Finite Group Databasebeta

← #32982645

Subgroups — #32982645

Order 18688.

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

normal subgroups only

This group has 422 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#22620879C211yesmaximal, characteristic
2#22618868C211yesmaximal, characteristic
3#22610905C211yesmaximal, characteristic
4#22608496C211yesmaximal, characteristic
5#22607791C211yesmaximal, characteristic
6#22607761C211yesmaximal, characteristic
7C73×#1549C211yesmaximal, characteristic
8#17989144C2211yescharacteristic
9#17988966C412yescomplemented
10#17988937C412yescomplemented
11#17988794—22no—
12#17988743—22no—
13#17988716—22no—
14#17988716—22no—
15#17988640C2211yescharacteristic
16#17988615—21no—
17#17988545—21no—
18C2×#14556502C2211yescharacteristic
19#17988453C2211yescharacteristic
20#17988349—21no—
21#17988349—21no—
22#17988325—22no—
23C2×C4⋊Q8×C73C2211yescharacteristic
24#431×C73C2211yescharacteristic
25#423×C73C2211yescharacteristic
26#332×C73—22no—
27C2×#11924018—24no—
28C22×#10596831D411yescharacteristic
29C22×#10596831D411yescharacteristic
30#14556591C2×C411yescharacteristic
31#14556591C2×C411yescharacteristic
32#14556583—21no—
33#14556570—21no—
34#14556570D412yes—
35C4×Dic146—21no—
36C4×Dic146D412yes—
37#14556550—21no—
38C2×C4⋊Dic73—21no—
39C4⋊#10596831—21no—
40#14556520—24no—
41#14556515—24no—
42C22⋊#10596831—21no—
43#14556502—21no—
44#14556502—21no—
45#14556501—22no—
46C4×#10596831C2×C412yes—
47C8×Dic73—22no—
48#14556498—21no—
49#14556498—21no—
50C22×Q8×C73—21no—
51C2×M16×C73—21no—
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×C42×C73C2311yescharacteristic
59C4⋊C8×C73—21no—
60C4⋊C8×C73—21no—
61C4⋊C8×C73D412yes—
62C22⋊C8×C73—21no—
63C4×C8×C73—21no—
64C2×Dic146—21no—
65C2×Dic146C22⋊C412yes—
66C22×Dic73—41no—
67#11924018—84no—
68C2×#10596831—21no—
69C2×#10596831—21no—
70C2×#10596831—22no—
71C2×#10596831—21no—
72C2×#10596831—21no—
73C2×#10596831—21no—
74C2×#10596831Q1612yescomplemented
75C4⋊Dic73—21no—
76C4⋊Dic73—21no—
77C4⋊Dic73—21no—
78C4⋊Dic73—41no—
79C4⋊Dic73—41no—
80C4⋊Dic73C22⋊C412yes—
81#11924007—22no—
82#11924007—22no—
83C22⋊Dic73—22no—
84C4×Dic73—22no—
85C2×Q8×C73—22no—
86C2×Q8×C73—41no—
87C2×Q8×C73—82no—
88C22×C4×C73—41no—
89C22×C4×C73C2×D411yescharacteristic
90C22×C4×C73C2×D411yescharacteristic
91C22×C4×C73#4111yescharacteristic
92M16×C73—41no—
93M16×C73—41no—
94C2×C8×C73—22no—
95C2×C8×C73—41no—
96C4⋊C4×C73—22no—
97C4⋊C4×C73—22no—
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×C73C2×C22⋊C411yescharacteristic
112C2×C4×C73C4×D411yescharacteristic
113C2×C4×C73C4×D411yescharacteristic
114C2×C4×C73#12011yescharacteristic
115C2×C4×C73#12111yescharacteristic
116C2×C4×C73#12311yescharacteristic
117C2×C4×C73C2×Q1611yescharacteristic
118C2×C4×C73#13511yescharacteristic
119C2×C4×C73#13711yescharacteristic
120C2×C4×C73#13711yescharacteristic
121C2×Dic73—22no—
122C2×Dic73—22no—
123C2×Dic73—81no—
124C2×Dic73—81no—
125C2×Dic73—81no—
126Dic146—42no—
127Dic146—81no—
128C8×C73—42no—
129C8×C73—81no—
130#10596831—22no—
131#10596831—42no—
132#10596831—41no—
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×C73C4×Q1611yescharacteristic
144C4×C73#44111yescharacteristic
145C4×C73#45111yescharacteristic
146C4×C73#45211yescharacteristic
147Dic73—42no—
148Dic73—42no—
149Dic73—161no—
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#1993—731no—
159#1549D7311yescharacteristic
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#431C2×D7311yescharacteristic
174#424—1461no—
175#423C2×D7311yescharacteristic
176#405—1461no—
177C2×#97—731no—
178#382—731no—
179#340—1461no—
180#340—1461no—
181#332—22no—
182#328—1462no—
183C22×Q8—21no—
184C2×Q16—1464no—
185C2×M16—21no—
186C22×C8—731no—
187C22×C8—731no—
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—1461no—
203C2×C42C22×D7311yescharacteristic
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—1464no—
207C4⋊C8—1461no—
208C4⋊C8C22⋊D7312yes—
209#103—1464no—
210C22⋊C8—21no—
211C22⋊C8—1461no—
212#97—1462no—
213#97—1461no—
214#97—1461no—
215C4×C8—21no—
216C4×C8—732no—
217C4×C8—1462no—
218#95—1461no—
219#95—1461no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×Q8—82no—
223C2×Q8—732no—
224C2×Q8—1461no—
225C22×C4—41no—
226C22×C4—2921no—
227C22×C4D4×D7311yescharacteristic
228C22×C4D4×D7311yescharacteristic
229C22×C4#1192402811yescharacteristic
230Q16—5844no—
231M16—41no—
232M16—41no—
233C2×C8—22no—
234C2×C8—41no—
235C2×C8—732no—
236C2×C8—1462no—
237C2×C8—1461no—
238C2×C8—1461no—
239C2×C8—1461no—
240C2×C8—1461no—
241C2×C8—1461no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—2921no—
247C4⋊C4—2921no—
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
258C42C2×C22⋊D7311yescharacteristic
259C42#1192402711yescharacteristic
260C23#1455656411yescharacteristic
261Q8—44no—
262Q8—82no—
263Q8—2922no—
264Q8—5841no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—82no—
269C2×C4—5841no—
270C2×C4—5841no—
271C2×C4—5841no—
272C2×C4—1462no—
273C2×C4—1462no—
274C2×C4C22⋊C4×D7311yescharacteristic
275C2×C4#1455655111yescharacteristic
276C2×C4C4×C22⋊D7311yescharacteristic
277C2×C4#1455655911yescharacteristic
278C2×C4#1455656211yescharacteristic
279C2×C4#1455657311yescharacteristic
280C2×C4Q16×D7311yescharacteristic
281C2×C4#1455661211yescharacteristic
282C2×C4#1455662011yescharacteristic
283C2×C4#1455662311yescharacteristic
284C8—42no—
285C8—81no—
286C8—2922no—
287C8—2921no—
288C8—1462no—
289C22—21no—
290C22—21no—
291C22#1798846311yescharacteristic
292C22#1798879611yescharacteristic
293C22#1798881211yescharacteristic
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—82no—
299C4—2922no—
300C4—2922no—
301C4—11681no—
302C4#1798867911yescharacteristic
303C4#1798869111yescharacteristic
304C4#1798874111yescharacteristic
305C4#1798874611yescharacteristic
306C2—21no—
307C2—21no—
308C2#2261026911yescharacteristic
309C2#2261040711yescharacteristic
310C2#2261045511yescharacteristic
subgroupG/H
centerC22#17988463
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1549C2
socleC22×C73#386
prime radicalC23×C73#121
cosocleC2×C42×C73C23
hypercenter#1549D73
hypocenterC73#9290
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
9. #17988966C4semidirect
10. #17988937C4semidirect
74. C2×#10596831Q16semidirect
163. C73#9290semidirect

Maximal subgroups

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

subgroupindex
#226208792
#226188682
#226109052
#226084962
#226077912
#226077612
C73×#15492
#929073