Finite Group Databasebeta

← C23×#12225 #19159851

Subgroups — C23×#12225 #19159851

Order 5888.

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

normal subgroups only

This group has 336 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×#2809C211yesmaximal, characteristic
2C23×#2028C211yesmaximal, characteristic
3C23×#1617C212yesmaximal
4C23×#1353C212yesmaximal
5C23×#1319C211yesmaximal, characteristic
6C23×#544C2211yescharacteristic
7C23×#484—22no—
8C23×#419—21no—
9C23×#400C2212yes—
10C23×#398—22no—
11C23×#391—21no—
12C23×#391C2211yescharacteristic
13C23×#389—21no—
14C23×#387—22no—
15C23×#343C2211yescharacteristic
16C23×#339—21no—
17C23×#328—22no—
18C23×#324C2212yes—
19C22×Q8×C23D411yescharacteristic
20C2×M16×C23—21no—
21C23×C4⋊Q8—41no—
22C23×#125—41no—
23C23×C22⋊Q8—22no—
24C23×#117—21no—
25C2×C4⋊C4×C23—22no—
26C2×C4⋊C4×C23—21no—
27C2×C4⋊C4×C23—21no—
28C2×C4⋊C4×C23—21no—
29C2×C4⋊C4×C23D412yes—
30C2×C4⋊C4×C23C2311yescharacteristic
31C2×C42×C23—21no—
32C2×C42×C23D411yescharacteristic
33C23×#107—42no—
34C23×#103—41no—
35C23×#103—41no—
36C23×#103—42no—
37C23×C22⋊C8—22no—
38C23×#95—22no—
39C23×#95—22no—
40C23×#95—22no—
41C23×#95—22no—
42C23×#95D412yes—
43C2×Q8×C23—21no—
44C2×Q8×C23—22no—
45C2×Q8×C23—41no—
46C2×Q8×C23—41no—
47C2×Q8×C23Q1612yes—
48C22×C4×C23—21no—
49C22×C4×C23—21no—
50C22×C4×C23—22no—
51C22×C4×C23—22no—
52C22×C4×C23C2×D411yescharacteristic
53C22×C4×C23C2×D412yes—
54M16×C23—41no—
55M16×C23—41no—
56C2×C8×C23—41no—
57C2×C8×C23—42no—
58C4⋊C4×C23—21no—
59C4⋊C4×C23—21no—
60C4⋊C4×C23—22no—
61C4⋊C4×C23—41no—
62C4⋊C4×C23—42no—
63C4⋊C4×C23—44no—
64C4⋊C4×C23—42no—
65C4⋊C4×C23—42no—
66C4⋊C4×C23—81no—
67C4⋊C4×C23—81no—
68C22⋊C4×C23—41no—
69C42×C23—41no—
70C42×C23—41no—
71C42×C23—81no—
72#12225C2311yesmaximal, characteristic, direct product factor, Sylow
73C23×C23#12011yescharacteristic
74Q8×C23—42no—
75Q8×C23—81no—
76Q8×C23—82no—
77C2×C4×C23—21no—
78C2×C4×C23—21no—
79C2×C4×C23—22no—
80C2×C4×C23—22no—
81C2×C4×C23—22no—
82C2×C4×C23—41no—
83C2×C4×C23—41no—
84C2×C4×C23—41no—
85C2×C4×C23—41no—
86C2×C4×C23—42no—
87C2×C4×C23—42no—
88C2×C4×C23—42no—
89C2×C4×C23—42no—
90C2×C4×C23—42no—
91C2×C4×C23—42no—
92C2×C4×C23—41no—
93C2×C4×C23—41no—
94C2×C4×C23—42no—
95C2×C4×C23C2×Q1611yescharacteristic
96C2×C4×C23#13711yescharacteristic
97C8×C23—81no—
98C8×C23—82no—
99#2809C2×C2311yescharacteristic
100#2028C2×C2311yescharacteristic
101#1617C2×C2312yes—
102#1353C2×C2312yes—
103#1319C2×C2311yescharacteristic
104C22×C23—21no—
105C22×C23—21no—
106C22×C23#45111yescharacteristic
107C22×C23#45511yescharacteristic
108C22×C23B(4,𝔽2)11yescharacteristic
109C4×C23—21no—
110C4×C23—21no—
111C4×C23—42no—
112C4×C23—42no—
113C4×C23—42no—
114C4×C23—82no—
115C4×C23—81no—
116C4×C23—81no—
117C4×C23—82no—
118#544C22×C2311yescharacteristic
119#484—22no—
120#419—21no—
121#400C22×C2312yes—
122#398—22no—
123#391—21no—
124#391C22×C2311yescharacteristic
125#389—21no—
126#387—22no—
127#343C22×C2311yescharacteristic
128#339—21no—
129#328—22no—
130#324C22×C2312yes—
131C2×C23—21no—
132C2×C23—21no—
133C2×C23#160211yescharacteristic
134C2×C23#220111yescharacteristic
135C2×C23#220411yescharacteristic
136C22×Q8D4×C2311yescharacteristic
137C2×M16—21no—
138C4⋊Q8—41no—
139#125—41no—
140C22⋊Q8—22no—
141#117—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4D4×C2312yes—
147C2×C4⋊C4C23×C2311yescharacteristic
148C2×C42—21no—
149C2×C42D4×C2311yescharacteristic
150#107—42no—
151#103—41no—
152#103—41no—
153#103—42no—
154C22⋊C8—22no—
155#95—22no—
156#95—22no—
157#95—22no—
158#95—22no—
159#95D4×C2312yes—
160C23#1222511yescharacteristic, direct product factor, Sylow
161C2×Q8—21no—
162C2×Q8—22no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8Q16×C2312yes—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4C2×D4×C2311yescharacteristic
171C22×C4C2×D4×C2312yes—
172M16—41no—
173M16—41no—
174C2×C8—42no—
175C2×C8—41no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—22no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—44no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—81no—
185C4⋊C4—81no—
186C22⋊C4—41no—
187C42—41no—
188C42—41no—
189C42—81no—
190C23C23×#12011yescharacteristic
191Q8—42no—
192Q8—81no—
193Q8—82no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4C2×Q16×C2311yescharacteristic
213C2×C4C23×#13711yescharacteristic
214C8—81no—
215C8—82no—
216C22—21no—
217C22—21no—
218C22C23×#45111yescharacteristic
219C22C23×#45511yescharacteristic
220C22C23×B(4,𝔽2)11yescharacteristic
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—82no—
227C4—81no—
228C4—82no—
229C4—81no—
230C2—21no—
231C2—21no—
232C2C23×#160211yescharacteristic
233C2C23×#220111yescharacteristic
234C2C23×#220411yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC23×C23#120
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
72. #12225C23direct (N × H)
160. C23#12225direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#28092
C23×#20282
C23×#16172
C23×#13532
C23×#13192
#1222523