Finite Group Databasebeta

← C23×#10217 #19157843

Subgroups — C23×#10217 #19157843

Order 5888.

430 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 98 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×#3255C211yesmaximal, characteristic
2C23×#3237C212yesmaximal
3C23×#1995C211yesmaximal, characteristic
4C23×#1506C212yesmaximal
5C23×#1307C211yesmaximal, characteristic
6C23×#533C2211yescharacteristic
7C23×#533C2212yes—
8C23×#501—22no—
9C23×#491—22no—
10C23×#485—24no—
11C23×#477—22no—
12C23×#475—22no—
13C23×#475—22no—
14C23×#470—22no—
15C23×#462—22no—
16C23×#432C2212yes—
17C23×#423C2211yescharacteristic
18C23×#376C2211yescharacteristic
19C23×#339—22no—
20C23×#328—24no—
21C2×Q16×C23—22no—
22C2×M16×C23D412yes—
23C23×C4⋊Q8D412yes—
24C23×C4⋊Q8D412yes—
25C23×C4⋊Q8D412yes—
26C23×#125—22no—
27C23×#125—21no—
28C23×C22⋊Q8—22no—
29C23×C22⋊Q8—22no—
30C23×C22⋊Q8—22no—
31C4×Q8×C23—22no—
32C4×Q8×C23—22no—
33C4×Q8×C23—22no—
34C23×#117—21no—
35C23×#117—22no—
36C2×C42×C23—21no—
37C2×C42×C23C2311yescharacteristic
38C23×#107—24no—
39C23×#106—22no—
40C23×#106—24no—
41C23×C4⋊C8—22no—
42C23×C4⋊C8—22no—
43C23×#103—24no—
44C23×#103—24no—
45C23×#103—24no—
46C23×C22⋊C8D414yes—
47C23×#97—22no—
48C23×#95—22no—
49C2×Q8×C23—22no—
50C2×Q8×C23—22no—
51C2×Q8×C23—22no—
52C22×C4×C23—41no—
53C22×C4×C23C2×D411yescharacteristic
54C22×C4×C23C2×D412yes—
55Q16×C23—82no—
56M16×C23—42no—
57C2×C8×C23—22no—
58C2×C8×C23—24no—
59C4⋊C4×C23—22no—
60C4⋊C4×C23—22no—
61C4⋊C4×C23—22no—
62C4⋊C4×C23—22no—
63C4⋊C4×C23—22no—
64C4⋊C4×C23—22no—
65C4⋊C4×C23—42no—
66C4⋊C4×C23—42no—
67C4⋊C4×C23—42no—
68C22⋊C4×C23—41no—
69C22⋊C4×C23—42no—
70C42×C23—22no—
71C42×C23—41no—
72C42×C23—41no—
73C42×C23—42no—
74C42×C23C2×D412yes—
75C42×C23C2×D411yescharacteristic
76C42×C23#4111yescharacteristic
77#10217C2311yesmaximal, characteristic, direct product factor, Sylow
78C23×C23#12711yescharacteristic
79Q8×C23—44no—
80Q8×C23—42no—
81Q8×C23—42no—
82Q8×C23—82no—
83C2×C4×C23—21no—
84C2×C4×C23—21no—
85C2×C4×C23—42no—
86C2×C4×C23—42no—
87C2×C4×C23—42no—
88C2×C4×C23—42no—
89C2×C4×C23—42no—
90C2×C4×C23—82no—
91C2×C4×C23#12012yes—
92C2×C4×C23#12011yescharacteristic
93C2×C4×C23#12112yes—
94C2×C4×C23#12111yescharacteristic
95C8×C23—44no—
96C8×C23—42no—
97#3255C2×C2311yescharacteristic
98#3237C2×C2312yes—
99#1995C2×C2311yescharacteristic
100#1506C2×C2312yes—
101#1307C2×C2311yescharacteristic
102C22×C23—21no—
103C22×C23—22no—
104C22×C23#39211yescharacteristic
105C4×C23—21no—
106C4×C23—22no—
107C4×C23—22no—
108C4×C23—82no—
109C4×C23—82no—
110C4×C23—82no—
111C4×C23—82no—
112C4×C23#45511yescharacteristic
113C4×C23#45611yescharacteristic
114#533C22×C2311yescharacteristic
115#533C22×C2312yes—
116#501—22no—
117#491—22no—
118#485—24no—
119#477—22no—
120#475—22no—
121#475—22no—
122#470—22no—
123#462—22no—
124#432C22×C2312yes—
125#423C22×C2311yescharacteristic
126#376C22×C2311yescharacteristic
127#339—22no—
128#328—24no—
129C2×C23—41no—
130C2×C23#200711yescharacteristic
131C2×C23#200912yes—
132C2×Q16—22no—
133C2×M16D4×C2312yes—
134C4⋊Q8D4×C2312yes—
135C4⋊Q8D4×C2312yes—
136C4⋊Q8D4×C2312yes—
137#125—22no—
138#125—21no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145#117—21no—
146#117—22no—
147C2×C42—21no—
148C2×C42C23×C2311yescharacteristic
149#107—24no—
150#106—22no—
151#106—24no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154#103—24no—
155#103—24no—
156#103—24no—
157C22⋊C8D4×C2314yes—
158#97—22no—
159#95—22no—
160C23#1021711yescharacteristic, direct product factor, Sylow
161C2×Q8—22no—
162C2×Q8—22no—
163C2×Q8—22no—
164C22×C4—41no—
165C22×C4C2×D4×C2311yescharacteristic
166C22×C4C2×D4×C2312yes—
167Q16—82no—
168M16—42no—
169C2×C8—24no—
170C2×C8—22no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C22⋊C4—41no—
181C22⋊C4—42no—
182C42—22no—
183C42—41no—
184C42—41no—
185C42—42no—
186C42C2×D4×C2311yescharacteristic
187C42C2×D4×C2312yes—
188C42#41×C2311yescharacteristic
189C23C23×#12711yescharacteristic
190Q8—44no—
191Q8—42no—
192Q8—42no—
193Q8—82no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—82no—
202C2×C4C23×#12012yes—
203C2×C4C23×#12011yescharacteristic
204C2×C4C23×#12112yes—
205C2×C4C23×#12111yescharacteristic
206C8—44no—
207C8—42no—
208C22—21no—
209C22—22no—
210C22C23×#39211yescharacteristic
211C4—21no—
212C4—22no—
213C4—22no—
214C4—82no—
215C4—82no—
216C4—82no—
217C4—82no—
218C4C23×#45511yescharacteristic
219C4C23×#45611yescharacteristic
220C2—41no—
221C2C23×#200711yescharacteristic
222C2C23×#200912yes—
subgroupG/H
center, socleC22×C23#392
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC23×C23#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
77. #10217C23direct (N × H)
160. C23#10217direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32552
C23×#32372
C23×#19952
C23×#15062
C23×#13072
#1021723