Finite Group Databasebeta

← C23×#12764 #19160390

Subgroups — C23×#12764 #19160390

Order 5888.

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

normal subgroups only

This group has 768 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
1C2×C23×#563C211yesmaximal, characteristic
2C2×C23×#354C211yesmaximal, characteristic
3C2×C23×#352C212yesmaximal
4C23×#1890C212yesmaximal
5C23×#1885C211yesmaximal, characteristic
6C23×#563—22no—
7C23×#563—24no—
8C23×#563C414yescomplemented
9C2×C23×C4⋊Q8—21no—
10C2×C23×#126—22no—
11C2×C23×#125—21no—
12C2×C23×#124C2211yescharacteristic
13C2×C23×#123C2212yes—
14C2×C23×C22⋊Q8—22no—
15C23×#518C2211yescharacteristic
16C23×#419—21no—
17C2×C23×#100C2211yescharacteristic
18C2×C23×#99C2212yes—
19C23×#381—22no—
20C23×#354—22no—
21C23×#352—24no—
22C23×#342—22no—
23C23×#339—21no—
24C22×Q8×C23—21no—
25C22×Q8×C23D411yescharacteristic
26C22×D4×C23C2311yescharacteristic
27C23×C4×C23D412yes—
28C2×M16×C23—21no—
29C23×C4⋊Q8—24no—
30C23×C4⋊Q8—24no—
31C23×#126—28no—
32C23×#126—28no—
33C23×#125—24no—
34C23×#125—24no—
35C23×#124—22no—
36C23×#124C2×C412yes—
37C23×#124D412yes—
38C23×#123—22no—
39C23×#123—22no—
40C23×#123C2×C414yes—
41C23×#123D414yes—
42C23×C22⋊Q8—28no—
43C23×C22⋊Q8—28no—
44C4×D4×C23—42no—
45C23×#117—21no—
46C2×C4⋊C4×C23—21no—
47C2×C4⋊C4×C23—22no—
48C2×C4⋊C4×C23—22no—
49C2×C4⋊C4×C23—21no—
50C2×C4⋊C4×C23—22no—
51C2×C22⋊C4×C23—22no—
52C2×C22⋊C4×C23—22no—
53C2×C22⋊C4×C23—22no—
54C2×C22⋊C4×C23—22no—
55C2×C22⋊C4×C23D412yes—
56C2×C42×C23—21no—
57C2×C42×C23D411yescharacteristic
58C23×#103—42no—
59C23×#100—22no—
60C23×#99—22no—
61C23×#99—22no—
62C23×#95—44no—
63C24×C23C2×D412yes—
64C2×Q8×C23—24no—
65C2×Q8×C23—21no—
66C2×Q8×C23—41no—
67C2×Q8×C23—41no—
68C2×Q8×C23—81no—
69C2×Q8×C23—81no—
70C2×Q8×C23C22⋊C412yes—
71C2×D4×C23—41no—
72C2×D4×C23—41no—
73C2×D4×C23C22×C411yescharacteristic
74C2×D4×C23C2×D411yescharacteristic
75C2×D4×C23C2×D412yes—
76C22×C4×C23—21no—
77C22×C4×C23—22no—
78C22×C4×C23—21no—
79C22×C4×C23—22no—
80C22×C4×C23—22no—
81C22×C4×C23—24no—
82C22×C4×C23—42no—
83C22×C4×C23—42no—
84C22×C4×C23—42no—
85C22×C4×C23C22⋊C414yes—
86C22×C4×C23C2×D411yescharacteristic
87M16×C23—41no—
88M16×C23—41no—
89C2×C8×C23—41no—
90C4⋊C4×C23—24no—
91C4⋊C4×C23—28no—
92C4⋊C4×C23—28no—
93C4⋊C4×C23—24no—
94C4⋊C4×C23—24no—
95C4⋊C4×C23—24no—
96C4⋊C4×C23—22no—
97C22⋊C4×C23—24no—
98C22⋊C4×C23—24no—
99C22⋊C4×C23—24no—
100C22⋊C4×C23—24no—
101C22⋊C4×C23—22no—
102C22⋊C4×C23—42no—
103C22⋊C4×C23—41no—
104C22⋊C4×C23—42no—
105C22⋊C4×C23—42no—
106C22⋊C4×C23C22⋊C414yes—
107C42×C23—24no—
108C42×C23—21no—
109C42×C23—41no—
110C42×C23C22⋊C412yes—
111#12764C2311yesmaximal, characteristic, direct product factor, Sylow
112C23×C23—22no—
113C23×C23—22no—
114C23×C23—44no—
115C23×C23C2×C22⋊C412yes—
116C23×C23#12012yes—
117C23×C23#12011yescharacteristic
118Q8×C23—42no—
119Q8×C23—81no—
120Q8×C23—84no—
121D4×C23—41no—
122D4×C23—42no—
123D4×C23—41no—
124C2×C4×C23—21no—
125C2×C4×C23—21no—
126C2×C4×C23—22no—
127C2×C4×C23—24no—
128C2×C4×C23—22no—
129C2×C4×C23—24no—
130C2×C4×C23—44no—
131C2×C4×C23—42no—
132C2×C4×C23—42no—
133C2×C4×C23—41no—
134C2×C4×C23—42no—
135C2×C4×C23—41no—
136C2×C4×C23—44no—
137C2×C4×C23—42no—
138C2×C4×C23—82no—
139C2×C4×C23—84no—
140C2×C4×C23—82no—
141C2×C4×C23—84no—
142C2×C4×C23C2×C22⋊C411yescharacteristic
143C2×C4×C23#12011yescharacteristic
144C8×C23—81no—
145C2×#563C2×C2311yescharacteristic
146C2×#354C2×C2311yescharacteristic
147C2×#352C2×C2312yes—
148#1890C2×C2312yes—
149#1885C2×C2311yescharacteristic
150C22×C23—22no—
151C22×C23—22no—
152C22×C23—22no—
153C22×C23—42no—
154C22×C23—42no—
155C22×C23—44no—
156C22×C23—82no—
157C22×C23#37911yescharacteristic
158C22×C23B(4,𝔽2)11yescharacteristic
159C22×C23#45811yescharacteristic
160C4×C23—21no—
161C4×C23—21no—
162C4×C23—44no—
163C4×C23—42no—
164C4×C23—44no—
165C4×C23—42no—
166C4×C23—82no—
167C4×C23—162no—
168#563—22no—
169#563—24no—
170#563C4×C2314yescomplemented
171C2×C4⋊Q8—21no—
172C2×#126—22no—
173C2×#125—21no—
174C2×#124C22×C2311yescharacteristic
175C2×#123C22×C2312yes—
176C2×C22⋊Q8—22no—
177#518C22×C2311yescharacteristic
178#419—21no—
179C2×#100C22×C2311yescharacteristic
180C2×#99C22×C2312yes—
181#381—22no—
182#354—22no—
183#352—24no—
184#342—22no—
185#339—21no—
186C2×C23—21no—
187C2×C23—21no—
188C2×C23—42no—
189C2×C23—42no—
190C2×C23#179011yescharacteristic
191C2×C23#220112yes—
192C22×Q8—21no—
193C22×Q8D4×C2311yescharacteristic
194C22×D4C23×C2311yescharacteristic
195C23×C4D4×C2312yes—
196C2×M16—21no—
197C4⋊Q8—24no—
198C4⋊Q8—24no—
199#126—28no—
200#126—28no—
201#125—24no—
202#125—24no—
203#124—22no—
204#124C2×C4×C2312yes—
205#124D4×C2312yes—
206#123—22no—
207#123—22no—
208#123C2×C4×C2314yes—
209#123D4×C2314yes—
210C22⋊Q8—28no—
211C22⋊Q8—28no—
212C4×D4—42no—
213#117—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4D4×C2312yes—
224C2×C42—21no—
225C2×C42D4×C2311yescharacteristic
226#103—42no—
227#100—22no—
228#99—22no—
229#99—22no—
230#95—44no—
231C23#1276411yescharacteristic, direct product factor, Sylow
232C24C2×D4×C2312yes—
233C2×Q8—24no—
234C2×Q8—21no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—81no—
238C2×Q8—81no—
239C2×Q8C22⋊C4×C2312yes—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4C22×C4×C2311yescharacteristic
243C2×D4C2×D4×C2311yescharacteristic
244C2×D4C2×D4×C2312yes—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4C22⋊C4×C2314yes—
255C22×C4C2×D4×C2311yescharacteristic
256M16—41no—
257M16—41no—
258C2×C8—41no—
259C4⋊C4—24no—
260C4⋊C4—28no—
261C4⋊C4—28no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—22no—
271C22⋊C4—42no—
272C22⋊C4—41no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4C22⋊C4×C2314yes—
276C42—24no—
277C42—21no—
278C42—41no—
279C42C22⋊C4×C2312yes—
280C23—22no—
281C23—22no—
282C23—44no—
283C23C2×C22⋊C4×C2312yes—
284C23C23×#12011yescharacteristic
285C23C23×#12012yes—
286Q8—42no—
287Q8—81no—
288Q8—84no—
289D4—41no—
290D4—42no—
291D4—41no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—82no—
307C2×C4—84no—
308C2×C4—82no—
309C2×C4—84no—
310C2×C4C2×C22⋊C4×C2311yescharacteristic
311C2×C4C23×#12011yescharacteristic
312C8—81no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—42no—
317C22—42no—
318C22—44no—
319C22—82no—
320C22C23×#37911yescharacteristic
321C22C23×B(4,𝔽2)11yescharacteristic
322C22C23×#45811yescharacteristic
323C4—21no—
324C4—21no—
325C4—44no—
326C4—42no—
327C4—44no—
328C4—42no—
329C4—82no—
330C4—162no—
331C2—21no—
332C2—21no—
333C2—42no—
334C2—42no—
335C2C23×#179011yescharacteristic
336C2C23×#220112yes—
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutatorC2×D4C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC22×D4×C23C23
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C22×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
8. C23×#563C4semidirect
111. #12764C23direct (N × H)
170. #563C4×C23semidirect
231. C23#12764direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5632
C2×C23×#3542
C2×C23×#3522
C23×#18902
C23×#18852
#1276423