Finite Group Databasebeta

← C23×#13676 #19161302

Subgroups — C23×#13676 #19161302

Order 5888.

394 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 58 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×#3291C211yesmaximal, characteristic
2C23×#2198C212yesmaximal, complemented
3C23×#1630C211yesmaximal, characteristic, complemented
4C23×#1409C211yesmaximal, characteristic, complemented
5C23×#1402C212yesmaximal, complemented
6C23×#577—22no—
7C23×#559C2211yescharacteristic
8C23×#492C2211yescharacteristic
9C23×#477—21no—
10C23×#471—22no—
11C23×#458—22no—
12C23×#456—21no—
13C23×#453C2212yes—
14C23×#443—21no—
15C23×#353C2212yes—
16C23×#352—22no—
17C23×#331C2211yescharacteristic
18C23×Q8⋊C8—21no—
19C23×2−1+4—21no—
20C23×H(5,𝔽2)D412yes—
21C23×#137—22no—
22C23×#137—42no—
23C23×AGL(1,ℤ/8)—22no—
24C23×AGL(1,ℤ/8)—42no—
25C23×#135—21no—
26C23×#135—42no—
27C2×SD16×C23—22no—
28C2×SD16×C23—42no—
29C23×#131—22no—
30C23×C4⋊Q8D411yescharacteristic
31C23×#127C2311yescharacteristic
32C23×#124—41no—
33C23×#123—22no—
34C23×#123—42no—
35C4×Q8×C23—21no—
36C4×D4×C23—41no—
37C23×#108—21no—
38C23×#107—41no—
39C23×C4⋊C8—21no—
40C23×#104—22no—
41C23×#103—41no—
42C23×#101—21no—
43C23×#100D412yes—
44C23×#99—22no—
45C23×#99—42no—
46C4×C8×C23D411yescharacteristic
47#41×C23—22no—
48#41×C23—42no—
49#41×C23—42no—
50#41×C23—81no—
51C2×Q8×C23—21no—
52C2×Q8×C23—21no—
53C2×Q8×C23—42no—
54C2×D4×C23—41no—
55C2×D4×C23—42no—
56C2×D4×C23—42no—
57C2×D4×C23C2×D412yes—
58C22×C4×C23—22no—
59Q16×C23—21no—
60Q16×C23—42no—
61SD16×C23—22no—
62SD16×C23—42no—
63SD16×C23—42no—
64SD16×C23—42no—
65SD16×C23—42no—
66D8×C23—21no—
67D8×C23—42no—
68M16×C23—22no—
69M16×C23—42no—
70C2×C8×C23—21no—
71C2×C8×C23—42no—
72C2×C8×C23—41no—
73C4⋊C4×C23—21no—
74C4⋊C4×C23—41no—
75C4⋊C4×C23—41no—
76C4⋊C4×C23—82no—
77C22⋊C4×C23—22no—
78C22⋊C4×C23—42no—
79C22⋊C4×C23—42no—
80C22⋊C4×C23—82no—
81C42×C23—41no—
82C42×C23C2×D411yescharacteristic
83#13676C2311yesmaximal, characteristic, direct product factor, Sylow
84C23×C23—22no—
85C23×C23—42no—
86Q8×C23—22no—
87Q8×C23—42no—
88Q8×C23—41no—
89Q8×C23—41no—
90Q8×C23—81no—
91D4×C23—22no—
92D4×C23—42no—
93D4×C23—42no—
94D4×C23—81no—
95D4×C23—81no—
96D4×C23—82no—
97D4×C23—81no—
98C2×C4×C23—21no—
99C2×C4×C23—42no—
100C2×C4×C23—42no—
101C2×C4×C23—41no—
102C2×C4×C23—41no—
103C2×C4×C23—41no—
104C2×C4×C23—82no—
105C2×C4×C23—82no—
106C2×C4×C23—82no—
107C2×C4×C23#12011yescharacteristic
108C8×C23—21no—
109C8×C23—21no—
110C8×C23—42no—
111C8×C23—81no—
112#3291C2×C2311yescharacteristic
113#2198C2×C2312yescomplemented
114#1630C2×C2311yescharacteristic, complemented
115#1409C2×C2311yescharacteristic, complemented
116#1402C2×C2312yescomplemented
117C22×C23—42no—
118C22×C23—42no—
119C22×C23—82no—
120C22×C23—162no—
121C22×C23B(4,𝔽2)11yescharacteristic
122C4×C23—21no—
123C4×C23—21no—
124C4×C23—21no—
125C4×C23—42no—
126C4×C23—81no—
127C4×C23—81no—
128C4×C23—81no—
129C4×C23—162no—
130#577—22no—
131#559C22×C2311yescharacteristic
132#492C22×C2311yescharacteristic
133#477—21no—
134#471—22no—
135#458—22no—
136#456—21no—
137#453C22×C2312yes—
138#443—21no—
139#353C22×C2312yes—
140#352—22no—
141#331C22×C2311yescharacteristic
142Q8⋊C8—21no—
143C2×C23—21no—
144C2×C23—82no—
145C2×C23—82no—
146C2×C23#219611yescharacteristic
1472−1+4—21no—
148H(5,𝔽2)D4×C2312yes—
149#137—22no—
150#137—42no—
151AGL(1,ℤ/8)—22no—
152AGL(1,ℤ/8)—42no—
153#135—21no—
154#135—42no—
155C2×SD16—22no—
156C2×SD16—42no—
157#131—22no—
158C4⋊Q8D4×C2311yescharacteristic
159#127C23×C2311yescharacteristic
160#124—41no—
161#123—22no—
162#123—42no—
163C4×Q8—21no—
164C4×D4—41no—
165#108—21no—
166#107—41no—
167C4⋊C8—21no—
168#104—22no—
169#103—41no—
170#101—21no—
171#100D4×C2312yes—
172#99—22no—
173#99—42no—
174C4×C8D4×C2311yescharacteristic
175C23#1367611yescharacteristic, direct product factor, Sylow
176#41—22no—
177#41—42no—
178#41—42no—
179#41—81no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—42no—
183C2×D4—41no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4C2×D4×C2312yes—
187C22×C4—22no—
188Q16—21no—
189Q16—42no—
190SD16—22no—
191SD16—42no—
192SD16—42no—
193SD16—42no—
194SD16—42no—
195D8—21no—
196D8—42no—
197M16—22no—
198M16—42no—
199C2×C8—21no—
200C2×C8—42no—
201C2×C8—41no—
202C4⋊C4—21no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—82no—
206C22⋊C4—22no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—82no—
210C42—41no—
211C42C2×D4×C2311yescharacteristic
212C23—22no—
213C23—42no—
214Q8—22no—
215Q8—42no—
216Q8—41no—
217Q8—41no—
218Q8—81no—
219D4—22no—
220D4—42no—
221D4—42no—
222D4—81no—
223D4—81no—
224D4—82no—
225D4—81no—
226C2×C4—21no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4C23×#12011yescharacteristic
236C8—21no—
237C8—21no—
238C8—42no—
239C8—81no—
240C22—42no—
241C22—42no—
242C22—82no—
243C22—162no—
244C22C23×B(4,𝔽2)11yescharacteristic
245C4—21no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—81no—
250C4—81no—
251C4—81no—
252C4—162no—
253C2—21no—
254C2—82no—
255C2—82no—
256C2C23×#219611yescharacteristic
subgroupG/H
center, socleC2×C23#2196
commutator, Frattini, cosocle#127C23×C23
prime radicalC23×#3291C2
LCSG ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C22×C23 ◁ C2×C4×C23 ◁ C23×#127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2198C2semidirect
3. C23×#1630C2semidirect
4. C23×#1409C2semidirect
5. C23×#1402C2semidirect
83. #13676C23direct (N × H)
113. #2198C2×C23semidirect
114. #1630C2×C23semidirect
115. #1409C2×C23semidirect
116. #1402C2×C23semidirect
175. C23#13676direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32912
C23×#21982
C23×#16302
C23×#14092
C23×#14022
#1367623