Finite Group Databasebeta

← C23×#13682 #19161308

Subgroups — C23×#13682 #19161308

Order 5888.

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

normal subgroups only

This group has 384 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×#3293C211yesmaximal, characteristic, complemented
2C23×#2204C212yesmaximal, complemented
3C23×#1634C211yesmaximal, characteristic, complemented
4C23×#1411C211yesmaximal, characteristic, complemented
5C23×#1405C212yesmaximal, complemented
6C23×#578—22no—
7C23×#555C2211yescharacteristic
8C23×#494C2211yescharacteristic
9C23×#476—21no—
10C23×#473—22no—
11C23×#456—21no—
12C23×#456C2212yes—
13C23×#454—22no—
14C23×#443—21no—
15C23×#356C2212yes—
16C23×#355—22no—
17C23×#333C2211yescharacteristic
18C23×D4⋊C8—21no—
19C23×2−1+4—21no—
20C23×2−1+4D412yes—
21C23×#137—22no—
22C23×#137—22no—
23C23×#137—42no—
24C23×#137—42no—
25C23×AGL(1,ℤ/8)—42no—
26C23×#135—21no—
27C23×#135—42no—
28C2×Q16×C23—22no—
29C2×Q16×C23—42no—
30C23×#131—22no—
31C23×C4⋊Q8D411yescharacteristic
32C23×C4⋊Q8C2311yescharacteristic
33C23×#124—22no—
34C23×#121—41no—
35C4×D4×C23—21no—
36C23×#117—41no—
37C23×#108—21no—
38C23×#106—41no—
39C23×C4⋊C8—21no—
40C23×#104—22no—
41C23×#104—42no—
42C23×#102—41no—
43C23×#101—21no—
44C23×#101—22no—
45C23×#101D412yes—
46C4×C8×C23D411yescharacteristic
47#41×C23—22no—
48#41×C23—42no—
49#41×C23—42no—
50#41×C23—42no—
51#41×C23—81no—
52C2×Q8×C23—21no—
53C2×Q8×C23—42no—
54C2×Q8×C23—42no—
55C2×Q8×C23C2×D412yes—
56C2×D4×C23—21no—
57C2×D4×C23—22no—
58C22×C4×C23—41no—
59Q16×C23—21no—
60Q16×C23—42no—
61Q16×C23—42no—
62Q16×C23—42no—
63Q16×C23—42no—
64SD16×C23—22no—
65SD16×C23—42no—
66SD16×C23—42no—
67SD16×C23—82no—
68D8×C23—21no—
69D8×C23—82no—
70M16×C23—22no—
71M16×C23—42no—
72M16×C23—42no—
73C2×C8×C23—21no—
74C2×C8×C23—42no—
75C2×C8×C23—41no—
76C4⋊C4×C23—21no—
77C4⋊C4×C23—41no—
78C4⋊C4×C23—41no—
79C22⋊C4×C23—42no—
80C22⋊C4×C23—41no—
81C42×C23—22no—
82C42×C23C2×D411yescharacteristic
83#13682C2311yesmaximal, characteristic, direct product factor, Sylow
84C23×C23—41no—
85Q8×C23—22no—
86Q8×C23—42no—
87Q8×C23—42no—
88Q8×C23—42no—
89Q8×C23—81no—
90D4×C23—22no—
91D4×C23—41no—
92D4×C23—41no—
93D4×C23—42no—
94D4×C23—42no—
95D4×C23—82no—
96D4×C23—81no—
97C2×C4×C23—21no—
98C2×C4×C23—22no—
99C2×C4×C23—42no—
100C2×C4×C23—41no—
101C2×C4×C23—41no—
102C2×C4×C23—81no—
103C2×C4×C23—81no—
104C2×C4×C23—82no—
105C2×C4×C23—82no—
106C2×C4×C23#12011yescharacteristic
107C8×C23—21no—
108C8×C23—21no—
109C8×C23—42no—
110C8×C23—81no—
111C8×C23—82no—
112#3293C2×C2311yescharacteristic, complemented
113#2204C2×C2312yescomplemented
114#1634C2×C2311yescharacteristic, complemented
115#1411C2×C2311yescharacteristic, complemented
116#1405C2×C2312yescomplemented
117C22×C23—42no—
118C22×C23—81no—
119C22×C23—81no—
120C22×C23—81no—
121C22×C23B(4,𝔽2)11yescharacteristic
122C4×C23—21no—
123C4×C23—21no—
124C4×C23—21no—
125C4×C23—42no—
126C4×C23—42no—
127C4×C23—81no—
128C4×C23—81no—
129#578—22no—
130#555C22×C2311yescharacteristic
131#494C22×C2311yescharacteristic
132#476—21no—
133#473—22no—
134#456—21no—
135#456C22×C2312yes—
136#454—22no—
137#443—21no—
138#356C22×C2312yes—
139#355—22no—
140#333C22×C2311yescharacteristic
141D4⋊C8—21no—
142C2×C23—21no—
143C2×C23—82no—
144C2×C23—161no—
145C2×C23#219611yescharacteristic
1462−1+4—21no—
1472−1+4D4×C2312yes—
148#137—22no—
149#137—22no—
150#137—42no—
151#137—42no—
152AGL(1,ℤ/8)—42no—
153#135—21no—
154#135—42no—
155C2×Q16—22no—
156C2×Q16—42no—
157#131—22no—
158C4⋊Q8D4×C2311yescharacteristic
159C4⋊Q8C23×C2311yescharacteristic
160#124—22no—
161#121—41no—
162C4×D4—21no—
163#117—41no—
164#108—21no—
165#106—41no—
166C4⋊C8—21no—
167#104—22no—
168#104—42no—
169#102—41no—
170#101—21no—
171#101—22no—
172#101D4×C2312yes—
173C4×C8D4×C2311yescharacteristic
174C23#1368211yescharacteristic, direct product factor, Sylow
175#41—22no—
176#41—42no—
177#41—42no—
178#41—42no—
179#41—81no—
180C2×Q8—21no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8C2×D4×C2312yes—
184C2×D4—21no—
185C2×D4—22no—
186C22×C4—41no—
187Q16—21no—
188Q16—42no—
189Q16—42no—
190Q16—42no—
191Q16—42no—
192SD16—22no—
193SD16—42no—
194SD16—42no—
195SD16—82no—
196D8—21no—
197D8—82no—
198M16—22no—
199M16—42no—
200M16—42no—
201C2×C8—21no—
202C2×C8—42no—
203C2×C8—41no—
204C4⋊C4—21no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C22⋊C4—42no—
208C22⋊C4—41no—
209C42—22no—
210C42C2×D4×C2311yescharacteristic
211C23—41no—
212Q8—22no—
213Q8—42no—
214Q8—42no—
215Q8—42no—
216Q8—81no—
217D4—22no—
218D4—41no—
219D4—41no—
220D4—42no—
221D4—42no—
222D4—82no—
223D4—81no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4C23×#12011yescharacteristic
234C8—21no—
235C8—21no—
236C8—42no—
237C8—81no—
238C8—82no—
239C22—42no—
240C22—81no—
241C22—81no—
242C22—81no—
243C22C23×B(4,𝔽2)11yescharacteristic
244C4—21no—
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—81no—
250C4—81no—
251C2—21no—
252C2—82no—
253C2—161no—
254C2C23×#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×#3293C2semidirect
2. C23×#2204C2semidirect
3. C23×#1634C2semidirect
4. C23×#1411C2semidirect
5. C23×#1405C2semidirect
83. #13682C23direct (N × H)
112. #3293C2×C23semidirect
113. #2204C2×C23semidirect
114. #1634C2×C23semidirect
115. #1411C2×C23semidirect
116. #1405C2×C23semidirect
174. C23#13682direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32932
C23×#22042
C23×#16342
C23×#14112
C23×#14052
#1368223