Finite Group Databasebeta

← C23×#10258 #19157884

Subgroups — C23×#10258 #19157884

Order 5888.

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

normal subgroups only

This group has 526 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×#3249C211yesmaximal, characteristic, complemented
2C23×#3077C212yesmaximal
3C23×#1954C211yesmaximal, characteristic
4C23×#1493C212yesmaximal, complemented
5C23×#1292C211yesmaximal, characteristic, complemented
6C23×#533C2212yes—
7C2×C4×D4×C23C2211yescharacteristic
8C23×#478—24no—
9C23×#477—22no—
10C23×#476—22no—
11C23×#474—22no—
12C23×#467—22no—
13C23×#462—22no—
14C23×#428—22no—
15C23×#423C2212yes—
16C2×C23×C4⋊C8C2211yescharacteristic
17C23×#418—22no—
18C2×C23×C22⋊C8—22no—
19C2×C4×C8×C23C2211yescharacteristic
20C23×Q8⋊C8—24no—
21C22×D4×C23D411yescharacteristic
22C23×C4×C23—21no—
23C2×Q16×C23—22no—
24C2×M16×C23—22no—
25C22×C8×C23—21no—
26C22×C8×C23D412yes—
27C23×C4⋊Q8D412yes—
28C23×C4⋊Q8D412yes—
29C23×#125—22no—
30C23×C22⋊Q8—22no—
31C23×C22⋊Q8—22no—
32C4×Q8×C23—22no—
33C4×Q8×C23—22no—
34C4×D4×C23—22no—
35C4×D4×C23—22no—
36C23×#117—22no—
37C2×C4⋊C4×C23D411yescharacteristic
38C2×C22⋊C4×C23—21no—
39C2×C42×C23C2311yescharacteristic
40C23×#107—24no—
41C23×#107—22no—
42C23×#106—24no—
43C23×C4⋊C8—22no—
44C23×C4⋊C8—22no—
45C23×C4⋊C8D414yes—
46C23×#103—24no—
47C23×#102—24no—
48C23×#102—24no—
49C23×C22⋊C8—44no—
50C4×C8×C23—22no—
51C24×C23—21no—
52C2×Q8×C23—22no—
53C2×Q8×C23—22no—
54C2×D4×C23—22no—
55C2×D4×C23—21no—
56C2×D4×C23—22no—
57C2×D4×C23Q1612yescomplemented
58C22×C4×C23—21no—
59C22×C4×C23—42no—
60C22×C4×C23—42no—
61C22×C4×C23—41no—
62C22×C4×C23—41no—
63C22×C4×C23C2×D411yescharacteristic
64C22×C4×C23C2×D411yescharacteristic
65C22×C4×C23#4111yescharacteristic
66Q16×C23—82no—
67M16×C23—42no—
68M16×C23—42no—
69C2×C8×C23—22no—
70C2×C8×C23—24no—
71C2×C8×C23—22no—
72C2×C8×C23—41no—
73C2×C8×C23—41no—
74C2×C8×C23—42no—
75C4⋊C4×C23—22no—
76C4⋊C4×C23—22no—
77C4⋊C4×C23—22no—
78C4⋊C4×C23—22no—
79C4⋊C4×C23—22no—
80C4⋊C4×C23—42no—
81C4⋊C4×C23—42no—
82C22⋊C4×C23—42no—
83C22⋊C4×C23—42no—
84C42×C23—42no—
85C42×C23C2×D412yes—
86C42×C23C2×D412yes—
87#10258C2311yesmaximal, characteristic, direct product factor, Sylow
88C23×C23—22no—
89C23×C23—42no—
90C23×C23—42no—
91C23×C23—41no—
92C23×C23—41no—
93C23×C23#12111yescharacteristic
94Q8×C23—44no—
95Q8×C23—42no—
96Q8×C23—42no—
97D4×C23—24no—
98D4×C23—42no—
99C2×C4×C23—21no—
100C2×C4×C23—21no—
101C2×C4×C23—41no—
102C2×C4×C23—42no—
103C2×C4×C23—42no—
104C2×C4×C23—42no—
105C2×C4×C23—41no—
106C2×C4×C23—82no—
107C2×C4×C23—81no—
108C2×C4×C23—81no—
109C2×C4×C23#12012yes—
110C2×C4×C23#12011yescharacteristic
111C2×C4×C23#12112yes—
112C2×C4×C23#12711yescharacteristic
113C2×C4×C23C2×Q1611yescharacteristic
114C2×C4×C23#13511yescharacteristic
115C2×C4×C23#13712yes—
116C8×C23—42no—
117C8×C23—44no—
118C8×C23—42no—
119#3249C2×C2311yescharacteristic, complemented
120#3077C2×C2312yes—
121#1954C2×C2311yescharacteristic
122#1493C2×C2312yescomplemented
123#1292C2×C2311yescharacteristic, complemented
124C22×C23—21no—
125C22×C23—21no—
126C22×C23—44no—
127C22×C23—42no—
128C22×C23—82no—
129C22×C23—81no—
130C22×C23—81no—
131C22×C23#39211yescharacteristic
132C22×C23#46711yescharacteristic
133C22×C23#47011yescharacteristic
134C4×C23—21no—
135C4×C23—22no—
136C4×C23—81no—
137C4×C23—82no—
138C4×C23—82no—
139C4×C23#45112yes—
140C4×C23#45212yes—
141C4×C23#45311yescharacteristic
142C4×C23#45411yescharacteristic
143#533C22×C2312yes—
144C2×C4×D4C22×C2311yescharacteristic
145#478—24no—
146#477—22no—
147#476—22no—
148#474—22no—
149#467—22no—
150#462—22no—
151#428—22no—
152#423C22×C2312yes—
153C2×C4⋊C8C22×C2311yescharacteristic
154#418—22no—
155C2×C22⋊C8—22no—
156C2×C4×C8C22×C2311yescharacteristic
157Q8⋊C8—24no—
158C2×C23—21no—
159C2×C23—21no—
160C2×C23—82no—
161C2×C23#200111yescharacteristic
162C2×C23#200611yescharacteristic
163C2×C23#201711yescharacteristic
164C22×D4D4×C2311yescharacteristic
165C23×C4—21no—
166C2×Q16—22no—
167C2×M16—22no—
168C22×C8—21no—
169C22×C8D4×C2312yes—
170C4⋊Q8D4×C2312yes—
171C4⋊Q8D4×C2312yes—
172#125—22no—
173C22⋊Q8—22no—
174C22⋊Q8—22no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×D4—22no—
178C4×D4—22no—
179#117—22no—
180C2×C4⋊C4D4×C2311yescharacteristic
181C2×C22⋊C4—21no—
182C2×C42C23×C2311yescharacteristic
183#107—24no—
184#107—22no—
185#106—24no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8D4×C2314yes—
189#103—24no—
190#102—24no—
191#102—24no—
192C22⋊C8—44no—
193C4×C8—22no—
194C23#1025811yescharacteristic, direct product factor, Sylow
195C24—21no—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4Q16×C2312yescomplemented
202C22×C4—21no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4C2×D4×C2311yescharacteristic
208C22×C4C2×D4×C2311yescharacteristic
209C22×C4#41×C2311yescharacteristic
210Q16—82no—
211M16—42no—
212M16—42no—
213C2×C8—22no—
214C2×C8—24no—
215C2×C8—22no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—42no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C42—42no—
229C42C2×D4×C2312yes—
230C42C2×D4×C2312yes—
231C23—22no—
232C23—42no—
233C23—42no—
234C23—41no—
235C23—41no—
236C23C23×#12111yescharacteristic
237Q8—44no—
238Q8—42no—
239Q8—42no—
240D4—24no—
241D4—42no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4C23×#12012yes—
253C2×C4C23×#12011yescharacteristic
254C2×C4C23×#12112yes—
255C2×C4C23×#12711yescharacteristic
256C2×C4C2×Q16×C2311yescharacteristic
257C2×C4C23×#13511yescharacteristic
258C2×C4C23×#13712yes—
259C8—42no—
260C8—44no—
261C8—42no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—42no—
266C22—82no—
267C22—81no—
268C22—81no—
269C22C23×#39211yescharacteristic
270C22C23×#46711yescharacteristic
271C22C23×#47011yescharacteristic
272C4—21no—
273C4—22no—
274C4—81no—
275C4—82no—
276C4—82no—
277C4C23×#45112yes—
278C4C23×#45212yes—
279C4C23×#45311yescharacteristic
280C4C23×#45411yescharacteristic
281C2—21no—
282C2—21no—
283C2—82no—
284C2C23×#200111yescharacteristic
285C2C23×#200611yescharacteristic
286C2C23×#201711yescharacteristic
subgroupG/H
center, socleC22×C23#392
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC22×D4×C23D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3249C2semidirect
4. C23×#1493C2semidirect
5. C23×#1292C2semidirect
57. C2×D4×C23Q16semidirect
87. #10258C23direct (N × H)
119. #3249C2×C23semidirect
122. #1493C2×C23semidirect
123. #1292C2×C23semidirect
194. C23#10258direct (N × H)
201. C2×D4Q16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#32492
C23×#30772
C23×#19542
C23×#14932
C23×#12922
#1025823