Finite Group Databasebeta

← #32983836

Subgroups — #32983836

Order 18688.

542 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 127 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#22619012C211yesmaximal, characteristic, complemented
2#22614836C211yesmaximal, characteristic
3#22608355C212yesmaximal
4#22607791C212yesmaximal, complemented
5C73×#1801C211yesmaximal, characteristic, complemented
6C2×#14556567—21no—
7C2×C4×D292C2211yescharacteristic
8Q8×#10596831—24no—
9#17988640—22no—
10#17988640C2212yes—
11#17988517C412yes—
12#17988504C412yes—
13#17988478—22no—
14#17988453C2211yescharacteristic
15#17988444—21no—
16#17988314—24no—
17C2×C4×Q8×C73C2211yescharacteristic
18#423×C73C2212yes—
19C2×#101×C73—21no—
20#343×C73—21no—
21C22×D292C2×C411yescharacteristic
22C22×C4×D73—21no—
23#14556567—42no—
24#14556567—42no—
25C4×D292—22no—
26C4×D292—22no—
27C2×C4⋊Dic73—21no—
28C2×C4⋊Dic73C2×C411yescharacteristic
29C2×#11924007—22no—
30C2×#11924005—22no—
31C2×#11924005—22no—
32C2×#11924005—21no—
33C4⋊#10596831—24no—
34C4⋊#10596831—22no—
35C22⋊#10596831—22no—
36C4×#10596831—24no—
37C4×#10596831D414yes—
38#14556498—22no—
39#14556498—21no—
40#14556498C2×C412yes—
41C22×Q8×C73D411yescharacteristic
42C2×M16×C73—22no—
43C4×Q8×C73—22no—
44C4×Q8×C73—22no—
45C2×C4⋊C4×C73C2×C412yes—
46C2×C4⋊C4×C73D411yescharacteristic
47C2×C42×C73D412yes—
48C2×C42×C73C2311yescharacteristic
49C4⋊C8×C73—22no—
50C4⋊C8×C73—22no—
51#101×C73—42no—
52C23×D73—21no—
53C2×D292—22no—
54C2×D292—42no—
55C22×Dic73—22no—
56C22×Dic73—21no—
57C2×C4×D73—42no—
58C2×C4×D73—41no—
59C2×C4×D73—41no—
60C2×C4×D73—42no—
61C2×#10596831—24no—
62C4⋊Dic73—22no—
63C4⋊Dic73—41no—
64C4⋊Dic73—41no—
65#11924007—44no—
66#11924005—44no—
67#11924005—44no—
68#11924005—41no—
69#11924005—41no—
70C2×Q8×C73—22no—
71C2×Q8×C73—22no—
72C2×Q8×C73D812yes—
73C2×Q8×C73SD1612yes—
74C22×C4×C73C22⋊C412yes—
75C22×C4×C73C22⋊C412yes—
76C22×C4×C73C22×C411yescharacteristic
77C22×C4×C73C2×D411yescharacteristic
78C22×C4×C73C2×D411yescharacteristic
79M16×C73—42no—
80M16×C73—42no—
81C2×C8×C73—42no—
82C4⋊C4×C73—24no—
83C4⋊C4×C73—22no—
84C42×C73—22no—
85C42×C73—22no—
86C42×C73C2×D412yes—
87C42×C73#4112yes—
88C23×C73C2×C22⋊C411yescharacteristic
89C22×D73—22no—
90C22×D73—41no—
91C22×D73—41no—
92C22×D73—41no—
93C22×D73—41no—
94C22×D73—41no—
95C22×D73—41no—
96Q8×C73—24no—
97Q8×C73—24no—
98C2×C4×C73—22no—
99C2×C4×C73—22no—
100C2×C4×C73—22no—
101C2×C4×C73—41no—
102C2×C4×C73—41no—
103C2×C4×C73—42no—
104C2×C4×C73#10214yes—
105C2×C4×C73C4×D412yes—
106C2×C4×C73#12011yescharacteristic
107C2×C4×C73#12112yes—
108C2×C4×C73#12311yescharacteristic
109C2×C4×C73C2×D811yescharacteristic
110C2×C4×C73C2×SD1611yescharacteristic
111C2×C4×C73#13512yes—
112C2×Dic73—42no—
113C2×Dic73—42no—
114C2×Dic73—41no—
115C2×Dic73—42no—
116C2×Dic73—41no—
117C2×Dic73—41no—
118D292—24no—
119D292—42no—
120C4×D73—82no—
121C4×D73—82no—
122C8×C73—82no—
123#10596831—44no—
124C22×C73—21no—
125C22×C73—21no—
126C22×C73#38611yescharacteristic
127C22×C73C2×#10211yescharacteristic
128C22×C73#41611yescharacteristic
129C2×D73—42no—
130C2×D73—42no—
131C2×D73—42no—
132C2×D73—81no—
133C2×D73—81no—
134C2×D73—81no—
135C2×D73—81no—
136C4×C73—22no—
137C4×C73—24no—
138C4×C73—21no—
139C4×C73—21no—
140C4×C73—42no—
141C4×C73#46512yes—
142C4×C73#46612yes—
143Dic73—82no—
144Dic73—81no—
145#9486—731nomaximal, Sylow
146C2×C73—21no—
147C2×C73—21no—
148C2×C73#189111yescharacteristic
149C2×C73#189311yescharacteristic
150C2×C73#191211yescharacteristic
151D73—82no—
152#2982—731no—
153#2536—731no—
154#1801D7311yescharacteristic, complemented
155#1486—732no—
156#1286—732no—
157C73#948611yescharacteristic, complemented, Sylow
158C2×C22⋊Q8—1461no—
159C2×C4×Q8C2×D7311yescharacteristic
160C2×C4×D4—731no—
161C8×Q8—1464no—
162#431—732no—
163#431—1462no—
164#423C2×D7312yes—
165C2×#101—21no—
166#398—732no—
167#394—732no—
168#388—1462no—
169#382—731no—
170#380—1461no—
171#343—21no—
172D4⋊C8—1464no—
173C22×Q8C22⋊D7311yescharacteristic
174C22×D4—731no—
175C23×C4—1461no—
176C2×M16—22no—
177C22⋊Q8—2922no—
178C22⋊Q8—2922no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×D4—1462no—
182C4×D4—1462no—
183C2×C4⋊C4—731no—
184C2×C4⋊C4—1462no—
185C2×C4⋊C4—1461no—
186C2×C4⋊C4C4×D7312yes—
187C2×C4⋊C4C22⋊D7311yescharacteristic
188C2×C22⋊C4—1462no—
189C2×C22⋊C4—1462no—
190C2×C22⋊C4—1461no—
191C2×C42D29212yes—
192C2×C42C22×D7311yescharacteristic
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—1464no—
196C4⋊C8—1462no—
197#101—42no—
198C22⋊C8—1462no—
199C4×C8—734no—
200C4×C8—1464no—
201#95—732no—
202#95—1462no—
203#95—1461no—
204C24—1461no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8#1192401312yes—
208C2×Q8#1192401512yes—
209C2×D4—2922no—
210C2×D4—1462no—
211C22×C4—2922no—
212C22×C4—2921no—
213C22×C4—2921no—
214C22×C4—2922no—
215C22×C4—1462no—
216C22×C4—1461no—
217C22×C4#1192400512yes—
218C22×C4#1192400512yes—
219C22×C4C2×C4×D7311yescharacteristic
220C22×C4C2×C22⋊D7311yescharacteristic
221C22×C4C2×D29211yescharacteristic
222M16—42no—
223M16—42no—
224C2×C8—42no—
225C2×C8—1464no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—2924no—
229C4⋊C4—2921no—
230C4⋊C4—2921no—
231C4⋊C4—1462no—
232C22⋊C4—2924no—
233C22⋊C4—2924no—
234C22⋊C4—2921no—
235C22⋊C4—2921no—
236C42—22no—
237C42—22no—
238C42D4×D7312yes—
239C42#1192402712yes—
240C23—2921no—
241C23—2921no—
242C23—2921no—
243C23—2921no—
244C23—2921no—
245C23—2921no—
246C23—1462no—
247C23C2×#1192400511yescharacteristic
248Q8—24no—
249Q8—24no—
250D4—2922no—
251D4—1464no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—2922no—
259C2×C4—2922no—
260C2×C4—2921no—
261C2×C4—2922no—
262C2×C4—2921no—
263C2×C4—2921no—
264C2×C4—5842no—
265C2×C4—5842no—
266C2×C4#1455651114yes—
267C2×C4#1455655112yes—
268C2×C4#1455656111yescharacteristic
269C2×C4#1455656312yes—
270C2×C4#1455657511yescharacteristic
271C2×C4C2×#1192401311yescharacteristic
272C2×C4C2×#1192401511yescharacteristic
273C2×C4#1455661112yes—
274C8—82no—
275C8—2924no—
276C22—21no—
277C22—21no—
278C22—2922no—
279C22—2922no—
280C22—2922no—
281C22—5841no—
282C22—5841no—
283C22—5841no—
284C22—5841no—
285C22#1798846411yescharacteristic
286C22C2×#1455651111yescharacteristic
287C22#1798858111yescharacteristic
288C4—21no—
289C4—21no—
290C4—22no—
291C4—24no—
292C4—42no—
293C4—5842no—
294C4—5841no—
295C4#1798881812yes—
296C4#1798882112yes—
297C2—21no—
298C2—21no—
299C2—5842no—
300C2#2261034711yescharacteristic
301C2#2261035711yescharacteristic
302C2#2261046611yescharacteristic
subgroupG/H
centerC22#17988464
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1801C2
socleC22×C73#386
prime radicalC22×D292C2×C4
cosocleC2×C42×C73C23
hypercenter#1801D73
hypocenterC73#9486
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
1. #22619012C2semidirect
4. #22607791C2semidirect
5. C73×#1801C2semidirect
154. #1801D73semidirect
157. C73#9486semidirect

Maximal subgroups

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

subgroupindex
#226190122
#226148362
#226083552
#226077912
C73×#18012
#948673