Finite Group Databasebeta

← C23×#7610 #19155236

Subgroups — C23×#7610 #19155236

Order 5888.

612 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#389C211yesmaximal, characteristic
2C23×#1817C211yesmaximal, characteristic
3C23×#1728C211yesmaximal, characteristic
4C23×#1271C214yesmaximal, complemented
5C22×M16×C23—21no—
6C22×C4⋊C4×C23—21no—
7C22×C4⋊C4×C23C411yescharacteristic
8C22×C4⋊C4×C23C411yescharacteristic
9C22×C42×C23C2211yescharacteristic
10C2×C23×C22⋊C8—21no—
11C23×#389—21no—
12C23×#389—21no—
13C23×#389C412yes—
14C23×#389C2212yes—
15C2×C23×#95—21no—
16C23×#336—21no—
17C23×#336—21no—
18C23×#336C2212yes—
19C23×#336C2212yes—
20C23×C4×C23—21no—
21C23×C4×C23—21no—
22C23×C4×C23C2×C411yescharacteristic
23C23×C4×C23D411yescharacteristic
24C23×C4×C23D411yescharacteristic
25C2×M16×C23—24no—
26C2×M16×C23—42no—
27C2×M16×C23—42no—
28C22×C8×C23—22no—
29C22×C8×C23—22no—
30C2×C4⋊C4×C23—21no—
31C2×C4⋊C4×C23—24no—
32C2×C4⋊C4×C23—21no—
33C2×C4⋊C4×C23—22no—
34C2×C4⋊C4×C23—22no—
35C2×C4⋊C4×C23—42no—
36C2×C4⋊C4×C23—42no—
37C2×C4⋊C4×C23—42no—
38C2×C4⋊C4×C23C2×C412yes—
39C2×C4⋊C4×C23C2×C412yes—
40C2×C42×C23—42no—
41C2×C42×C23C2×C411yescharacteristic
42C2×C42×C23D411yescharacteristic
43C2×C42×C23D411yescharacteristic
44C2×C42×C23C2311yescharacteristic
45C23×C22⋊C8—42no—
46C23×#95—22no—
47C23×#95—22no—
48C24×C23C22⋊C411yescharacteristic
49C22×C4×C23—22no—
50C22×C4×C23—22no—
51C22×C4×C23—22no—
52C22×C4×C23—22no—
53C22×C4×C23—22no—
54C22×C4×C23—24no—
55C22×C4×C23—22no—
56C22×C4×C23—22no—
57C22×C4×C23—22no—
58C22×C4×C23—22no—
59C22×C4×C23—41no—
60C22×C4×C23—42no—
61C22×C4×C23—42no—
62C22×C4×C23—42no—
63C22×C4×C23—42no—
64C22×C4×C23—42no—
65C22×C4×C23C22⋊C411yescharacteristic
66C22×C4×C23C22⋊C411yescharacteristic
67C22×C4×C23C22⋊C411yescharacteristic
68C22×C4×C23D812yes—
69C22×C4×C23SD1612yes—
70C22×C4×C23C22×C411yescharacteristic
71C22×C4×C23C2×D411yescharacteristic
72C22×C4×C23C2×D411yescharacteristic
73M16×C23—48no—
74C2×C8×C23—24no—
75C2×C8×C23—42no—
76C2×C8×C23—42no—
77C2×C8×C23—42no—
78C2×C8×C23—42no—
79C2×C8×C23—42no—
80C4⋊C4×C23—24no—
81C4⋊C4×C23—22no—
82C4⋊C4×C23—42no—
83C4⋊C4×C23—42no—
84C4⋊C4×C23—44no—
85C4⋊C4×C23—42no—
86C4⋊C4×C23—42no—
87C4⋊C4×C23M1614yescomplemented
88C42×C23—41no—
89C42×C23—41no—
90C42×C23—41no—
91C42×C23—41no—
92#7610C2311yesmaximal, characteristic, direct product factor, Sylow
93C23×C23—21no—
94C23×C23—21no—
95C23×C23—21no—
96C23×C23—21no—
97C23×C23#9912yes—
98C23×C23#10212yes—
99C23×C23#10412yes—
100C23×C23C2×C22⋊C411yescharacteristic
101C2×C4×C23—28no—
102C2×C4×C23—24no—
103C2×C4×C23—24no—
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—24no—
115C2×C4×C23—21no—
116C2×C4×C23—21no—
117C2×C4×C23—21no—
118C2×C4×C23—44no—
119C2×C4×C23—48no—
120C2×C4×C23—42no—
121C2×C4×C23—42no—
122C2×C4×C23—42no—
123C2×C4×C23—42no—
124C2×C4×C23—42no—
125C2×C4×C23—84no—
126C2×C4×C23#10112yes—
127C2×C4×C23#10212yes—
128C2×C4×C23C2×M1612yes—
129C2×C4×C23C2×D811yescharacteristic
130C2×C4×C23C2×SD1611yescharacteristic
131C2×C4×C23#13712yes—
132C8×C23—44no—
133C8×C23—82no—
134C2×#389C2×C2311yescharacteristic
135#1817C2×C2311yescharacteristic
136#1728C2×C2311yescharacteristic
137#1271C2×C2314yescomplemented
138C22×C23—22no—
139C22×C23—22no—
140C22×C23—22no—
141C22×C23—22no—
142C22×C23—22no—
143C22×C23—22no—
144C22×C23#32714yes—
145C22×C23#40711yescharacteristic
146C22×C23C2×#9911yescharacteristic
147C22×C23C2×#10111yescharacteristic
148C22×C23C2×#10211yescharacteristic
149C22×C23#41911yescharacteristic
150C22×C23C2×#10411yescharacteristic
151C22×C23#42111yescharacteristic
152C4×C23—22no—
153C4×C23—22no—
154C4×C23—21no—
155C4×C23—22no—
156C4×C23—21no—
157C4×C23—22no—
158C4×C23—44no—
159C4×C23—41no—
160C4×C23—82no—
161C22×M16—21no—
162C22×C4⋊C4—21no—
163C22×C4⋊C4C4×C2311yescharacteristic
164C22×C4⋊C4C4×C2311yescharacteristic
165C22×C42C22×C2311yescharacteristic
166C2×C22⋊C8—21no—
167#389—21no—
168#389—21no—
169#389C4×C2312yes—
170#389C22×C2312yes—
171C2×#95—21no—
172#336—21no—
173#336—21no—
174#336C22×C2312yes—
175#336C22×C2312yes—
176C2×C23—24no—
177C2×C23#146311yescharacteristic
178C2×C23#148611yescharacteristic
179C2×C23#149111yescharacteristic
180C2×C23C2×#32711yescharacteristic
181C2×C23#151111yescharacteristic
182C2×C23#152811yescharacteristic
183C2×C23#154611yescharacteristic
184C23×C4—21no—
185C23×C4—21no—
186C23×C4C2×C4×C2311yescharacteristic
187C23×C4D4×C2311yescharacteristic
188C23×C4D4×C2311yescharacteristic
189C2×M16—24no—
190C2×M16—42no—
191C2×M16—42no—
192C22×C8—22no—
193C22×C8—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4C2×C4×C2312yes—
203C2×C4⋊C4C2×C4×C2312yes—
204C2×C42—42no—
205C2×C42C2×C4×C2311yescharacteristic
206C2×C42D4×C2311yescharacteristic
207C2×C42D4×C2311yescharacteristic
208C2×C42C23×C2311yescharacteristic
209C22⋊C8—42no—
210#95—22no—
211#95—22no—
212C23#761011yescharacteristic, direct product factor, Sylow
213C24C22⋊C4×C2311yescharacteristic
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4C22⋊C4×C2311yescharacteristic
231C22×C4C22⋊C4×C2311yescharacteristic
232C22×C4C22⋊C4×C2311yescharacteristic
233C22×C4D8×C2312yes—
234C22×C4SD16×C2312yes—
235C22×C4C22×C4×C2311yescharacteristic
236C22×C4C2×D4×C2311yescharacteristic
237C22×C4C2×D4×C2311yescharacteristic
238M16—48no—
239C2×C8—24no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C4⋊C4—24no—
246C4⋊C4—22no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4M16×C2314yescomplemented
253C42—41no—
254C42—41no—
255C42—41no—
256C42—41no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23C23×#9912yes—
262C23C23×#10212yes—
263C23C23×#10412yes—
264C23C2×C22⋊C4×C2311yescharacteristic
265C2×C4—28no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—21no—
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—24no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—44no—
283C2×C4—48no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—84no—
290C2×C4C23×#10112yes—
291C2×C4C23×#10212yes—
292C2×C4C2×M16×C2312yes—
293C2×C4C2×D8×C2311yescharacteristic
294C2×C4C2×SD16×C2311yescharacteristic
295C2×C4C23×#13712yes—
296C8—44no—
297C8—82no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22C23×#32714yes—
305C22C23×#40711yescharacteristic
306C22C2×C23×#9911yescharacteristic
307C22C2×C23×#10111yescharacteristic
308C22C2×C23×#10211yescharacteristic
309C22C23×#41911yescharacteristic
310C22C2×C23×#10411yescharacteristic
311C22C23×#42111yescharacteristic
312C4—22no—
313C4—22no—
314C4—21no—
315C4—22no—
316C4—21no—
317C4—22no—
318C4—44no—
319C4—41no—
320C4—82no—
321C2—24no—
322C2C23×#146311yescharacteristic
323C2C23×#148611yescharacteristic
324C2C23×#149111yescharacteristic
325C2C2×C23×#32711yescharacteristic
326C2C23×#151111yescharacteristic
327C2C23×#152811yescharacteristic
328C2C23×#154611yescharacteristic
subgroupG/H
center, socleC23×C23C2×C22⋊C4
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC24×C23C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C22×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×#1271C2semidirect
87. C4⋊C4×C23M16semidirect
92. #7610C23direct (N × H)
137. #1271C2×C23semidirect
212. C23#7610direct (N × H)
252. C4⋊C4M16×C23semidirect

Maximal subgroups

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

subgroupindex
C2×C23×#3892
C23×#18172
C23×#17282
C23×#12712
#761023