Finite Group Databasebeta

← C23×#7729 #19155355

Subgroups — C23×#7729 #19155355

Order 5888.

346 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2863C211yesmaximal, characteristic
2C23×#1839C211yesmaximal, characteristic
3C23×#1839C211yesmaximal, characteristic
4C23×#1272C212yesmaximal
5C23×#1271C212yesmaximal
6C23×#391C411yescharacteristic
7C23×#391C411yescharacteristic
8C23×#391C2211yescharacteristic
9C23×#389C411yescharacteristic
10C23×#389C411yescharacteristic
11C23×#389C2211yescharacteristic
12C23×#387—21no—
13C23×#387—21no—
14C23×#387—21no—
15C23×#383—21no—
16C43×C23C2211yescharacteristic
17C23×#336—21no—
18C23×#336—21no—
19C23×#336—21no—
20C23×#336—21no—
21C23×#336C2212yes—
22C23×#336C2212yes—
23C22×Q8×C23C2×C411yescharacteristic
24C22×C8×C23—22no—
25C22×C8×C23—22no—
26C2×C4⋊C4×C23—21no—
27C2×C4⋊C4×C23—21no—
28C2×C4⋊C4×C23—21no—
29C2×C4⋊C4×C23C2×C411yescharacteristic
30C2×C4⋊C4×C23C2×C411yescharacteristic
31C2×C4⋊C4×C23C2×C411yescharacteristic
32C2×C42×C23C2×C411yescharacteristic
33C2×C42×C23C2×C411yescharacteristic
34C2×C42×C23D411yescharacteristic
35C2×C42×C23D411yescharacteristic
36C2×C42×C23D411yescharacteristic
37C2×C42×C23D411yescharacteristic
38C2×C42×C23C2311yescharacteristic
39C23×#95—21no—
40C23×#95—21no—
41C23×#95—21no—
42C23×#95—21no—
43C23×#95—21no—
44C23×#95—21no—
45C23×#95—21no—
46C23×#95—21no—
47C23×#95—21no—
48C2×Q8×C23—22no—
49C2×Q8×C23—41no—
50C2×Q8×C23—41no—
51C22×C4×C23—21no—
52C22×C4×C23—21no—
53C22×C4×C23—21no—
54C22×C4×C23—21no—
55C22×C4×C23C22⋊C411yescharacteristic
56C22×C4×C23C22⋊C411yescharacteristic
57C22×C4×C23C22⋊C411yescharacteristic
58C22×C4×C23C22⋊C411yescharacteristic
59C22×C4×C23C22×C411yescharacteristic
60C22×C4×C23C2×D411yescharacteristic
61C22×C4×C23C2×D411yescharacteristic
62C2×C8×C23—42no—
63C2×C8×C23—42no—
64C2×C8×C23—42no—
65C2×C8×C23—42no—
66C2×C8×C23—42no—
67C2×C8×C23—42no—
68C4⋊C4×C23—22no—
69C4⋊C4×C23—41no—
70C4⋊C4×C23—41no—
71C4⋊C4×C23—42no—
72C4⋊C4×C23—42no—
73C4⋊C4×C23—42no—
74C42×C23—22no—
75C42×C23—22no—
76C42×C23—22no—
77C42×C23—41no—
78C42×C23—41no—
79C42×C23—41no—
80C42×C23—41no—
81#7729C2311yesmaximal, characteristic, direct product factor, Sylow
82C23×C23C2×C22⋊C411yescharacteristic
83Q8×C23—82no—
84C2×C4×C23—21no—
85C2×C4×C23—21no—
86C2×C4×C23—21no—
87C2×C4×C23—21no—
88C2×C4×C23—21no—
89C2×C4×C23—21no—
90C2×C4×C23—21no—
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—21no—
100C2×C4×C23—21no—
101C2×C4×C23—21no—
102C2×C4×C23—41no—
103C2×C4×C23—41no—
104C2×C4×C23—41no—
105C2×C4×C23—41no—
106C2×C4×C23—41no—
107C2×C4×C23—41no—
108C2×C4×C23—41no—
109C2×C4×C23—41no—
110C2×C4×C23—41no—
111C2×C4×C23—41no—
112C2×C4×C23—41no—
113C2×C4×C23—41no—
114C2×C4×C23#13112yes—
115C2×C4×C23#13512yes—
116C2×C4×C23#13712yes—
117C8×C23—82no—
118C8×C23—82no—
119#2863C2×C2311yescharacteristic
120#1839C2×C2311yescharacteristic
121#1839C2×C2311yescharacteristic
122#1272C2×C2312yes—
123#1271C2×C2312yes—
124C22×C23#40811yescharacteristic
125C22×C23#41011yescharacteristic
126C22×C23#41311yescharacteristic
127C22×C23#41611yescharacteristic
128C22×C23#41911yescharacteristic
129C22×C23#42111yescharacteristic
130C22×C23#42111yescharacteristic
131C4×C23—22no—
132C4×C23—22no—
133C4×C23—22no—
134C4×C23—41no—
135C4×C23—41no—
136C4×C23—41no—
137C4×C23—41no—
138C4×C23—81no—
139C4×C23—81no—
140C4×C23—81no—
141C4×C23—81no—
142#391C4×C2311yescharacteristic
143#391C4×C2311yescharacteristic
144#391C22×C2311yescharacteristic
145#389C4×C2311yescharacteristic
146#389C4×C2311yescharacteristic
147#389C22×C2311yescharacteristic
148#387—21no—
149#387—21no—
150#387—21no—
151#383—21no—
152C43C22×C2311yescharacteristic
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336C22×C2312yes—
158#336C22×C2312yes—
159C2×C23#147311yescharacteristic
160C2×C23#148211yescharacteristic
161C2×C23#149711yescharacteristic
162C2×C23#151511yescharacteristic
163C2×C23#152111yescharacteristic
164C2×C23#153511yescharacteristic
165C2×C23#155411yescharacteristic
166C22×Q8C2×C4×C2311yescharacteristic
167C22×C8—22no—
168C22×C8—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4C2×C4×C2311yescharacteristic
173C2×C4⋊C4C2×C4×C2311yescharacteristic
174C2×C4⋊C4C2×C4×C2311yescharacteristic
175C2×C42C2×C4×C2311yescharacteristic
176C2×C42C2×C4×C2311yescharacteristic
177C2×C42D4×C2311yescharacteristic
178C2×C42D4×C2311yescharacteristic
179C2×C42D4×C2311yescharacteristic
180C2×C42D4×C2311yescharacteristic
181C2×C42C23×C2311yescharacteristic
182#95—21no—
183#95—21no—
184#95—21no—
185#95—21no—
186#95—21no—
187#95—21no—
188#95—21no—
189#95—21no—
190#95—21no—
191C23#772911yescharacteristic, direct product factor, Sylow
192C2×Q8—22no—
193C2×Q8—41no—
194C2×Q8—41no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4C22⋊C4×C2311yescharacteristic
200C22×C4C22⋊C4×C2311yescharacteristic
201C22×C4C22⋊C4×C2311yescharacteristic
202C22×C4C22⋊C4×C2311yescharacteristic
203C22×C4C22×C4×C2311yescharacteristic
204C22×C4C2×D4×C2311yescharacteristic
205C22×C4C2×D4×C2311yescharacteristic
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C4⋊C4—22no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C42—22no—
219C42—22no—
220C42—22no—
221C42—41no—
222C42—41no—
223C42—41no—
224C42—41no—
225C23C2×C22⋊C4×C2311yescharacteristic
226Q8—82no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
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—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4C23×#13112yes—
258C2×C4C23×#13512yes—
259C2×C4C23×#13712yes—
260C8—82no—
261C8—82no—
262C22C23×#40811yescharacteristic
263C22C23×#41011yescharacteristic
264C22C23×#41311yescharacteristic
265C22C23×#41611yescharacteristic
266C22C23×#41911yescharacteristic
267C22C23×#42111yescharacteristic
268C22C23×#42111yescharacteristic
269C4—22no—
270C4—22no—
271C4—22no—
272C4—41no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—81no—
277C4—81no—
278C4—81no—
279C4—81no—
280C2C23×#147311yescharacteristic
281C2C23×#148211yescharacteristic
282C2C23×#149711yescharacteristic
283C2C23×#151511yescharacteristic
284C2C23×#152111yescharacteristic
285C2C23×#153511yescharacteristic
286C2C23×#155411yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C23C2×C22⋊C4
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C43×C23 ◁ G

Complements

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

NH ≅ G/Ntype
81. #7729C23direct (N × H)
191. C23#7729direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#28632
C23×#18392
C23×#12722
C23×#12712
#772923