Finite Group Databasebeta

← C23×#12220 #19159846

Subgroups — C23×#12220 #19159846

Order 5888.

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

normal subgroups only

This group has 460 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×#2683C211yesmaximal, characteristic
2C23×#2027C211yesmaximal, characteristic
3C23×#1605C212yesmaximal, complemented
4C23×#1354C212yesmaximal, complemented
5C23×#1318C211yesmaximal, characteristic
6C23×#544C2211yescharacteristic
7C2×C23×C22⋊Q8—22no—
8C23×#451—22no—
9C23×#418—21no—
10C23×#396—22no—
11C23×#395C2212yes—
12C23×#390C2211yescharacteristic
13C23×#389—21no—
14C23×#386—21no—
15C23×#343C2211yescharacteristic
16C23×#339—21no—
17C23×#328—22no—
18C23×#324C2212yes—
19C22×Q8×C23D412yes—
20C22×D4×C23D411yescharacteristic
21C23×C4×C23—21no—
22C2×Q16×C23—42no—
23C2×M16×C23—21no—
24C23×C4⋊Q8—41no—
25C23×#125—41no—
26C23×C22⋊Q8—22no—
27C23×C22⋊Q8—48no—
28C23×#117—21no—
29C2×C4⋊C4×C23—21no—
30C2×C4⋊C4×C23—22no—
31C2×C4⋊C4×C23—22no—
32C2×C4⋊C4×C23—21no—
33C2×C4⋊C4×C23C2311yescharacteristic
34C2×C22⋊C4×C23—22no—
35C2×C22⋊C4×C23—22no—
36C2×C22⋊C4×C23—21no—
37C2×C42×C23D411yescharacteristic
38C23×#103—42no—
39C23×#102—41no—
40C23×#102—41no—
41C23×C22⋊C8—22no—
42C23×#95—22no—
43C23×#95D412yes—
44C24×C23—21no—
45C2×Q8×C23—22no—
46C2×Q8×C23—44no—
47C2×Q8×C23—42no—
48C2×D4×C23—21no—
49C2×D4×C23—41no—
50C2×D4×C23—41no—
51C2×D4×C23Q1612yescomplemented
52C22×C4×C23—21no—
53C22×C4×C23—22no—
54C22×C4×C23—22no—
55C22×C4×C23—44no—
56C22×C4×C23—81no—
57C22×C4×C23C2×D411yescharacteristic
58C22×C4×C23C2×D412yes—
59Q16×C23—84no—
60M16×C23—41no—
61M16×C23—41no—
62C2×C8×C23—41no—
63C2×C8×C23—42no—
64C4⋊C4×C23—24no—
65C4⋊C4×C23—21no—
66C4⋊C4×C23—21no—
67C4⋊C4×C23—22no—
68C4⋊C4×C23—41no—
69C4⋊C4×C23—44no—
70C4⋊C4×C23—44no—
71C4⋊C4×C23—42no—
72C4⋊C4×C23—81no—
73C22⋊C4×C23—44no—
74C22⋊C4×C23—44no—
75C22⋊C4×C23—41no—
76C22⋊C4×C23—81no—
77C42×C23—41no—
78C42×C23—41no—
79#12220C2311yesmaximal, characteristic, direct product factor, Sylow
80C23×C23—22no—
81C23×C23—42no—
82C23×C23—42no—
83C23×C23—41no—
84C23×C23—41no—
85C23×C23#12011yescharacteristic
86Q8×C23—44no—
87Q8×C23—82no—
88Q8×C23—82no—
89D4×C23—42no—
90D4×C23—81no—
91C2×C4×C23—21no—
92C2×C4×C23—21no—
93C2×C4×C23—22no—
94C2×C4×C23—24no—
95C2×C4×C23—22no—
96C2×C4×C23—42no—
97C2×C4×C23—41no—
98C2×C4×C23—41no—
99C2×C4×C23—42no—
100C2×C4×C23—42no—
101C2×C4×C23—42no—
102C2×C4×C23—42no—
103C2×C4×C23—42no—
104C2×C4×C23—41no—
105C2×C4×C23—41no—
106C2×C4×C23—84no—
107C2×C4×C23C2×Q1611yescharacteristic
108C2×C4×C23#13711yescharacteristic
109C8×C23—81no—
110C8×C23—82no—
111#2683C2×C2311yescharacteristic
112#2027C2×C2311yescharacteristic
113#1605C2×C2312yescomplemented
114#1354C2×C2312yescomplemented
115#1318C2×C2311yescharacteristic
116C22×C23—21no—
117C22×C23—21no—
118C22×C23—44no—
119C22×C23—42no—
120C22×C23—82no—
121C22×C23—81no—
122C22×C23—81no—
123C22×C23#45111yescharacteristic
124C22×C23#45511yescharacteristic
125C22×C23B(4,𝔽2)11yescharacteristic
126C4×C23—21no—
127C4×C23—21no—
128C4×C23—42no—
129C4×C23—42no—
130C4×C23—44no—
131C4×C23—82no—
132C4×C23—81no—
133C4×C23—81no—
134#544C22×C2311yescharacteristic
135C2×C22⋊Q8—22no—
136#451—22no—
137#418—21no—
138#396—22no—
139#395C22×C2312yes—
140#390C22×C2311yescharacteristic
141#389—21no—
142#386—21no—
143#343C22×C2311yescharacteristic
144#339—21no—
145#328—22no—
146#324C22×C2312yes—
147C2×C23—21no—
148C2×C23—21no—
149C2×C23—82no—
150C2×C23#160211yescharacteristic
151C2×C23#219611yescharacteristic
152C2×C23#220311yescharacteristic
153C22×Q8D4×C2312yes—
154C22×D4D4×C2311yescharacteristic
155C23×C4—21no—
156C2×Q16—42no—
157C2×M16—21no—
158C4⋊Q8—41no—
159#125—41no—
160C22⋊Q8—22no—
161C22⋊Q8—48no—
162#117—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4C23×C2311yescharacteristic
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—21no—
171C2×C42D4×C2311yescharacteristic
172#103—42no—
173#102—41no—
174#102—41no—
175C22⋊C8—22no—
176#95—22no—
177#95D4×C2312yes—
178C23#1222011yescharacteristic, direct product factor, Sylow
179C24—21no—
180C2×Q8—22no—
181C2×Q8—44no—
182C2×Q8—42no—
183C2×D4—21no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4Q16×C2312yescomplemented
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—44no—
191C22×C4—81no—
192C22×C4C2×D4×C2311yescharacteristic
193C22×C4C2×D4×C2312yes—
194Q16—84no—
195M16—41no—
196M16—41no—
197C2×C8—42no—
198C2×C8—41no—
199C4⋊C4—24no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—22no—
203C4⋊C4—41no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—42no—
207C4⋊C4—81no—
208C22⋊C4—44no—
209C22⋊C4—44no—
210C22⋊C4—41no—
211C22⋊C4—81no—
212C42—41no—
213C42—41no—
214C23—22no—
215C23—42no—
216C23—42no—
217C23—41no—
218C23—41no—
219C23C23×#12011yescharacteristic
220Q8—44no—
221Q8—82no—
222Q8—82no—
223D4—42no—
224D4—81no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—84no—
241C2×C4C2×Q16×C2311yescharacteristic
242C2×C4C23×#13711yescharacteristic
243C8—81no—
244C8—82no—
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—81no—
252C22C23×#45111yescharacteristic
253C22C23×#45511yescharacteristic
254C22C23×B(4,𝔽2)11yescharacteristic
255C4—21no—
256C4—21no—
257C4—42no—
258C4—42no—
259C4—44no—
260C4—82no—
261C4—81no—
262C4—81no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2C23×#160211yescharacteristic
267C2C23×#219611yescharacteristic
268C2C23×#220311yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC22×D4×C23D4
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C2×C4⋊C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1605C2semidirect
4. C23×#1354C2semidirect
51. C2×D4×C23Q16semidirect
79. #12220C23direct (N × H)
113. #1605C2×C23semidirect
114. #1354C2×C23semidirect
178. C23#12220direct (N × H)
186. C2×D4Q16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#26832
C23×#20272
C23×#16052
C23×#13542
C23×#13182
#1222023