Finite Group Databasebeta

← #34013219

Subgroups — #34013219

Order 18688.

670 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 80 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#22625760C212yesmaximal
2#22625025C214yesmaximal, complemented
3#22612421C211yesmaximal, characteristic
4C2×#14556650C2214yes—
5C2×#14556649—22no—
6#17989530—22no—
7C22×#11924028C2211yescharacteristic
8C2×#14556550C411yescharacteristic, complemented
9#17988493C412yescomplemented
10#17988489C2212yes—
11#17988466—22no—
12C4×#11924005—22no—
13C22×C42×C73C411yescharacteristic, complemented
14C2×C22⋊#64408—24no—
15#14556650—28no—
16#14556649—44no—
17(C2×#64408)⋊C4—44no—
18C2×#11924028—24no—
19C2×#11924028—42no—
20C2×#11924028—42no—
21C22×Dic146C2×C411yescharacteristic
22C22×D292C2311yescharacteristic
23C22×C22⋊D73D412yes—
24C22×C4×D73D412yes—
25#14556550—42no—
26C2×C4⋊Dic73C2×C412yes—
27C2×C22⋊Dic73—21no—
28C2×#11924005—24no—
29C2×C4×Dic73—21no—
30#14556498—24no—
31C23×C4×C73—21no—
32C23×C4×C73C2×C411yescharacteristic
33C2×C42×C73—21no—
34C2×C42×C73—42no—
35C2×C42×C73C2×C412yes—
36C22×#64408—44no—
37C22⋊#64408—44no—
38C22⋊#64408—44no—
39C23×D73C2×D412yes—
40#11924028—416no—
41C2×Dic146—22no—
42C2×Dic146—41no—
43C2×Dic146—41no—
44C2×D292—41no—
45C2×D292—41no—
46C2×D292#4114yes—
47C2×C22⋊D73—24no—
48C2×C22⋊D73—24no—
49C2×C22⋊D73—42no—
50C22×Dic73—22no—
51C22×Dic73C22⋊C412yes—
52C2×C4×D73—24no—
53C2×C4×D73—24no—
54C2×C4×D73—42no—
55C4⋊Dic73—42no—
56C22⋊Dic73—42no—
57#11924005—48no—
58C4×Dic73—42no—
59C24×C73C22⋊C411yescharacteristic
60C22×C4×C73—22no—
61C22×C4×C73—22no—
62C22×C4×C73—22no—
63C22×C4×C73—22no—
64C22×C4×C73—44no—
65C22×C4×C73—42no—
66C22×C4×C73C22⋊C411yescharacteristic
67C22×C4×C73C22×C411yescharacteristic
68C42×C73—42no—
69C42×C73—42no—
70C23×C73—21no—
71C23×C73—21no—
72C23×C73—42no—
73C23×C73#9912yes—
74C23×C73C2×C22⋊C411yescharacteristic
75C22×D73—22no—
76C22×D73—22no—
77C22×D73—44no—
78C22×D73#12314yes—
79C2×#64408—84no—
80C2×#64408—88no—
81C2×C4×C73—28no—
82C2×C4×C73—21no—
83C2×C4×C73—21no—
84C2×C4×C73—21no—
85C2×C4×C73—21no—
86C2×C4×C73—24no—
87C2×C4×C73—42no—
88C2×C4×C73—48no—
89C2×C4×C73—42no—
90C2×C4×C73#9912yes—
91C2×C4×C73#11712yes—
92C22⋊D73—48no—
93C2×Dic73—22no—
94C2×Dic73—22no—
95C2×Dic73—22no—
96C2×Dic73—44no—
97C2×Dic73—42no—
98D292—44no—
99C4×D73—48no—
100Dic146—44no—
101C22×C73—22no—
102C22×C73—22no—
103C22×C73—42no—
104C22×C73—42no—
105C22×C73—42no—
106C22×C73#38111yescharacteristic
107C22×C73C2×#9911yescharacteristic
108C22×C73C2×#9911yescharacteristic
109C2×D73—24no—
110C2×D73—44no—
111C2×D73—44no—
112C2×D73—82no—
113#64408—164no—
114C4×C73—22no—
115C4×C73—22no—
116C4×C73—44no—
117Dic73—42no—
118Dic73—82no—
119#12857—731nomaximal, Sylow
120C2×C73—21no—
121C2×C73—21no—
122C2×C73—42no—
123C2×C73#181911yescharacteristic
124C2×C73#212612yes—
125D73—44no—
126#2297—731no—
127#1782—732no—
128#1393—734no—
129C73#1285711yescharacteristic, complemented, Sylow
130C22×#41—731no—
131C2×#117—731no—
132C22×C42#6440811yescharacteristic, complemented
133C2×#99—734no—
134C2×#99—1462no—
135#391—732no—
136#390—732no—
137#386—1462no—
138#381—1462no—
139C4×C22⋊C4—1462no—
140C2×#41—2922no—
141C2×#41—2922no—
142C2×#41—1464no—
143C22×Q8—731no—
144C22×D4—732no—
145C22×D4—731no—
146C23×C4—21no—
147C23×C4—732no—
148C23×C4C2×#6440811yescharacteristic
149#117—2922no—
150C2×C4⋊C4—732no—
151C2×C22⋊C4—1464no—
152C2×C22⋊C4—1464no—
153C2×C22⋊C4—1461no—
154C2×C42—21no—
155C2×C42—42no—
156C2×C42—1461no—
157C2×C42C2×#6440812yes—
158#99—2924no—
159#99—1468no—
160#95—2924no—
161#95—1464no—
162C24—732no—
163C24C22⋊#6440811yescharacteristic
164#41—29216no—
165C2×Q8—2921no—
166C2×Q8—2921no—
167C2×Q8—1462no—
168C2×D4—2921no—
169C2×D4—2921no—
170C2×D4—2922no—
171C2×D4—734no—
172C2×D4—1464no—
173C2×D4—1464no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—44no—
179C22×C4—42no—
180C22×C4—2922no—
181C22×C4—2924no—
182C22×C4—732no—
183C22×C4—1464no—
184C22×C4—1462no—
185C22×C4—1464no—
186C22×C4C22⋊#6440811yescharacteristic
187C22×C4C22×#6440811yescharacteristic
188C4⋊C4—2922no—
189C22⋊C4—2928no—
190C22⋊C4—2922no—
191C22⋊C4—2924no—
192C22⋊C4—2924no—
193C42—42no—
194C42—42no—
195C42—2922no—
196C23—21no—
197C23—21no—
198C23—42no—
199C23—2924no—
200C23—734no—
201C23—1462no—
202C23—1462no—
203C23#1455664912yes—
204C23C2×C22⋊#6440811yescharacteristic
205Q8—2924no—
206D4—2924no—
207D4—2928no—
208C2×C4—28no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—24no—
214C2×C4—42no—
215C2×C4—48no—
216C2×C4—42no—
217C2×C4—2928no—
218C2×C4—2924no—
219C2×C4—2922no—
220C2×C4—5848no—
221C2×C4—5844no—
222C2×C4—1462no—
223C2×C4—1462no—
224C2×C4—1462no—
225C2×C4#1455665012yes—
226C2×C4#1455667012yes—
227C22—22no—
228C22—22no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—2924no—
233C22—2924no—
234C22—5842no—
235C22—1464no—
236C22#1798953011yescharacteristic
237C22C2×#1455664911yescharacteristic
238C22C2×#1455665011yescharacteristic
239C4—22no—
240C4—22no—
241C4—44no—
242C4—2922no—
243C4—5842no—
244C4—11684no—
245C2—21no—
246C2—21no—
247C2—42no—
248C2—2924no—
249C2#2262582611yescharacteristic
250C2#2262620412yes—
subgroupG/H
centerC22C2×#14556650
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC22×C42×C73C4
socleC22×C73C2×#99
prime radicalC22×#11924028C22
cosocleC22×D292C23
hypercenterC22×C42#64408
hypocenterC73#12857
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. #22625025C2semidirect
8. C2×#14556550C4semidirect
9. #17988493C4semidirect
13. C22×C42×C73C4semidirect
129. C73#12857semidirect
132. C22×C42#64408semidirect

Maximal subgroups

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

subgroupindex
#226257602
#226250252
#226124212
#1285773