Finite Group Databasebeta

← C23×#10347 #19157973

Subgroups — C23×#10347 #19157973

Order 5888.

500 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2646C211yesmaximal, characteristic
2C23×#1824C212yesmaximal
3C23×#1445C211yesmaximal, characteristic
4C23×#1294C212yesmaximal
5C23×#1280C211yesmaximal, characteristic
6C22×C4⋊C4×C23C2211yescharacteristic
7C2×C23×C22⋊C8C2212yes—
8C23×#400—22no—
9C23×#398—22no—
10C23×#385—22no—
11C23×#380C2212yes—
12C2×C23×#95—22no—
13C2×C23×#95C2211yescharacteristic
14C2×C23×#95C2211yescharacteristic
15C23×#340—22no—
16C23×#340—22no—
17C23×C4×C23—21no—
18C23×C4×C23—21no—
19C23×C4×C23D412yes—
20C23×C4×C23C2311yescharacteristic
21C22×C8×C23—22no—
22C2×C4⋊C4×C23—22no—
23C2×C4⋊C4×C23—22no—
24C2×C4⋊C4×C23—21no—
25C2×C4⋊C4×C23—22no—
26C2×C4⋊C4×C23—22no—
27C2×C4⋊C4×C23D412yes—
28C2×C4⋊C4×C23D412yes—
29C2×C4⋊C4×C23Q812yes—
30C2×C22⋊C4×C23—22no—
31C23×C22⋊C8—22no—
32C23×C22⋊C8—22no—
33C23×#95—22no—
34C23×#95—22no—
35C23×#95—22no—
36C23×#95—22no—
37C23×#95—22no—
38C23×#95—44no—
39C24×C23C2×D411yescharacteristic
40C22×C4×C23—22no—
41C22×C4×C23—22no—
42C22×C4×C23—22no—
43C22×C4×C23—22no—
44C22×C4×C23—22no—
45C22×C4×C23—22no—
46C22×C4×C23—22no—
47C22×C4×C23—42no—
48C22×C4×C23—42no—
49C22×C4×C23—42no—
50C22×C4×C23—41no—
51C22×C4×C23—42no—
52C22×C4×C23—42no—
53C22×C4×C23—42no—
54C22×C4×C23—41no—
55C22×C4×C23—41no—
56C22×C4×C23—41no—
57C22×C4×C23—42no—
58C22×C4×C23C2×D412yes—
59C22×C4×C23C2×Q811yescharacteristic
60C22×C4×C23#4112yes—
61C22×C4×C23#4111yescharacteristic
62C2×C8×C23—42no—
63C2×C8×C23—42no—
64C2×C8×C23—42no—
65C4⋊C4×C23—24no—
66C4⋊C4×C23—42no—
67C4⋊C4×C23—42no—
68C4⋊C4×C23—42no—
69C4⋊C4×C23—82no—
70C4⋊C4×C23—82no—
71C22⋊C4×C23—42no—
72C22⋊C4×C23—42no—
73#10347C2311yesmaximal, characteristic, direct product factor, Sylow
74C23×C23—21no—
75C23×C23—21no—
76C23×C23—22no—
77C23×C23#12011yescharacteristic
78C23×C23C22⋊Q811yescharacteristic
79C23×C23#12411yescharacteristic
80C23×C23#13512yes—
81C23×C23#13712yes—
82C2×C4×C23—22no—
83C2×C4×C23—22no—
84C2×C4×C23—41no—
85C2×C4×C23—41no—
86C2×C4×C23—42no—
87C2×C4×C23—44no—
88C2×C4×C23—42no—
89C2×C4×C23—44no—
90C2×C4×C23—44no—
91C2×C4×C23—42no—
92C2×C4×C23—42no—
93C2×C4×C23—42no—
94C2×C4×C23—42no—
95C2×C4×C23—42no—
96C2×C4×C23—42no—
97C2×C4×C23—42no—
98C2×C4×C23—42no—
99C2×C4×C23—42no—
100C2×C4×C23—42no—
101C2×C4×C23—82no—
102C2×C4×C23—81no—
103C2×C4×C23—82no—
104C2×C4×C23—81no—
105C2×C4×C23—81no—
106C2×C4×C23—81no—
107C2×C4×C23—82no—
108C2×C4×C23C22⋊Q812yes—
109C2×C4×C23#12412yes—
110C8×C23—82no—
111#2646C2×C2311yescharacteristic
112#1824C2×C2312yes—
113#1445C2×C2311yescharacteristic
114#1294C2×C2312yes—
115#1280C2×C2311yescharacteristic
116C22×C23—22no—
117C22×C23—22no—
118C22×C23—22no—
119C22×C23—22no—
120C22×C23—21no—
121C22×C23—21no—
122C22×C23—21no—
123C22×C23—21no—
124C22×C23#39311yescharacteristic
125C22×C23#45212yes—
126C22×C23#45311yescharacteristic
127C22×C23#45411yescharacteristic
128C22×C23#45812yes—
129C22×C23#47912yes—
130C22×C23#48811yescharacteristic
131C22×C23#49111yescharacteristic
132C4×C23—22no—
133C4×C23—22no—
134C4×C23—82no—
135C4×C23—82no—
136C4×C23—82no—
137C4×C23—82no—
138C4×C23—82no—
139C22×C4⋊C4C22×C2311yescharacteristic
140C2×C22⋊C8C22×C2312yes—
141#400—22no—
142#398—22no—
143#385—22no—
144#380C22×C2312yes—
145C2×#95—22no—
146C2×#95C22×C2311yescharacteristic
147C2×#95C22×C2311yescharacteristic
148#340—22no—
149#340—22no—
150C2×C23—21no—
151C2×C23—22no—
152C2×C23—21no—
153C2×C23#161512yes—
154C2×C23#161812yes—
155C2×C23#202611yescharacteristic
156C2×C23#202711yescharacteristic
157C2×C23#203011yescharacteristic
158C23×C4—21no—
159C23×C4—21no—
160C23×C4D4×C2312yes—
161C23×C4C23×C2311yescharacteristic
162C22×C8—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4D4×C2312yes—
169C2×C4⋊C4D4×C2312yes—
170C2×C4⋊C4Q8×C2312yes—
171C2×C22⋊C4—22no—
172C22⋊C8—22no—
173C22⋊C8—22no—
174#95—22no—
175#95—22no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—44no—
180C23#1034711yescharacteristic, direct product factor, Sylow
181C24C2×D4×C2311yescharacteristic
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4C2×D4×C2312yes—
201C22×C4C2×Q8×C2311yescharacteristic
202C22×C4#41×C2312yes—
203C22×C4#41×C2311yescharacteristic
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C4⋊C4—24no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—82no—
212C4⋊C4—82no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C23—21no—
216C23—21no—
217C23—22no—
218C23C23×#12011yescharacteristic
219C23C23×C22⋊Q811yescharacteristic
220C23C23×#12411yescharacteristic
221C23C23×#13512yes—
222C23C23×#13712yes—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—81no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—82no—
249C2×C4C23×C22⋊Q812yes—
250C2×C4C23×#12412yes—
251C8—82no—
252C22—21no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—22no—
260C22C23×#39311yescharacteristic
261C22C23×#45212yes—
262C22C23×#45311yescharacteristic
263C22C23×#45411yescharacteristic
264C22C23×#45812yes—
265C22C23×#47912yes—
266C22C23×#48811yescharacteristic
267C22C23×#49111yescharacteristic
268C4—22no—
269C4—22no—
270C4—82no—
271C4—82no—
272C4—82no—
273C4—82no—
274C4—82no—
275C2—21no—
276C2—22no—
277C2—21no—
278C2C23×#161512yes—
279C2C23×#161812yes—
280C2C23×#202611yescharacteristic
281C2C23×#202711yescharacteristic
282C2C23×#203011yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutator, Frattini, cosocleC23×C4C23×C23
prime radicalC24×C23C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
73. #10347C23direct (N × H)
180. C23#10347direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#26462
C23×#18242
C23×#14452
C23×#12942
C23×#12802
#1034723