Finite Group Databasebeta

← #34009798

Subgroups — #34009798

Order 18688.

526 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 112 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
1#22625162C212yesmaximal
2#22624857C214yesmaximal
3#22612517C211yesmaximal, characteristic
4#17989603—22no—
5#17989432—22no—
6#17989430C2214yes—
7C2×#14556550C411yescharacteristic
8C2×#14556550C2211yescharacteristic
9#17988494—22no—
10#17988494—22no—
11#17988450C412yes—
12#17988450C2212yes—
13C23×Q8×C73C411yescharacteristic
14C4⋊#10596833—42no—
15C22⋊#10596833—24no—
16C4×#10596833—42no—
17#14556550—42no—
18#14556550—42no—
19C2×C4⋊Dic73—21no—
20C2×C4⋊Dic73D412yes—
21C2×C22⋊Dic73C2×C412yes—
22C2×C22⋊Dic73C2×C411yescharacteristic
23C2×C22⋊Dic73C2311yescharacteristic
24C2×C4×Dic73—21no—
25C2×C4×Dic73D412yes—
26#14556498—24no—
27#14556498—24no—
28C22×Q8×C73—21no—
29C22×Q8×C73—21no—
30C22×Q8×C73—22no—
31C22×Q8×C73—42no—
32C22×Q8×C73—42no—
33C22×Q8×C73C814yescomplemented
34C23×C4×C73C2×C411yescharacteristic
35C23×C4×C73C2×C412yes—
36C2×#10596833—44no—
37C22×Dic73—22no—
38C22×Dic73C22⋊C412yes—
39C22×Dic73C2×D412yes—
40C4⋊Dic73—42no—
41C4⋊Dic73—42no—
42C22⋊Dic73—41no—
43C22⋊Dic73—41no—
44C22⋊Dic73—42no—
45C4×Dic73—22no—
46C4×Dic73—22no—
47C4×Dic73—42no—
48C24×C73C22×C411yescharacteristic
49C2×Q8×C73—24no—
50C2×Q8×C73—24no—
51C2×Q8×C73—48no—
52C2×Q8×C73—42no—
53C2×Q8×C73—42no—
54C2×Q8×C73—48no—
55C2×Q8×C73—42no—
56C2×Q8×C73—42no—
57C22×C4×C73—22no—
58C22×C4×C73—22no—
59C22×C4×C73—22no—
60C22×C4×C73—22no—
61C22×C4×C73—42no—
62C22×C4×C73—42no—
63C22×C4×C73C22⋊C411yescharacteristic
64C22×C4×C73C22⋊C411yescharacteristic
65C22×C4×C73C2×C812yes—
66C22×C4×C73C2×C814yes—
67C22×C4×C73M1612yes—
68C23×C73—21no—
69C23×C73—21no—
70C23×C73—42no—
71C23×C73C2×C22⋊C411yescharacteristic
72C23×C73C22×C811yescharacteristic
73C23×C73C2×M1611yescharacteristic
74Q8×C73—48no—
75Q8×C73—48no—
76C2×C4×C73—21no—
77C2×C4×C73—21no—
78C2×C4×C73—24no—
79C2×C4×C73—24no—
80C2×C4×C73—24no—
81C2×C4×C73—44no—
82C2×C4×C73—42no—
83C2×C4×C73—42no—
84C2×C4×C73—44no—
85C2×C4×C73C22⋊C812yes—
86C2×C4×C73C22⋊C812yes—
87C2×C4×C73#9912yes—
88C2×C4×C73#10012yes—
89C2×Dic73—22no—
90C2×Dic73—22no—
91C2×Dic73—42no—
92C2×Dic73—42no—
93C2×Dic73—44no—
94C2×Dic73—42no—
95#10596833—84no—
96C22×C73—22no—
97C22×C73—21no—
98C22×C73—21no—
99C22×C73—41no—
100C22×C73—41no—
101C22×C73—42no—
102C22×C73—42no—
103C22×C73C2×C22⋊C811yescharacteristic
104C22×C73C2×#9911yescharacteristic
105C22×C73C2×#10011yescharacteristic
106C4×C73—22no—
107C4×C73—44no—
108C4×C73#32314yes—
109Dic73—42no—
110Dic73—42no—
111Dic73—82no—
112#11428—731nomaximal, Sylow
113C2×C73—21no—
114C2×C73—21no—
115C2×C73—41no—
116C2×C73—41no—
117C2×C73C2×#32311yescharacteristic
118C2×C73#212312yes—
119#2310—731no—
120#1458—732no—
121#1317—734no—
122C73#1142811yescharacteristic, complemented, Sylow
123C23×Q8#6440811yescharacteristic
124C2×#117—731no—
125C2×#117—731no—
126#431—1462no—
127#391—1462no—
128#391—1462no—
129#381—732no—
130#381—732no—
131#324—1462no—
132#323—734no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8#1059683314yescomplemented
139C23×C4C2×#6440812yes—
140C23×C4C2×#6440811yescharacteristic
141#117—2922no—
142#117—2922no—
143C2×C4⋊C4—732no—
144C2×C4⋊C4—1461no—
145C2×C22⋊C4—731no—
146C2×C22⋊C4—731no—
147C2×C22⋊C4—732no—
148C2×C42—732no—
149C2×C42—1461no—
150C4⋊C8—2922no—
151C22⋊C8—1464no—
152C4×C8—2922no—
153#95—1464no—
154#95—1464no—
155C24C22×#6440811yescharacteristic
156C2×Q8—24no—
157C2×Q8—24no—
158C2×Q8—48no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—48no—
162C2×Q8—42no—
163C2×Q8—42no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—732no—
171C22×C4—732no—
172C22×C4—1462no—
173C22×C4C22⋊#6440811yescharacteristic
174C22×C4C22⋊#6440811yescharacteristic
175C22×C4C2×#1059683312yes—
176C22×C4C2×#1059683314yes—
177C22×C4#1192403712yes—
178C2×C8—2924no—
179C4⋊C4—2922no—
180C4⋊C4—2922no—
181C22⋊C4—2922no—
182C22⋊C4—2921no—
183C22⋊C4—2921no—
184C42—2922no—
185C42—1462no—
186C42—1462no—
187C23—21no—
188C23—21no—
189C23—42no—
190C23C2×C22⋊#6440811yescharacteristic
191C23C22×#1059683311yescharacteristic
192C23C2×#1192403711yescharacteristic
193Q8—48no—
194Q8—48no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—44no—
204C2×C4—2922no—
205C2×C4—2922no—
206C2×C4—2924no—
207C2×C4—2922no—
208C2×C4—1462no—
209C2×C4—1462no—
210C2×C4C22⋊#1059683312yes—
211C2×C4C22⋊#1059683312yes—
212C2×C4#1455664912yes—
213C2×C4#1455665112yes—
214C8—5844no—
215C22—22no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—42no—
221C22—42no—
222C22C2×C22⋊#1059683311yescharacteristic
223C22C2×#1455664911yescharacteristic
224C22C2×#1455665111yescharacteristic
225C4—22no—
226C4—44no—
227C4—2922no—
228C4—2922no—
229C4—5842no—
230C4#1798943014yes—
231C2—21no—
232C2—21no—
233C2—41no—
234C2—41no—
235C2C2×#1798943011yescharacteristic
236C2#2262619312yes—
subgroupG/H
centerC22C2×#14556649
commutatorC23×C73C22×C8
FrattiniC24C22×#64408
FittingC23×Q8×C73C4
socleC22×C73C2×#99
prime radicalC24×C73C22×C4
cosocleC2×C22⋊Dic73C23
hypercenterC23×Q8#64408
hypocenterC73#11428
LCSG ▷ C23×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
33. C22×Q8×C73C8semidirect
122. C73#11428semidirect
138. C22×Q8#10596833semidirect

Maximal subgroups

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

subgroupindex
#226251622
#226248572
#226125172
#1142873