Finite Group Databasebeta

← #32982651

Subgroups — #32982651

Order 18688.

422 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 109 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#22620879C211yesmaximal, characteristic
2#22618871C211yesmaximal, characteristic
3#22608616C211yesmaximal, characteristic
4#22608494C211yesmaximal, characteristic
5#22607789C211yesmaximal, characteristic
6#22607762C211yesmaximal, characteristic
7C73×#1993C211yesmaximal, characteristic
8#17989144C2211yescharacteristic
9#17988966C412yescomplemented
10#17988937C412yescomplemented
11#17988794—22no—
12#17988743—22no—
13#17988715—22no—
14#17988715—22no—
15#17988640C2211yescharacteristic
16#17988640C2211yescharacteristic
17#17988613—21no—
18#17988611C2211yescharacteristic
19#17988542—21no—
20#17988333—22no—
21#17988323—22no—
22C2×C4⋊Q8×C73C2211yescharacteristic
23C2×#97×C73C2211yescharacteristic
24#382×C73C2211yescharacteristic
25#340×C73—21no—
26#340×C73—21no—
27C2×#11924018—24no—
28C2×#11924012—21no—
29#14556591C2×C411yescharacteristic
30#14556591C2×C411yescharacteristic
31#14556583—21no—
32#14556570—21no—
33#14556570D412yes—
34C4×Dic146—21no—
35C4×Dic146D412yes—
36#14556550—21no—
37C4⋊#10596831—21no—
38C4⋊#10596831—21no—
39C4⋊#10596831—21no—
40C4⋊#10596831D412yes—
41#14556520—24no—
42#14556515—24no—
43C22⋊#10596831—21no—
44C22⋊#10596831—21no—
45#14556501—22no—
46C4×#10596831—21no—
47C4×#10596831C2×C412yes—
48C8×Dic73—22no—
49C22×Q8×C73—21no—
50C22×C8×C73D411yescharacteristic
51C22×C8×C73D411yescharacteristic
52C4⋊Q8×C73—21no—
53C4⋊Q8×C73—21no—
54C4⋊Q8×C73—21no—
55C4⋊Q8×C73C2×C412yes—
56C2×C4⋊C4×C73—21no—
57C2×C4⋊C4×C73—21no—
58C2×C4⋊C4×C73—21no—
59C2×C42×C73C2311yescharacteristic
60#97×C73—21no—
61#97×C73—21no—
62#95×C73—21no—
63#95×C73—21no—
64C2×Dic146—21no—
65C2×Dic146C22⋊C412yes—
66#11924018—84no—
67#11924012—41no—
68#11924012—41no—
69C2×#10596831—22no—
70C2×#10596831—22no—
71C2×#10596831—41no—
72C4⋊Dic73—21no—
73C4⋊Dic73—21no—
74C4⋊Dic73—21no—
75C4⋊Dic73C22⋊C412yes—
76#11924007—22no—
77#11924007—22no—
78C22⋊Dic73—22no—
79C4×Dic73—22no—
80C2×Q8×C73—22no—
81C2×Q8×C73—41no—
82C2×Q8×C73—82no—
83C22×C4×C73—41no—
84C22×C4×C73—41no—
85C22×C4×C73C2×D411yescharacteristic
86C22×C4×C73C2×D411yescharacteristic
87C22×C4×C73#4111yescharacteristic
88C2×C8×C73—21no—
89C2×C8×C73—21no—
90C2×C8×C73—21no—
91C2×C8×C73—21no—
92C2×C8×C73—21no—
93C2×C8×C73Q1612yescomplemented
94C4⋊C4×C73—22no—
95C4⋊C4×C73—22no—
96C4⋊C4×C73—41no—
97C4⋊C4×C73—41no—
98C4⋊C4×C73—41no—
99C4⋊C4×C73—41no—
100C42×C73C22×C411yescharacteristic
101C42×C73C2×D411yescharacteristic
102C42×C73C2×D411yescharacteristic
103C42×C73#4111yescharacteristic
104C23×C73#12111yescharacteristic
105Q8×C73—44no—
106Q8×C73—82no—
107C2×C4×C73—41no—
108C2×C4×C73—41no—
109C2×C4×C73—42no—
110C2×C4×C73—82no—
111C2×C4×C73—81no—
112C2×C4×C73—81no—
113C2×C4×C73—81no—
114C2×C4×C73C2×C22⋊C411yescharacteristic
115C2×C4×C73C4×D411yescharacteristic
116C2×C4×C73C4×D411yescharacteristic
117C2×C4×C73#12011yescharacteristic
118C2×C4×C73#12111yescharacteristic
119C2×C4×C73#12311yescharacteristic
120C2×C4×C73C2×Q1611yescharacteristic
121C2×C4×C73#13511yescharacteristic
122C2×C4×C73#13711yescharacteristic
123C2×C4×C73#13711yescharacteristic
124C2×Dic73—22no—
125C2×Dic73—22no—
126Dic146—42no—
127Dic146—81no—
128C8×C73—22no—
129C8×C73—41no—
130#10596831—42no—
131#10596831—42no—
132#10596831—81no—
133C22×C73—21no—
134C22×C73—21no—
135C22×C73#38611yescharacteristic
136C22×C73#46211yescharacteristic
137C22×C73#46411yescharacteristic
138C4×C73—21no—
139C4×C73—21no—
140C4×C73—21no—
141C4×C73—21no—
142C4×C73—82no—
143C4×C73—161no—
144C4×C73C4×Q1611yescharacteristic
145C4×C73#44111yescharacteristic
146C4×C73#45111yescharacteristic
147C4×C73#45211yescharacteristic
148Dic73—42no—
149Dic73—42no—
150#9290—731nomaximal, Sylow
151C2×C73—21no—
152C2×C73—21no—
153C2×C73#187811yescharacteristic
154C2×C73#190211yescharacteristic
155C2×C73#191011yescharacteristic
156#3209—731no—
157#2961—731no—
158#1993D7311yescharacteristic
159#1549—731no—
160#1518—731no—
161#1286—731no—
162#1275—731no—
163C73#929011yescharacteristic, complemented, Sylow
164#533—731no—
165C2×C4⋊Q8C2×D7311yescharacteristic
166#494—732no—
167#487—732no—
168#462—1462no—
169#451—1462no—
170#446—1462no—
171#446—1462no—
172#431—731no—
173#431—731no—
174#424—1461no—
175#423—731no—
176#405—1461no—
177C2×#97C2×D7311yescharacteristic
178#382C2×D7311yescharacteristic
179#340—21no—
180#340—21no—
181#332—1462no—
182#328—1462no—
183C22×Q8—21no—
184C2×Q16—1464no—
185C2×M16—1461no—
186C22×C8C22⋊D7311yescharacteristic
187C22×C8C22⋊D7311yescharacteristic
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—731no—
192C4⋊Q8—731no—
193C4⋊Q8C4×D7312yes—
194#125—1461no—
195C22⋊Q8—732no—
196C22⋊Q8—1461no—
197C4×Q8—732no—
198C4×Q8—1461no—
199#117—1461no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C42C22×D7311yescharacteristic
204C4⋊C8—732no—
205C4⋊C8—1464no—
206C4⋊C8—1461no—
207C4⋊C8—1461no—
208C4⋊C8—1461no—
209#103—1464no—
210C22⋊C8—1461no—
211C22⋊C8—1461no—
212#97—21no—
213#97—21no—
214#97—1462no—
215C4×C8—732no—
216C4×C8—1462no—
217C4×C8—1461no—
218#95—21no—
219#95—21no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×Q8—82no—
223C2×Q8—732no—
224C2×Q8—1461no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4D4×D7311yescharacteristic
228C22×C4C2×C22⋊D7311yescharacteristic
229C22×C4#1192402711yescharacteristic
230Q16—5844no—
231M16—2921no—
232M16—2921no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—2921no—
239C2×C8—1462no—
240C2×C8—1462no—
241C2×C8#1192401812yescomplemented
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—732no—
249C4⋊C4—1461no—
250C4⋊C4—1461no—
251C4⋊C4—1461no—
252C4⋊C4—1462no—
253C4⋊C4—1462no—
254C22⋊C4—1462no—
255C42—1462no—
256C42C2×C4×D7311yescharacteristic
257C42D4×D7311yescharacteristic
258C42D4×D7311yescharacteristic
259C42#1192402811yescharacteristic
260C23#1455656211yescharacteristic
261Q8—44no—
262Q8—82no—
263Q8—2922no—
264Q8—5841no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—1462no—
273C2×C4—1462no—
274C2×C4C22⋊C4×D7311yescharacteristic
275C2×C4#1455655111yescharacteristic
276C2×C4C4×C22⋊D7311yescharacteristic
277C2×C4#1455655911yescharacteristic
278C2×C4#1455656411yescharacteristic
279C2×C4#1455657311yescharacteristic
280C2×C4C2×#1192401811yescharacteristic
281C2×C4#1455661111yescharacteristic
282C2×C4#1455662111yescharacteristic
283C2×C4#1455662211yescharacteristic
284C8—22no—
285C8—41no—
286C8—2922no—
287C8—2922no—
288C8—5841no—
289C22—21no—
290C22—21no—
291C22#1798846311yescharacteristic
292C22#1798879411yescharacteristic
293C22#1798880611yescharacteristic
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—82no—
299C4—161no—
300C4—2922no—
301C4—2922no—
302C4C4×#1192401811yescharacteristic
303C4#1798869211yescharacteristic
304C4#1798874211yescharacteristic
305C4#1798874711yescharacteristic
306C2—21no—
307C2—21no—
308C2#2261027511yescharacteristic
309C2#2261040111yescharacteristic
310C2#2261045511yescharacteristic
subgroupG/H
centerC22#17988463
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1993C2
socleC22×C73#386
prime radicalC23×C73#121
cosocleC2×C42×C73C23
hypercenter#1993D73
hypocenterC73#9290
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
9. #17988966C4semidirect
10. #17988937C4semidirect
93. C2×C8×C73Q16semidirect
163. C73#9290semidirect
241. C2×C8#11924018semidirect

Maximal subgroups

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

subgroupindex
#226208792
#226188712
#226086162
#226084942
#226077892
#226077622
C73×#19932
#929073