Finite Group Databasebeta

← #34010066

Subgroups — #34010066

Order 18688.

598 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 88 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#22625179C212yesmaximal
2#22624861C214yesmaximal, complemented
3C2×#17988494C211yesmaximal, characteristic
4#17989557—22no—
5#17989432—22no—
6#17989432C2214yes—
7C22×C4⋊Dic73C2211yescharacteristic
8C22×C4×Dic73C411yescharacteristic
9#17988494—22no—
10#17988494C412yes—
11#17988494C2212yes—
12C2×#14556498—22no—
13C23×Q8×C73C411yescharacteristic
14C2×#11924037—42no—
15C22⋊#10596833—24no—
16C22⋊#10596833—42no—
17C23×Dic73—21no—
18C23×Dic73D412yes—
19C2×C4⋊Dic73—41no—
20C2×C4⋊Dic73—41no—
21C2×C4⋊Dic73C2×C411yescharacteristic
22C2×C4⋊Dic73D412yes—
23C2×C4⋊Dic73C2311yescharacteristic
24C2×C4×Dic73—21no—
25C2×C4×Dic73—42no—
26C2×C4×Dic73C2×C412yes—
27#14556498—24no—
28#14556498—24no—
29C22×Q8×C73—24no—
30C22×Q8×C73—21no—
31C22×Q8×C73—42no—
32C22×Q8×C73—41no—
33C22×Q8×C73—41no—
34C22×Q8×C73C2×C412yes—
35C23×C4×C73—21no—
36C23×C4×C73C2×C411yescharacteristic
37#11924037—84no—
38C2×#10596833—44no—
39C22×Dic73—24no—
40C22×Dic73—22no—
41C22×Dic73—22no—
42C22×Dic73—22no—
43C22×Dic73—42no—
44C22×Dic73—42no—
45C22×Dic73—44no—
46C22×Dic73C22⋊C412yes—
47C22×Dic73C2×D412yes—
48C4⋊Dic73—41no—
49C4⋊Dic73—42no—
50C4⋊Dic73—41no—
51C4×Dic73—42no—
52C4×Dic73—42no—
53C24×C73C22⋊C411yescharacteristic
54C2×Q8×C73—24no—
55C2×Q8×C73—22no—
56C2×Q8×C73—48no—
57C2×Q8×C73—48no—
58C2×Q8×C73—42no—
59C2×Q8×C73—42no—
60C2×Q8×C73—42no—
61C2×Q8×C73—42no—
62C2×Q8×C73M1614yescomplemented
63C22×C4×C73—21no—
64C22×C4×C73—21no—
65C22×C4×C73—22no—
66C22×C4×C73—22no—
67C22×C4×C73—24no—
68C22×C4×C73—22no—
69C22×C4×C73—42no—
70C22×C4×C73—41no—
71C22×C4×C73—41no—
72C22×C4×C73C22⋊C411yescharacteristic
73C22×C4×C73C22×C411yescharacteristic
74C23×C73—22no—
75C23×C73—22no—
76C23×C73—21no—
77C23×C73—21no—
78C23×C73#9911yescharacteristic
79C23×C73#9911yescharacteristic
80C23×C73C2×C22⋊C411yescharacteristic
81Q8×C73—44no—
82Q8×C73—48no—
83Q8×C73—44no—
84C2×C4×C73—24no—
85C2×C4×C73—24no—
86C2×C4×C73—21no—
87C2×C4×C73—21no—
88C2×C4×C73—21no—
89C2×C4×C73—21no—
90C2×C4×C73—24no—
91C2×C4×C73—44no—
92C2×C4×C73—44no—
93C2×C4×C73—42no—
94C2×C4×C73—42no—
95C2×C4×C73#10012yes—
96C2×C4×C73C2×M1612yes—
97C2×Dic73—24no—
98C2×Dic73—22no—
99C2×Dic73—22no—
100C2×Dic73—48no—
101C2×Dic73—42no—
102C2×Dic73—44no—
103C2×Dic73—42no—
104C2×Dic73—42no—
105C2×Dic73—42no—
106C2×Dic73—82no—
107#10596833—84no—
108C22×C73—24no—
109C22×C73—24no—
110C22×C73—21no—
111C22×C73—21no—
112C22×C73—21no—
113C22×C73—21no—
114C22×C73—21no—
115C22×C73—21no—
116C22×C73#35412yes—
117C22×C73#35612yes—
118C22×C73#40711yescharacteristic
119C22×C73C2×#9911yescharacteristic
120C22×C73C2×#10011yescharacteristic
121C4×C73—22no—
122C4×C73—22no—
123C4×C73—44no—
124Dic73—44no—
125Dic73—42no—
126Dic73—42no—
127#11506—731nomaximal, Sylow
128C2×C73—21no—
129C2×C73—22no—
130C2×C73—22no—
131C2×C73—21no—
132C2×C73#146211yescharacteristic
133C2×C73C2×#35411yescharacteristic
134C2×C73C2×#35611yescharacteristic
135C2×#391—731no—
136#1463—732no—
137#1319—734no—
138C73#1150611yescharacteristic, complemented, Sylow
139C23×Q8#6440811yescharacteristic
140C22×C4⋊C4—731no—
141C22×C42—731no—
142#407—1462no—
143#391—732no—
144#391—732no—
145#391—1462no—
146C2×#95—1462no—
147#324—734no—
148#324—1462no—
149C22×Q8—24no—
150C22×Q8—21no—
151C22×Q8—42no—
152C22×Q8—41no—
153C22×Q8—41no—
154C22×Q8C2×#6440812yes—
155C23×C4—21no—
156C23×C4—732no—
157C23×C4—1461no—
158C23×C4C2×#6440811yescharacteristic
159C2×M16—2922no—
160C2×C4⋊C4—2921no—
161C2×C4⋊C4—2921no—
162C2×C4⋊C4—732no—
163C2×C4⋊C4—731no—
164C2×C4⋊C4—731no—
165C2×C42—2922no—
166C2×C42—732no—
167C2×C42—1461no—
168C22⋊C8—2922no—
169C22⋊C8—1464no—
170#95—1464no—
171#95—1464no—
172C24C22⋊#6440811yescharacteristic
173C2×Q8—24no—
174C2×Q8—22no—
175C2×Q8—48no—
176C2×Q8—48no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8#1192403714yescomplemented
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—2922no—
192C22×C4—2922no—
193C22×C4—2924no—
194C22×C4—732no—
195C22×C4—732no—
196C22×C4—1464no—
197C22×C4—1462no—
198C22×C4—1462no—
199C22×C4—1462no—
200C22×C4C22⋊#6440811yescharacteristic
201C22×C4C22×#6440811yescharacteristic
202M16—5844no—
203C2×C8—2924no—
204C4⋊C4—2922no—
205C4⋊C4—2921no—
206C4⋊C4—2921no—
207C42—2922no—
208C42—2922no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—21no—
213C23#1455664911yescharacteristic
214C23#1455664911yescharacteristic
215C23C2×C22⋊#6440811yescharacteristic
216Q8—44no—
217Q8—44no—
218Q8—48no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—24no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—2928no—
231C2×C4—2922no—
232C2×C4—2924no—
233C2×C4—2922no—
234C2×C4—2922no—
235C2×C4—2922no—
236C2×C4—5842no—
237C2×C4—1464no—
238C2×C4—1462no—
239C2×C4—1462no—
240C2×C4#1455665112yes—
241C2×C4C2×#1192403712yes—
242C8—5844no—
243C22—24no—
244C22—24no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22#1798949312yes—
252C22#1798949712yes—
253C22#1798955711yescharacteristic
254C22C2×#1455664911yescharacteristic
255C22C2×#1455665111yescharacteristic
256C4—22no—
257C4—22no—
258C4—44no—
259C4—2924no—
260C4—2922no—
261C4—2922no—
262C2—21no—
263C2—22no—
264C2—22no—
265C2—21no—
266C2#2262517611yescharacteristic
267C2C2×#1798949311yescharacteristic
268C2C2×#1798949711yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC23×Q8×C73C4
socleC22×C73C2×#99
prime radicalC24×C73C22⋊C4
cosocleC2×C4⋊Dic73C23
hypercenterC23×Q8#64408
hypocenterC73#11506
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
2. #22624861C2semidirect
62. C2×Q8×C73M16semidirect
138. C73#11506semidirect
181. C2×Q8#11924037semidirect

Maximal subgroups

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

subgroupindex
#226251792
#226248612
C2×#179884942
#1150673