Finite Group Databasebeta

← C23×#10566 #19158192

Subgroups — C23×#10566 #19158192

Order 5888.

490 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 114 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×#3189C211yesmaximal, characteristic
2C23×#1996C212yesmaximal
3C23×#1892C212yesmaximal
4C23×#1850C211yesmaximal, characteristic
5C23×#1289C211yesmaximal, characteristic
6C23×C8×C23C2211yescharacteristic
7C2×C23×C22⋊Q8C2212yes—
8C23×#484—22no—
9C23×#467—21no—
10C23×#467—21no—
11C23×#451—22no—
12C2×C23×C4⋊C8—21no—
13C2×C23×#103—22no—
14C2×C23×C22⋊C8C2211yescharacteristic
15C23×#380C2212yes—
16C4×C22⋊C4×C23C2211yescharacteristic
17C23×#342—22no—
18C23×#342—22no—
19C23×#340—22no—
20C23×#336—21no—
21C22×Q8×C23D412yes—
22C23×C4×C23C2311yescharacteristic
23C2×Q16×C23—21no—
24C2×Q16×C23—21no—
25C22×C8×C23—21no—
26C22×C8×C23—22no—
27C22×C8×C23—42no—
28C22×C8×C23D411yescharacteristic
29C22×C8×C23D411yescharacteristic
30C23×C22⋊Q8—22no—
31C23×C22⋊Q8—22no—
32C23×C22⋊Q8—22no—
33C23×C22⋊Q8—22no—
34C2×C4⋊C4×C23—22no—
35C2×C4⋊C4×C23—22no—
36C2×C4⋊C4×C23D412yes—
37C2×C22⋊C4×C23—22no—
38C2×C22⋊C4×C23—22no—
39C2×C22⋊C4×C23D411yescharacteristic
40C2×C22⋊C4×C23D411yescharacteristic
41C2×C42×C23—21no—
42C23×#107—21no—
43C23×#107—21no—
44C23×C4⋊C8—42no—
45C23×#103—22no—
46C23×#103—22no—
47C23×#103—44no—
48C23×C22⋊C8—22no—
49C23×#95—21no—
50C23×#95—22no—
51C24×C23C2×D411yescharacteristic
52C2×Q8×C23—22no—
53C2×Q8×C23—22no—
54C2×Q8×C23—42no—
55C2×Q8×C23—42no—
56C22×C4×C23—21no—
57C22×C4×C23—21no—
58C22×C4×C23—22no—
59C22×C4×C23—22no—
60C22×C4×C23—41no—
61C22×C4×C23—41no—
62C22×C4×C23—42no—
63C22×C4×C23C2×D412yes—
64C22×C4×C23C2×D411yescharacteristic
65C22×C4×C23#4111yescharacteristic
66C22×C4×C23#4112yes—
67Q16×C23—81no—
68Q16×C23—81no—
69C2×C8×C23—24no—
70C2×C8×C23—22no—
71C2×C8×C23—21no—
72C2×C8×C23—21no—
73C2×C8×C23—41no—
74C2×C8×C23—42no—
75C2×C8×C23—41no—
76C2×C8×C23—41no—
77C2×C8×C23—41no—
78C4⋊C4×C23—22no—
79C4⋊C4×C23—22no—
80C4⋊C4×C23—42no—
81C4⋊C4×C23—42no—
82C4⋊C4×C23—42no—
83C4⋊C4×C23—42no—
84C22⋊C4×C23—21no—
85C22⋊C4×C23—22no—
86C22⋊C4×C23—42no—
87C22⋊C4×C23—42no—
88C22⋊C4×C23—42no—
89C22⋊C4×C23—42no—
90C22⋊C4×C23Q1612yescomplemented
91C42×C23—41no—
92C42×C23—41no—
93#10566C2311yesmaximal, characteristic, direct product factor, Sylow
94C23×C23—41no—
95C23×C23—41no—
96C23×C23#12112yes—
97C23×C23#12111yescharacteristic
98C23×C23C2×Q1611yescharacteristic
99C23×C23#13511yescharacteristic
100C23×C23#13712yes—
101Q8×C23—44no—
102Q8×C23—42no—
103Q8×C23—42no—
104C2×C4×C23—21no—
105C2×C4×C23—22no—
106C2×C4×C23—21no—
107C2×C4×C23—22no—
108C2×C4×C23—22no—
109C2×C4×C23—41no—
110C2×C4×C23—42no—
111C2×C4×C23—41no—
112C2×C4×C23—41no—
113C2×C4×C23—41no—
114C2×C4×C23—41no—
115C2×C4×C23—42no—
116C2×C4×C23—42no—
117C2×C4×C23—41no—
118C2×C4×C23—82no—
119C2×C4×C23—82no—
120C2×C4×C23—82no—
121C2×C4×C23—82no—
122C2×C4×C23—82no—
123C2×C4×C23#12011yescharacteristic
124C2×C4×C23#12312yes—
125C2×C4×C23#12411yescharacteristic
126C8×C23—42no—
127C8×C23—41no—
128C8×C23—41no—
129#3189C2×C2311yescharacteristic
130#1996C2×C2312yes—
131#1892C2×C2312yes—
132#1850C2×C2311yescharacteristic
133#1289C2×C2311yescharacteristic
134C22×C23—21no—
135C22×C23—22no—
136C22×C23—21no—
137C22×C23—22no—
138C22×C23—41no—
139C22×C23—41no—
140C22×C23—42no—
141C22×C23#39411yescharacteristic
142C22×C23#46212yes—
143C22×C23#46412yes—
144C22×C23#46711yescharacteristic
145C22×C23#47011yescharacteristic
146C4×C23—22no—
147C4×C23—21no—
148C4×C23—42no—
149C4×C23—81no—
150C4×C23—82no—
151C4×C23—82no—
152C4×C23—162no—
153C4×C23B(4,𝔽2)11yescharacteristic
154C4×C23#45811yescharacteristic
155C23×C8C22×C2311yescharacteristic
156C2×C22⋊Q8C22×C2312yes—
157#484—22no—
158#467—21no—
159#467—21no—
160#451—22no—
161C2×C4⋊C8—21no—
162C2×#103—22no—
163C2×C22⋊C8C22×C2311yescharacteristic
164#380C22×C2312yes—
165C4×C22⋊C4C22×C2311yescharacteristic
166#342—22no—
167#342—22no—
168#340—22no—
169#336—21no—
170C2×C23—21no—
171C2×C23—21no—
172C2×C23—22no—
173C2×C23—22no—
174C2×C23#204411yescharacteristic
175C2×C23#205311yescharacteristic
176C2×C23#205611yescharacteristic
177C22×Q8D4×C2312yes—
178C23×C4C23×C2311yescharacteristic
179C2×Q16—21no—
180C2×Q16—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—42no—
184C22×C8D4×C2311yescharacteristic
185C22×C8D4×C2311yescharacteristic
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4D4×C2312yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4D4×C2311yescharacteristic
196C2×C22⋊C4D4×C2311yescharacteristic
197C2×C42—21no—
198#107—21no—
199#107—21no—
200C4⋊C8—42no—
201#103—22no—
202#103—22no—
203#103—44no—
204C22⋊C8—22no—
205#95—21no—
206#95—22no—
207C23#1056611yescharacteristic, direct product factor, Sylow
208C24C2×D4×C2311yescharacteristic
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—42no—
212C2×Q8—42no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4C2×D4×C2312yes—
221C22×C4C2×D4×C2311yescharacteristic
222C22×C4#41×C2311yescharacteristic
223C22×C4#41×C2312yes—
224Q16—81no—
225Q16—81no—
226C2×C8—24no—
227C2×C8—22no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—41no—
231C2×C8—42no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C22⋊C4—21no—
242C22⋊C4—22no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4Q16×C2312yescomplemented
248C42—41no—
249C42—41no—
250C23—41no—
251C23—41no—
252C23C23×#12112yes—
253C23C23×#12111yescharacteristic
254C23C2×Q16×C2311yescharacteristic
255C23C23×#13511yescharacteristic
256C23C23×#13712yes—
257Q8—44no—
258Q8—42no—
259Q8—42no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4C23×#12011yescharacteristic
280C2×C4C23×#12312yes—
281C2×C4C23×#12411yescharacteristic
282C8—42no—
283C8—41no—
284C8—41no—
285C22—21no—
286C22—22no—
287C22—21no—
288C22—22no—
289C22—41no—
290C22—41no—
291C22—42no—
292C22C23×#39411yescharacteristic
293C22C23×#46212yes—
294C22C23×#46412yes—
295C22C23×#46711yescharacteristic
296C22C23×#47011yescharacteristic
297C4—22no—
298C4—21no—
299C4—42no—
300C4—81no—
301C4—82no—
302C4—82no—
303C4—162no—
304C4C23×B(4,𝔽2)11yescharacteristic
305C4C23×#45811yescharacteristic
306C2—21no—
307C2—21no—
308C2—22no—
309C2—22no—
310C2C23×#204411yescharacteristic
311C2C23×#205311yescharacteristic
312C2C23×#205611yescharacteristic
subgroupG/H
center, socleC22×C23#394
commutator, Frattini, cosocleC23×C4C23×C23
prime radicalC24×C23C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
90. C22⋊C4×C23Q16semidirect
93. #10566C23direct (N × H)
207. C23#10566direct (N × H)
247. C22⋊C4Q16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#31892
C23×#19962
C23×#18922
C23×#18502
C23×#12892
#1056623