Finite Group Databasebeta

← C23×#10389 #19158015

Subgroups — C23×#10389 #19158015

Order 5888.

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

normal subgroups only

This group has 406 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×#2608C211yesmaximal, characteristic
2C23×#1826C212yesmaximal
3C23×#1820C211yesmaximal, characteristic
4C23×#1306C211yesmaximal, characteristic
5C23×#1284C212yesmaximal
6C2×C23×#117C2211yescharacteristic
7C23×#407C2212yes—
8C23×#396—22no—
9C23×#389—22no—
10C23×#387—22no—
11C23×#381C2211yescharacteristic
12C23×#381C2211yescharacteristic
13C4×C22⋊C4×C23C2212yes—
14C23×#342—22no—
15C23×#339—22no—
16C23×#339—22no—
17C23×C4×C23C2311yescharacteristic
18C2×M16×C23—22no—
19C23×#117—22no—
20C23×#117—22no—
21C2×C4⋊C4×C23—22no—
22C2×C4⋊C4×C23—22no—
23C2×C4⋊C4×C23—22no—
24C2×C4⋊C4×C23Q812yes—
25C2×C22⋊C4×C23—21no—
26C2×C22⋊C4×C23—22no—
27C2×C22⋊C4×C23—21no—
28C2×C22⋊C4×C23D412yes—
29C2×C42×C23—21no—
30C2×C42×C23D412yes—
31C2×C42×C23D412yes—
32C23×C22⋊C8—22no—
33C23×C22⋊C8—22no—
34C23×#95—22no—
35C23×#95—22no—
36C23×#95—22no—
37C23×#95—22no—
38C24×C23C2×D411yescharacteristic
39C22×C4×C23—22no—
40C22×C4×C23—22no—
41C22×C4×C23—22no—
42C22×C4×C23—22no—
43C22×C4×C23—42no—
44C22×C4×C23—41no—
45C22×C4×C23C2×D412yes—
46C22×C4×C23C2×Q811yescharacteristic
47C22×C4×C23#4112yes—
48C22×C4×C23#4111yescharacteristic
49M16×C23—42no—
50M16×C23—42no—
51C2×C8×C23—42no—
52C4⋊C4×C23—24no—
53C4⋊C4×C23—44no—
54C4⋊C4×C23—44no—
55C4⋊C4×C23—82no—
56C22⋊C4×C23—44no—
57C22⋊C4×C23—41no—
58C22⋊C4×C23—41no—
59C22⋊C4×C23—42no—
60C22⋊C4×C23—42no—
61C42×C23—42no—
62C42×C23—42no—
63C42×C23—42no—
64#10389C2311yesmaximal, characteristic, direct product factor, Sylow
65C23×C23—22no—
66C23×C23—42no—
67C23×C23#12011yescharacteristic
68C23×C23C22⋊Q811yescharacteristic
69C23×C23#12411yescharacteristic
70C2×C4×C23—22no—
71C2×C4×C23—22no—
72C2×C4×C23—24no—
73C2×C4×C23—41no—
74C2×C4×C23—41no—
75C2×C4×C23—42no—
76C2×C4×C23—42no—
77C2×C4×C23—42no—
78C2×C4×C23—42no—
79C2×C4×C23—42no—
80C2×C4×C23—42no—
81C2×C4×C23—42no—
82C2×C4×C23—42no—
83C2×C4×C23—42no—
84C2×C4×C23—42no—
85C2×C4×C23—82no—
86C2×C4×C23—81no—
87C2×C4×C23—82no—
88C2×C4×C23C22⋊Q812yes—
89C2×C4×C23#12412yes—
90C8×C23—82no—
91#2608C2×C2311yescharacteristic
92#1826C2×C2312yes—
93#1820C2×C2311yescharacteristic
94#1306C2×C2311yescharacteristic
95#1284C2×C2312yes—
96C22×C23—22no—
97C22×C23—21no—
98C22×C23—21no—
99C22×C23—22no—
100C22×C23—22no—
101C22×C23—82no—
102C22×C23#39311yescharacteristic
103C22×C23#45812yes—
104C4×C23—22no—
105C4×C23—22no—
106C4×C23—44no—
107C4×C23—82no—
108C4×C23—82no—
109C4×C23—82no—
110C4×C23—161no—
111C2×#117C22×C2311yescharacteristic
112#407C22×C2312yes—
113#396—22no—
114#389—22no—
115#387—22no—
116#381C22×C2311yescharacteristic
117#381C22×C2311yescharacteristic
118C4×C22⋊C4C22×C2312yes—
119#342—22no—
120#339—22no—
121#339—22no—
122C2×C23—22no—
123C2×C23—41no—
124C2×C23—41no—
125C2×C23#203011yescharacteristic
126C2×C23#203112yes—
127C23×C4C23×C2311yescharacteristic
128C2×M16—22no—
129#117—22no—
130#117—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4Q8×C2312yes—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4D4×C2312yes—
139C2×C42—21no—
140C2×C42D4×C2312yes—
141C2×C42D4×C2312yes—
142C22⋊C8—22no—
143C22⋊C8—22no—
144#95—22no—
145#95—22no—
146#95—22no—
147#95—22no—
148C23#1038911yescharacteristic, direct product factor, Sylow
149C24C2×D4×C2311yescharacteristic
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—42no—
155C22×C4—41no—
156C22×C4C2×D4×C2312yes—
157C22×C4C2×Q8×C2311yescharacteristic
158C22×C4#41×C2312yes—
159C22×C4#41×C2311yescharacteristic
160M16—42no—
161M16—42no—
162C2×C8—42no—
163C4⋊C4—24no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C4⋊C4—82no—
167C22⋊C4—44no—
168C22⋊C4—41no—
169C22⋊C4—41no—
170C22⋊C4—42no—
171C22⋊C4—42no—
172C42—42no—
173C42—42no—
174C42—42no—
175C23—22no—
176C23—42no—
177C23C23×#12011yescharacteristic
178C23C23×C22⋊Q811yescharacteristic
179C23C23×#12411yescharacteristic
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—82no—
196C2×C4—81no—
197C2×C4—82no—
198C2×C4C23×C22⋊Q812yes—
199C2×C4C23×#12412yes—
200C8—82no—
201C22—22no—
202C22—21no—
203C22—21no—
204C22—22no—
205C22—22no—
206C22—82no—
207C22C23×#39311yescharacteristic
208C22C23×#45812yes—
209C4—22no—
210C4—22no—
211C4—44no—
212C4—82no—
213C4—82no—
214C4—82no—
215C4—161no—
216C2—22no—
217C2—41no—
218C2—41no—
219C2C23×#203011yescharacteristic
220C2C23×#203112yes—
subgroupG/H
center, socleC22×C23#393
commutator, Frattini, cosocleC23×C4C23×C23
prime radicalC24×C23C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
64. #10389C23direct (N × H)
148. C23#10389direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#26082
C23×#18262
C23×#18202
C23×#13062
C23×#12842
#1038923