Finite Group Databasebeta

← C23×#7132 #19154758

Subgroups — C23×#7132 #19154758

Order 5888.

368 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 90 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×#1445C211yesmaximal, characteristic
2C23×#1280C212yesmaximal
3C22×C4⋊C4×C23C411yescharacteristic
4C2×C23×C22⋊C8—22no—
5C2×C23×#95—22no—
6C2×C23×#95C411yescharacteristic
7C2×C23×#95C2211yescharacteristic
8C23×C4×C23—21no—
9C23×C4×C23—21no—
10C23×C4×C23C2×C411yescharacteristic
11C23×C4×C23D412yes—
12C22×C8×C23—42no—
13C2×C4⋊C4×C23—21no—
14C2×C4⋊C4×C23—41no—
15C2×C4⋊C4×C23—41no—
16C2×C4⋊C4×C23C812yescomplemented
17C23×C22⋊C8—24no—
18C23×C22⋊C8—42no—
19C23×#95—41no—
20C23×#95—41no—
21C23×#95—44no—
22C24×C23C22⋊C411yescharacteristic
23C22×C4×C23—21no—
24C22×C4×C23—22no—
25C22×C4×C23—21no—
26C22×C4×C23—22no—
27C22×C4×C23—22no—
28C22×C4×C23—22no—
29C22×C4×C23—42no—
30C22×C4×C23—42no—
31C22×C4×C23—42no—
32C22×C4×C23—42no—
33C22×C4×C23—41no—
34C22×C4×C23—42no—
35C22×C4×C23—42no—
36C22×C4×C23—41no—
37C22×C4×C23—41no—
38C22×C4×C23—41no—
39C22×C4×C23—42no—
40C22×C4×C23C2×C811yescharacteristic
41C22×C4×C23M1611yescharacteristic
42C22×C4×C23SD1612yes—
43C22×C4×C23Q1612yes—
44C2×C8×C23—44no—
45C2×C8×C23—82no—
46C2×C8×C23—82no—
47C4⋊C4×C23—42no—
48C4⋊C4×C23—42no—
49#7132C2311yesmaximal, characteristic, direct product factor, Sylow
50C23×C23—22no—
51C23×C23—21no—
52C23×C23—21no—
53C23×C23C22⋊C811yescharacteristic
54C23×C23#9911yescharacteristic
55C23×C23#10011yescharacteristic
56C23×C23#10312yes—
57C23×C23#10412yes—
58C2×C4×C23—21no—
59C2×C4×C23—21no—
60C2×C4×C23—22no—
61C2×C4×C23—22no—
62C2×C4×C23—42no—
63C2×C4×C23—42no—
64C2×C4×C23—42no—
65C2×C4×C23—42no—
66C2×C4×C23—41no—
67C2×C4×C23—41no—
68C2×C4×C23—42no—
69C2×C4×C23—42no—
70C2×C4×C23—42no—
71C2×C4×C23—44no—
72C2×C4×C23—44no—
73C2×C4×C23—42no—
74C2×C4×C23—82no—
75C2×C4×C23—81no—
76C2×C4×C23—82no—
77C2×C4×C23—82no—
78C2×C4×C23—81no—
79C2×C4×C23—81no—
80C2×C4×C23—81no—
81C8×C23—84no—
82#1445C2×C2311yescharacteristic
83#1280C2×C2312yes—
84C22×C23—21no—
85C22×C23—21no—
86C22×C23—21no—
87C22×C23—21no—
88C22×C23—22no—
89C22×C23—22no—
90C22×C23—22no—
91C22×C23—22no—
92C22×C23#32311yescharacteristic
93C22×C23Q8⋊C812yes—
94C22×C23#32812yes—
95C22×C23#32911yescharacteristic
96C22×C23#33311yescharacteristic
97C22×C23#35212yes—
98C22×C23#35311yescharacteristic
99C22×C23#35511yescharacteristic
100C4×C23—41no—
101C4×C23—41no—
102C4×C23—42no—
103C4×C23—42no—
104C4×C23—82no—
105C4×C23—82no—
106C22×C4⋊C4C4×C2311yescharacteristic
107C2×C22⋊C8—22no—
108C2×#95—22no—
109C2×#95C4×C2311yescharacteristic
110C2×#95C22×C2311yescharacteristic
111C2×C23—21no—
112C2×C23—22no—
113C2×C23—21no—
114C2×C23#127211yescharacteristic
115C2×C23#131711yescharacteristic
116C2×C23#131811yescharacteristic
117C2×C23#135212yes—
118C2×C23#135312yes—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4C2×C4×C2311yescharacteristic
122C23×C4D4×C2312yes—
123C22×C8—42no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4C8×C2312yescomplemented
128C22⋊C8—24no—
129C22⋊C8—42no—
130#95—41no—
131#95—41no—
132#95—44no—
133C23#713211yescharacteristic, direct product factor, Sylow
134C24C22⋊C4×C2311yescharacteristic
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—41no—
147C22×C4—42no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—42no—
152C22×C4C2×C8×C2311yescharacteristic
153C22×C4M16×C2311yescharacteristic
154C22×C4SD16×C2312yes—
155C22×C4Q16×C2312yes—
156C2×C8—44no—
157C2×C8—82no—
158C2×C8—82no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C23—22no—
162C23—21no—
163C23—21no—
164C23C23×C22⋊C811yescharacteristic
165C23C23×#9911yescharacteristic
166C23C23×#10011yescharacteristic
167C23C23×#10312yes—
168C23C23×#10412yes—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—44no—
183C2×C4—44no—
184C2×C4—42no—
185C2×C4—82no—
186C2×C4—81no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C8—84no—
193C22—21no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—22no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22C23×#32311yescharacteristic
202C22C23×Q8⋊C812yes—
203C22C23×#32812yes—
204C22C23×#32911yescharacteristic
205C22C23×#33311yescharacteristic
206C22C23×#35212yes—
207C22C23×#35311yescharacteristic
208C22C23×#35511yescharacteristic
209C4—41no—
210C4—41no—
211C4—42no—
212C4—42no—
213C4—82no—
214C4—82no—
215C2—21no—
216C2—22no—
217C2—21no—
218C2C23×#127211yescharacteristic
219C2C23×#131711yescharacteristic
220C2C23×#131811yescharacteristic
221C2C23×#135212yes—
222C2C23×#135312yes—
subgroupG/H
center, socleC23×C23#99
commutatorC22×C4C2×C8×C23
Frattini, cosocleC2×#95C22×C23
prime radicalC24×C23C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C23 ◁ C24×C23 ◁ C2×C23×#95 ◁ G

Complements

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

NH ≅ G/Ntype
16. C2×C4⋊C4×C23C8semidirect
49. #7132C23direct (N × H)
127. C2×C4⋊C4C8×C23semidirect
133. C23#7132direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#14452
C23×#12802
#713223