Finite Group Databasebeta

← C23×#11807 #19159433

Subgroups — C23×#11807 #19159433

Order 5888.

388 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2621C211yesmaximal, characteristic
2C23×#1822C211yesmaximal, characteristic
3C23×#1464C211yesmaximal, characteristic
4C23×#1346C214yesmaximal, complemented
5C2×C23×C4⋊Q8C411yescharacteristic
6C2×C23×C22⋊Q8—21no—
7C2×C23×#117C2211yescharacteristic
8C23×#432—21no—
9C23×#407—21no—
10C23×#398—21no—
11C23×#398C2212yes—
12C23×#397—21no—
13C23×#397C412yes—
14C23×#391—21no—
15C23×#382C411yescharacteristic
16C23×#343—21no—
17C23×#343C2212yes—
18C23×#324C2212yes—
19C23×#323—21no—
20C22×Q8×C23—21no—
21C23×C4×C23D411yescharacteristic
22C2×M16×C23—22no—
23C23×C4⋊Q8—41no—
24C23×C4⋊Q8—41no—
25C23×C22⋊Q8—44no—
26C23×#117—41no—
27C23×#117—41no—
28C2×C4⋊C4×C23—21no—
29C2×C4⋊C4×C23—21no—
30C2×C4⋊C4×C23—22no—
31C2×C4⋊C4×C23C2×C412yes—
32C2×C4⋊C4×C23D411yescharacteristic
33C2×C4⋊C4×C23C2311yescharacteristic
34C2×C22⋊C4×C23—22no—
35C2×C22⋊C4×C23C2×C411yescharacteristic
36C2×C22⋊C4×C23D411yescharacteristic
37C2×C42×C23C2×C411yescharacteristic
38C2×C42×C23D411yescharacteristic
39C23×C4⋊C8—41no—
40C23×C22⋊C8—22no—
41C23×C22⋊C8—41no—
42C23×C22⋊C8—41no—
43C23×#97—41no—
44C23×#95—21no—
45C23×#95—22no—
46C23×#95—22no—
47C23×#95C2×C412yes—
48C24×C23C22⋊C411yescharacteristic
49C2×Q8×C23—42no—
50C2×Q8×C23—42no—
51C2×Q8×C23—41no—
52C2×Q8×C23—41no—
53C22×C4×C23—21no—
54C22×C4×C23—21no—
55C22×C4×C23—22no—
56C22×C4×C23—22no—
57C22×C4×C23—22no—
58C22×C4×C23—41no—
59C22×C4×C23C22⋊C411yescharacteristic
60C22×C4×C23C22⋊C411yescharacteristic
61C22×C4×C23C22⋊C411yescharacteristic
62C22×C4×C23C22×C411yescharacteristic
63C22×C4×C23C2×D411yescharacteristic
64C22×C4×C23C2×D411yescharacteristic
65M16×C23—42no—
66M16×C23—42no—
67C2×C8×C23—42no—
68C2×C8×C23—42no—
69C4⋊C4×C23—41no—
70C4⋊C4×C23—41no—
71C4⋊C4×C23—42no—
72C4⋊C4×C23—42no—
73C4⋊C4×C23—42no—
74C4⋊C4×C23—44no—
75C22⋊C4×C23—41no—
76C22⋊C4×C23—41no—
77C22⋊C4×C23—44no—
78C42×C23—21no—
79C42×C23—21no—
80C42×C23—21no—
81C42×C23—21no—
82#11807C2311yesmaximal, characteristic, direct product factor, Sylow
83C23×C23—21no—
84C23×C23—21no—
85C23×C23—21no—
86C23×C23—41no—
87C23×C23—41no—
88C23×C23C2×C22⋊C411yescharacteristic
89Q8×C23—82no—
90Q8×C23—82no—
91C2×C4×C23—21no—
92C2×C4×C23—21no—
93C2×C4×C23—21no—
94C2×C4×C23—21no—
95C2×C4×C23—21no—
96C2×C4×C23—21no—
97C2×C4×C23—21no—
98C2×C4×C23—21no—
99C2×C4×C23—42no—
100C2×C4×C23—42no—
101C2×C4×C23—41no—
102C2×C4×C23—41no—
103C2×C4×C23—41no—
104C2×C4×C23—41no—
105C2×C4×C23—42no—
106C2×C4×C23—42no—
107C2×C4×C23—42no—
108C2×C4×C23—42no—
109C2×C4×C23—42no—
110C2×C4×C23—42no—
111C2×C4×C23#9911yescharacteristic
112C2×C4×C23#9911yescharacteristic
113C2×C4×C23AGL(1,ℤ/8)12yes—
114C8×C23—82no—
115C8×C23—82no—
116#2621C2×C2311yescharacteristic
117#1822C2×C2311yescharacteristic
118#1464C2×C2311yescharacteristic
119#1346C2×C2314yescomplemented
120C22×C23—21no—
121C22×C23—21no—
122C22×C23—42no—
123C22×C23—41no—
124C22×C23—41no—
125C22×C23—41no—
126C22×C23—81no—
127C22×C23C2×#9911yescharacteristic
128C22×C23#41811yescharacteristic
129C22×C23#42111yescharacteristic
130C4×C23—22no—
131C4×C23—21no—
132C4×C23—21no—
133C4×C23—41no—
134C4×C23—41no—
135C4×C23—41no—
136C4×C23—41no—
137C4×C23—82no—
138C4×C23—82no—
139C2×C4⋊Q8C4×C2311yescharacteristic
140C2×C22⋊Q8—21no—
141C2×#117C22×C2311yescharacteristic
142#432—21no—
143#407—21no—
144#398—21no—
145#398C22×C2312yes—
146#397—21no—
147#397C4×C2312yes—
148#391—21no—
149#382C4×C2311yescharacteristic
150#343—21no—
151#343C22×C2312yes—
152#324C22×C2312yes—
153#323—21no—
154C2×C23—21no—
155C2×C23—21no—
156C2×C23—81no—
157C2×C23#151011yescharacteristic
158C2×C23#212311yescharacteristic
159C2×C23#213511yescharacteristic
160C22×Q8—21no—
161C23×C4D4×C2311yescharacteristic
162C2×M16—22no—
163C4⋊Q8—41no—
164C4⋊Q8—41no—
165C22⋊Q8—44no—
166#117—41no—
167#117—41no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4C2×C4×C2312yes—
172C2×C4⋊C4D4×C2311yescharacteristic
173C2×C4⋊C4C23×C2311yescharacteristic
174C2×C22⋊C4—22no—
175C2×C22⋊C4C2×C4×C2311yescharacteristic
176C2×C22⋊C4D4×C2311yescharacteristic
177C2×C42C2×C4×C2311yescharacteristic
178C2×C42D4×C2311yescharacteristic
179C4⋊C8—41no—
180C22⋊C8—22no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183#97—41no—
184#95—21no—
185#95—22no—
186#95—22no—
187#95C2×C4×C2312yes—
188C23#1180711yescharacteristic, direct product factor, Sylow
189C24C22⋊C4×C2311yescharacteristic
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—41no—
193C2×Q8—41no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—41no—
200C22×C4C22⋊C4×C2311yescharacteristic
201C22×C4C22⋊C4×C2311yescharacteristic
202C22×C4C22⋊C4×C2311yescharacteristic
203C22×C4C22×C4×C2311yescharacteristic
204C22×C4C2×D4×C2311yescharacteristic
205C22×C4C2×D4×C2311yescharacteristic
206M16—42no—
207M16—42no—
208C2×C8—42no—
209C2×C8—42no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—44no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C22⋊C4—44no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—41no—
227C23—41no—
228C23C2×C22⋊C4×C2311yescharacteristic
229Q8—82no—
230Q8—82no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4C23×#9911yescharacteristic
252C2×C4C23×#9911yescharacteristic
253C2×C4C23×AGL(1,ℤ/8)12yes—
254C8—82no—
255C8—82no—
256C22—21no—
257C22—21no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—81no—
263C22C2×C23×#9911yescharacteristic
264C22C23×#41811yescharacteristic
265C22C23×#42111yescharacteristic
266C4—22no—
267C4—21no—
268C4—21no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—82no—
274C4—82no—
275C2—21no—
276C2—21no—
277C2—81no—
278C2C23×#151011yescharacteristic
279C2C23×#212311yescharacteristic
280C2C23×#213511yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC24×C23C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C23×#117 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×#1346C2semidirect
82. #11807C23direct (N × H)
119. #1346C2×C23semidirect
188. C23#11807direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#26212
C23×#18222
C23×#14642
C23×#13462
#1180723