Finite Group Databasebeta

← C23×#12289 #19159915

Subgroups — C23×#12289 #19159915

Order 5888.

310 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 82 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×#3249C211yesmaximal, characteristic
2C23×#2235C212yesmaximal
3C23×#2086C211yesmaximal, characteristic
4C23×#1348C212yesmaximal
5C23×#1322C211yesmaximal, characteristic
6C23×#533—21no—
7C23×#477—22no—
8C23×#467—21no—
9C23×#467C2212yes—
10C23×#462—22no—
11C2×C23×#107C2211yescharacteristic
12C2×C23×C4⋊C8C2211yescharacteristic
13C23×#389—21no—
14C23×#367—22no—
15C23×#358—22no—
16C23×C22⋊C16C2212yes—
17C23×#340—21no—
18C23×#340C2211yescharacteristic
19C2×Q16×C23—21no—
20C2×Q16×C23D412yes—
21C22×C8×C23D411yescharacteristic
22C22×C8×C23C2311yescharacteristic
23C23×C4⋊Q8—21no—
24C23×C4⋊Q8—21no—
25C23×#125—41no—
26C23×C22⋊Q8—22no—
27C23×C22⋊Q8—41no—
28C4×Q8×C23—22no—
29C4×Q8×C23—41no—
30C23×#117—41no—
31C2×C4⋊C4×C23—21no—
32C2×C4⋊C4×C23—21no—
33C2×C4⋊C4×C23—41no—
34C2×C42×C23D411yescharacteristic
35C2×C16×C23—22no—
36C23×#107—21no—
37C23×#107—22no—
38C23×#107D412yes—
39C23×C4⋊C8—22no—
40C23×#103—22no—
41C23×#103—22no—
42C23×#95—41no—
43C2×Q8×C23—22no—
44C2×Q8×C23—41no—
45C22×C4×C23—41no—
46C22×C4×C23—41no—
47C22×C4×C23D811yescharacteristic
48C22×C4×C23D811yescharacteristic
49C22×C4×C23C2×D411yescharacteristic
50Q16×C23—42no—
51Q16×C23—81no—
52C2×C8×C23—21no—
53C2×C8×C23—41no—
54C2×C8×C23—41no—
55C2×C8×C23C2×D412yes—
56C4⋊C4×C23—22no—
57C4⋊C4×C23—22no—
58C4⋊C4×C23—41no—
59C4⋊C4×C23—41no—
60C4⋊C4×C23—41no—
61C4⋊C4×C23—42no—
62C4⋊C4×C23—41no—
63C4⋊C4×C23—42no—
64C4⋊C4×C23—41no—
65C4⋊C4×C23—41no—
66C4⋊C4×C23—82no—
67C22⋊C4×C23—42no—
68C42×C23—21no—
69C42×C23—21no—
70C42×C23—42no—
71C16×C23—42no—
72#12289C2311yesmaximal, characteristic, direct product factor, Sylow
73C23×C23C2×D811yescharacteristic
74Q8×C23—42no—
75Q8×C23—42no—
76Q8×C23—82no—
77C2×C4×C23—21no—
78C2×C4×C23—21no—
79C2×C4×C23—21no—
80C2×C4×C23—21no—
81C2×C4×C23—21no—
82C2×C4×C23—41no—
83C2×C4×C23—42no—
84C2×C4×C23—42no—
85C2×C4×C23—42no—
86C2×C4×C23—81no—
87C2×C4×C23—81no—
88C2×C4×C23—81no—
89C2×C4×C23—81no—
90C2×C4×C23—81no—
91C2×C4×C23SD3212yes—
92C2×C4×C23#12011yescharacteristic
93C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
94C8×C23—22no—
95C8×C23—41no—
96#3249C2×C2311yescharacteristic
97#2235C2×C2312yes—
98#2086C2×C2311yescharacteristic
99#1348C2×C2312yes—
100#1322C2×C2311yescharacteristic
101C22×C23—21no—
102C22×C23—21no—
103C22×C23#44711yescharacteristic
104C22×C23C2×SD3211yescharacteristic
105C22×C23#51011yescharacteristic
106C4×C23—21no—
107C4×C23—22no—
108C4×C23—41no—
109C4×C23—82no—
110C4×C23—82no—
111C4×C23—82no—
112C4×C23—161no—
113C4×C23#45611yescharacteristic
114C4×C23#45811yescharacteristic
115#533—21no—
116#477—22no—
117#467—21no—
118#467C22×C2312yes—
119#462—22no—
120C2×#107C22×C2311yescharacteristic
121C2×C4⋊C8C22×C2311yescharacteristic
122#389—21no—
123#367—22no—
124#358—22no—
125C22⋊C16C22×C2312yes—
126#340—21no—
127#340C22×C2311yescharacteristic
128C2×C23—21no—
129C2×C23—21no—
130C2×C23#161011yescharacteristic
131C2×C23#218511yescharacteristic
132C2×C23#219111yescharacteristic
133C2×Q16—21no—
134C2×Q16D4×C2312yes—
135C22×C8D4×C2311yescharacteristic
136C22×C8C23×C2311yescharacteristic
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139#125—41no—
140C22⋊Q8—22no—
141C22⋊Q8—41no—
142C4×Q8—22no—
143C4×Q8—41no—
144#117—41no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—41no—
148C2×C42D4×C2311yescharacteristic
149C2×C16—22no—
150#107—21no—
151#107—22no—
152#107D4×C2312yes—
153C4⋊C8—22no—
154#103—22no—
155#103—22no—
156#95—41no—
157C23#1228911yescharacteristic, direct product factor, Sylow
158C2×Q8—22no—
159C2×Q8—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4D8×C2311yescharacteristic
163C22×C4D8×C2311yescharacteristic
164C22×C4C2×D4×C2311yescharacteristic
165Q16—42no—
166Q16—81no—
167C2×C8—21no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8C2×D4×C2312yes—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—42no—
177C4⋊C4—41no—
178C4⋊C4—42no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—82no—
182C22⋊C4—42no—
183C42—21no—
184C42—21no—
185C42—42no—
186C16—42no—
187C23C2×D8×C2311yescharacteristic
188Q8—42no—
189Q8—42no—
190Q8—82no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4C23×SD3212yes—
206C2×C4C23×#12011yescharacteristic
207C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
208C8—22no—
209C8—41no—
210C22—21no—
211C22—21no—
212C22C23×#44711yescharacteristic
213C22C2×C23×SD3211yescharacteristic
214C22C23×#51011yescharacteristic
215C4—22no—
216C4—21no—
217C4—41no—
218C4—82no—
219C4—82no—
220C4—82no—
221C4—161no—
222C4C23×#45611yescharacteristic
223C4C23×#45811yescharacteristic
224C2—21no—
225C2—21no—
226C2C23×#161011yescharacteristic
227C2C23×#218511yescharacteristic
228C2C23×#219111yescharacteristic
subgroupG/H
center, socleC22×C23#447
commutator, Frattini, cosocleC22×C8C23×C23
prime radicalC23×C23C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C23×C4⋊C8 ◁ G

Complements

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

NH ≅ G/Ntype
72. #12289C23direct (N × H)
157. C23#12289direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32492
C23×#22352
C23×#20862
C23×#13482
C23×#13222
#1228923