Finite Group Databasebeta

← C23×#13604 #19161230

Subgroups — C23×#13604 #19161230

Order 5888.

322 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 90 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×#3142C211yesmaximal, characteristic
2C23×#2132C211yesmaximal, characteristic
3C23×#2126C211yesmaximal, characteristic
4C23×#1414C214yesmaximal, complemented
5C23×#532C2211yescharacteristic
6C2×C23×#125C411yescharacteristic, complemented
7C23×#491—21no—
8C23×#490—21no—
9C23×#488C412yescomplemented
10C23×#488C2212yes—
11C23×#417—21no—
12C23×#417—21no—
13C23×#413—21no—
14C23×#410—21no—
15C4×M16×C23C411yescharacteristic, complemented
16C23×#356—21no—
17C23×#355C2212yes—
18C23×#354C2212yes—
19C23×#353—21no—
20C2×#41×C23D411yescharacteristic
21C2×#41×C23D411yescharacteristic
22C2×M16×C23—21no—
23C2×M16×C23—41no—
24C23×C4⋊Q8D411yescharacteristic
25C23×#127D411yescharacteristic
26C23×#125—21no—
27C23×#125—41no—
28C23×#125C2×C412yes—
29C23×#124C2×C411yescharacteristic
30C23×#124C2311yescharacteristic
31C23×#121—41no—
32C4×D4×C23—41no—
33C23×#117—41no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23—41no—
36C2×C42×C23C2×C411yescharacteristic
37C23×#103—22no—
38C23×#103—22no—
39C23×#102—22no—
40C23×#102—22no—
41C23×#101—22no—
42C23×#100—41no—
43C23×#99—22no—
44C23×#99—41no—
45C23×#97—21no—
46C4×C8×C23C2×C412yes—
47#41×C23—41no—
48#41×C23—41no—
49#41×C23—41no—
50#41×C23—41no—
51C2×Q8×C23C22⋊C411yescharacteristic
52C2×Q8×C23C2×D411yescharacteristic
53C2×D4×C23—21no—
54C2×D4×C23—21no—
55C2×D4×C23—41no—
56C2×D4×C23C22⋊C411yescharacteristic
57C2×D4×C23C2×D411yescharacteristic
58C22×C4×C23—21no—
59C22×C4×C23—21no—
60C22×C4×C23—21no—
61C22×C4×C23—41no—
62C22×C4×C23C22⋊C411yescharacteristic
63M16×C23—41no—
64M16×C23—41no—
65M16×C23—42no—
66M16×C23—81no—
67C2×C8×C23—22no—
68C2×C8×C23—81no—
69C4⋊C4×C23—22no—
70C4⋊C4×C23—41no—
71C4⋊C4×C23—42no—
72C4⋊C4×C23—42no—
73C4⋊C4×C23—41no—
74C4⋊C4×C23—81no—
75C22⋊C4×C23—41no—
76C22⋊C4×C23—41no—
77C22⋊C4×C23—42no—
78C42×C23—21no—
79C42×C23—81no—
80C42×C23C22⋊C411yescharacteristic
81C42×C23C22×C411yescharacteristic
82#13604C2311yesmaximal, characteristic, direct product factor, Sylow
83C23×C23—21no—
84C23×C23—21no—
85C23×C23#9911yescharacteristic
86Q8×C23—41no—
87Q8×C23—41no—
88D4×C23—41no—
89D4×C23—41no—
90D4×C23—81no—
91D4×C23—81no—
92D4×C23—82no—
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—41no—
100C2×C4×C23—41no—
101C2×C4×C23—41no—
102C2×C4×C23—41no—
103C2×C4×C23—42no—
104C2×C4×C23—81no—
105C2×C4×C23—82no—
106C2×C4×C23—82no—
107C2×C4×C23#9911yescharacteristic
108C2×C4×C23C2×C22⋊C411yescharacteristic
109C8×C23—42no—
110C8×C23—82no—
111#3142C2×C2311yescharacteristic
112#2132C2×C2311yescharacteristic
113#2126C2×C2311yescharacteristic
114#1414C2×C2314yescomplemented
115C22×C23—21no—
116C22×C23—41no—
117C22×C23—41no—
118C22×C23—41no—
119C22×C23—81no—
120C22×C23—81no—
121C22×C23C2×#9911yescharacteristic
122C4×C23—21no—
123C4×C23—41no—
124C4×C23—41no—
125C4×C23—41no—
126C4×C23—41no—
127C4×C23—82no—
128C4×C23—162no—
129C4×C23#35411yescharacteristic
130C4×C23#35411yescharacteristic
131#532C22×C2311yescharacteristic
132C2×#125C4×C2311yescharacteristic, complemented
133#491—21no—
134#490—21no—
135#488C4×C2312yescomplemented
136#488C22×C2312yes—
137#417—21no—
138#417—21no—
139#413—21no—
140#410—21no—
141C4×M16C4×C2311yescharacteristic, complemented
142#356—21no—
143#355C22×C2312yes—
144#354C22×C2312yes—
145#353—21no—
146C2×C23—21no—
147C2×C23—41no—
148C2×C23—81no—
149C2×C23—81no—
150C2×C23C2×#35411yescharacteristic
151C2×#41D4×C2311yescharacteristic
152C2×#41D4×C2311yescharacteristic
153C2×M16—21no—
154C2×M16—41no—
155C4⋊Q8D4×C2311yescharacteristic
156#127D4×C2311yescharacteristic
157#125—21no—
158#125—41no—
159#125C2×C4×C2312yes—
160#124C2×C4×C2311yescharacteristic
161#124C23×C2311yescharacteristic
162#121—41no—
163C4×D4—41no—
164#117—41no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—41no—
167C2×C42C2×C4×C2311yescharacteristic
168#103—22no—
169#103—22no—
170#102—22no—
171#102—22no—
172#101—22no—
173#100—41no—
174#99—22no—
175#99—41no—
176#97—21no—
177C4×C8C2×C4×C2312yes—
178C23#1360411yescharacteristic, direct product factor, Sylow
179#41—41no—
180#41—41no—
181#41—41no—
182#41—41no—
183C2×Q8C22⋊C4×C2311yescharacteristic
184C2×Q8C2×D4×C2311yescharacteristic
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—41no—
188C2×D4C22⋊C4×C2311yescharacteristic
189C2×D4C2×D4×C2311yescharacteristic
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—41no—
194C22×C4C22⋊C4×C2311yescharacteristic
195M16—41no—
196M16—41no—
197M16—42no—
198M16—81no—
199C2×C8—22no—
200C2×C8—81no—
201C4⋊C4—22no—
202C4⋊C4—41no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—41no—
206C4⋊C4—81no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—42no—
210C42—21no—
211C42—81no—
212C42C22⋊C4×C2311yescharacteristic
213C42C22×C4×C2311yescharacteristic
214C23—21no—
215C23—21no—
216C23C23×#9911yescharacteristic
217Q8—41no—
218Q8—41no—
219D4—41no—
220D4—41no—
221D4—81no—
222D4—81no—
223D4—82no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—81no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4C23×#9911yescharacteristic
239C2×C4C2×C22⋊C4×C2311yescharacteristic
240C8—42no—
241C8—82no—
242C22—21no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—81no—
247C22—81no—
248C22C2×C23×#9911yescharacteristic
249C4—21no—
250C4—41no—
251C4—41no—
252C4—41no—
253C4—41no—
254C4—82no—
255C4—162no—
256C4C23×#35411yescharacteristic
257C4C23×#35411yescharacteristic
258C2—21no—
259C2—41no—
260C2—81no—
261C2—81no—
262C2C2×C23×#35411yescharacteristic
subgroupG/H
center, socleC2×C23C2×#354
commutatorC42C22×C4×C23
Frattini, cosocle#124C23×C23
prime radicalC23×#532C22
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C2×C4×C23 ◁ C22×C4×C23 ◁ C23×#532 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×#1414C2semidirect
6. C2×C23×#125C4semidirect
9. C23×#488C4semidirect
15. C4×M16×C23C4semidirect
82. #13604C23direct (N × H)
114. #1414C2×C23semidirect
132. C2×#125C4×C23semidirect
135. #488C4×C23semidirect
141. C4×M16C4×C23semidirect
178. C23#13604direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31422
C23×#21322
C23×#21262
C23×#14142
#1360423