Finite Group Databasebeta

← C23×#11570 #19159196

Subgroups — C23×#11570 #19159196

Order 5888.

480 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2542C211yesmaximal, characteristic
2C23×#1465C211yesmaximal, characteristic
3C23×#1459C211yesmaximal, characteristic
4C23×#1325C214yesmaximal, complemented
5C2×C23×C4⋊Q8C411yescharacteristic
6C2×C23×C22⋊Q8—21no—
7C2×C23×C22⋊Q8—21no—
8C2×C4×D4×C23C2211yescharacteristic
9C23×#407—21no—
10C2×C23×C22⋊C8—21no—
11C23×#399—21no—
12C23×#399—21no—
13C23×#390C412yes—
14C23×#384C2212yes—
15C4×C4⋊C4×C23C411yescharacteristic
16C23×#324C2212yes—
17C23×#324C2212yes—
18C23×#323—21no—
19C23×#323—21no—
20C22×Q8×C23—21no—
21C22×D4×C23C2×C411yescharacteristic
22C23×C4×C23D411yescharacteristic
23C23×C4×C23D411yescharacteristic
24C2×M16×C23—41no—
25C22×C8×C23—41no—
26C23×C4⋊Q8—41no—
27C23×C4⋊Q8—41no—
28C23×C22⋊Q8—44no—
29C23×C22⋊Q8—44no—
30C4×D4×C23—41no—
31C4×D4×C23—41no—
32C2×C4⋊C4×C23—21no—
33C2×C4⋊C4×C23—21no—
34C2×C4⋊C4×C23—22no—
35C2×C4⋊C4×C23—22no—
36C2×C4⋊C4×C23C2×C412yes—
37C2×C4⋊C4×C23C2311yescharacteristic
38C2×C22⋊C4×C23—22no—
39C2×C22⋊C4×C23—22no—
40C2×C22⋊C4×C23D411yescharacteristic
41C2×C22⋊C4×C23D411yescharacteristic
42C2×C42×C23C2×C412yes—
43C2×C42×C23C2×C411yescharacteristic
44C23×C22⋊C8—22no—
45C23×C22⋊C8—22no—
46C23×C22⋊C8—41no—
47C23×C22⋊C8—41no—
48C23×#95—21no—
49C24×C23C22⋊C411yescharacteristic
50C24×C23C22⋊C411yescharacteristic
51C2×Q8×C23—42no—
52C2×Q8×C23—42no—
53C2×Q8×C23—41no—
54C2×Q8×C23—41no—
55C2×D4×C23—22no—
56C2×D4×C23—22no—
57C2×D4×C23—41no—
58C22×C4×C23—22no—
59C22×C4×C23—21no—
60C22×C4×C23—21no—
61C22×C4×C23—21no—
62C22×C4×C23—21no—
63C22×C4×C23—22no—
64C22×C4×C23—42no—
65C22×C4×C23—42no—
66C22×C4×C23C22⋊C411yescharacteristic
67C22×C4×C23C22⋊C411yescharacteristic
68C22×C4×C23C22×C411yescharacteristic
69C22×C4×C23C2×D411yescharacteristic
70C22×C4×C23C2×D411yescharacteristic
71M16×C23—82no—
72C2×C8×C23—42no—
73C2×C8×C23—42no—
74C2×C8×C23—82no—
75C4⋊C4×C23—24no—
76C4⋊C4×C23—41no—
77C4⋊C4×C23—42no—
78C4⋊C4×C23—44no—
79C4⋊C4×C23—44no—
80C4⋊C4×C23—42no—
81C22⋊C4×C23—41no—
82C22⋊C4×C23—41no—
83C22⋊C4×C23—44no—
84C22⋊C4×C23—44no—
85C42×C23—21no—
86C42×C23—21no—
87C42×C23—24no—
88#11570C2311yesmaximal, characteristic, direct product factor, Sylow
89C23×C23—21no—
90C23×C23—21no—
91C23×C23—21no—
92C23×C23—21no—
93C23×C23—21no—
94C23×C23—41no—
95C23×C23—41no—
96C23×C23—41no—
97C23×C23—41no—
98C23×C23#10112yes—
99C23×C23C2×C22⋊C411yescharacteristic
100Q8×C23—82no—
101Q8×C23—82no—
102D4×C23—44no—
103C2×C4×C23—21no—
104C2×C4×C23—21no—
105C2×C4×C23—22no—
106C2×C4×C23—24no—
107C2×C4×C23—21no—
108C2×C4×C23—21no—
109C2×C4×C23—21no—
110C2×C4×C23—42no—
111C2×C4×C23—42no—
112C2×C4×C23—42no—
113C2×C4×C23—42no—
114C2×C4×C23—41no—
115C2×C4×C23—41no—
116C2×C4×C23—41no—
117C2×C4×C23—42no—
118C2×C4×C23—42no—
119C2×C4×C23—42no—
120C2×C4×C23—81no—
121C2×C4×C23—81no—
122C2×C4×C23#9911yescharacteristic
123C2×C4×C23#9911yescharacteristic
124C2×C4×C23#13112yes—
125C8×C23—82no—
126C8×C23—82no—
127#2542C2×C2311yescharacteristic
128#1465C2×C2311yescharacteristic
129#1459C2×C2311yescharacteristic
130#1325C2×C2314yescomplemented
131C22×C23—21no—
132C22×C23—21no—
133C22×C23—22no—
134C22×C23—42no—
135C22×C23—42no—
136C22×C23—41no—
137C22×C23—41no—
138C22×C23—41no—
139C22×C23—41no—
140C22×C23—41no—
141C22×C23—81no—
142C22×C23—81no—
143C22×C23#40811yescharacteristic
144C22×C23C2×#9911yescharacteristic
145C22×C23C2×#10111yescharacteristic
146C4×C23—21no—
147C4×C23—22no—
148C4×C23—44no—
149C4×C23—41no—
150C4×C23—41no—
151C4×C23—41no—
152C4×C23—82no—
153C4×C23#35312yes—
154C2×C4⋊Q8C4×C2311yescharacteristic
155C2×C22⋊Q8—21no—
156C2×C22⋊Q8—21no—
157C2×C4×D4C22×C2311yescharacteristic
158#407—21no—
159C2×C22⋊C8—21no—
160#399—21no—
161#399—21no—
162#390C4×C2312yes—
163#384C22×C2312yes—
164C4×C4⋊C4C4×C2311yescharacteristic
165#324C22×C2312yes—
166#324C22×C2312yes—
167#323—21no—
168#323—21no—
169C2×C23—21no—
170C2×C23—21no—
171C2×C23—42no—
172C2×C23—81no—
173C2×C23#147211yescharacteristic
174C2×C23C2×#35311yescharacteristic
175C2×C23#213511yescharacteristic
176C22×Q8—21no—
177C22×D4C2×C4×C2311yescharacteristic
178C23×C4D4×C2311yescharacteristic
179C23×C4D4×C2311yescharacteristic
180C2×M16—41no—
181C22×C8—41no—
182C4⋊Q8—41no—
183C4⋊Q8—41no—
184C22⋊Q8—44no—
185C22⋊Q8—44no—
186C4×D4—41no—
187C4×D4—41no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4C2×C4×C2312yes—
193C2×C4⋊C4C23×C2311yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4D4×C2311yescharacteristic
197C2×C22⋊C4D4×C2311yescharacteristic
198C2×C42C2×C4×C2312yes—
199C2×C42C2×C4×C2311yescharacteristic
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204#95—21no—
205C23#1157011yescharacteristic, direct product factor, Sylow
206C24C22⋊C4×C2311yescharacteristic
207C24C22⋊C4×C2311yescharacteristic
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—41no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4C22⋊C4×C2311yescharacteristic
224C22×C4C22⋊C4×C2311yescharacteristic
225C22×C4C22×C4×C2311yescharacteristic
226C22×C4C2×D4×C2311yescharacteristic
227C22×C4C2×D4×C2311yescharacteristic
228M16—82no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—82no—
232C4⋊C4—24no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—42no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—44no—
241C22⋊C4—44no—
242C42—21no—
243C42—21no—
244C42—24no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23C23×#10112yes—
255C23C2×C22⋊C4×C2311yescharacteristic
256Q8—82no—
257Q8—82no—
258D4—44no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4C23×#9911yescharacteristic
279C2×C4C23×#9911yescharacteristic
280C2×C4C23×#13112yes—
281C8—82no—
282C8—82no—
283C22—21no—
284C22—21no—
285C22—22no—
286C22—42no—
287C22—42no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22—81no—
295C22C23×#40811yescharacteristic
296C22C2×C23×#9911yescharacteristic
297C22C2×C23×#10111yescharacteristic
298C4—22no—
299C4—21no—
300C4—44no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—82no—
305C4C23×#35312yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—81no—
310C2C23×#147211yescharacteristic
311C2C2×C23×#35311yescharacteristic
312C2C23×#213511yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC22×D4×C23C2×C4
LCSG ▷ C22×C4 ▷ 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
4. C23×#1325C2semidirect
88. #11570C23direct (N × H)
130. #1325C2×C23semidirect
205. C23#11570direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25422
C23×#14652
C23×#14592
C23×#13252
#1157023