Finite Group Databasebeta

← C23×#13217 #19160843

Subgroups — C23×#13217 #19160843

Order 5888.

536 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2486C211yesmaximal, characteristic
2C2×C23×#352C214yesmaximal
3C23×#1799C211yesmaximal, characteristic
4C4×C23×#100C211yesmaximal, characteristic
5C2×C23×#123C2212yes—
6C2×C4×D4×C23C2211yescharacteristic
7C22×C4⋊C4×C23C411yescharacteristic, complemented
8C2×C23×#100C2212yes—
9C2×C23×#99C2212yes—
10C4×M16×C23—21no—
11C23×#398—21no—
12C23×#397—21no—
13C23×#389—21no—
14C23×#381C412yescomplemented
15C23×#380—21no—
16C23×#380—21no—
17C23×#380C411yescharacteristic, complemented
18C23×#352—28no—
19C23×#343—21no—
20C23×#342—21no—
21C22×D4×C23C2311yescharacteristic
22C23×C4×C23C2×C412yes—
23C23×C4×C23C2×C411yescharacteristic
24C23×C4×C23D411yescharacteristic
25C2×M16×C23—22no—
26C23×#123—24no—
27C23×#123—24no—
28C4×D4×C23—41no—
29C4×D4×C23—41no—
30C2×C4⋊C4×C23—21no—
31C2×C4⋊C4×C23—21no—
32C2×C4⋊C4×C23—21no—
33C2×C4⋊C4×C23—22no—
34C2×C4⋊C4×C23—41no—
35C2×C4⋊C4×C23—41no—
36C2×C4⋊C4×C23—41no—
37C2×C4⋊C4×C23D411yescharacteristic
38C2×C22⋊C4×C23—22no—
39C2×C22⋊C4×C23—22no—
40C2×C22⋊C4×C23C2×C412yes—
41C2×C22⋊C4×C23C2×C411yescharacteristic
42C2×C22⋊C4×C23D411yescharacteristic
43C2×C42×C23D411yescharacteristic
44C23×#100—22no—
45C23×#100D414yes—
46C23×#99—24no—
47C23×#97—41no—
48C4×C8×C23—41no—
49C23×#95—22no—
50C23×#95—21no—
51C23×#95—22no—
52C23×#95—41no—
53C24×C23C22×C411yescharacteristic
54C24×C23C2×D411yescharacteristic
55C2×D4×C23—41no—
56C2×D4×C23—41no—
57C2×D4×C23C2×D412yes—
58C2×D4×C23#4112yes—
59C22×C4×C23—21no—
60C22×C4×C23—21no—
61C22×C4×C23—21no—
62C22×C4×C23—21no—
63C22×C4×C23—22no—
64C22×C4×C23—22no—
65C22×C4×C23—24no—
66C22×C4×C23—42no—
67C22×C4×C23—42no—
68C22×C4×C23—42no—
69C22×C4×C23—42no—
70C22×C4×C23—42no—
71C22×C4×C23C22⋊C411yescharacteristic
72C22×C4×C23C22⋊C411yescharacteristic
73C22×C4×C23C22⋊C411yescharacteristic
74C22×C4×C23C22⋊C411yescharacteristic
75C22×C4×C23C2×D411yescharacteristic
76M16×C23—24no—
77M16×C23—42no—
78C2×C8×C23—42no—
79C4⋊C4×C23—41no—
80C4⋊C4×C23—42no—
81C4⋊C4×C23—42no—
82C4⋊C4×C23—41no—
83C4⋊C4×C23—41no—
84C4⋊C4×C23—44no—
85C4⋊C4×C23—82no—
86C22⋊C4×C23—24no—
87C22⋊C4×C23—41no—
88C22⋊C4×C23—41no—
89C22⋊C4×C23—42no—
90C22⋊C4×C23—42no—
91C22⋊C4×C23—42no—
92C22⋊C4×C23—42no—
93C42×C23—21no—
94C42×C23—21no—
95#13217C2311yesmaximal, characteristic, direct product factor, Sylow
96C23×C23—21no—
97C23×C23—21no—
98C23×C23—21no—
99C23×C23—21no—
100C23×C23—41no—
101C23×C23—41no—
102C23×C23—41no—
103C23×C23—41no—
104C23×C23C2×C22⋊C411yescharacteristic
105C23×C23C4×D412yes—
106C23×C23#12411yescharacteristic
107C23×C23#12711yescharacteristic
108D4×C23—42no—
109D4×C23—42no—
110C2×C4×C23—21no—
111C2×C4×C23—21no—
112C2×C4×C23—22no—
113C2×C4×C23—21no—
114C2×C4×C23—21no—
115C2×C4×C23—21no—
116C2×C4×C23—42no—
117C2×C4×C23—42no—
118C2×C4×C23—42no—
119C2×C4×C23—41no—
120C2×C4×C23—41no—
121C2×C4×C23—41no—
122C2×C4×C23—42no—
123C2×C4×C23—42no—
124C2×C4×C23—42no—
125C2×C4×C23—42no—
126C2×C4×C23—42no—
127C2×C4×C23—42no—
128C2×C4×C23—81no—
129C2×C4×C23—84no—
130C2×C4×C23—82no—
131C2×C4×C23—82no—
132C2×C4×C23—82no—
133C2×C4×C23#9911yescharacteristic
134C2×C4×C23#9911yescharacteristic
135C2×C4×C23#12112yes—
136C8×C23—44no—
137#2486C2×C2311yescharacteristic
138C2×#352C2×C2314yes—
139#1799C2×C2311yescharacteristic
140C4×#100C2×C2311yescharacteristic
141C22×C23—21no—
142C22×C23—21no—
143C22×C23—22no—
144C22×C23—21no—
145C22×C23—21no—
146C22×C23—42no—
147C22×C23—41no—
148C22×C23—41no—
149C22×C23—42no—
150C22×C23—41no—
151C22×C23—41no—
152C22×C23—41no—
153C22×C23—41no—
154C22×C23—81no—
155C22×C23#39011yescharacteristic
156C22×C23C2×#9911yescharacteristic
157C22×C23#41011yescharacteristic
158C4×C23—21no—
159C4×C23—22no—
160C4×C23—41no—
161C4×C23—41no—
162C4×C23—41no—
163C4×C23—82no—
164C4×C23—82no—
165C4×C23—162no—
166C4×C23#35212yescomplemented
167C2×#123C22×C2312yes—
168C2×C4×D4C22×C2311yescharacteristic
169C22×C4⋊C4C4×C2311yescharacteristic, complemented
170C2×#100C22×C2312yes—
171C2×#99C22×C2312yes—
172C4×M16—21no—
173#398—21no—
174#397—21no—
175#389—21no—
176#381C4×C2312yescomplemented
177#380—21no—
178#380—21no—
179#380C4×C2311yescharacteristic, complemented
180#352—28no—
181#343—21no—
182#342—21no—
183C2×C23—21no—
184C2×C23—21no—
185C2×C23—41no—
186C2×C23—42no—
187C2×C23—41no—
188C2×C23#196411yescharacteristic
189C2×C23C2×#35211yescharacteristic
190C2×C23#212311yescharacteristic
191C22×D4C23×C2311yescharacteristic
192C23×C4C2×C4×C2312yes—
193C23×C4C2×C4×C2311yescharacteristic
194C23×C4D4×C2311yescharacteristic
195C2×M16—22no—
196#123—24no—
197#123—24no—
198C4×D4—41no—
199C4×D4—41no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4D4×C2311yescharacteristic
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4C2×C4×C2312yes—
211C2×C22⋊C4C2×C4×C2311yescharacteristic
212C2×C22⋊C4D4×C2311yescharacteristic
213C2×C42D4×C2311yescharacteristic
214#100—22no—
215#100D4×C2314yes—
216#99—24no—
217#97—41no—
218C4×C8—41no—
219#95—22no—
220#95—21no—
221#95—22no—
222#95—41no—
223C23#1321711yescharacteristic, direct product factor, Sylow
224C24C22×C4×C2311yescharacteristic
225C24C2×D4×C2311yescharacteristic
226C2×D4—41no—
227C2×D4—41no—
228C2×D4C2×D4×C2312yes—
229C2×D4#41×C2312yes—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4C22⋊C4×C2311yescharacteristic
243C22×C4C22⋊C4×C2311yescharacteristic
244C22×C4C22⋊C4×C2311yescharacteristic
245C22×C4C22⋊C4×C2311yescharacteristic
246C22×C4C2×D4×C2311yescharacteristic
247M16—24no—
248M16—42no—
249C2×C8—42no—
250C4⋊C4—41no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—44no—
256C4⋊C4—82no—
257C22⋊C4—24no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C42—21no—
265C42—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23C2×C22⋊C4×C2311yescharacteristic
275C23C4×D4×C2312yes—
276C23C23×#12411yescharacteristic
277C23C23×#12711yescharacteristic
278D4—42no—
279D4—42no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4—84no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4C23×#9911yescharacteristic
304C2×C4C23×#9911yescharacteristic
305C2×C4C23×#12112yes—
306C8—44no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22C23×#39011yescharacteristic
322C22C2×C23×#9911yescharacteristic
323C22C23×#41011yescharacteristic
324C4—22no—
325C4—21no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—82no—
330C4—82no—
331C4—162no—
332C4C23×#35212yescomplemented
333C2—21no—
334C2—21no—
335C2—41no—
336C2—42no—
337C2—41no—
338C2C23×#196411yescharacteristic
339C2C2×C23×#35211yescharacteristic
340C2C23×#212311yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC24C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC22×D4×C23C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C4×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
7. C22×C4⋊C4×C23C4semidirect
14. C23×#381C4semidirect
17. C23×#380C4semidirect
95. #13217C23direct (N × H)
166. C4×C23#352semidirect
169. C22×C4⋊C4C4×C23semidirect
176. #381C4×C23semidirect
179. #380C4×C23semidirect
223. C23#13217direct (N × H)
332. C4C23×#352semidirect

Maximal subgroups

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

subgroupindex
C23×#24862
C2×C23×#3522
C23×#17992
C4×C23×#1002
#1321723