Finite Group Databasebeta

← C23×#13611 #19161237

Subgroups — C23×#13611 #19161237

Order 5888.

316 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 86 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×#3221C211yesmaximal, characteristic
2C23×#2135C211yesmaximal, characteristic, complemented
3C23×#2133C211yesmaximal, characteristic, complemented
4C23×#1413C212yesmaximal, complemented
5C23×#1410C212yesmaximal, complemented
6C23×#540C411yescharacteristic
7C23×#536C2211yescharacteristic
8C23×#496C411yescharacteristic
9C23×#492C2211yescharacteristic
10C23×#491C2211yescharacteristic
11C23×#488C411yescharacteristic
12C23×#463—21no—
13C23×#461—21no—
14C23×#449—21no—
15C23×#448—21no—
16C23×#433C411yescharacteristic
17C23×#413—21no—
18C2×C23×#101—21no—
19C23×#356—21no—
20C23×#356C2212yes—
21C23×#356C2212yes—
22C23×#355—21no—
23C2×#41×C23D411yescharacteristic
24C22×Q8×C23D411yescharacteristic
25C2×SD16×C23—21no—
26C2×SD16×C23—21no—
27C2×SD16×C23—21no—
28C2×D8×C23—21no—
29C2×M16×C23—41no—
30C2×M16×C23—41no—
31C23×C4⋊Q8D411yescharacteristic
32C23×C4⋊Q8C2311yescharacteristic
33C23×#127C2×C411yescharacteristic
34C23×#125C2×C411yescharacteristic
35C23×#124C2×C411yescharacteristic
36C23×#124D411yescharacteristic
37C23×#123—41no—
38C23×#123—41no—
39C23×C22⋊Q8—41no—
40C23×#121—21no—
41C23×#121—41no—
42C4×D4×C23—21no—
43C23×#117C2×C411yescharacteristic
44C23×C4⋊C8—21no—
45C23×#103—21no—
46C23×#103—21no—
47C23×#103—21no—
48C23×#102—21no—
49C23×#101—22no—
50C23×#101—22no—
51C23×#101—41no—
52C23×#100—41no—
53C23×C22⋊C8—21no—
54C23×#97C2×C411yescharacteristic
55C4×C8×C23C2×C411yescharacteristic
56#41×C23—81no—
57C2×Q8×C23—21no—
58C2×Q8×C23—81no—
59C2×Q8×C23C22⋊C411yescharacteristic
60C2×Q8×C23C2×D411yescharacteristic
61C2×Q8×C23C2×D411yescharacteristic
62C2×D4×C23—21no—
63C2×D4×C23—21no—
64C2×D4×C23—41no—
65C2×D4×C23—41no—
66C2×D4×C23C22⋊C411yescharacteristic
67C22×C4×C23—21no—
68C22×C4×C23—21no—
69C22×C4×C23—41no—
70C22×C4×C23C22⋊C411yescharacteristic
71SD16×C23—81no—
72SD16×C23—81no—
73SD16×C23—81no—
74D8×C23—81no—
75M16×C23—42no—
76M16×C23—42no—
77M16×C23—81no—
78M16×C23—81no—
79C2×C8×C23—21no—
80C2×C8×C23—21no—
81C2×C8×C23—81no—
82C2×C8×C23—81no—
83C4⋊C4×C23—21no—
84C4⋊C4×C23—21no—
85C4⋊C4×C23—41no—
86C4⋊C4×C23—41no—
87C4⋊C4×C23—41no—
88C22⋊C4×C23—21no—
89C22⋊C4×C23—41no—
90C22⋊C4×C23—41no—
91C22⋊C4×C23—41no—
92C22⋊C4×C23—41no—
93C42×C23C22⋊C411yescharacteristic
94C42×C23C22×C411yescharacteristic
95#13611C2311yesmaximal, characteristic, direct product factor, Sylow
96C23×C23—21no—
97C23×C23—41no—
98C23×C23#9911yescharacteristic
99Q8×C23—41no—
100Q8×C23—41no—
101Q8×C23—41no—
102Q8×C23—81no—
103D4×C23—41no—
104D4×C23—41no—
105D4×C23—41no—
106D4×C23—81no—
107D4×C23—161no—
108D4×C23—161no—
109C2×C4×C23—21no—
110C2×C4×C23—21no—
111C2×C4×C23—21no—
112C2×C4×C23—21no—
113C2×C4×C23—21no—
114C2×C4×C23—41no—
115C2×C4×C23—41no—
116C2×C4×C23—41no—
117C2×C4×C23—41no—
118C2×C4×C23—41no—
119C2×C4×C23—41no—
120C2×C4×C23—161no—
121C2×C4×C23#9911yescharacteristic
122C2×C4×C23C2×C22⋊C411yescharacteristic
123C8×C23—41no—
124C8×C23—41no—
125C8×C23—82no—
126C8×C23—82no—
127#3221C2×C2311yescharacteristic
128#2135C2×C2311yescharacteristic, complemented
129#2133C2×C2311yescharacteristic, complemented
130#1413C2×C2312yescomplemented
131#1410C2×C2312yescomplemented
132C22×C23—21no—
133C22×C23—41no—
134C22×C23—41no—
135C22×C23—81no—
136C22×C23—81no—
137C22×C23—81no—
138C22×C23—81no—
139C22×C23C2×#9911yescharacteristic
140C4×C23—21no—
141C4×C23—21no—
142C4×C23—41no—
143C4×C23—41no—
144C4×C23—41no—
145C4×C23—41no—
146C4×C23—41no—
147C4×C23—81no—
148#540C4×C2311yescharacteristic
149#536C22×C2311yescharacteristic
150#496C4×C2311yescharacteristic
151#492C22×C2311yescharacteristic
152#491C22×C2311yescharacteristic
153#488C4×C2311yescharacteristic
154#463—21no—
155#461—21no—
156#449—21no—
157#448—21no—
158#433C4×C2311yescharacteristic
159#413—21no—
160C2×#101—21no—
161#356—21no—
162#356C22×C2312yes—
163#356C22×C2312yes—
164#355—21no—
165C2×C23—21no—
166C2×C23—41no—
167C2×C23—81no—
168C2×C23—161no—
169C2×C23#212711yescharacteristic
170C2×#41D4×C2311yescharacteristic
171C22×Q8D4×C2311yescharacteristic
172C2×SD16—21no—
173C2×SD16—21no—
174C2×SD16—21no—
175C2×D8—21no—
176C2×M16—41no—
177C2×M16—41no—
178C4⋊Q8D4×C2311yescharacteristic
179C4⋊Q8C23×C2311yescharacteristic
180#127C2×C4×C2311yescharacteristic
181#125C2×C4×C2311yescharacteristic
182#124C2×C4×C2311yescharacteristic
183#124D4×C2311yescharacteristic
184#123—41no—
185#123—41no—
186C22⋊Q8—41no—
187#121—21no—
188#121—41no—
189C4×D4—21no—
190#117C2×C4×C2311yescharacteristic
191C4⋊C8—21no—
192#103—21no—
193#103—21no—
194#103—21no—
195#102—21no—
196#101—22no—
197#101—22no—
198#101—41no—
199#100—41no—
200C22⋊C8—21no—
201#97C2×C4×C2311yescharacteristic
202C4×C8C2×C4×C2311yescharacteristic
203C23#1361111yescharacteristic, direct product factor, Sylow
204#41—81no—
205C2×Q8—21no—
206C2×Q8—81no—
207C2×Q8C22⋊C4×C2311yescharacteristic
208C2×Q8C2×D4×C2311yescharacteristic
209C2×Q8C2×D4×C2311yescharacteristic
210C2×D4—21no—
211C2×D4—21no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4C22⋊C4×C2311yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—41no—
218C22×C4C22⋊C4×C2311yescharacteristic
219SD16—81no—
220SD16—81no—
221SD16—81no—
222D8—81no—
223M16—42no—
224M16—42no—
225M16—81no—
226M16—81no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—81no—
230C2×C8—81no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C22⋊C4—21no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—41no—
241C42C22⋊C4×C2311yescharacteristic
242C42C22×C4×C2311yescharacteristic
243C23—21no—
244C23—41no—
245C23C23×#9911yescharacteristic
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—81no—
250D4—41no—
251D4—41no—
252D4—41no—
253D4—81no—
254D4—161no—
255D4—161no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—161no—
268C2×C4C23×#9911yescharacteristic
269C2×C4C2×C22⋊C4×C2311yescharacteristic
270C8—41no—
271C8—41no—
272C8—82no—
273C8—82no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22C2×C23×#9911yescharacteristic
282C4—21no—
283C4—21no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—81no—
290C2—21no—
291C2—41no—
292C2—81no—
293C2—161no—
294C2C23×#212711yescharacteristic
subgroupG/H
center, socleC2×C23#2127
commutatorC42C22×C4×C23
Frattini, cosocleC4⋊Q8C23×C23
prime radicalC23×#3221C2
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C22×C23 ◁ C22×C4×C23 ◁ C23×#536 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2135C2semidirect
3. C23×#2133C2semidirect
4. C23×#1413C2semidirect
5. C23×#1410C2semidirect
95. #13611C23direct (N × H)
128. #2135C2×C23semidirect
129. #2133C2×C23semidirect
130. #1413C2×C23semidirect
131. #1410C2×C23semidirect
203. C23#13611direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32212
C23×#21352
C23×#21332
C23×#14132
C23×#14102
#1361123