Finite Group Databasebeta

← C23×#13683 #19161309

Subgroups — C23×#13683 #19161309

Order 5888.

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

normal subgroups only

This group has 392 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×#2205C212yesmaximal, complemented
3C23×#1623C211yesmaximal, characteristic, complemented
4C23×#1410C211yesmaximal, characteristic, complemented
5C23×#1405C212yesmaximal, complemented
6C23×#577—22no—
7Q82×C23C2211yescharacteristic, complemented
8C23×#492C2211yescharacteristic
9C23×#471—22no—
10C23×#460—21no—
11C23×#456—22no—
12C23×#456C2212yes—
13C23×#453—21no—
14C23×#443—21no—
15C23×#356—22no—
16C23×#356C2212yescomplemented
17C23×#333C2211yescharacteristic, complemented
18C23×Q8⋊C8—21no—
19C23×2−1+4D412yes—
20C23×H(5,𝔽2)—21no—
21C23×#137—22no—
22C23×#137—42no—
23C23×#137—42no—
24C23×AGL(1,ℤ/8)—22no—
25C23×AGL(1,ℤ/8)—42no—
26C23×#135—21no—
27C23×#135—42no—
28C2×SD16×C23—22no—
29C2×SD16×C23—42no—
30C2×SD16×C23—41no—
31C23×#131—22no—
32C23×C4⋊Q8—22no—
33C23×C4⋊Q8—41no—
34C23×C4⋊Q8C2311yescharacteristic
35C23×#127D411yescharacteristic
36C4×Q8×C23—21no—
37C4×Q8×C23—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—21no—
52C2×Q8×C23—22no—
53C2×Q8×C23—42no—
54C2×Q8×C23C2×D412yes—
55C2×D4×C23—21no—
56C2×D4×C23—41no—
57C2×D4×C23—42no—
58C2×D4×C23—81no—
59Q16×C23—21no—
60Q16×C23—42no—
61Q16×C23—82no—
62SD16×C23—22no—
63SD16×C23—42no—
64SD16×C23—42no—
65SD16×C23—42no—
66SD16×C23—42no—
67SD16×C23—82no—
68SD16×C23—82no—
69D8×C23—21no—
70D8×C23—42no—
71M16×C23—22no—
72M16×C23—42no—
73M16×C23—42no—
74C2×C8×C23—21no—
75C2×C8×C23—42no—
76C2×C8×C23—41no—
77C4⋊C4×C23—21no—
78C4⋊C4×C23—41no—
79C4⋊C4×C23—42no—
80C4⋊C4×C23—41no—
81C42×C23—22no—
82C42×C23—41no—
83C42×C23C2×D411yescharacteristic
84#13683C2311yesmaximal, characteristic, direct product factor, Sylow
85C23×C23—41no—
86C23×C23—81no—
87Q8×C23—22no—
88Q8×C23—22no—
89Q8×C23—42no—
90Q8×C23—41no—
91Q8×C23—42no—
92Q8×C23—82no—
93D4×C23—22no—
94D4×C23—42no—
95D4×C23—42no—
96D4×C23—42no—
97D4×C23—81no—
98D4×C23—81no—
99D4×C23—82no—
100C2×C4×C23—21no—
101C2×C4×C23—22no—
102C2×C4×C23—42no—
103C2×C4×C23—41no—
104C2×C4×C23—41no—
105C2×C4×C23—82no—
106C2×C4×C23—81no—
107C2×C4×C23#12011yescharacteristic
108C8×C23—21no—
109C8×C23—21no—
110C8×C23—42no—
111C8×C23—81no—
112C8×C23—82no—
113#3291C2×C2311yescharacteristic
114#2205C2×C2312yescomplemented
115#1623C2×C2311yescharacteristic, complemented
116#1410C2×C2311yescharacteristic, complemented
117#1405C2×C2312yescomplemented
118C22×C23—42no—
119C22×C23—81no—
120C22×C23—161no—
121C22×C23—321no—
122C22×C23B(4,𝔽2)11yescharacteristic
123C4×C23—21no—
124C4×C23—21no—
125C4×C23—21no—
126C4×C23—42no—
127C4×C23—42no—
128C4×C23—42no—
129C4×C23—81no—
130#577—22no—
131Q82C22×C2311yescharacteristic, complemented
132#492C22×C2311yescharacteristic
133#471—22no—
134#460—21no—
135#456—22no—
136#456C22×C2312yes—
137#453—21no—
138#443—21no—
139#356—22no—
140#356C22×C2312yescomplemented
141#333C22×C2311yescharacteristic, complemented
142Q8⋊C8—21no—
143C2×C23—21no—
144C2×C23—82no—
145C2×C23—161no—
146C2×C23#219611yescharacteristic
1472−1+4D4×C2312yes—
148H(5,𝔽2)—21no—
149#137—22no—
150#137—42no—
151#137—42no—
152AGL(1,ℤ/8)—22no—
153AGL(1,ℤ/8)—42no—
154#135—21no—
155#135—42no—
156C2×SD16—22no—
157C2×SD16—42no—
158C2×SD16—41no—
159#131—22no—
160C4⋊Q8—22no—
161C4⋊Q8—41no—
162C4⋊Q8C23×C2311yescharacteristic
163#127D4×C2311yescharacteristic
164C4×Q8—21no—
165C4×Q8—41no—
166#108—21no—
167C4⋊C8—21no—
168#104—22no—
169#104—42no—
170#102—41no—
171#101—22no—
172#101D4×C2312yes—
173#100—21no—
174C4×C8D4×C2311yescharacteristic
175C23#1368311yescharacteristic, direct product factor, Sylow
176#41—22no—
177#41—42no—
178#41—42no—
179#41—42no—
180C2×Q8—21no—
181C2×Q8—22no—
182C2×Q8—42no—
183C2×Q8C2×D4×C2312yes—
184C2×D4—21no—
185C2×D4—41no—
186C2×D4—42no—
187C2×D4—81no—
188Q16—21no—
189Q16—42no—
190Q16—82no—
191SD16—22no—
192SD16—42no—
193SD16—42no—
194SD16—42no—
195SD16—42no—
196SD16—82no—
197SD16—82no—
198D8—21no—
199D8—42no—
200M16—22no—
201M16—42no—
202M16—42no—
203C2×C8—21no—
204C2×C8—42no—
205C2×C8—41no—
206C4⋊C4—21no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C42—22no—
211C42—41no—
212C42C2×D4×C2311yescharacteristic
213C23—41no—
214C23—81no—
215Q8—22no—
216Q8—22no—
217Q8—42no—
218Q8—41no—
219Q8—42no—
220Q8—82no—
221D4—22no—
222D4—42no—
223D4—42no—
224D4—42no—
225D4—81no—
226D4—81no—
227D4—82no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—82no—
234C2×C4—81no—
235C2×C4C23×#12011yescharacteristic
236C8—21no—
237C8—21no—
238C8—42no—
239C8—81no—
240C8—82no—
241C22—42no—
242C22—81no—
243C22—161no—
244C22—321no—
245C22C23×B(4,𝔽2)11yescharacteristic
246C4—21no—
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—42no—
252C4—81no—
253C2—21no—
254C2—82no—
255C2—161no—
256C2C23×#219611yescharacteristic
subgroupG/H
center, socleC2×C23#2196
commutator, Frattini, cosocleC4⋊Q8C23×C23
prime radicalC23×#3291C2
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
2. C23×#2205C2semidirect
3. C23×#1623C2semidirect
4. C23×#1410C2semidirect
5. C23×#1405C2semidirect
7. Q82×C23C22semidirect
16. C23×#356C22semidirect
17. C23×#333C22semidirect
84. #13683C23direct (N × H)
114. #2205C2×C23semidirect
115. #1623C2×C23semidirect
116. #1410C2×C23semidirect
117. #1405C2×C23semidirect
131. Q82C22×C23semidirect
140. #356C22×C23semidirect
141. #333C22×C23semidirect
175. C23#13683direct (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×#22052
C23×#16232
C23×#14102
C23×#14052
#1368323