Finite Group Databasebeta

← C23×#13681 #19161307

Subgroups — C23×#13681 #19161307

Order 5888.

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

normal subgroups only

This group has 398 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, complemented
2C23×#2204C212yesmaximal, complemented
3C23×#1624C211yesmaximal, characteristic, complemented
4C23×#1410C211yesmaximal, characteristic, complemented
5C23×#1405C212yesmaximal, complemented
6C23×#577—22no—
7C23×#555C2211yescharacteristic, complemented
8C23×#492C2211yescharacteristic
9C23×#471—22no—
10C23×#459—21no—
11C23×#456C2212yes—
12C23×#454—22no—
13C23×#453—21no—
14C23×#443—21no—
15C23×#356C2212yescomplemented
16C23×#355—22no—
17C23×#333C2211yescharacteristic, complemented
18C23×D4⋊C8—21no—
19C23×2−1+4D412yes—
20C23×H(5,𝔽2)—21no—
21C23×#137—22no—
22C23×#137—42no—
23C23×AGL(1,ℤ/8)—22no—
24C23×AGL(1,ℤ/8)—42no—
25C23×AGL(1,ℤ/8)—42no—
26C23×#135—21no—
27C23×#135—42no—
28C2×SD16×C23—22no—
29C2×SD16×C23—42no—
30C2×D8×C23—41no—
31C23×#131—22no—
32C23×C4⋊Q8C2311yescharacteristic
33C23×#127D411yescharacteristic
34C23×#124—22no—
35C23×#121—41no—
36C4×D4×C23—21no—
37C23×#117—41no—
38C23×#108—21no—
39C23×C4⋊C8—21no—
40C23×#104—22no—
41C23×#104—42no—
42C23×#102—41no—
43C23×#101—22no—
44C23×#101D412yes—
45C23×#100—21no—
46C4×C8×C23D411yescharacteristic
47#41×C23—22no—
48#41×C23—42no—
49#41×C23—42no—
50#41×C23—42no—
51C2×Q8×C23—42no—
52C2×Q8×C23C2×D412yes—
53C2×D4×C23—21no—
54C2×D4×C23—22no—
55C2×D4×C23—21no—
56C2×D4×C23—41no—
57C2×D4×C23—42no—
58C2×D4×C23—81no—
59C22×C4×C23—41no—
60Q16×C23—21no—
61Q16×C23—42no—
62SD16×C23—22no—
63SD16×C23—42no—
64SD16×C23—42no—
65SD16×C23—42no—
66SD16×C23—42no—
67SD16×C23—82no—
68D8×C23—21no—
69D8×C23—42no—
70D8×C23—82no—
71D8×C23—82no—
72M16×C23—22no—
73M16×C23—42no—
74M16×C23—42no—
75C2×C8×C23—21no—
76C2×C8×C23—42no—
77C2×C8×C23—41no—
78C4⋊C4×C23—21no—
79C4⋊C4×C23—41no—
80C22⋊C4×C23—42no—
81C22⋊C4×C23—41no—
82C42×C23—22no—
83C42×C23C2×D411yescharacteristic
84#13681C2311yesmaximal, characteristic, direct product factor, Sylow
85C23×C23—41no—
86C23×C23—41no—
87C23×C23—81no—
88Q8×C23—22no—
89Q8×C23—42no—
90Q8×C23—42no—
91D4×C23—22no—
92D4×C23—42no—
93D4×C23—41no—
94D4×C23—41no—
95D4×C23—42no—
96D4×C23—42no—
97D4×C23—81no—
98D4×C23—81no—
99D4×C23—82no—
100D4×C23—82no—
101C2×C4×C23—21no—
102C2×C4×C23—22no—
103C2×C4×C23—42no—
104C2×C4×C23—41no—
105C2×C4×C23—81no—
106C2×C4×C23—81no—
107C2×C4×C23—82no—
108C2×C4×C23—81no—
109C2×C4×C23#12011yescharacteristic
110C8×C23—21no—
111C8×C23—21no—
112C8×C23—42no—
113C8×C23—81no—
114C8×C23—82no—
115#3291C2×C2311yescharacteristic, complemented
116#2204C2×C2312yescomplemented
117#1624C2×C2311yescharacteristic, complemented
118#1410C2×C2311yescharacteristic, complemented
119#1405C2×C2312yescomplemented
120C22×C23—42no—
121C22×C23—81no—
122C22×C23—81no—
123C22×C23—81no—
124C22×C23—81no—
125C22×C23—161no—
126C22×C23—321no—
127C22×C23B(4,𝔽2)11yescharacteristic
128C4×C23—21no—
129C4×C23—21no—
130C4×C23—21no—
131C4×C23—42no—
132C4×C23—42no—
133C4×C23—81no—
134#577—22no—
135#555C22×C2311yescharacteristic, complemented
136#492C22×C2311yescharacteristic
137#471—22no—
138#459—21no—
139#456C22×C2312yes—
140#454—22no—
141#453—21no—
142#443—21no—
143#356C22×C2312yescomplemented
144#355—22no—
145#333C22×C2311yescharacteristic, complemented
146D4⋊C8—21no—
147C2×C23—21no—
148C2×C23—82no—
149C2×C23—161no—
150C2×C23—161no—
151C2×C23#219611yescharacteristic
1522−1+4D4×C2312yes—
153H(5,𝔽2)—21no—
154#137—22no—
155#137—42no—
156AGL(1,ℤ/8)—22no—
157AGL(1,ℤ/8)—42no—
158AGL(1,ℤ/8)—42no—
159#135—21no—
160#135—42no—
161C2×SD16—22no—
162C2×SD16—42no—
163C2×D8—41no—
164#131—22no—
165C4⋊Q8C23×C2311yescharacteristic
166#127D4×C2311yescharacteristic
167#124—22no—
168#121—41no—
169C4×D4—21no—
170#117—41no—
171#108—21no—
172C4⋊C8—21no—
173#104—22no—
174#104—42no—
175#102—41no—
176#101—22no—
177#101D4×C2312yes—
178#100—21no—
179C4×C8D4×C2311yescharacteristic
180C23#1368111yescharacteristic, direct product factor, Sylow
181#41—22no—
182#41—42no—
183#41—42no—
184#41—42no—
185C2×Q8—42no—
186C2×Q8C2×D4×C2312yes—
187C2×D4—21no—
188C2×D4—22no—
189C2×D4—21no—
190C2×D4—41no—
191C2×D4—42no—
192C2×D4—81no—
193C22×C4—41no—
194Q16—21no—
195Q16—42no—
196SD16—22no—
197SD16—42no—
198SD16—42no—
199SD16—42no—
200SD16—42no—
201SD16—82no—
202D8—21no—
203D8—42no—
204D8—82no—
205D8—82no—
206M16—22no—
207M16—42no—
208M16—42no—
209C2×C8—21no—
210C2×C8—42no—
211C2×C8—41no—
212C4⋊C4—21no—
213C4⋊C4—41no—
214C22⋊C4—42no—
215C22⋊C4—41no—
216C42—22no—
217C42C2×D4×C2311yescharacteristic
218C23—41no—
219C23—41no—
220C23—81no—
221Q8—22no—
222Q8—42no—
223Q8—42no—
224D4—22no—
225D4—42no—
226D4—41no—
227D4—41no—
228D4—42no—
229D4—42no—
230D4—81no—
231D4—81no—
232D4—82no—
233D4—82no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—82no—
241C2×C4—81no—
242C2×C4C23×#12011yescharacteristic
243C8—21no—
244C8—21no—
245C8—42no—
246C8—81no—
247C8—82no—
248C22—42no—
249C22—81no—
250C22—81no—
251C22—81no—
252C22—81no—
253C22—161no—
254C22—321no—
255C22C23×B(4,𝔽2)11yescharacteristic
256C4—21no—
257C4—21no—
258C4—21no—
259C4—42no—
260C4—42no—
261C4—81no—
262C2—21no—
263C2—82no—
264C2—161no—
265C2—161no—
266C2C23×#219611yescharacteristic
subgroupG/H
center, socleC2×C23#2196
commutator, Frattini, cosocleC4⋊Q8C23×C23
LCSG ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C22×C23 ◁ C2×C4×C23 ◁ C23×C4⋊Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3291C2semidirect
2. C23×#2204C2semidirect
3. C23×#1624C2semidirect
4. C23×#1410C2semidirect
5. C23×#1405C2semidirect
7. C23×#555C22semidirect
15. C23×#356C22semidirect
17. C23×#333C22semidirect
84. #13681C23direct (N × H)
115. #3291C2×C23semidirect
116. #2204C2×C23semidirect
117. #1624C2×C23semidirect
118. #1410C2×C23semidirect
119. #1405C2×C23semidirect
135. #555C22×C23semidirect
143. #356C22×C23semidirect
145. #333C22×C23semidirect
180. C23#13681direct (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×#22042
C23×#16242
C23×#14102
C23×#14052
#1368123