Finite Group Databasebeta

← C23×#7427 #19155053

Subgroups — C23×#7427 #19155053

Order 5888.

214 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 42 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×#2030C211yesmaximal, characteristic
2C23×#1317C211yesmaximal, characteristic
3C23×#1306C211yesmaximal, characteristic
4C23×#543C411yescharacteristic
5C23×#407—21no—
6C23×#381—21no—
7C23×#381C2211yescharacteristic
8C23×#342—21no—
9C23×#342C411yescharacteristic
10C23×#323—21no—
11C23×C4×C23D411yescharacteristic
12C2×M16×C23—41no—
13C23×C22⋊Q8—42no—
14C23×C22⋊Q8—41no—
15C2×C22⋊C4×C23—21no—
16C2×C22⋊C4×C23—21no—
17C2×C22⋊C4×C23C2×C411yescharacteristic
18C2×C22⋊C4×C23D411yescharacteristic
19C23×C22⋊C8—22no—
20C23×C22⋊C8—41no—
21C23×C22⋊C8—41no—
22C23×#95—41no—
23C23×#95—42no—
24C24×C23C22⋊C411yescharacteristic
25C2×Q8×C23—41no—
26C22×C4×C23—21no—
27C22×C4×C23—21no—
28C22×C4×C23—22no—
29C22×C4×C23—21no—
30C22×C4×C23—21no—
31C22×C4×C23—41no—
32C22×C4×C23—41no—
33C22×C4×C23—41no—
34C22×C4×C23SD1611yescharacteristic
35C22×C4×C23Q1611yescharacteristic
36M16×C23—81no—
37M16×C23—81no—
38C2×C8×C23—42no—
39C2×C8×C23—81no—
40C4⋊C4×C23—42no—
41C4⋊C4×C23—41no—
42C22⋊C4×C23—41no—
43C22⋊C4×C23—41no—
44C22⋊C4×C23—41no—
45C22⋊C4×C23—41no—
46C22⋊C4×C23—41no—
47C22⋊C4×C23—41no—
48#7427C2311yesmaximal, characteristic, direct product factor, Sylow
49C23×C23—21no—
50C23×C23—21no—
51C23×C23—81no—
52C23×C23#9911yescharacteristic
53C23×C23#10311yescharacteristic
54C23×C23#10411yescharacteristic
55Q8×C23—161no—
56C2×C4×C23—21no—
57C2×C4×C23—21no—
58C2×C4×C23—41no—
59C2×C4×C23—41no—
60C2×C4×C23—41no—
61C2×C4×C23—41no—
62C2×C4×C23—42no—
63C2×C4×C23—41no—
64C2×C4×C23—41no—
65C2×C4×C23—81no—
66C2×C4×C23—81no—
67C2×C4×C23—81no—
68C2×C4×C23—82no—
69C2×C4×C23—81no—
70C2×C4×C23—82no—
71C8×C23—82no—
72C8×C23—161no—
73#2030C2×C2311yescharacteristic
74#1317C2×C2311yescharacteristic
75#1306C2×C2311yescharacteristic
76C22×C23—21no—
77C22×C23—21no—
78C22×C23—41no—
79C22×C23—41no—
80C22×C23—41no—
81C22×C23—42no—
82C22×C23—81no—
83C22×C23#32811yescharacteristic
84C22×C23AΣL(1,𝔽16)11yescharacteristic
85C22×C23#35411yescharacteristic
86C4×C23—41no—
87C4×C23—41no—
88C4×C23—81no—
89C4×C23—82no—
90C4×C23—81no—
91C4×C23—161no—
92#543C4×C2311yescharacteristic
93#407—21no—
94#381—21no—
95#381C22×C2311yescharacteristic
96#342—21no—
97#342C4×C2311yescharacteristic
98#323—21no—
99C2×C23—21no—
100C2×C23—21no—
101C2×C23—21no—
102C2×C23—41no—
103C2×C23—41no—
104C2×C23#135111yescharacteristic
105C23×C4D4×C2311yescharacteristic
106C2×M16—41no—
107C22⋊Q8—42no—
108C22⋊Q8—41no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4C2×C4×C2311yescharacteristic
112C2×C22⋊C4D4×C2311yescharacteristic
113C22⋊C8—22no—
114C22⋊C8—41no—
115C22⋊C8—41no—
116#95—41no—
117#95—42no—
118C23#742711yescharacteristic, direct product factor, Sylow
119C24C22⋊C4×C2311yescharacteristic
120C2×Q8—41no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4SD16×C2311yescharacteristic
130C22×C4Q16×C2311yescharacteristic
131M16—81no—
132M16—81no—
133C2×C8—42no—
134C2×C8—81no—
135C4⋊C4—42no—
136C4⋊C4—41no—
137C22⋊C4—41no—
138C22⋊C4—41no—
139C22⋊C4—41no—
140C22⋊C4—41no—
141C22⋊C4—41no—
142C22⋊C4—41no—
143C23—21no—
144C23—21no—
145C23—81no—
146C23C23×#9911yescharacteristic
147C23C23×#10311yescharacteristic
148C23C23×#10411yescharacteristic
149Q8—161no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—81no—
160C2×C4—81no—
161C2×C4—81no—
162C2×C4—82no—
163C2×C4—81no—
164C2×C4—82no—
165C8—82no—
166C8—161no—
167C22—21no—
168C22—21no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22—42no—
173C22—81no—
174C22C23×#32811yescharacteristic
175C22C23×AΣL(1,𝔽16)11yescharacteristic
176C22C23×#35411yescharacteristic
177C4—41no—
178C4—41no—
179C4—81no—
180C4—82no—
181C4—81no—
182C4—161no—
183C2—21no—
184C2—21no—
185C2—21no—
186C2—41no—
187C2—41no—
188C2C23×#135111yescharacteristic
subgroupG/H
center, socleC2×C23#1351
commutatorC2×C22⋊C4C2×C4×C23
Frattini, cosocle#381C22×C23
prime radicalC24×C23C22⋊C4
LCSG ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C23×C23 ◁ C24×C23 ◁ C23×#381 ◁ G

Complements

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

NH ≅ G/Ntype
48. #7427C23direct (N × H)
118. C23#7427direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#20302
C23×#13172
C23×#13062
#742723