Finite Group Databasebeta

← C23×#12167 #19159793

Subgroups — C23×#12167 #19159793

Order 5888.

354 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 74 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×#3259C211yesmaximal, characteristic
2C23×#2185C212yesmaximal, complemented
3C23×#2047C211yesmaximal, characteristic, complemented
4C23×#1349C212yesmaximal, complemented
5C23×#1323C211yesmaximal, characteristic, complemented
6C22×Q16×C23C2211yescharacteristic
7C23×#540—21no—
8C2×C23×SD32—22no—
9C23×#478—22no—
10C23×#469—21no—
11C23×#466C2212yes—
12C23×#459—22no—
13C23×#424C2211yescharacteristic
14C23×#419—21no—
15C2×C23×#101—21no—
16C23×#358—22no—
17C23×C22⋊C16C2212yes—
18C23×#341C2211yescharacteristic
19C22×Q8×C23—21no—
20C2×Q16×C23—22no—
21C2×Q16×C23—42no—
22C2×D8×C23—21no—
23C2×D8×C23D412yes—
24C2×M16×C23—21no—
25C2×M16×C23D411yescharacteristic
26C22×C8×C23C2311yescharacteristic
27C23×#127—21no—
28C23×#125—21no—
29C23×#123—42no—
30C23×#121—22no—
31C23×#121—42no—
32C4×D4×C23—22no—
33C23×#117D411yescharacteristic
34C23×SD32—44no—
35C2×C16×C23—22no—
36C23×#107D412yes—
37C23×#106—21no—
38C23×C4⋊C8—22no—
39C23×#103—42no—
40C23×#102—22no—
41C23×#102—22no—
42C23×#101—42no—
43C2×Q8×C23—22no—
44C2×Q8×C23—41no—
45C2×Q8×C23—41no—
46C2×Q8×C23—41no—
47C2×Q8×C23—41no—
48C2×Q8×C23—41no—
49C2×D4×C23—22no—
50C2×D4×C23—42no—
51C2×D4×C23—42no—
52C22×C4×C23—42no—
53C22×C4×C23—41no—
54C22×C4×C23C2×D411yescharacteristic
55Q16×C23—24no—
56Q16×C23—42no—
57D8×C23—42no—
58D8×C23—81no—
59M16×C23—41no—
60M16×C23—41no—
61M16×C23—41no—
62C2×C8×C23—21no—
63C2×C8×C23—41no—
64C2×C8×C23—41no—
65C2×C8×C23C2×D412yes—
66C4⋊C4×C23—22no—
67C4⋊C4×C23—42no—
68C4⋊C4×C23D812yes—
69C22⋊C4×C23—21no—
70C22⋊C4×C23—42no—
71C22⋊C4×C23—42no—
72C42×C23—21no—
73C16×C23—42no—
74#12167C2311yesmaximal, characteristic, direct product factor, Sylow
75C23×C23—42no—
76C23×C23AGL(1,ℤ/8)11yescharacteristic
77Q8×C23—42no—
78Q8×C23—42no—
79Q8×C23—81no—
80Q8×C23—81no—
81D4×C23—42no—
82D4×C23—42no—
83D4×C23—162no—
84D4×C23—162no—
85C2×C4×C23—21no—
86C2×C4×C23—21no—
87C2×C4×C23—22no—
88C2×C4×C23—42no—
89C2×C4×C23—42no—
90C2×C4×C23—81no—
91C2×C4×C23—81no—
92C2×C4×C23—162no—
93C2×C4×C23#12011yescharacteristic
94C2×C4×C23C2×D811yescharacteristic
95C8×C23—22no—
96C8×C23—41no—
97C8×C23—81no—
98#3259C2×C2311yescharacteristic
99#2185C2×C2312yescomplemented
100#2047C2×C2311yescharacteristic, complemented
101#1349C2×C2312yescomplemented
102#1323C2×C2311yescharacteristic, complemented
103C22×C23—21no—
104C22×C23—22no—
105C22×C23—82no—
106C22×C23—82no—
107C22×C23—82no—
108C22×C23#44711yescharacteristic
109C4×C23—21no—
110C4×C23—42no—
111C4×C23—82no—
112C4×C23—82no—
113C4×C23#45411yescharacteristic
114C4×C23B(4,𝔽2)11yescharacteristic
115C22×Q16C22×C2311yescharacteristic
116#540—21no—
117C2×SD32—22no—
118#478—22no—
119#469—21no—
120#466C22×C2312yes—
121#459—22no—
122#424C22×C2311yescharacteristic
123#419—21no—
124C2×#101—21no—
125#358—22no—
126C22⋊C16C22×C2312yes—
127#341C22×C2311yescharacteristic
128C2×C23—41no—
129C2×C23—162no—
130C2×C23#159811yescharacteristic
131C2×C23#219412yes—
132C22×Q8—21no—
133C2×Q16—22no—
134C2×Q16—42no—
135C2×D8—21no—
136C2×D8D4×C2312yes—
137C2×M16—21no—
138C2×M16D4×C2311yescharacteristic
139C22×C8C23×C2311yescharacteristic
140#127—21no—
141#125—21no—
142#123—42no—
143#121—22no—
144#121—42no—
145C4×D4—22no—
146#117D4×C2311yescharacteristic
147SD32—44no—
148C2×C16—22no—
149#107D4×C2312yes—
150#106—21no—
151C4⋊C8—22no—
152#103—42no—
153#102—22no—
154#102—22no—
155#101—42no—
156C23#1216711yescharacteristic, direct product factor, Sylow
157C2×Q8—22no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×D4—22no—
164C2×D4—42no—
165C2×D4—42no—
166C22×C4—42no—
167C22×C4—41no—
168C22×C4C2×D4×C2311yescharacteristic
169Q16—24no—
170Q16—42no—
171D8—42no—
172D8—81no—
173M16—41no—
174M16—41no—
175M16—41no—
176C2×C8—21no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8C2×D4×C2312yes—
180C4⋊C4—22no—
181C4⋊C4—42no—
182C4⋊C4D8×C2312yes—
183C22⋊C4—21no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C42—21no—
187C16—42no—
188C23—42no—
189C23C23×AGL(1,ℤ/8)11yescharacteristic
190Q8—42no—
191Q8—42no—
192Q8—81no—
193Q8—81no—
194D4—42no—
195D4—42no—
196D4—162no—
197D4—162no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—162no—
206C2×C4C23×#12011yescharacteristic
207C2×C4C2×D8×C2311yescharacteristic
208C8—22no—
209C8—41no—
210C8—81no—
211C22—22no—
212C22—21no—
213C22—82no—
214C22—82no—
215C22—82no—
216C22C23×#44711yescharacteristic
217C4—21no—
218C4—42no—
219C4—82no—
220C4—82no—
221C4C23×#45411yescharacteristic
222C4C23×B(4,𝔽2)11yescharacteristic
223C2—41no—
224C2—162no—
225C2C23×#159811yescharacteristic
226C2C23×#219412yes—
subgroupG/H
center, socleC22×C23#447
commutator, Frattini, cosocleC22×C8C23×C23
prime radicalC23×#3259C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C23×#424 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2185C2semidirect
3. C23×#2047C2semidirect
4. C23×#1349C2semidirect
5. C23×#1323C2semidirect
74. #12167C23direct (N × H)
99. #2185C2×C23semidirect
100. #2047C2×C23semidirect
101. #1349C2×C23semidirect
102. #1323C2×C23semidirect
156. C23#12167direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32592
C23×#21852
C23×#20472
C23×#13492
C23×#13232
#1216723