Finite Group Databasebeta

← #32980021

Subgroups — #32980021

Order 18688.

418 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 107 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#22610730C211yesmaximal, characteristic
2#22608717C212yesmaximal, complemented
3#22608252C211yesmaximal, characteristic
4#22607829C212yesmaximal, complemented
5C73×#2973C211yesmaximal, characteristic
6#17988640C2212yes—
7C2×C4⋊#10596831C2211yescharacteristic
8C2×C22⋊#10596831—21no—
9C2×C22⋊#10596831—21no—
10C2×#14556502C2211yescharacteristic
11#17988339—22no—
12#17988337—22no—
13#17988311C412yes—
14#17988311C412yes—
15C2×C4×D4×C73C2211yescharacteristic
16#435×C73—22no—
17#435×C73—22no—
18#432×C73C2212yes—
19#407×C73—21no—
20C2×C22⋊C8×C73—21no—
21C22×#10596831—21no—
22C22×#10596831—21no—
23C4⋊#10596831—22no—
24C4⋊#10596831—22no—
25C22⋊#10596831—42no—
26C22⋊#10596831—42no—
27C22⋊#10596831C2×C414yes—
28#14556502—22no—
29C4×#10596831—22no—
30C22×D4×C73Q811yescharacteristic
31C23×C4×C73D411yescharacteristic
32C23×C4×C73D411yescharacteristic
33C2×M16×C73—21no—
34C2×M16×C73—21no—
35C22×C8×C73—21no—
36C22×C8×C73—21no—
37C4×D4×C73—22no—
38C4×D4×C73—22no—
39C2×C4⋊C4×C73Q811yescharacteristic
40C2×C22⋊C4×C73C2×C411yescharacteristic
41C2×C22⋊C4×C73C2×C411yescharacteristic
42C2×C42×C73C2311yescharacteristic
43C4⋊C8×C73D412yes—
44C4⋊C8×C73D412yes—
45C22⋊C8×C73—22no—
46C22⋊C8×C73—22no—
47C22⋊C8×C73—22no—
48#97×C73—22no—
49C2×#10596831—22no—
50C2×#10596831—24no—
51C2×#10596831—41no—
52C2×#10596831—41no—
53C2×#10596831—41no—
54C2×#10596831—41no—
55C2×#10596831—41no—
56C24×C73C4⋊C411yescharacteristic
57C24×C73C4⋊C411yescharacteristic
58C2×D4×C73—22no—
59C2×D4×C73—22no—
60C2×D4×C73—41no—
61C22×C4×C73—21no—
62C22×C4×C73—21no—
63C22×C4×C73—21no—
64C22×C4×C73—21no—
65C22×C4×C73—41no—
66C22×C4×C73—41no—
67C22×C4×C73—41no—
68C22×C4×C73—41no—
69C22×C4×C73C4⋊C411yescharacteristic
70C22×C4×C73C4⋊C411yescharacteristic
71C22×C4×C73C22×C411yescharacteristic
72C22×C4×C73C2×D411yescharacteristic
73C22×C4×C73C2×Q811yescharacteristic
74M16×C73—81no—
75M16×C73—81no—
76C2×C8×C73—22no—
77C2×C8×C73—22no—
78C2×C8×C73—81no—
79C2×C8×C73—81no—
80C4⋊C4×C73—22no—
81C22⋊C4×C73—22no—
82C22⋊C4×C73—22no—
83C42×C73C2×D412yes—
84C42×C73#4112yes—
85C23×C73—21no—
86C23×C73—21no—
87C23×C73—21no—
88C23×C73—21no—
89C23×C73—21no—
90C23×C73—41no—
91C23×C73—41no—
92C23×C73—41no—
93C23×C73—41no—
94C23×C73#10812yes—
95C23×C73C2×C4⋊C411yescharacteristic
96D4×C73—44no—
97C2×C4×C73—21no—
98C2×C4×C73—21no—
99C2×C4×C73—21no—
100C2×C4×C73—21no—
101C2×C4×C73—21no—
102C2×C4×C73—21no—
103C2×C4×C73—41no—
104C2×C4×C73—41no—
105C2×C4×C73—41no—
106C2×C4×C73—41no—
107C2×C4×C73—41no—
108C2×C4×C73—41no—
109C2×C4×C73—41no—
110C2×C4×C73—81no—
111C2×C4×C73—81no—
112C2×C4×C73C4×D412yes—
113C2×C4×C73#12011yescharacteristic
114C2×C4×C73C22⋊Q812yes—
115C2×C4×C73#12311yescharacteristic
116C8×C73—42no—
117C8×C73—42no—
118#10596831—44no—
119#10596831—42no—
120#10596831—81no—
121C22×C73—21no—
122C22×C73—21no—
123C22×C73—22no—
124C22×C73—42no—
125C22×C73—42no—
126C22×C73—41no—
127C22×C73—41no—
128C22×C73—41no—
129C22×C73—41no—
130C22×C73—41no—
131C22×C73—81no—
132C22×C73—81no—
133C22×C73#38511yescharacteristic
134C22×C73C2×#10811yescharacteristic
135C22×C73#43011yescharacteristic
136C4×C73—22no—
137C4×C73—21no—
138C4×C73—22no—
139C4×C73—41no—
140C4×C73—41no—
141C4×C73#45311yescharacteristic
142C4×C73#45511yescharacteristic
143#8864—731nomaximal, Sylow
144C2×C73—21no—
145C2×C73—21no—
146C2×C73—42no—
147C2×C73—81no—
148C2×C73#185511yescharacteristic
149C2×C73#185611yescharacteristic
150C2×C73#186811yescharacteristic
151#2973D7311yescharacteristic
152#1955—731no—
153#1573—732no—
154#1460—731no—
155#1302—732no—
156C73#886411yescharacteristic, complemented, Sylow
157C2×C4×D4C2×D7311yescharacteristic
158#435—22no—
159#435—22no—
160#432C2×D7312yes—
161#431—732no—
162C2×C4⋊C8—731no—
163#407—21no—
164C2×C22⋊C8—21no—
165C2×C22⋊C8—1461no—
166C2×C22⋊C8—1461no—
167C2×#97—731no—
168#335—1462no—
169#334—1462no—
170#323—732no—
171#323—732no—
172C22×D4Dic14611yescharacteristic
173C23×C4C22⋊D7311yescharacteristic
174C23×C4C22⋊D7311yescharacteristic
175C2×M16—21no—
176C2×M16—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—1461no—
180C22×C8—1461no—
181C4×D4—22no—
182C4×D4—22no—
183C2×C4⋊C4Dic14611yescharacteristic
184C2×C22⋊C4C4×D7311yescharacteristic
185C2×C22⋊C4C4×D7311yescharacteristic
186C2×C42C22×D7311yescharacteristic
187C4⋊C8—1462no—
188C4⋊C8—1462no—
189C4⋊C8C22⋊D7312yes—
190C4⋊C8C22⋊D7312yes—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—2922no—
195C22⋊C8—2922no—
196C22⋊C8—734no—
197#97—22no—
198#97—1462no—
199C4×C8—1462no—
200C24#1192400711yescharacteristic
201C24#1192400711yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4#1192400711yescharacteristic
214C22×C4#1192400711yescharacteristic
215C22×C4C2×C4×D7311yescharacteristic
216C22×C4C2×C22⋊D7311yescharacteristic
217C22×C4C2×Dic14611yescharacteristic
218M16—81no—
219M16—81no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—81no—
223C2×C8—81no—
224C2×C8—2921no—
225C2×C8—2921no—
226C2×C8—2921no—
227C2×C8—2921no—
228C2×C8—2921no—
229C2×C8—1462no—
230C2×C8—1464no—
231C4⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C42C2×C22⋊D7312yes—
235C42#1192402812yes—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23#1455652612yes—
246C23C2×#1192400711yescharacteristic
247D4—44no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4C4×C22⋊D7312yes—
264C2×C4#1455656011yescharacteristic
265C2×C4#1455657012yes—
266C2×C4#1455657411yescharacteristic
267C8—42no—
268C8—42no—
269C8—2922no—
270C8—2924no—
271C8—5841no—
272C22—21no—
273C22—21no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—81no—
283C22—81no—
284C22#1798846011yescharacteristic
285C22C2×#1455652611yescharacteristic
286C22#1798863511yescharacteristic
287C4—22no—
288C4—22no—
289C4—21no—
290C4—41no—
291C4—41no—
292C4#1798875511yescharacteristic
293C4#1798876511yescharacteristic
294C2—21no—
295C2—21no—
296C2—42no—
297C2—81no—
298C2#2261013611yescharacteristic
299C2#2261014111yescharacteristic
300C2#2261021711yescharacteristic
subgroupG/H
centerC2×C4#14556560
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#2973C2
socleC22×C73#385
prime radicalC22×D4×C73Q8
cosocleC2×C42×C73C23
hypercenter#2973D73
hypocenterC73#8864
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #22608717C2semidirect
4. #22607829C2semidirect
156. C73#8864semidirect

Maximal subgroups

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

subgroupindex
#226107302
#226087172
#226082522
#226078292
C73×#29732
#886473