Finite Group Databasebeta

← C23×#12221 #19159847

Subgroups — C23×#12221 #19159847

Order 5888.

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

normal subgroups only

This group has 392 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×#2804C211yesmaximal, characteristic
2C23×#2027C211yesmaximal, characteristic
3C23×#1617C212yesmaximal, complemented
4C23×#1353C212yesmaximal, complemented
5C23×#1318C211yesmaximal, characteristic
6C23×#544C2211yescharacteristic
7C23×#484—22no—
8C23×#418—21no—
9C23×#400C2212yes—
10C23×#393—22no—
11C23×#391—21no—
12C23×#390C2211yescharacteristic
13C23×#386—21no—
14C23×#385—22no—
15C23×#343C2211yescharacteristic
16C23×#339—21no—
17C23×#328—22no—
18C23×#324C2212yes—
19C22×Q8×C23—21no—
20C22×D4×C23D411yescharacteristic
21C23×C4×C23—21no—
22C2×M16×C23—21no—
23C23×C4⋊Q8—41no—
24C23×#125—41no—
25C23×C22⋊Q8—22no—
26C23×#117—21no—
27C2×C4⋊C4×C23—21no—
28C2×C4⋊C4×C23D412yes—
29C2×C4⋊C4×C23C2311yescharacteristic
30C2×C22⋊C4×C23—22no—
31C2×C22⋊C4×C23—22no—
32C2×C22⋊C4×C23—21no—
33C2×C42×C23D411yescharacteristic
34C23×#107—42no—
35C23×#103—42no—
36C23×#102—41no—
37C23×#102—41no—
38C23×C22⋊C8—22no—
39C23×#95—22no—
40C23×#95—22no—
41C23×#95—22no—
42C23×#95D412yes—
43C24×C23—21no—
44C2×Q8×C23—22no—
45C2×Q8×C23—42no—
46C2×Q8×C23—81no—
47C2×Q8×C23—81no—
48C2×D4×C23—21no—
49C2×D4×C23—41no—
50C2×D4×C23—41no—
51C2×D4×C23Q1612yes—
52C22×C4×C23—21no—
53C22×C4×C23—22no—
54C22×C4×C23—22no—
55C22×C4×C23—82no—
56C22×C4×C23—81no—
57C22×C4×C23C2×D411yescharacteristic
58C22×C4×C23C2×D412yes—
59M16×C23—41no—
60M16×C23—41no—
61C2×C8×C23—41no—
62C2×C8×C23—42no—
63C4⋊C4×C23—21no—
64C4⋊C4×C23—21no—
65C4⋊C4×C23—22no—
66C4⋊C4×C23—41no—
67C4⋊C4×C23—42no—
68C4⋊C4×C23—42no—
69C4⋊C4×C23—81no—
70C22⋊C4×C23—44no—
71C22⋊C4×C23—41no—
72C22⋊C4×C23—82no—
73C22⋊C4×C23—81no—
74C42×C23—41no—
75C42×C23—41no—
76#12221C2311yesmaximal, characteristic, direct product factor, Sylow
77C23×C23—22no—
78C23×C23—42no—
79C23×C23—42no—
80C23×C23—41no—
81C23×C23—41no—
82C23×C23#12011yescharacteristic
83Q8×C23—82no—
84Q8×C23—84no—
85D4×C23—42no—
86D4×C23—81no—
87C2×C4×C23—21no—
88C2×C4×C23—21no—
89C2×C4×C23—22no—
90C2×C4×C23—22no—
91C2×C4×C23—41no—
92C2×C4×C23—41no—
93C2×C4×C23—42no—
94C2×C4×C23—42no—
95C2×C4×C23—42no—
96C2×C4×C23—42no—
97C2×C4×C23—42no—
98C2×C4×C23—42no—
99C2×C4×C23—41no—
100C2×C4×C23—41no—
101C2×C4×C23—42no—
102C2×C4×C23—84no—
103C2×C4×C23C2×Q1611yescharacteristic
104C2×C4×C23#13711yescharacteristic
105C8×C23—81no—
106C8×C23—82no—
107#2804C2×C2311yescharacteristic
108#2027C2×C2311yescharacteristic
109#1617C2×C2312yescomplemented
110#1353C2×C2312yescomplemented
111#1318C2×C2311yescharacteristic
112C22×C23—21no—
113C22×C23—21no—
114C22×C23—44no—
115C22×C23—42no—
116C22×C23—82no—
117C22×C23—81no—
118C22×C23—81no—
119C22×C23#45111yescharacteristic
120C22×C23#45511yescharacteristic
121C22×C23B(4,𝔽2)11yescharacteristic
122C4×C23—21no—
123C4×C23—21no—
124C4×C23—42no—
125C4×C23—42no—
126C4×C23—82no—
127C4×C23—81no—
128C4×C23—81no—
129C4×C23—82no—
130#544C22×C2311yescharacteristic
131#484—22no—
132#418—21no—
133#400C22×C2312yes—
134#393—22no—
135#391—21no—
136#390C22×C2311yescharacteristic
137#386—21no—
138#385—22no—
139#343C22×C2311yescharacteristic
140#339—21no—
141#328—22no—
142#324C22×C2312yes—
143C2×C23—21no—
144C2×C23—21no—
145C2×C23—82no—
146C2×C23#160211yescharacteristic
147C2×C23#219811yescharacteristic
148C2×C23#220211yescharacteristic
149C22×Q8—21no—
150C22×D4D4×C2311yescharacteristic
151C23×C4—21no—
152C2×M16—21no—
153C4⋊Q8—41no—
154#125—41no—
155C22⋊Q8—22no—
156#117—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4D4×C2312yes—
159C2×C4⋊C4C23×C2311yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C42D4×C2311yescharacteristic
164#107—42no—
165#103—42no—
166#102—41no—
167#102—41no—
168C22⋊C8—22no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95D4×C2312yes—
173C23#1222111yescharacteristic, direct product factor, Sylow
174C24—21no—
175C2×Q8—22no—
176C2×Q8—42no—
177C2×Q8—81no—
178C2×Q8—81no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4Q16×C2312yes—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—82no—
187C22×C4—81no—
188C22×C4C2×D4×C2311yescharacteristic
189C22×C4C2×D4×C2312yes—
190M16—41no—
191M16—41no—
192C2×C8—42no—
193C2×C8—41no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—22no—
197C4⋊C4—41no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—81no—
201C22⋊C4—44no—
202C22⋊C4—41no—
203C22⋊C4—82no—
204C22⋊C4—81no—
205C42—41no—
206C42—41no—
207C23—22no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23C23×#12011yescharacteristic
213Q8—82no—
214Q8—84no—
215D4—42no—
216D4—81no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—84no—
233C2×C4C2×Q16×C2311yescharacteristic
234C2×C4C23×#13711yescharacteristic
235C8—81no—
236C8—82no—
237C22—21no—
238C22—21no—
239C22—44no—
240C22—42no—
241C22—82no—
242C22—81no—
243C22—81no—
244C22C23×#45111yescharacteristic
245C22C23×#45511yescharacteristic
246C22C23×B(4,𝔽2)11yescharacteristic
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—82no—
252C4—81no—
253C4—82no—
254C4—81no—
255C2—21no—
256C2—21no—
257C2—82no—
258C2C23×#160211yescharacteristic
259C2C23×#219811yescharacteristic
260C2C23×#220211yescharacteristic
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×#1617C2semidirect
4. C23×#1353C2semidirect
76. #12221C23direct (N × H)
109. #1617C2×C23semidirect
110. #1353C2×C23semidirect
173. C23#12221direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#28042
C23×#20272
C23×#16172
C23×#13532
C23×#13182
#1222123