Finite Group Databasebeta

← C23×#12967 #19160593

Subgroups — C23×#12967 #19160593

Order 5888.

516 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2379C211yesmaximal, characteristic
2C2×C23×#354C211yesmaximal, characteristic
3C23×#1885C211yesmaximal, characteristic
4C23×#1867C212yesmaximal
5C23×#1390C212yesmaximal
6C2×C23×#124C2211yescharacteristic
7C23×#518C2211yescharacteristic
8C23×#419—21no—
9C2×C23×#100C2211yescharacteristic
10C2×C23×#99C2212yes—
11C23×#391—21no—
12C23×#388—22no—
13C23×#386C2212yes—
14C23×#385—22no—
15C23×#385—22no—
16C23×#382—21no—
17C23×#354C414yescomplemented
18C23×#342—22no—
19C23×#339—21no—
20C22×Q8×C23D411yescharacteristic
21C22×D4×C23C2311yescharacteristic
22C23×C4×C23D412yes—
23C2×M16×C23—21no—
24C23×#124—22no—
25C23×#124C2×C412yes—
26C23×#124Q812yes—
27C4×D4×C23—41no—
28C4×D4×C23—41no—
29C23×#117—21no—
30C2×C4⋊C4×C23—21no—
31C2×C4⋊C4×C23—21no—
32C2×C4⋊C4×C23—21no—
33C2×C4⋊C4×C23—42no—
34C2×C22⋊C4×C23—22no—
35C2×C22⋊C4×C23—22no—
36C2×C22⋊C4×C23—22no—
37C2×C22⋊C4×C23—22no—
38C2×C22⋊C4×C23D412yes—
39C2×C42×C23—21no—
40C2×C42×C23D411yescharacteristic
41C23×#103—41no—
42C23×#103—41no—
43C23×#100—22no—
44C23×#99—22no—
45C23×#99C2×C414yes—
46C23×#95—22no—
47C23×#95—22no—
48C23×#95—22no—
49C23×#95—42no—
50C24×C23C2×D412yes—
51C2×Q8×C23—21no—
52C2×Q8×C23—41no—
53C2×Q8×C23—41no—
54C2×Q8×C23C4⋊C412yescomplemented
55C2×D4×C23—41no—
56C2×D4×C23—41no—
57C2×D4×C23C22×C411yescharacteristic
58C2×D4×C23C2×Q811yescharacteristic
59C2×D4×C23#4112yes—
60C22×C4×C23—21no—
61C22×C4×C23—21no—
62C22×C4×C23—22no—
63C22×C4×C23—22no—
64C22×C4×C23—22no—
65C22×C4×C23—22no—
66C22×C4×C23—42no—
67C22×C4×C23—42no—
68C22×C4×C23—82no—
69C22×C4×C23C2×D411yescharacteristic
70M16×C23—41no—
71M16×C23—41no—
72C2×C8×C23—41no—
73C4⋊C4×C23—21no—
74C4⋊C4×C23—21no—
75C4⋊C4×C23—81no—
76C4⋊C4×C23—81no—
77C4⋊C4×C23—81no—
78C4⋊C4×C23—84no—
79C22⋊C4×C23—24no—
80C22⋊C4×C23—24no—
81C22⋊C4×C23—24no—
82C22⋊C4×C23—42no—
83C22⋊C4×C23—42no—
84C22⋊C4×C23—41no—
85C22⋊C4×C23—42no—
86C22⋊C4×C23—82no—
87C42×C23—21no—
88C42×C23—41no—
89C42×C23—81no—
90C42×C23C4⋊C412yescomplemented
91#12967C2311yesmaximal, characteristic, direct product factor, Sylow
92C23×C23—22no—
93C23×C23—22no—
94C23×C23—42no—
95C23×C23—42no—
96C23×C23C4×D412yes—
97C23×C23#12011yescharacteristic
98C23×C23C22⋊Q812yes—
99Q8×C23—42no—
100Q8×C23—81no—
101D4×C23—41no—
102D4×C23—41no—
103D4×C23—42no—
104C2×C4×C23—21no—
105C2×C4×C23—21no—
106C2×C4×C23—22no—
107C2×C4×C23—22no—
108C2×C4×C23—41no—
109C2×C4×C23—41no—
110C2×C4×C23—41no—
111C2×C4×C23—41no—
112C2×C4×C23—44no—
113C2×C4×C23—42no—
114C2×C4×C23—42no—
115C2×C4×C23—42no—
116C2×C4×C23—42no—
117C2×C4×C23—44no—
118C2×C4×C23—41no—
119C2×C4×C23—41no—
120C2×C4×C23—42no—
121C2×C4×C23—42no—
122C2×C4×C23—82no—
123C2×C4×C23—82no—
124C2×C4×C23—82no—
125C2×C4×C23—82no—
126C2×C4×C23—82no—
127C2×C4×C23C2×C4⋊C411yescharacteristic
128C2×C4×C23#12311yescharacteristic
129C8×C23—81no—
130#2379C2×C2311yescharacteristic
131C2×#354C2×C2311yescharacteristic
132#1885C2×C2311yescharacteristic
133#1867C2×C2312yes—
134#1390C2×C2312yes—
135C22×C23—21no—
136C22×C23—21no—
137C22×C23—22no—
138C22×C23—22no—
139C22×C23—42no—
140C22×C23—42no—
141C22×C23—42no—
142C22×C23—42no—
143C22×C23—82no—
144C22×C23#38511yescharacteristic
145C22×C23B(4,𝔽2)11yescharacteristic
146C22×C23#45811yescharacteristic
147C4×C23—21no—
148C4×C23—21no—
149C4×C23—42no—
150C4×C23—42no—
151C4×C23—82no—
152C4×C23—84no—
153C4×C23—82no—
154C4×C23—81no—
155C4×C23—81no—
156C2×#124C22×C2311yescharacteristic
157#518C22×C2311yescharacteristic
158#419—21no—
159C2×#100C22×C2311yescharacteristic
160C2×#99C22×C2312yes—
161#391—21no—
162#388—22no—
163#386C22×C2312yes—
164#385—22no—
165#385—22no—
166#382—21no—
167#354C4×C2314yescomplemented
168#342—22no—
169#339—21no—
170C2×C23—21no—
171C2×C23—21no—
172C2×C23—42no—
173C2×C23—42no—
174C2×C23#186711yescharacteristic
175C2×C23#219911yescharacteristic
176C2×C23#220111yescharacteristic
177C22×Q8D4×C2311yescharacteristic
178C22×D4C23×C2311yescharacteristic
179C23×C4D4×C2312yes—
180C2×M16—21no—
181#124—22no—
182#124C2×C4×C2312yes—
183#124Q8×C2312yes—
184C4×D4—41no—
185C4×D4—41no—
186#117—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—42no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4D4×C2312yes—
196C2×C42—21no—
197C2×C42D4×C2311yescharacteristic
198#103—41no—
199#103—41no—
200#100—22no—
201#99—22no—
202#99C2×C4×C2314yes—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—42no—
207C23#1296711yescharacteristic, direct product factor, Sylow
208C24C2×D4×C2312yes—
209C2×Q8—21no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8C4⋊C4×C2312yescomplemented
213C2×D4—41no—
214C2×D4—41no—
215C2×D4C22×C4×C2311yescharacteristic
216C2×D4C2×Q8×C2311yescharacteristic
217C2×D4#41×C2312yes—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—82no—
227C22×C4C2×D4×C2311yescharacteristic
228M16—41no—
229M16—41no—
230C2×C8—41no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—81no—
234C4⋊C4—81no—
235C4⋊C4—81no—
236C4⋊C4—84no—
237C22⋊C4—24no—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—41no—
243C22⋊C4—42no—
244C22⋊C4—82no—
245C42—21no—
246C42—41no—
247C42—81no—
248C42C4⋊C4×C2312yescomplemented
249C23—22no—
250C23—22no—
251C23—42no—
252C23—42no—
253C23C4×D4×C2312yes—
254C23C23×#12011yescharacteristic
255C23C23×C22⋊Q812yes—
256Q8—42no—
257Q8—81no—
258D4—41no—
259D4—41no—
260D4—42no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4C2×C4⋊C4×C2311yescharacteristic
285C2×C4C23×#12311yescharacteristic
286C8—81no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—82no—
296C22C23×#38511yescharacteristic
297C22C23×B(4,𝔽2)11yescharacteristic
298C22C23×#45811yescharacteristic
299C4—21no—
300C4—21no—
301C4—42no—
302C4—42no—
303C4—82no—
304C4—84no—
305C4—82no—
306C4—81no—
307C4—81no—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2C23×#186711yescharacteristic
313C2C23×#219911yescharacteristic
314C2C23×#220111yescharacteristic
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
17. C23×#354C4semidirect
54. C2×Q8×C23C4⋊C4semidirect
90. C42×C23C4⋊C4semidirect
91. #12967C23direct (N × H)
167. #354C4×C23semidirect
207. C23#12967direct (N × H)
212. C2×Q8C4⋊C4×C23semidirect
248. C42C4⋊C4×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#23792
C2×C23×#3542
C23×#18852
C23×#18672
C23×#13902
#1296723