Finite Group Databasebeta

← #34001330

Subgroups — #34001330

Order 18688.

528 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 122 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#22625646C212yesmaximal, complemented
2#22624751C212yesmaximal, complemented
3#22624748C212yesmaximal
4C4×#14556580C211yesmaximal, characteristic
5C2×C4×#10596833—22no—
6#17989453—22no—
7#17989453C2214yes—
8C2×#14556580C411yescharacteristic
9C2×C4×D292C411yescharacteristic
10#17988494C2211yescharacteristic
11#17988488C2211yescharacteristic
12#17988477—22no—
13#17988452C411yescharacteristic
14C4×#11924005—22no—
15C42×Dic73C2211yescharacteristic
16C2×C4×Q8×C73C411yescharacteristic
17C22×#10596833—24no—
18C4×#10596833—28no—
19C22×D292C2×C411yescharacteristic
20C22×C4×D73—21no—
21#14556580—22no—
22#14556580C814yescomplemented
23C4×D292—42no—
24C2×C4⋊Dic73C2×C411yescharacteristic
25C2×#11924005—22no—
26C2×#11924005—22no—
27C2×#11924005—21no—
28C2×C4×Dic73C2×C411yescharacteristic
29C2×C4×Dic73D412yes—
30C2×C4×Dic73D412yes—
31C2×C4×Dic73C2311yescharacteristic
32#14556498—22no—
33#14556498—22no—
34C22×Q8×C73C2×C411yescharacteristic
35C4×Q8×C73—42no—
36C2×C4⋊C4×C73—21no—
37C2×C4⋊C4×C73C2×C411yescharacteristic
38C2×C42×C73—21no—
39C2×C42×C73C2×C411yescharacteristic
40C2×#10596833—28no—
41C2×#10596833—44no—
42C2×#10596833—44no—
43C23×D73—21no—
44C2×D292—21no—
45C2×D292—41no—
46C2×D292—41no—
47C2×D292C2×C812yes—
48C22×Dic73—21no—
49C22×Dic73C22⋊C412yes—
50C22×Dic73C2×D412yes—
51C2×C4×D73—42no—
52C2×C4×D73—44no—
53C4⋊Dic73—41no—
54#11924005—28no—
55#11924005—42no—
56#11924005—44no—
57C4×Dic73—24no—
58C4×Dic73—24no—
59C4×Dic73—41no—
60C4×Dic73C2×C812yes—
61C4×Dic73M1612yes—
62C2×Q8×C73—21no—
63C2×Q8×C73—41no—
64C2×Q8×C73—41no—
65C2×Q8×C73C2×C812yes—
66C22×C4×C73—21no—
67C22×C4×C73—21no—
68C22×C4×C73C22⋊C411yescharacteristic
69C22×C4×C73C22⋊C411yescharacteristic
70C22×C4×C73C22×C411yescharacteristic
71C4⋊C4×C73—41no—
72C4⋊C4×C73—42no—
73C42×C73—22no—
74C42×C73—42no—
75C23×C73C2×C22⋊C411yescharacteristic
76C22×D73—22no—
77C22×D73—41no—
78C22×D73—41no—
79C22×D73—42no—
80C22×D73—42no—
81Q8×C73—42no—
82Q8×C73—42no—
83C2×C4×C73—21no—
84C2×C4×C73—21no—
85C2×C4×C73—22no—
86C2×C4×C73—21no—
87C2×C4×C73—22no—
88C2×C4×C73—41no—
89C2×C4×C73—41no—
90C2×C4×C73—41no—
91C2×C4×C73—42no—
92C2×C4×C73#10012yes—
93C2×C4×C73#10414yes—
94C2×C4×C73C22×C811yescharacteristic
95C2×C4×C73C2×M1611yescharacteristic
96C2×Dic73—22no—
97C2×Dic73—22no—
98C2×Dic73—22no—
99C2×Dic73—22no—
100C2×Dic73—41no—
101C2×Dic73—42no—
102C2×Dic73C22⋊C814yes—
103C2×Dic73#13514yes—
104D292—42no—
105D292—42no—
106C4×D73—84no—
107#10596833—48no—
108C22×C73C2×C22⋊C811yescharacteristic
109C22×C73#41011yescharacteristic
110C22×C73C2×#10011yescharacteristic
111C22×C73#41612yes—
112C22×C73C2×#10412yes—
113C2×D73—42no—
114C2×D73—44no—
115C2×D73—81no—
116C2×D73—82no—
117C2×D73—81no—
118C4×C73—21no—
119C4×C73—21no—
120C4×C73—41no—
121C4×C73—42no—
122C4×C73—81no—
123C4×C73#32914yes—
124Dic73—24no—
125Dic73—24no—
126Dic73—81no—
127#7567—731nomaximal, Sylow
128C2×C73#146011yescharacteristic
129C2×C73#147612yes—
130C2×C73#150012yes—
131C2×C73C2×#32911yescharacteristic
132C2×C73#154311yescharacteristic
133D73—82no—
134C4×#124—731no—
135#1744—732no—
136#1272—732no—
137#1270—732no—
138C73#756711yescharacteristic, complemented, Sylow
139C2×#124—731no—
140C2×C4×Q8#6440811yescharacteristic
141C2×C4×D4—731no—
142C2×C4×C8—1462no—
143#391—731no—
144#390—731no—
145#388—1462no—
146#382—731no—
147C4×C22⋊C4—1462no—
148C43—731no—
149#336—734no—
150#336—1462no—
151C22×Q8C2×#6440811yescharacteristic
152C22×D4—731no—
153C23×C4—1461no—
154C22×C8—1464no—
155#124—734no—
156#124—1462no—
157C4×Q8—42no—
158C4×D4—2922no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—731no—
161C2×C4⋊C4C2×#6440811yescharacteristic
162C2×C22⋊C4—1462no—
163C2×C22⋊C4—1461no—
164C2×C22⋊C4—1462no—
165C2×C42—21no—
166C2×C42—731no—
167C2×C42—731no—
168C2×C42—732no—
169C2×C42—732no—
170C2×C42C2×#6440811yescharacteristic
171C4×C8—1468no—
172#95—1462no—
173#95—1462no—
174C24—1461no—
175C2×Q8—21no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8#1192403612yes—
179C2×D4—2921no—
180C2×D4—2921no—
181C2×D4—732no—
182C2×D4—1461no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—2922no—
186C22×C4—2924no—
187C22×C4—732no—
188C22×C4—732no—
189C22×C4—1461no—
190C22×C4C22⋊#6440811yescharacteristic
191C22×C4C22⋊#6440811yescharacteristic
192C22×C4C22×#6440811yescharacteristic
193C2×C8—2924no—
194C2×C8—2924no—
195C2×C8—1468no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—2921no—
199C22⋊C4—2922no—
200C22⋊C4—2924no—
201C22⋊C4—1468no—
202C42—22no—
203C42—42no—
204C42—2921no—
205C42—732no—
206C42—732no—
207C42—1464no—
208C42—1464no—
209C23—2921no—
210C23—2921no—
211C23—2922no—
212C23—2922no—
213C23—1462no—
214C23C2×C22⋊#6440811yescharacteristic
215Q8—42no—
216Q8—42no—
217D4—2922no—
218D4—2922no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—2921no—
229C2×C4—2922no—
230C2×C4—5844no—
231C2×C4—734no—
232C2×C4—734no—
233C2×C4—1462no—
234C2×C4—1462no—
235C2×C4—1462no—
236C2×C4—1462no—
237C2×C4#1455665112yes—
238C2×C4#1455665614yes—
239C2×C4C2×#1192403611yescharacteristic
240C2×C4D73⋊M1611yescharacteristic
241C8—2928no—
242C22—2922no—
243C22—2924no—
244C22—5842no—
245C22—5841no—
246C22—5841no—
247C22#1798955511yescharacteristic
248C22#1798956611yescharacteristic
249C22C2×#1455665111yescharacteristic
250C22#1798957712yes—
251C22C2×#1455665612yes—
252C4—21no—
253C4—21no—
254C4—42no—
255C4—41no—
256C4—81no—
257C4—5841no—
258C4—1464no—
259C4—1464no—
260C4#1798944314yes—
261C2—5842no—
262C2#2262517011yescharacteristic
263C2#2262522512yes—
264C2#2262530412yes—
265C2C2×#1798944311yescharacteristic
266C2#2262543411yescharacteristic
subgroupG/H
centerC22×C4C22⋊#64408
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC2×C4×Q8×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC22×D292C2×C4
cosocleC2×C4×Dic73C23
hypercenterC2×C4×Q8#64408
hypocenterC73#7567
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #22625646C2semidirect
2. #22624751C2semidirect
22. #14556580C8semidirect
138. C73#7567semidirect

Maximal subgroups

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

subgroupindex
#226256462
#226247512
#226247482
C4×#145565802
#756773