Finite Group Databasebeta

← C23×#8551 #19156177

Subgroups — C23×#8551 #19156177

Order 5888.

1026 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 174 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
1C22×C23×C4⋊Q8C211yesmaximal, characteristic
2C23×#1784C212yesmaximal
3C23×#1463C214yesmaximal
4C23×Q8×C23C411yescharacteristic
5C23×Q8×C23C2211yescharacteristic
6C2×C23×C4⋊Q8—24no—
7C2×C23×C4⋊Q8—28no—
8C22×C4⋊C4×C23C412yes—
9C22×C4⋊C4×C23C2212yes—
10C22×C42×C23C411yescharacteristic
11C2×C23×#101—24no—
12C23×#407C2214yes—
13C23×#324—28no—
14C22×Q8×C23—24no—
15C22×Q8×C23—22no—
16C22×Q8×C23—24no—
17C22×Q8×C23—44no—
18C22×Q8×C23—44no—
19C22×Q8×C23D414yes—
20C23×C4×C23C2×C411yescharacteristic
21C23×C4×C23C2×C412yes—
22C23×C4×C23C2×C412yes—
23C23×C4×C23C2×C411yescharacteristic
24C23×C4×C23C2311yescharacteristic
25C2×M16×C23—24no—
26C23×C4⋊Q8—232no—
27C2×C4⋊C4×C23—24no—
28C2×C4⋊C4×C23—24no—
29C2×C4⋊C4×C23—24no—
30C2×C4⋊C4×C23—24no—
31C2×C4⋊C4×C23—24no—
32C2×C4⋊C4×C23D418yes—
33C2×C42×C23—22no—
34C2×C42×C23—24no—
35C23×#101—48no—
36C23×C22⋊C8—28no—
37C24×C23C22×C411yescharacteristic
38C2×Q8×C23—28no—
39C2×Q8×C23—28no—
40C2×Q8×C23—28no—
41C2×Q8×C23—44no—
42C2×Q8×C23—416no—
43C2×Q8×C23—416no—
44C2×Q8×C23—44no—
45C2×Q8×C23—44no—
46C22×C4×C23—22no—
47C22×C4×C23—22no—
48C22×C4×C23—22no—
49C22×C4×C23—24no—
50C22×C4×C23—28no—
51C22×C4×C23—22no—
52C22×C4×C23—22no—
53C22×C4×C23—22no—
54C22×C4×C23—42no—
55C22×C4×C23—42no—
56C22×C4×C23—44no—
57C22×C4×C23C22⋊C412yes—
58C22×C4×C23C22⋊C412yes—
59C22×C4×C23C22⋊C414yes—
60C22×C4×C23C22⋊C414yes—
61C22×C4×C23C2×D412yes—
62C22×C4×C23C2×D414yes—
63M16×C23—48no—
64C2×C8×C23—44no—
65C4⋊C4×C23—216no—
66C4⋊C4×C23—216no—
67C42×C23—28no—
68#8551C2311yesmaximal, characteristic, direct product factor, Sylow
69C23×C23—22no—
70C23×C23—24no—
71C23×C23C2×C22⋊C411yescharacteristic
72C23×C23C2×C22⋊C412yes—
73Q8×C23—416no—
74Q8×C23—416no—
75C2×C4×C23—28no—
76C2×C4×C23—28no—
77C2×C4×C23—28no—
78C2×C4×C23—24no—
79C2×C4×C23—22no—
80C2×C4×C23—24no—
81C2×C4×C23—22no—
82C2×C4×C23—24no—
83C2×C4×C23—22no—
84C2×C4×C23—24no—
85C2×C4×C23—24no—
86C2×C4×C23—44no—
87C2×C4×C23—44no—
88C2×C4×C23—44no—
89C2×C4×C23—44no—
90C2×C4×C23—44no—
91C2×C4×C23#9914yes—
92C2×C4×C23#10118yescomplemented
93C2×C4×C23#12014yes—
94C8×C23—84no—
95C22×C4⋊Q8C2×C2311yescharacteristic
96#1784C2×C2312yes—
97#1463C2×C2314yes—
98C22×C23—28no—
99C22×C23—22no—
100C22×C23—22no—
101C22×C23—22no—
102C22×C23#37911yescharacteristic
103C22×C23C2×#9912yes—
104C22×C23C2×#10114yes—
105C4×C23—24no—
106C4×C23—24no—
107C4×C23—28no—
108C4×C23—42no—
109C4×C23—44no—
110C23×Q8C4×C2311yescharacteristic
111C23×Q8C22×C2311yescharacteristic
112C2×C4⋊Q8—24no—
113C2×C4⋊Q8—28no—
114C22×C4⋊C4C4×C2312yes—
115C22×C4⋊C4C22×C2312yes—
116C22×C42C4×C2311yescharacteristic
117C2×#101—24no—
118#407C22×C2314yes—
119#324—28no—
120C2×C23—22no—
121C2×C23—24no—
122C2×C23#178111yescharacteristic
123C2×C23#178412yes—
124C22×Q8—24no—
125C22×Q8—22no—
126C22×Q8—24no—
127C22×Q8—44no—
128C22×Q8—44no—
129C22×Q8D4×C2314yes—
130C23×C4C2×C4×C2312yes—
131C23×C4C2×C4×C2311yescharacteristic
132C23×C4C2×C4×C2311yescharacteristic
133C23×C4C2×C4×C2312yes—
134C23×C4C23×C2311yescharacteristic
135C2×M16—24no—
136C4⋊Q8—232no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4D4×C2318yes—
143C2×C42—22no—
144C2×C42—24no—
145#101—48no—
146C22⋊C8—28no—
147C23#855111yescharacteristic, direct product factor, Sylow
148C24C22×C4×C2311yescharacteristic
149C2×Q8—28no—
150C2×Q8—28no—
151C2×Q8—28no—
152C2×Q8—44no—
153C2×Q8—416no—
154C2×Q8—416no—
155C2×Q8—44no—
156C2×Q8—44no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—28no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—44no—
168C22×C4C22⋊C4×C2312yes—
169C22×C4C22⋊C4×C2312yes—
170C22×C4C22⋊C4×C2314yes—
171C22×C4C22⋊C4×C2314yes—
172C22×C4C2×D4×C2312yes—
173C22×C4C2×D4×C2314yes—
174M16—48no—
175C2×C8—44no—
176C4⋊C4—216no—
177C4⋊C4—216no—
178C42—28no—
179C23—22no—
180C23—24no—
181C23C2×C22⋊C4×C2311yescharacteristic
182C23C2×C22⋊C4×C2312yes—
183Q8—416no—
184Q8—416no—
185C2×C4—28no—
186C2×C4—28no—
187C2×C4—28no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4C23×#9914yes—
202C2×C4C23×#10118yescomplemented
203C2×C4C23×#12014yes—
204C8—84no—
205C22—28no—
206C22—22no—
207C22—22no—
208C22—22no—
209C22C23×#37911yescharacteristic
210C22C2×C23×#9912yes—
211C22C2×C23×#10114yes—
212C4—24no—
213C4—24no—
214C4—28no—
215C4—42no—
216C4—44no—
217C2—22no—
218C2—24no—
219C2C23×#178111yescharacteristic
220C2C23×#178412yes—
subgroupG/H
center, socleC22×C23#379
commutatorC24C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC24×C23C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
68. #8551C23direct (N × H)
92. C2×C4×C23#101semidirect
147. C23#8551direct (N × H)
202. C2×C4C23×#101semidirect

Maximal subgroups

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

subgroupindex
C22×C23×C4⋊Q82
C23×#17842
C23×#14632
#855123