Finite Group Databasebeta

← C23×#12207 #19159833

Subgroups — C23×#12207 #19159833

Order 5888.

388 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2817C211yesmaximal, characteristic
2C23×#2030C211yesmaximal, characteristic
3C23×#1608C212yesmaximal
4C23×#1344C212yesmaximal
5C23×#1317C211yesmaximal, characteristic
6C23×#543C2211yescharacteristic
7C2×C23×C22⋊Q8—21no—
8C23×#483—22no—
9C23×#400—22no—
10C23×#397—21no—
11C23×#397C2212yes—
12C23×#395—22no—
13C23×#381—21no—
14C23×#381C2211yescharacteristic
15C23×#342—21no—
16C23×#342C2211yescharacteristic
17C23×#328—22no—
18C23×#323C2212yes—
19C22×Q8×C23—21no—
20C23×C4×C23D411yescharacteristic
21C23×C22⋊Q8—22no—
22C23×C22⋊Q8—42no—
23C23×C22⋊Q8—42no—
24C23×C22⋊Q8—42no—
25C2×C4⋊C4×C23—22no—
26C2×C4⋊C4×C23—21no—
27C2×C4⋊C4×C23—21no—
28C2×C4⋊C4×C23D412yes—
29C2×C22⋊C4×C23—21no—
30C2×C22⋊C4×C23—22no—
31C2×C22⋊C4×C23—21no—
32C2×C22⋊C4×C23D411yescharacteristic
33C2×C22⋊C4×C23C2311yescharacteristic
34C23×#106—42no—
35C23×#103—42no—
36C23×C22⋊C8—22no—
37C23×#95—22no—
38C23×#95—21no—
39C23×#95—22no—
40C23×#95—22no—
41C23×#95—42no—
42C23×#95D412yes—
43C24×C23C2×D411yescharacteristic
44C2×Q8×C23—22no—
45C2×Q8×C23—42no—
46C2×Q8×C23—81no—
47C2×Q8×C23—81no—
48C22×C4×C23—21no—
49C22×C4×C23—21no—
50C22×C4×C23—22no—
51C22×C4×C23—22no—
52C22×C4×C23—21no—
53C22×C4×C23—22no—
54C22×C4×C23—41no—
55C22×C4×C23—41no—
56C22×C4×C23—41no—
57C22×C4×C23SD1612yes—
58C22×C4×C23C2×D412yes—
59C2×C8×C23—42no—
60C4⋊C4×C23—22no—
61C4⋊C4×C23—42no—
62C4⋊C4×C23—42no—
63C4⋊C4×C23—44no—
64C4⋊C4×C23—42no—
65C4⋊C4×C23—81no—
66C22⋊C4×C23—41no—
67C22⋊C4×C23—41no—
68C22⋊C4×C23—41no—
69C22⋊C4×C23—42no—
70C22⋊C4×C23—42no—
71C22⋊C4×C23—41no—
72C22⋊C4×C23—41no—
73C22⋊C4×C23—41no—
74#12207C2311yesmaximal, characteristic, direct product factor, Sylow
75C23×C23—21no—
76C23×C23—21no—
77C23×C23—42no—
78C23×C23#12011yescharacteristic
79C23×C23C2×SD1611yescharacteristic
80C23×C23AGL(1,ℤ/8)11yescharacteristic
81Q8×C23—82no—
82Q8×C23—84no—
83C2×C4×C23—22no—
84C2×C4×C23—22no—
85C2×C4×C23—41no—
86C2×C4×C23—41no—
87C2×C4×C23—42no—
88C2×C4×C23—42no—
89C2×C4×C23—44no—
90C2×C4×C23—42no—
91C2×C4×C23—42no—
92C2×C4×C23—42no—
93C2×C4×C23—41no—
94C2×C4×C23—42no—
95C2×C4×C23—42no—
96C2×C4×C23—42no—
97C2×C4×C23—81no—
98C2×C4×C23—82no—
99C2×C4×C23—81no—
100C2×C4×C23—82no—
101C8×C23—82no—
102#2817C2×C2311yescharacteristic
103#2030C2×C2311yescharacteristic
104#1608C2×C2312yes—
105#1344C2×C2312yes—
106#1317C2×C2311yescharacteristic
107C22×C23—22no—
108C22×C23—21no—
109C22×C23—21no—
110C22×C23—41no—
111C22×C23—41no—
112C22×C23—42no—
113C22×C23—81no—
114C22×C23#45011yescharacteristic
115C22×C23#45511yescharacteristic
116C22×C23B(4,𝔽2)11yescharacteristic
117C4×C23—42no—
118C4×C23—42no—
119C4×C23—82no—
120C4×C23—81no—
121C4×C23—82no—
122C4×C23—82no—
123C4×C23—161no—
124#543C22×C2311yescharacteristic
125C2×C22⋊Q8—21no—
126#483—22no—
127#400—22no—
128#397—21no—
129#397C22×C2312yes—
130#395—22no—
131#381—21no—
132#381C22×C2311yescharacteristic
133#342—21no—
134#342C22×C2311yescharacteristic
135#328—22no—
136#323C22×C2312yes—
137C2×C23—21no—
138C2×C23—21no—
139C2×C23—41no—
140C2×C23—41no—
141C2×C23#160111yescharacteristic
142C2×C23#220112yes—
143C22×Q8—21no—
144C23×C4D4×C2311yescharacteristic
145C22⋊Q8—22no—
146C22⋊Q8—42no—
147C22⋊Q8—42no—
148C22⋊Q8—42no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4D4×C2312yes—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4D4×C2311yescharacteristic
157C2×C22⋊C4C23×C2311yescharacteristic
158#106—42no—
159#103—42no—
160C22⋊C8—22no—
161#95—22no—
162#95—21no—
163#95—22no—
164#95—22no—
165#95—42no—
166#95D4×C2312yes—
167C23#1220711yescharacteristic, direct product factor, Sylow
168C24C2×D4×C2311yescharacteristic
169C2×Q8—22no—
170C2×Q8—42no—
171C2×Q8—81no—
172C2×Q8—81no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4SD16×C2312yes—
183C22×C4C2×D4×C2312yes—
184C2×C8—42no—
185C4⋊C4—22no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—44no—
189C4⋊C4—42no—
190C4⋊C4—81no—
191C22⋊C4—41no—
192C22⋊C4—41no—
193C22⋊C4—41no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—41no—
197C22⋊C4—41no—
198C22⋊C4—41no—
199C23—21no—
200C23—21no—
201C23—42no—
202C23C23×#12011yescharacteristic
203C23C2×SD16×C2311yescharacteristic
204C23C23×AGL(1,ℤ/8)11yescharacteristic
205Q8—82no—
206Q8—84no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—81no—
222C2×C4—82no—
223C2×C4—81no—
224C2×C4—82no—
225C8—82no—
226C22—22no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—81no—
233C22C23×#45011yescharacteristic
234C22C23×#45511yescharacteristic
235C22C23×B(4,𝔽2)11yescharacteristic
236C4—42no—
237C4—42no—
238C4—82no—
239C4—81no—
240C4—82no—
241C4—82no—
242C4—161no—
243C2—21no—
244C2—21no—
245C2—41no—
246C2—41no—
247C2C23×#160111yescharacteristic
248C2C23×#220112yes—
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C23
prime radicalC24×C23C2×D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C2×C22⋊C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
74. #12207C23direct (N × H)
167. C23#12207direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#28172
C23×#20302
C23×#16082
C23×#13442
C23×#13172
#1220723