Finite Group Databasebeta

← C23×#7375 #19155001

Subgroups — C23×#7375 #19155001

Order 5888.

228 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 50 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×#1856C211yesmaximal, characteristic
2C23×#1318C211yesmaximal, characteristic
3C23×#1270C211yesmaximal, characteristic
4C23×#518C411yescharacteristic
5C23×#428—21no—
6C2×C23×#100C411yescharacteristic
7C23×#390C2211yescharacteristic
8C23×#339—21no—
9C23×#336—21no—
10C23×#324—21no—
11C22×D4×C23C2×C411yescharacteristic
12C2×M16×C23—21no—
13C22×C8×C23—41no—
14C4×D4×C23—41no—
15C4×D4×C23—41no—
16C23×#117—21no—
17C2×C4⋊C4×C23D411yescharacteristic
18C2×C22⋊C4×C23—41no—
19C2×C22⋊C4×C23—41no—
20C2×C42×C23D411yescharacteristic
21C23×#107—41no—
22C23×#106—41no—
23C23×#100—21no—
24C23×#100C812yescomplemented
25C23×C22⋊C8—41no—
26C24×C23—21no—
27C2×D4×C23—21no—
28C2×D4×C23—41no—
29C2×D4×C23—41no—
30C2×D4×C23C2×C811yescharacteristic
31C2×D4×C23M1611yescharacteristic
32C22×C4×C23—21no—
33C22×C4×C23—21no—
34C22×C4×C23—41no—
35C22×C4×C23—41no—
36C22×C4×C23C22⋊C411yescharacteristic
37M16×C23—22no—
38M16×C23—41no—
39C2×C8×C23—41no—
40C2×C8×C23—42no—
41C2×C8×C23—81no—
42C2×C8×C23—81no—
43C2×C8×C23—81no—
44C4⋊C4×C23—21no—
45C4⋊C4×C23—21no—
46C4⋊C4×C23—41no—
47C22⋊C4×C23—42no—
48C22⋊C4×C23—41no—
49C22⋊C4×C23—81no—
50C22⋊C4×C23—81no—
51C42×C23—41no—
52C42×C23—41no—
53#7375C2311yesmaximal, characteristic, direct product factor, Sylow
54C23×C23—21no—
55C23×C23—21no—
56C23×C23—41no—
57C23×C23—41no—
58C23×C23—81no—
59C23×C23—81no—
60C23×C23#9911yescharacteristic
61D4×C23—41no—
62D4×C23—41no—
63D4×C23—42no—
64C2×C4×C23—21no—
65C2×C4×C23—21no—
66C2×C4×C23—21no—
67C2×C4×C23—21no—
68C2×C4×C23—41no—
69C2×C4×C23—41no—
70C2×C4×C23—41no—
71C2×C4×C23—41no—
72C2×C4×C23—41no—
73C2×C4×C23—81no—
74C2×C4×C23—161no—
75C2×C4×C23—161no—
76C2×C4×C23C22⋊C811yescharacteristic
77C2×C4×C23#10111yescharacteristic
78C8×C23—42no—
79C8×C23—82no—
80C8×C23—161no—
81#1856C2×C2311yescharacteristic
82#1318C2×C2311yescharacteristic
83#1270C2×C2311yescharacteristic
84C22×C23—21no—
85C22×C23—21no—
86C22×C23—41no—
87C22×C23—41no—
88C22×C23—81no—
89C22×C23—81no—
90C22×C23—81no—
91C22×C23—81no—
92C22×C23—161no—
93C22×C23#32411yescharacteristic
94C22×C23AΣL(1,𝔽16)11yescharacteristic
95C22×C23#35211yescharacteristic
96C4×C23—21no—
97C4×C23—21no—
98C4×C23—41no—
99C4×C23—41no—
100C4×C23—81no—
101C4×C23—81no—
102C4×C23—81no—
103#518C4×C2311yescharacteristic
104#428—21no—
105C2×#100C4×C2311yescharacteristic
106#390C22×C2311yescharacteristic
107#339—21no—
108#336—21no—
109#324—21no—
110C2×C23—21no—
111C2×C23—21no—
112C2×C23—81no—
113C2×C23—81no—
114C2×C23#132611yescharacteristic
115C2×C23#140211yescharacteristic
116C2×C23#140311yescharacteristic
117C22×D4C2×C4×C2311yescharacteristic
118C2×M16—21no—
119C22×C8—41no—
120C4×D4—41no—
121C4×D4—41no—
122#117—21no—
123C2×C4⋊C4D4×C2311yescharacteristic
124C2×C22⋊C4—41no—
125C2×C22⋊C4—41no—
126C2×C42D4×C2311yescharacteristic
127#107—41no—
128#106—41no—
129#100—21no—
130#100C8×C2312yescomplemented
131C22⋊C8—41no—
132C23#737511yescharacteristic, direct product factor, Sylow
133C24—21no—
134C2×D4—21no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4C2×C8×C2311yescharacteristic
138C2×D4M16×C2311yescharacteristic
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4C22⋊C4×C2311yescharacteristic
144M16—22no—
145M16—41no—
146C2×C8—41no—
147C2×C8—42no—
148C2×C8—81no—
149C2×C8—81no—
150C2×C8—81no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—41no—
154C22⋊C4—42no—
155C22⋊C4—41no—
156C22⋊C4—81no—
157C22⋊C4—81no—
158C42—41no—
159C42—41no—
160C23—21no—
161C23—21no—
162C23—41no—
163C23—41no—
164C23—81no—
165C23—81no—
166C23C23×#9911yescharacteristic
167D4—41no—
168D4—41no—
169D4—42no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—81no—
180C2×C4—161no—
181C2×C4—161no—
182C2×C4C23×C22⋊C811yescharacteristic
183C2×C4C23×#10111yescharacteristic
184C8—42no—
185C8—82no—
186C8—161no—
187C22—21no—
188C22—21no—
189C22—41no—
190C22—41no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22—161no—
196C22C23×#32411yescharacteristic
197C22C23×AΣL(1,𝔽16)11yescharacteristic
198C22C23×#35211yescharacteristic
199C4—21no—
200C4—21no—
201C4—41no—
202C4—41no—
203C4—81no—
204C4—81no—
205C4—81no—
206C2—21no—
207C2—21no—
208C2—81no—
209C2—81no—
210C2C23×#132611yescharacteristic
211C2C23×#140211yescharacteristic
212C2C23×#140311yescharacteristic
subgroupG/H
center, socleC22×C23AΣL(1,𝔽16)
commutatorC2×D4C2×C8×C23
Frattini, cosocle#390C22×C23
prime radicalC22×D4×C23C2×C4
LCSG ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C22×C4×C23 ◁ C23×#390 ◁ G

Complements

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

NH ≅ G/Ntype
24. C23×#100C8semidirect
53. #7375C23direct (N × H)
130. #100C8×C23semidirect
132. C23#7375direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#18562
C23×#13182
C23×#12702
#737523