Finite Group Databasebeta

← C23×#8498 #19156124

Subgroups — C23×#8498 #19156124

Order 5888.

692 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 142 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×#3544C211yesmaximal, characteristic
2C23×#1960C211yesmaximal, characteristic, complemented
3C23×#1558C211yesmaximal, characteristic, complemented
4C23×#1532C212yesmaximal, complemented
5C23×#1512C212yesmaximal, complemented
6C23×#563—22no—
7Q82×C23C411yescharacteristic, complemented
8C23×#555C411yescharacteristic, complemented
9C23×#554C412yescomplemented
10C23×#552—22no—
11C23×#547—22no—
12C23×#544—22no—
13C23×#542—22no—
14C23×#536—22no—
15C2×C23×C4⋊Q8C2211yescharacteristic
16C23×#529C2212yes—
17C23×#432C2212yes—
18C2×C23×#103—22no—
19C2×C23×#97C2211yescharacteristic
20C23×#382C2211yescharacteristic
21C23×#333—22no—
22C23×#330—22no—
23C23×#329—22no—
24C23×#328—22no—
25C23×#327—22no—
26C2×#41×C23—22no—
27C22×Q8×C23D412yes—
28C22×C8×C23—21no—
29C23×C4⋊Q8—22no—
30C23×C4⋊Q8—22no—
31C23×C4⋊Q8—22no—
32C23×C4⋊Q8—22no—
33C23×C4⋊Q8C2×C411yescharacteristic
34C23×C4⋊Q8C2×C411yescharacteristic
35C23×C4⋊Q8D412yes—
36C23×#126—24no—
37C23×#126—22no—
38C23×#125—24no—
39C23×#125—22no—
40C23×#125D412yes—
41C23×#124—22no—
42C23×#124—22no—
43C23×#124D412yes—
44C23×#123—24no—
45C23×#123—24no—
46C23×#123—22no—
47C23×C22⋊Q8—24no—
48C23×C22⋊Q8—24no—
49C23×C22⋊Q8—22no—
50C23×C22⋊Q8—24no—
51C23×C22⋊Q8D412yes—
52C23×#121—22no—
53C23×#121D412yes—
54C4×Q8×C23—22no—
55C4×Q8×C23C2×C412yes—
56C4×D4×C23—22no—
57C4×D4×C23—22no—
58C4×D4×C23C2×C412yes—
59C23×#117—22no—
60C23×#117—22no—
61C23×#117—22no—
62C2×C4⋊C4×C23—22no—
63C2×C4⋊C4×C23—22no—
64C2×C4⋊C4×C23—21no—
65C2×C42×C23C2311yescharacteristic
66C23×C4⋊C8—22no—
67C23×#103—44no—
68C23×C22⋊C8—22no—
69C23×#97—22no—
70C23×#97—22no—
71C23×#95—22no—
72#41×C23—82no—
73#41×C23—82no—
74C2×Q8×C23—22no—
75C2×Q8×C23—22no—
76C2×Q8×C23—42no—
77C2×Q8×C23—42no—
78C2×Q8×C23C22⋊C412yes—
79C2×Q8×C23C22⋊C412yes—
80C2×Q8×C23C22⋊C412yes—
81C2×D4×C23—22no—
82C2×D4×C23—22no—
83C2×D4×C23C22⋊C412yes—
84C22×C4×C23—22no—
85C22×C4×C23—24no—
86C22×C4×C23—22no—
87C22×C4×C23—41no—
88C22×C4×C23C2×D411yescharacteristic
89C22×C4×C23C2×D412yes—
90C2×C8×C23—22no—
91C2×C8×C23—41no—
92C2×C8×C23—41no—
93C2×C8×C23—42no—
94C4⋊C4×C23—24no—
95C4⋊C4×C23—24no—
96C4⋊C4×C23—24no—
97C4⋊C4×C23—22no—
98C4⋊C4×C23—22no—
99C4⋊C4×C23—22no—
100C4⋊C4×C23—24no—
101C4⋊C4×C23—22no—
102C4⋊C4×C23—22no—
103C4⋊C4×C23—22no—
104C4⋊C4×C23—42no—
105C4⋊C4×C23C22⋊C412yes—
106C4⋊C4×C23C22⋊C412yes—
107C22⋊C4×C23—24no—
108C22⋊C4×C23—24no—
109C22⋊C4×C23—22no—
110C22⋊C4×C23—22no—
111C22⋊C4×C23—22no—
112C22⋊C4×C23—22no—
113C22⋊C4×C23—22no—
114C42×C23—22no—
115C42×C23—22no—
116C42×C23—22no—
117C42×C23C22×C411yescharacteristic
118C42×C23C2×D411yescharacteristic
119C42×C23C2×D412yes—
120#8498C2311yesmaximal, characteristic, direct product factor, Sylow
121C23×C23—22no—
122C23×C23#12011yescharacteristic
123Q8×C23—42no—
124Q8×C23—42no—
125Q8×C23—42no—
126Q8×C23—82no—
127Q8×C23—82no—
128D4×C23—42no—
129D4×C23—42no—
130D4×C23—42no—
131D4×C23—82no—
132C2×C4×C23—21no—
133C2×C4×C23—21no—
134C2×C4×C23—22no—
135C2×C4×C23—22no—
136C2×C4×C23—22no—
137C2×C4×C23—22no—
138C2×C4×C23—22no—
139C2×C4×C23—42no—
140C2×C4×C23—44no—
141C2×C4×C23—42no—
142C2×C4×C23—42no—
143C2×C4×C23—84no—
144C2×C4×C23—82no—
145C2×C4×C23—81no—
146C2×C4×C23C2×C22⋊C412yes—
147C2×C4×C23C2×C22⋊C411yescharacteristic
148C2×C4×C23#12012yes—
149C2×C4×C23#12011yescharacteristic
150C8×C23—42no—
151C8×C23—82no—
152#3544C2×C2311yescharacteristic
153#1960C2×C2311yescharacteristic, complemented
154#1558C2×C2311yescharacteristic, complemented
155#1532C2×C2312yescomplemented
156#1512C2×C2312yescomplemented
157C22×C23—22no—
158C22×C23—42no—
159C22×C23—42no—
160C22×C23—42no—
161C22×C23#37911yescharacteristic
162C22×C23#45311yescharacteristic
163C22×C23#45511yescharacteristic
164C4×C23—21no—
165C4×C23—21no—
166C4×C23—22no—
167C4×C23—22no—
168C4×C23—42no—
169C4×C23—42no—
170C4×C23—42no—
171C4×C23—42no—
172C4×C23—42no—
173C4×C23—161no—
174#563—22no—
175Q82C4×C2311yescharacteristic, complemented
176#555C4×C2311yescharacteristic, complemented
177#554C4×C2312yescomplemented
178#552—22no—
179#547—22no—
180#544—22no—
181#542—22no—
182#536—22no—
183C2×C4⋊Q8C22×C2311yescharacteristic
184#529C22×C2312yes—
185#432C22×C2312yes—
186C2×#103—22no—
187C2×#97C22×C2311yescharacteristic
188#382C22×C2311yescharacteristic
189#333—22no—
190#330—22no—
191#329—22no—
192#328—22no—
193#327—22no—
194C2×C23—21no—
195C2×C23—21no—
196C2×C23—82no—
197C2×C23#179211yescharacteristic
198C2×C23#179612yes—
199C2×#41—22no—
200C22×Q8D4×C2312yes—
201C22×C8—21no—
202C4⋊Q8—22no—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8C2×C4×C2311yescharacteristic
207C4⋊Q8C2×C4×C2311yescharacteristic
208C4⋊Q8D4×C2312yes—
209#126—24no—
210#126—22no—
211#125—24no—
212#125—22no—
213#125D4×C2312yes—
214#124—22no—
215#124—22no—
216#124D4×C2312yes—
217#123—24no—
218#123—24no—
219#123—22no—
220C22⋊Q8—24no—
221C22⋊Q8—24no—
222C22⋊Q8—24no—
223C22⋊Q8—22no—
224C22⋊Q8D4×C2312yes—
225#121—22no—
226#121D4×C2312yes—
227C4×Q8—22no—
228C4×Q8C2×C4×C2312yes—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4C2×C4×C2312yes—
232#117—22no—
233#117—22no—
234#117—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C42C23×C2311yescharacteristic
239C4⋊C8—22no—
240#103—44no—
241C22⋊C8—22no—
242#97—22no—
243#97—22no—
244#95—22no—
245C23#849811yescharacteristic, direct product factor, Sylow
246#41—82no—
247#41—82no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8C22⋊C4×C2312yes—
253C2×Q8C22⋊C4×C2312yes—
254C2×Q8C22⋊C4×C2312yes—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4C22⋊C4×C2312yes—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4C2×D4×C2311yescharacteristic
263C22×C4C2×D4×C2312yes—
264C2×C8—22no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—42no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—42no—
279C4⋊C4C22⋊C4×C2312yes—
280C4⋊C4C22⋊C4×C2312yes—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42C22×C4×C2311yescharacteristic
292C42C2×D4×C2312yes—
293C42C2×D4×C2311yescharacteristic
294C23—22no—
295C23C23×#12011yescharacteristic
296Q8—42no—
297Q8—42no—
298Q8—42no—
299Q8—82no—
300Q8—82no—
301D4—42no—
302D4—42no—
303D4—42no—
304D4—82no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—84no—
317C2×C4—82no—
318C2×C4—81no—
319C2×C4C2×C22⋊C4×C2312yes—
320C2×C4C2×C22⋊C4×C2311yescharacteristic
321C2×C4C23×#12012yes—
322C2×C4C23×#12011yescharacteristic
323C8—42no—
324C8—82no—
325C22—22no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22C23×#37911yescharacteristic
330C22C23×#45311yescharacteristic
331C22C23×#45511yescharacteristic
332C4—21no—
333C4—21no—
334C4—22no—
335C4—22no—
336C4—42no—
337C4—42no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—161no—
342C2—21no—
343C2—21no—
344C2—82no—
345C2C23×#179211yescharacteristic
346C2C23×#179612yes—
subgroupG/H
center, socleC22×C23#379
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3544C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1960C2semidirect
3. C23×#1558C2semidirect
4. C23×#1532C2semidirect
5. C23×#1512C2semidirect
7. Q82×C23C4semidirect
8. C23×#555C4semidirect
9. C23×#554C4semidirect
120. #8498C23direct (N × H)
153. #1960C2×C23semidirect
154. #1558C2×C23semidirect
155. #1532C2×C23semidirect
156. #1512C2×C23semidirect
175. Q82C4×C23semidirect
176. #555C4×C23semidirect
177. #554C4×C23semidirect
245. C23#8498direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35442
C23×#19602
C23×#15582
C23×#15322
C23×#15122
#849823