Finite Group Databasebeta

← C23×#12109 #19159735

Subgroups — C23×#12109 #19159735

Order 5888.

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

normal subgroups only

This group has 410 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×#3202C211yesmaximal, characteristic, complemented
2C23×#2187C212yesmaximal, complemented
3C23×#2018C211yesmaximal, characteristic
4C23×#1350C212yesmaximal, complemented
5C23×#1315C211yesmaximal, characteristic, complemented
6C2×C23×AGL(1,ℤ/8)—21no—
7C22×D8×C23C2211yescharacteristic
8C23×#536—21no—
9C2×C23×SD32—22no—
10C23×#485—22no—
11C23×#470—21no—
12C23×#467C2212yes—
13C23×#451—22no—
14C2×C23×#100—21no—
15C23×#408C2211yescharacteristic
16C23×#357—22no—
17C23×C22⋊C16C2212yes—
18C23×#341C2211yescharacteristic
19C2×#41×C23D411yescharacteristic
20C22×Q8×C23—21no—
21C22×D4×C23—21no—
22C23×AGL(1,ℤ/8)—42no—
23C23×AGL(1,ℤ/8)—42no—
24C2×Q16×C23—21no—
25C2×Q16×C23D412yes—
26C2×SD16×C23—41no—
27C2×D8×C23—22no—
28C2×D8×C23—42no—
29C2×D8×C23—41no—
30C2×M16×C23—21no—
31C2×M16×C23D411yescharacteristic
32C22×C8×C23C2311yescharacteristic
33C23×C4⋊Q8—41no—
34C23×#124—41no—
35C23×#123—41no—
36C23×#123—41no—
37C23×C22⋊Q8—22no—
38C23×C22⋊Q8—22no—
39C23×#117—21no—
40C23×SD32—44no—
41C2×C16×C23—22no—
42C23×#107D412yes—
43C23×#106—21no—
44C23×#103—22no—
45C23×#103—22no—
46C23×#100—42no—
47C23×C22⋊C8—22no—
48C24×C23—41no—
49#41×C23—41no—
50#41×C23—41no—
51C2×Q8×C23—22no—
52C2×Q8×C23—41no—
53C2×Q8×C23—81no—
54C2×Q8×C23—81no—
55C2×Q8×C23D811yescharacteristic
56C2×D4×C23—22no—
57C2×D4×C23—21no—
58C2×D4×C23—41no—
59C2×D4×C23—41no—
60C2×D4×C23—41no—
61C2×D4×C23—41no—
62C2×D4×C23—41no—
63C2×D4×C23D811yescharacteristic
64C22×C4×C23—21no—
65C22×C4×C23—41no—
66C22×C4×C23C2×D411yescharacteristic
67Q16×C23—42no—
68Q16×C23—81no—
69SD16×C23—44no—
70D8×C23—24no—
71D8×C23—42no—
72D8×C23—44no—
73M16×C23—22no—
74M16×C23—41no—
75M16×C23—41no—
76C2×C8×C23—21no—
77C2×C8×C23—41no—
78C2×C8×C23—41no—
79C2×C8×C23C2×D412yes—
80C4⋊C4×C23—22no—
81C4⋊C4×C23—42no—
82C4⋊C4×C23—42no—
83C22⋊C4×C23—41no—
84C22⋊C4×C23—42no—
85C22⋊C4×C23—42no—
86C42×C23—41no—
87C16×C23—42no—
88#12109C2311yesmaximal, characteristic, direct product factor, Sylow
89C23×C23—21no—
90C23×C23—42no—
91C23×C23—81no—
92C23×C23—81no—
93C23×C23—162no—
94C23×C23AGL(1,ℤ/8)11yescharacteristic
95Q8×C23—41no—
96Q8×C23—42no—
97Q8×C23—42no—
98Q8×C23—82no—
99D4×C23—41no—
100D4×C23—42no—
101D4×C23—42no—
102D4×C23—81no—
103D4×C23—81no—
104D4×C23—81no—
105C2×C4×C23—21no—
106C2×C4×C23—21no—
107C2×C4×C23—21no—
108C2×C4×C23—41no—
109C2×C4×C23—41no—
110C2×C4×C23—42no—
111C2×C4×C23—42no—
112C2×C4×C23—82no—
113C2×C4×C23#12011yescharacteristic
114C2×C4×C23C2×D811yescharacteristic
115C8×C23—22no—
116C8×C23—42no—
117C8×C23—41no—
118#3202C2×C2311yescharacteristic, complemented
119#2187C2×C2312yescomplemented
120#2018C2×C2311yescharacteristic
121#1350C2×C2312yescomplemented
122#1315C2×C2311yescharacteristic, complemented
123C22×C23—21no—
124C22×C23—22no—
125C22×C23—41no—
126C22×C23—42no—
127C22×C23—82no—
128C22×C23—162no—
129C22×C23—162no—
130C22×C23—162no—
131C22×C23#44711yescharacteristic
132C4×C23—21no—
133C4×C23—41no—
134C4×C23—82no—
135C4×C23—82no—
136C4×C23#45311yescharacteristic
137C4×C23B(4,𝔽2)11yescharacteristic
138C2×AGL(1,ℤ/8)—21no—
139C22×D8C22×C2311yescharacteristic
140#536—21no—
141C2×SD32—22no—
142#485—22no—
143#470—21no—
144#467C22×C2312yes—
145#451—22no—
146C2×#100—21no—
147#408C22×C2311yescharacteristic
148#357—22no—
149C22⋊C16C22×C2312yes—
150#341C22×C2311yescharacteristic
151C2×C23—41no—
152C2×C23—81no—
153C2×C23—162no—
154C2×C23#159511yescharacteristic
155C2×C23#219312yes—
156C2×#41D4×C2311yescharacteristic
157C22×Q8—21no—
158C22×D4—21no—
159AGL(1,ℤ/8)—42no—
160AGL(1,ℤ/8)—42no—
161C2×Q16—21no—
162C2×Q16D4×C2312yes—
163C2×SD16—41no—
164C2×D8—22no—
165C2×D8—42no—
166C2×D8—41no—
167C2×M16—21no—
168C2×M16D4×C2311yescharacteristic
169C22×C8C23×C2311yescharacteristic
170C4⋊Q8—41no—
171#124—41no—
172#123—41no—
173#123—41no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176#117—21no—
177SD32—44no—
178C2×C16—22no—
179#107D4×C2312yes—
180#106—21no—
181#103—22no—
182#103—22no—
183#100—42no—
184C22⋊C8—22no—
185C23#1210911yescharacteristic, direct product factor, Sylow
186C24—41no—
187#41—41no—
188#41—41no—
189C2×Q8—22no—
190C2×Q8—41no—
191C2×Q8—81no—
192C2×Q8—81no—
193C2×Q8D8×C2311yescharacteristic
194C2×D4—22no—
195C2×D4—21no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4D8×C2311yescharacteristic
202C22×C4—21no—
203C22×C4—41no—
204C22×C4C2×D4×C2311yescharacteristic
205Q16—42no—
206Q16—81no—
207SD16—44no—
208D8—24no—
209D8—42no—
210D8—44no—
211M16—22no—
212M16—41no—
213M16—41no—
214C2×C8—21no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8C2×D4×C2312yes—
218C4⋊C4—22no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C22⋊C4—41no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C42—41no—
225C16—42no—
226C23—21no—
227C23—42no—
228C23—81no—
229C23—81no—
230C23—162no—
231C23C23×AGL(1,ℤ/8)11yescharacteristic
232Q8—41no—
233Q8—42no—
234Q8—42no—
235Q8—82no—
236D4—41no—
237D4—42no—
238D4—42no—
239D4—81no—
240D4—81no—
241D4—81no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—82no—
250C2×C4C23×#12011yescharacteristic
251C2×C4C2×D8×C2311yescharacteristic
252C8—22no—
253C8—42no—
254C8—41no—
255C22—22no—
256C22—21no—
257C22—42no—
258C22—41no—
259C22—82no—
260C22—162no—
261C22—162no—
262C22—162no—
263C22C23×#44711yescharacteristic
264C4—21no—
265C4—41no—
266C4—82no—
267C4—82no—
268C4C23×#45311yescharacteristic
269C4C23×B(4,𝔽2)11yescharacteristic
270C2—41no—
271C2—81no—
272C2—162no—
273C2C23×#159511yescharacteristic
274C2C23×#219312yes—
subgroupG/H
center, socleC22×C23#447
commutator, Frattini, cosocleC22×C8C23×C23
prime radicalC23×#2018C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C23×#408 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3202C2semidirect
2. C23×#2187C2semidirect
4. C23×#1350C2semidirect
5. C23×#1315C2semidirect
88. #12109C23direct (N × H)
118. #3202C2×C23semidirect
119. #2187C2×C23semidirect
121. #1350C2×C23semidirect
122. #1315C2×C23semidirect
185. C23#12109direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32022
C23×#21872
C23×#20182
C23×#13502
C23×#13152
#1210923