Finite Group Databasebeta

← C23×#11818 #19159444

Subgroups — C23×#11818 #19159444

Order 5888.

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

normal subgroups only

This group has 368 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×#2628C211yesmaximal, characteristic
2C23×#1822C211yesmaximal, characteristic
3C23×#1464C211yesmaximal, characteristic
4C23×#1353C214yesmaximal, complemented
5C2×C23×C4⋊Q8C411yescharacteristic
6C2×C23×#117C2211yescharacteristic
7C23×#432—21no—
8C23×#407—21no—
9C23×#400C2212yes—
10C23×#399C412yes—
11C23×#395—21no—
12C23×#393—21no—
13C23×#389—21no—
14C23×#382C411yescharacteristic
15C23×#380—21no—
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×#117—41no—
26C23×#117—41no—
27C2×C4⋊C4×C23—21no—
28C2×C4⋊C4×C23—21no—
29C2×C4⋊C4×C23—22no—
30C2×C4⋊C4×C23C2×C412yes—
31C2×C4⋊C4×C23D411yescharacteristic
32C2×C4⋊C4×C23C2311yescharacteristic
33C2×C22⋊C4×C23—22no—
34C2×C22⋊C4×C23C2×C411yescharacteristic
35C2×C22⋊C4×C23D411yescharacteristic
36C2×C42×C23C2×C411yescharacteristic
37C2×C42×C23D411yescharacteristic
38C23×C4⋊C8—41no—
39C23×C22⋊C8—22no—
40C23×C22⋊C8—41no—
41C23×C22⋊C8—41no—
42C23×#97—41no—
43C23×#95—21no—
44C23×#95—22no—
45C23×#95—22no—
46C23×#95C2×C412yes—
47C24×C23C22⋊C411yescharacteristic
48C2×Q8×C23—42no—
49C2×Q8×C23—82no—
50C22×C4×C23—21no—
51C22×C4×C23—21no—
52C22×C4×C23—22no—
53C22×C4×C23—22no—
54C22×C4×C23—22no—
55C22×C4×C23—41no—
56C22×C4×C23C22⋊C411yescharacteristic
57C22×C4×C23C22⋊C411yescharacteristic
58C22×C4×C23C22⋊C411yescharacteristic
59C22×C4×C23C22×C411yescharacteristic
60C22×C4×C23C2×D411yescharacteristic
61C22×C4×C23C2×D411yescharacteristic
62M16×C23—42no—
63M16×C23—42no—
64C2×C8×C23—42no—
65C2×C8×C23—42no—
66C4⋊C4×C23—41no—
67C4⋊C4×C23—41no—
68C4⋊C4×C23—42no—
69C4⋊C4×C23—42no—
70C4⋊C4×C23—41no—
71C4⋊C4×C23—41no—
72C4⋊C4×C23—82no—
73C22⋊C4×C23—41no—
74C22⋊C4×C23—41no—
75C22⋊C4×C23—82no—
76C42×C23—21no—
77C42×C23—21no—
78C42×C23—21no—
79C42×C23—21no—
80#11818C2311yesmaximal, characteristic, direct product factor, Sylow
81C23×C23—21no—
82C23×C23—21no—
83C23×C23—21no—
84C23×C23—41no—
85C23×C23—41no—
86C23×C23C2×C22⋊C411yescharacteristic
87Q8×C23—84no—
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—42no—
97C2×C4×C23—42no—
98C2×C4×C23—41no—
99C2×C4×C23—41no—
100C2×C4×C23—41no—
101C2×C4×C23—41no—
102C2×C4×C23—42no—
103C2×C4×C23—42no—
104C2×C4×C23—42no—
105C2×C4×C23—42no—
106C2×C4×C23—42no—
107C2×C4×C23—42no—
108C2×C4×C23#9911yescharacteristic
109C2×C4×C23#9911yescharacteristic
110C2×C4×C23#13712yes—
111C8×C23—82no—
112C8×C23—82no—
113#2628C2×C2311yescharacteristic
114#1822C2×C2311yescharacteristic
115#1464C2×C2311yescharacteristic
116#1353C2×C2314yescomplemented
117C22×C23—21no—
118C22×C23—21no—
119C22×C23—42no—
120C22×C23—41no—
121C22×C23—41no—
122C22×C23—41no—
123C22×C23—81no—
124C22×C23C2×#9911yescharacteristic
125C22×C23#41911yescharacteristic
126C22×C23#42111yescharacteristic
127C4×C23—22no—
128C4×C23—21no—
129C4×C23—21no—
130C4×C23—41no—
131C4×C23—41no—
132C4×C23—41no—
133C4×C23—41no—
134C4×C23—82no—
135C4×C23—82no—
136C2×C4⋊Q8C4×C2311yescharacteristic
137C2×#117C22×C2311yescharacteristic
138#432—21no—
139#407—21no—
140#400C22×C2312yes—
141#399C4×C2312yes—
142#395—21no—
143#393—21no—
144#389—21no—
145#382C4×C2311yescharacteristic
146#380—21no—
147#343—21no—
148#343C22×C2312yes—
149#324C22×C2312yes—
150#323—21no—
151C2×C23—21no—
152C2×C23—21no—
153C2×C23—81no—
154C2×C23#151111yescharacteristic
155C2×C23#212311yescharacteristic
156C2×C23#213411yescharacteristic
157C22×Q8—21no—
158C23×C4D4×C2311yescharacteristic
159C2×M16—22no—
160C4⋊Q8—41no—
161C4⋊Q8—41no—
162#117—41no—
163#117—41no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4C2×C4×C2312yes—
168C2×C4⋊C4D4×C2311yescharacteristic
169C2×C4⋊C4C23×C2311yescharacteristic
170C2×C22⋊C4—22no—
171C2×C22⋊C4C2×C4×C2311yescharacteristic
172C2×C22⋊C4D4×C2311yescharacteristic
173C2×C42C2×C4×C2311yescharacteristic
174C2×C42D4×C2311yescharacteristic
175C4⋊C8—41no—
176C22⋊C8—22no—
177C22⋊C8—41no—
178C22⋊C8—41no—
179#97—41no—
180#95—21no—
181#95—22no—
182#95—22no—
183#95C2×C4×C2312yes—
184C23#1181811yescharacteristic, direct product factor, Sylow
185C24C22⋊C4×C2311yescharacteristic
186C2×Q8—42no—
187C2×Q8—82no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—41no—
194C22×C4C22⋊C4×C2311yescharacteristic
195C22×C4C22⋊C4×C2311yescharacteristic
196C22×C4C22⋊C4×C2311yescharacteristic
197C22×C4C22×C4×C2311yescharacteristic
198C22×C4C2×D4×C2311yescharacteristic
199C22×C4C2×D4×C2311yescharacteristic
200M16—42no—
201M16—42no—
202C2×C8—42no—
203C2×C8—42no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—82no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—82no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—41no—
222C23—41no—
223C23C2×C22⋊C4×C2311yescharacteristic
224Q8—84no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4C23×#9911yescharacteristic
246C2×C4C23×#9911yescharacteristic
247C2×C4C23×#13712yes—
248C8—82no—
249C8—82no—
250C22—21no—
251C22—21no—
252C22—42no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—81no—
257C22C2×C23×#9911yescharacteristic
258C22C23×#41911yescharacteristic
259C22C23×#42111yescharacteristic
260C4—22no—
261C4—21no—
262C4—21no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—41no—
267C4—82no—
268C4—82no—
269C2—21no—
270C2—21no—
271C2—81no—
272C2C23×#151111yescharacteristic
273C2C23×#212311yescharacteristic
274C2C23×#213411yescharacteristic
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×#1353C2semidirect
80. #11818C23direct (N × H)
116. #1353C2×C23semidirect
184. C23#11818direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#26282
C23×#18222
C23×#14642
C23×#13532
#1181823