Finite Group Databasebeta

← C23×#8521 #19156147

Subgroups — C23×#8521 #19156147

Order 5888.

690 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 142 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
1C2×C23×#541C211yesmaximal, characteristic
2C23×#1822C211yesmaximal, characteristic
3C23×#1820C211yesmaximal, characteristic
4C23×#1511C214yesmaximal
5C23×#541—28no—
6C2×C23×C4⋊Q8C411yescharacteristic, complemented
7C2×C23×#126—22no—
8C2×C23×#125—22no—
9C2×C23×#125C411yescharacteristic, complemented
10C2×C23×C22⋊Q8C2212yes—
11C2×C4×Q8×C23C412yescomplemented
12C2×C23×#117C2211yescharacteristic
13C23×#407C2212yes—
14C23×#381C2212yes—
15C23×#343—22no—
16C23×#342—22no—
17C23×#328—28no—
18C22×Q8×C23C2×C412yes—
19C23×C4×C23C2311yescharacteristic
20C2×M16×C23—22no—
21C23×C4⋊Q8—24no—
22C23×#126—216no—
23C23×#125—216no—
24C23×#125—24no—
25C23×C22⋊Q8—24no—
26C23×C22⋊Q8D418yes—
27C4×Q8×C23—24no—
28C4×Q8×C23—24no—
29C23×#117—24no—
30C2×C4⋊C4×C23—22no—
31C2×C4⋊C4×C23—22no—
32C2×C4⋊C4×C23—22no—
33C2×C4⋊C4×C23—22no—
34C2×C4⋊C4×C23C2×C412yes—
35C2×C4⋊C4×C23D412yes—
36C2×C22⋊C4×C23—22no—
37C2×C22⋊C4×C23—22no—
38C2×C22⋊C4×C23D412yes—
39C2×C42×C23—22no—
40C2×C42×C23C2×C411yescharacteristic
41C2×C42×C23C2×C411yescharacteristic
42C23×C22⋊C8—24no—
43C23×#95—24no—
44C24×C23C2×D411yescharacteristic
45C2×Q8×C23—22no—
46C2×Q8×C23—44no—
47C2×Q8×C23C22⋊C414yes—
48C22×C4×C23—22no—
49C22×C4×C23—22no—
50C22×C4×C23—42no—
51C22×C4×C23—42no—
52C22×C4×C23C22⋊C412yes—
53C22×C4×C23C22⋊C412yes—
54C22×C4×C23C22×C411yescharacteristic
55C22×C4×C23C2×D411yescharacteristic
56C22×C4×C23C2×D414yes—
57M16×C23—44no—
58C2×C8×C23—42no—
59C4⋊C4×C23—28no—
60C4⋊C4×C23—28no—
61C4⋊C4×C23—28no—
62C4⋊C4×C23—22no—
63C4⋊C4×C23—28no—
64C4⋊C4×C23—24no—
65C4⋊C4×C23C22⋊C414yes—
66C22⋊C4×C23—28no—
67C22⋊C4×C23—24no—
68C22⋊C4×C23—44no—
69C42×C23—28no—
70C42×C23—22no—
71C42×C23—22no—
72#8521C2311yesmaximal, characteristic, direct product factor, Sylow
73C23×C23—22no—
74C23×C23—42no—
75C23×C23C2×C22⋊C411yescharacteristic
76C23×C23#12012yes—
77Q8×C23—44no—
78Q8×C23—44no—
79C2×C4×C23—22no—
80C2×C4×C23—22no—
81C2×C4×C23—22no—
82C2×C4×C23—22no—
83C2×C4×C23—24no—
84C2×C4×C23—24no—
85C2×C4×C23—44no—
86C2×C4×C23—44no—
87C2×C4×C23—44no—
88C2×C4×C23—42no—
89C2×C4×C23—42no—
90C2×C4×C23—82no—
91C2×C4×C23—84no—
92C2×C4×C23C2×C22⋊C412yes—
93C2×C4×C23#12012yes—
94C8×C23—82no—
95C2×#541C2×C2311yescharacteristic
96#1822C2×C2311yescharacteristic
97#1820C2×C2311yescharacteristic
98#1511C2×C2314yes—
99C22×C23—22no—
100C22×C23—24no—
101C22×C23—22no—
102C22×C23—82no—
103C22×C23#37911yescharacteristic
104C22×C23#41012yes—
105C4×C23—22no—
106C4×C23—22no—
107C4×C23—42no—
108C4×C23—42no—
109C4×C23—44no—
110C4×C23—44no—
111C4×C23—162no—
112#541—28no—
113C2×C4⋊Q8C4×C2311yescharacteristic, complemented
114C2×#126—22no—
115C2×#125—22no—
116C2×#125C4×C2311yescharacteristic, complemented
117C2×C22⋊Q8C22×C2312yes—
118C2×C4×Q8C4×C2312yescomplemented
119C2×#117C22×C2311yescharacteristic
120#407C22×C2312yes—
121#381C22×C2312yes—
122#343—22no—
123#342—22no—
124#328—28no—
125C2×C23—22no—
126C2×C23—42no—
127C2×C23#178211yescharacteristic
128C2×C23#179612yes—
129C22×Q8C2×C4×C2312yes—
130C23×C4C23×C2311yescharacteristic
131C2×M16—22no—
132C4⋊Q8—24no—
133#126—216no—
134#125—216no—
135#125—24no—
136C22⋊Q8—24no—
137C22⋊Q8D4×C2318yes—
138C4×Q8—24no—
139C4×Q8—24no—
140#117—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4C2×C4×C2312yes—
146C2×C4⋊C4D4×C2312yes—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4D4×C2312yes—
150C2×C42—22no—
151C2×C42C2×C4×C2311yescharacteristic
152C2×C42C2×C4×C2311yescharacteristic
153C22⋊C8—24no—
154#95—24no—
155C23#852111yescharacteristic, direct product factor, Sylow
156C24C2×D4×C2311yescharacteristic
157C2×Q8—22no—
158C2×Q8—44no—
159C2×Q8C22⋊C4×C2314yes—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4C22⋊C4×C2312yes—
165C22×C4C22⋊C4×C2312yes—
166C22×C4C22×C4×C2311yescharacteristic
167C22×C4C2×D4×C2311yescharacteristic
168C22×C4C2×D4×C2314yes—
169M16—44no—
170C2×C8—42no—
171C4⋊C4—28no—
172C4⋊C4—28no—
173C4⋊C4—28no—
174C4⋊C4—22no—
175C4⋊C4—28no—
176C4⋊C4—24no—
177C4⋊C4C22⋊C4×C2314yes—
178C22⋊C4—28no—
179C22⋊C4—24no—
180C22⋊C4—44no—
181C42—28no—
182C42—22no—
183C42—22no—
184C23—22no—
185C23—42no—
186C23C2×C22⋊C4×C2311yescharacteristic
187C23C23×#12012yes—
188Q8—44no—
189Q8—44no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—82no—
202C2×C4—84no—
203C2×C4C2×C22⋊C4×C2312yes—
204C2×C4C23×#12012yes—
205C8—82no—
206C22—22no—
207C22—22no—
208C22—24no—
209C22—82no—
210C22C23×#37911yescharacteristic
211C22C23×#41012yes—
212C4—22no—
213C4—22no—
214C4—44no—
215C4—44no—
216C4—42no—
217C4—42no—
218C4—162no—
219C2—22no—
220C2—42no—
221C2C23×#178211yescharacteristic
222C2C23×#179612yes—
subgroupG/H
center, socleC22×C23#379
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC24×C23C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
6. C2×C23×C4⋊Q8C4semidirect
9. C2×C23×#125C4semidirect
11. C2×C4×Q8×C23C4semidirect
72. #8521C23direct (N × H)
113. C2×C4⋊Q8C4×C23semidirect
116. C2×#125C4×C23semidirect
118. C2×C4×Q8C4×C23semidirect
155. C23#8521direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5412
C23×#18222
C23×#18202
C23×#15112
#852123