Finite Group Databasebeta

← C23×#10017 #19157643

Subgroups — C23×#10017 #19157643

Order 5888.

528 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3104C212yesmaximal
2C23×#2399C211yesmaximal, characteristic
3C23×#1995C211yesmaximal, characteristic
4C23×#1487C212yesmaximal
5C23×#1274C211yesmaximal, characteristic
6C23×#533C2211yescharacteristic
7C2×C4×Q8×C23C2212yes—
8C23×#484—22no—
9C23×#477—22no—
10C23×#462—22no—
11C23×#451—22no—
12C4×Q16×C23—24no—
13C23×#431C2212yes—
14C23×#423C2211yescharacteristic
15C23×#398—22no—
16C23×#395—22no—
17C4×C4⋊C4×C23—22no—
18C23×#376C2211yescharacteristic
19C23×#339—22no—
20C23×Q8⋊C8—24no—
21C22×Q8×C23D412yes—
22C2×Q16×C23—24no—
23C2×M16×C23—21no—
24C23×C4⋊Q8D412yes—
25C23×#125—21no—
26C23×C22⋊Q8—22no—
27C4×Q8×C23—22no—
28C4×Q8×C23—22no—
29C4×Q8×C23—22no—
30C4×Q8×C23—22no—
31C4×Q8×C23—22no—
32C23×#117—21no—
33C2×C4⋊C4×C23—24no—
34C2×C4⋊C4×C23—22no—
35C2×C4⋊C4×C23D412yes—
36C2×C42×C23—22no—
37C2×C42×C23D412yes—
38C2×C42×C23C2311yescharacteristic
39C23×#107—24no—
40C23×C4⋊C8—22no—
41C23×C4⋊C8—22no—
42C23×#103—24no—
43C23×#103—24no—
44C23×C22⋊C8—22no—
45C4×C8×C23D414yes—
46C23×#95—22no—
47C23×#95—22no—
48C23×#95—22no—
49C2×Q8×C23—22no—
50C2×Q8×C23—22no—
51C2×Q8×C23—22no—
52C2×Q8×C23—44no—
53C22×C4×C23—22no—
54C22×C4×C23—22no—
55C22×C4×C23—22no—
56C22×C4×C23C2×D412yes—
57C22×C4×C23C2×D411yescharacteristic
58Q16×C23—48no—
59M16×C23—42no—
60C2×C8×C23—24no—
61C2×C8×C23—41no—
62C4⋊C4×C23—22no—
63C4⋊C4×C23—22no—
64C4⋊C4×C23—22no—
65C4⋊C4×C23—22no—
66C4⋊C4×C23—48no—
67C4⋊C4×C23—42no—
68C4⋊C4×C23—42no—
69C4⋊C4×C23—42no—
70C22⋊C4×C23—41no—
71C42×C23—22no—
72C42×C23—42no—
73C42×C23—42no—
74C42×C23—41no—
75C42×C23Q1614yescomplemented
76C42×C23C2×D411yescharacteristic
77C42×C23C2×D412yes—
78C42×C23#4111yescharacteristic
79#10017C2311yesmaximal, characteristic, direct product factor, Sylow
80C23×C23#12011yescharacteristic
81Q8×C23—24no—
82Q8×C23—24no—
83Q8×C23—42no—
84Q8×C23—42no—
85Q8×C23—44no—
86C2×C4×C23—24no—
87C2×C4×C23—24no—
88C2×C4×C23—42no—
89C2×C4×C23—44no—
90C2×C4×C23—42no—
91C2×C4×C23—42no—
92C2×C4×C23—44no—
93C2×C4×C23—42no—
94C2×C4×C23—42no—
95C2×C4×C23#12012yes—
96C2×C4×C23#12112yes—
97C2×C4×C23#12111yescharacteristic
98C2×C4×C23#12711yescharacteristic
99C2×C4×C23C2×Q1612yes—
100C2×C4×C23#13712yes—
101C8×C23—44no—
102C8×C23—81no—
103#3104C2×C2312yes—
104#2399C2×C2311yescharacteristic
105#1995C2×C2311yescharacteristic
106#1487C2×C2312yes—
107#1274C2×C2311yescharacteristic
108C22×C23—22no—
109C22×C23#39211yescharacteristic
110C22×C23#45112yes—
111C4×C23—22no—
112C4×C23—22no—
113C4×C23—44no—
114C4×C23—44no—
115C4×C23—82no—
116C4×C23—82no—
117C4×C23#46212yes—
118C4×C23#49411yescharacteristic
119C4×C23#49711yescharacteristic
120#533C22×C2311yescharacteristic
121C2×C4×Q8C22×C2312yes—
122#484—22no—
123#477—22no—
124#462—22no—
125#451—22no—
126C4×Q16—24no—
127#431C22×C2312yes—
128#423C22×C2311yescharacteristic
129#398—22no—
130#395—22no—
131C4×C4⋊C4—22no—
132#376C22×C2311yescharacteristic
133#339—22no—
134Q8⋊C8—24no—
135C2×C23—22no—
136C2×C23#201012yes—
137C2×C23#201611yescharacteristic
138C22×Q8D4×C2312yes—
139C2×Q16—24no—
140C2×M16—21no—
141C4⋊Q8D4×C2312yes—
142#125—21no—
143C22⋊Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149#117—21no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4D4×C2312yes—
153C2×C42—22no—
154C2×C42D4×C2312yes—
155C2×C42C23×C2311yescharacteristic
156#107—24no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159#103—24no—
160#103—24no—
161C22⋊C8—22no—
162C4×C8D4×C2314yes—
163#95—22no—
164#95—22no—
165#95—22no—
166C23#1001711yescharacteristic, direct product factor, Sylow
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—22no—
170C2×Q8—44no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4C2×D4×C2312yes—
175C22×C4C2×D4×C2311yescharacteristic
176Q16—48no—
177M16—42no—
178C2×C8—24no—
179C2×C8—41no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—48no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C22⋊C4—41no—
189C42—22no—
190C42—42no—
191C42—42no—
192C42—41no—
193C42Q16×C2314yescomplemented
194C42C2×D4×C2311yescharacteristic
195C42C2×D4×C2312yes—
196C42#41×C2311yescharacteristic
197C23C23×#12011yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—42no—
201Q8—42no—
202Q8—44no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4C23×#12012yes—
213C2×C4C23×#12112yes—
214C2×C4C23×#12111yescharacteristic
215C2×C4C23×#12711yescharacteristic
216C2×C4C2×Q16×C2312yes—
217C2×C4C23×#13712yes—
218C8—44no—
219C8—81no—
220C22—22no—
221C22C23×#39211yescharacteristic
222C22C23×#45112yes—
223C4—22no—
224C4—22no—
225C4—44no—
226C4—44no—
227C4—82no—
228C4—82no—
229C4C23×#46212yes—
230C4C23×#49411yescharacteristic
231C4C23×#49711yescharacteristic
232C2—22no—
233C2C23×#201012yes—
234C2C23×#201611yescharacteristic
subgroupG/H
centerC2×C4×C23#127
commutator, Frattini, cosocleC2×C42C23×C23
socleC22×C23#392
prime radicalC23×C23#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
75. C42×C23Q16semidirect
79. #10017C23direct (N × H)
166. C23#10017direct (N × H)
193. C42Q16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#31042
C23×#23992
C23×#19952
C23×#14872
C23×#12742
#1001723