39C23×#127—26no—
40C23×#127—26no—
41C23×#121—26no—
42C23×#121—26no—
43C23×#121—26no—
44C23×#121—26no—
45C4×Q8×C23—23no—
46C4×Q8×C23—21no—
47C4×D4×C23—26no—
48C4×D4×C23—26no—
49C2×C4⋊C4×C23C2×C413yes—
50C2×C4⋊C4×C23C2311yescharacteristic
51C2×C22⋊C4×C23C2×C413yes—
52C2×C22⋊C4×C23C2313yes—
53C2×C42×C23C2×C413yes—
54C23×C22⋊C8—26no—
55C23×C22⋊C8—26no—
56C23×C22⋊C8C2312yes—
57C23×C22⋊C8C2316yes—
58C24×C23C22×C413yes—
59C24×C23C22×C413yes—
60#41×C23—28no—
61#41×C23—412no—
62#41×C23—412no—
63#41×C23—44no—
64C2×Q8×C23—24no—
65C2×Q8×C23C22⋊C413yes—
66C2×Q8×C23C22⋊C411yescharacteristic
67C2×D4×C23—26no—
68C2×D4×C23—212no—
69C2×D4×C23—23no—
70C2×D4×C23—23no—
71C2×D4×C23—26no—
72C2×D4×C23—46no—
73C2×D4×C23—43no—
74C2×D4×C23—46no—
75C2×D4×C23—86no—
76C2×D4×C23C22⋊C416yes—
77C2×D4×C23C22⋊C416yes—
78C22×C4×C23—23no—
79C22×C4×C23—23no—
80C22×C4×C23—46no—
81C22×C4×C23—46no—
82C22×C4×C23C22×C413yes—
83C22×C4×C23C22×C411yescharacteristic
84C22×C4×C23C22×C413yes—
85C22×C4×C23C22×C411yescharacteristic
86C22×C4×C23C2×D416yes—
87C22×C4×C23C2×D416yes—
88C22×C4×C23C2411yescharacteristic
89M16×C23—412no—
90C2×C8×C23—22no—
91C2×C8×C23—26no—
92C2×C8×C23—412no—
93C4⋊C4×C23—23no—
94C4⋊C4×C23—23no—
95C4⋊C4×C23—22no—
96C22⋊C4×C23—26no—
97C22⋊C4×C23—26no—
98C42×C23—23no—
99C42×C23—23no—
100#17419C2311yesmaximal, characteristic, direct product factor, Sylow
101C23×C23—23no—
102C23×C23—23no—
103C23×C23—23no—
104C23×C23—23no—
105C23×C23—23no—
106C23×C23—43no—
107C23×C23—43no—
108C23×C23—43no—
109C23×C23—43no—
110C23×C23C2×C22⋊C416yes—
111C23×C23C23×C411yescharacteristic
112Q8×C23—26no—
113Q8×C23#9914yes—
114D4×C23—212no—
115D4×C23—46no—
116D4×C23—412no—
117D4×C23—46no—
118D4×C23—46no—
119D4×C23—812no—
120C2×C4×C23—21no—
121C2×C4×C23—21no—
122C2×C4×C23—212no—
123C2×C4×C23—26no—
124C2×C4×C23—23no—
125C2×C4×C23—21no—
126C2×C4×C23—41no—
127C2×C4×C23—46no—
128C2×C4×C23—46no—
129C2×C4×C23—46no—
130C2×C4×C23—41no—
131C2×C4×C23—43no—
132C2×C4×C23C2×C22⋊C413yes—
133C2×C4×C23C2×C22⋊C413yes—
134C2×C4×C23#13112yescomplemented
135C2×C4×C23C22×D412yes—
136C8×C23—42no—
137C8×C23—46no—
138C2×#550C2×C2311yescharacteristic
139#2880C2×C2312yescomplemented
140#1471C2×C2316yescomplemented
141#1465C2×C2313yescomplemented
142#1459C2×C2313yescomplemented
143C22×C23—21no—
144C22×C23—21no—
145C22×C23—26no—
146C22×C23—26no—
147C22×C23—26no—
148C22×C23—43no—
149C22×C23—43no—
150C22×C23—43no—
151C22×C23—83no—
152C22×C23—83no—
153C22×C23—83no—
154C22×C23—83no—
155C22×C23C22×C22⋊C411yescharacteristic
156C22×C23C2×#13111yescharacteristic
157C22×C23#56811yescharacteristic
158C4×C23—22no—
159C4×C23—22no—
160C4×C23—23no—
161C4×C23—41no—
162C4×C23—43no—
163C4×C23C2×#9916yes—
164C22×#41C4×C2311yescharacteristic
165C22×#41C22×C2311yescharacteristic
166C2×#131—22no—
167#550—22no—
168#550—26no—
169C2×#127C4×C2313yes—
170C2×#121C4×C2313yes—
171C2×#121C22×C2313yes—
172C2×C4×Q8C4×C2311yescharacteristic
173C2×C4×D4C22×C2313yes—
174#408—26no—
175#407C22×C2316yes—
176C2×C22⋊C8C22×C2316yes—
177#324C22×C2311yescharacteristic
178#324C22×C2313yes—
179#323C22×C2316yes—
180#323C22×C2316yes—
181C2×C23—21no—
182C2×C23—21no—
183C2×C23—46no—
184C2×C23—83no—
185C2×C23#288011yescharacteristic
186C2×C23C22×#9911yescharacteristic
187C2×C23#288811yescharacteristic
188C2×#41—23no—
189C2×#41—21no—
190C2×#41—44no—
191C2×#41—44no—
192C2×#41D4×C2312yes—
193C2×#41D4×C2316yes—
194C22×Q8C2×C4×C2311yescharacteristic
195C22×D4C2×C4×C2313yes—
196C22×D4C2×C4×C2313yes—
197C22×D4C2×C4×C2316yes—
198C22×D4C2×C4×C2313yes—
199C23×C4C2×C4×C2313yes—
200C23×C4C23×C2313yes—
201#131—48no—
202C2×M16—26no—
203C22×C8—26no—
204#127—26no—
205#127—26no—
206#121—26no—
207#121—26no—
208#121—26no—
209#121—26no—
210C4×Q8—23no—
211C4×Q8—21no—
212C4×D4—26no—
213C4×D4—26no—
214C2×C4⋊C4C2×C4×C2313yes—
215C2×C4⋊C4C23×C2311yescharacteristic
216C2×C22⋊C4C2×C4×C2313yes—
217C2×C22⋊C4C23×C2313yes—
218C2×C42C2×C4×C2313yes—
219C22⋊C8—26no—
220C22⋊C8—26no—
221C22⋊C8C23×C2312yes—
222C22⋊C8C23×C2316yes—
223C23#1741911yescharacteristic, direct product factor, Sylow
224C24C22×C4×C2313yes—
225C24C22×C4×C2313yes—
226#41—28no—
227#41—412no—
228#41—412no—
229#41—44no—
230C2×Q8—24no—
231C2×Q8C22⋊C4×C2313yes—
232C2×Q8C22⋊C4×C2311yescharacteristic
233C2×D4—26no—
234C2×D4—212no—
235C2×D4—23no—
236C2×D4—23no—
237C2×D4—26no—
238C2×D4—46no—
239C2×D4—43no—
240C2×D4—46no—
241C2×D4—86no—
242C2×D4C22⋊C4×C2316yes—
243C2×D4C22⋊C4×C2316yes—
244C22×C4—23no—
245C22×C4—23no—
246C22×C4—46no—
247C22×C4—46no—
248C22×C4C22×C4×C2313yes—
249C22×C4C22×C4×C2311yescharacteristic
250C22×C4C22×C4×C2313yes—
251C22×C4C22×C4×C2311yescharacteristic
252C22×C4C2×D4×C2316yes—
253C22×C4C2×D4×C2316yes—
254C22×C4C24×C2311yescharacteristic
255M16—412no—
256C2×C8—22no—
257C2×C8—26no—
258C2×C8—412no—
259C4⋊C4—23no—
260C4⋊C4—23no—
261C4⋊C4—22no—
262C22⋊C4—26no—
263C22⋊C4—26no—
264C42—23no—
265C42—23no—
266C23—23no—
267C23—23no—
268C23—23no—
269C23—23no—
270C23—23no—
271C23—43no—
272C23—43no—
273C23—43no—
274C23—43no—
275C23C2×C22⋊C4×C2316yes—
276C23C23×C4×C2311yescharacteristic
277Q8—26no—
278Q8C23×#9914yes—
279D4—212no—
280D4—46no—
281D4—412no—
282D4—46no—
283D4—46no—
284D4—812no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—212no—
288C2×C4—26no—
289C2×C4—23no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—46no—
293C2×C4—46no—
294C2×C4—46no—
295C2×C4—41no—
296C2×C4—43no—
297C2×C4C2×C22⋊C4×C2313yes—
298C2×C4C2×C22⋊C4×C2313yes—
299C2×C4C23×#13112yescomplemented
300C2×C4C22×D4×C2312yes—
301C8—42no—
302C8—46no—
303C22—21no—
304C22—21no—
305C22—26no—
306C22—26no—
307C22—26no—
308C22—43no—
309C22—43no—
310C22—43no—
311C22—83no—
312C22—83no—
313C22—83no—
314C22—83no—
315C22C22×C22⋊C4×C2311yescharacteristic
316C22C2×C23×#13111yescharacteristic
317C22C23×#56811yescharacteristic
318C4—22no—
319C4—22no—
320C4—23no—
321C4—41no—
322C4—43no—
323C4C2×C23×#9916yes—
324C2—21no—
325C2—21no—
326C2—46no—
327C2—83no—
328C2C23×#288011yescharacteristic
329C2C22×C23×#9911yescharacteristic
330C2C23×#288811yescharacteristic
subgroupG/H
center, socleC22×C23C22×C22⋊C4
commutatorC23C23×C4×C23
Frattini, cosocleC22×C4C24×C23
prime radicalC2×C23×#550C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×#41×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2880C2semidirect
3. C23×#1471C2semidirect
4. C23×#1465C2semidirect
5. C23×#1459C2semidirect
100. #17419C23direct (N × H)
134. C2×C4×C23#131semidirect
139. #2880C2×C23semidirect
140. #1471C2×C23semidirect
141. #1465C2×C23semidirect
142. #1459C2×C23semidirect
223. C23#17419direct (N × H)
299. C2×C4C23×#131semidirect

Maximal subgroups

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

subgroupindex
C2×C23×#5502
C23×#28802
C23×#14712
C23×#14652
C23×#14592
#1741923