Finite Group Databasebeta

← #32999320

Subgroups — #32999320

Order 18688.

328 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 59 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#22612148C211yesmaximal, characteristic, complemented
2#22612143C211yesmaximal, characteristic, complemented
3#22611447C211yesmaximal, characteristic, complemented
4#22607925C211yesmaximal, characteristic, complemented
5#22607922C211yesmaximal, characteristic, complemented
6#22607867C211yesmaximal, characteristic, complemented
7C73×#3197C211yesmaximal, characteristic
8#17988636C2211yescharacteristic
9#17988635—21no—
10#17988551—21no—
11#17988409—21no—
12#17988409—21no—
13#17988403—21no—
14#17988400—21no—
15#17988369C2211yescharacteristic
16#17988369C2211yescharacteristic
17#17988345C2211yescharacteristic
18#525×C73—21no—
19#485×C73—21no—
20#482×C73—21no—
21#470×C73C2211yescharacteristic
22#469×C73C2211yescharacteristic
23#468×C73—21no—
24#452×C73—21no—
25#449×C73—21no—
26#407×C73C2211yescharacteristic
27C2×#11924012—21no—
28C2×#11924012—21no—
29#14556531—21no—
30#14556531—21no—
31#14556527—21no—
32#14556527—21no—
33#14556526—42no—
34C22⋊#10596831—42no—
35C2×#41×C73—21no—
36C23×C4×C73D411yescharacteristic
37C2×Q16×C73D411yescharacteristic
38C2×SD16×C73—21no—
39C2×D8×C73D411yescharacteristic
40C2×M16×C73D411yescharacteristic
41C2×M16×C73C2311yescharacteristic
42#123×C73—41no—
43C22⋊Q8×C73—21no—
44C22⋊Q8×C73—21no—
45#121×C73—21no—
46#121×C73—21no—
47#120×C73—41no—
48C4×D4×C73—41no—
49C4×D4×C73—41no—
50#117×C73—21no—
51#107×C73—21no—
52#106×C73D411yescharacteristic
53#106×C73D411yescharacteristic
54#103×C73—21no—
55#103×C73—21no—
56#102×C73—21no—
57#102×C73—21no—
58C22⋊C8×C73—21no—
59C22⋊C8×C73—21no—
60#11924012—41no—
61#11924012—41no—
62#11924012—41no—
63#11924012—41no—
64C2×#10596831—22no—
65C2×#10596831—41no—
66#11924003—41no—
67#11924003—41no—
68C24×C73SD1611yescharacteristic
69#41×C73—81no—
70#41×C73—81no—
71C2×Q8×C73—21no—
72C2×D4×C73—21no—
73C2×D4×C73—41no—
74C2×D4×C73—41no—
75C2×D4×C73—41no—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—41no—
79C22×C4×C73—41no—
80C22×C4×C73—41no—
81C22×C4×C73SD1611yescharacteristic
82C22×C4×C73C2×D411yescharacteristic
83Q16×C73—41no—
84SD16×C73—81no—
85D8×C73—41no—
86M16×C73—41no—
87M16×C73—41no—
88C2×C8×C73—21no—
89C2×C8×C73C2×D411yescharacteristic
90C2×C8×C73C2×D411yescharacteristic
91C4⋊C4×C73—21no—
92C4⋊C4×C73—21no—
93C4⋊C4×C73—41no—
94C4⋊C4×C73—41no—
95C22⋊C4×C73—41no—
96C22⋊C4×C73—41no—
97C22⋊C4×C73—41no—
98C22⋊C4×C73—41no—
99C22⋊C4×C73—41no—
100C42×C73—41no—
101C23×C73—21no—
102C23×C73—41no—
103C23×C73—41no—
104C23×C73—81no—
105C23×C73C2×SD1611yescharacteristic
106Q8×C73—41no—
107Q8×C73—41no—
108D4×C73—41no—
109D4×C73—41no—
110D4×C73—82no—
111D4×C73—82no—
112D4×C73—82no—
113C2×C4×C73—21no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—41no—
117C2×C4×C73—41no—
118C2×C4×C73—41no—
119C2×C4×C73—41no—
120C2×C4×C73—41no—
121C2×C4×C73—41no—
122C2×C4×C73—81no—
123C2×C4×C73—82no—
124C2×C4×C73#12011yescharacteristic
125C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
126C8×C73—21no—
127C8×C73—21no—
128C8×C73—41no—
129#10596831—42no—
130#10596831—81no—
131C22×C73—21no—
132C22×C73—41no—
133C22×C73—41no—
134C22×C73—41no—
135C22×C73—42no—
136C22×C73—41no—
137C22×C73—81no—
138C22×C73—81no—
139C22×C73—81no—
140C22×C73—81no—
141C22×C73#45011yescharacteristic
142C4×C73—21no—
143C4×C73—41no—
144C4×C73—81no—
145C4×C73—81no—
146C4×C73—81no—
147C4×C73#45311yescharacteristic
148C4×C73#45811yescharacteristic
149#12300—731nomaximal, Sylow
150C2×C73—21no—
151C2×C73—41no—
152C2×C73—81no—
153C2×C73—161no—
154C2×C73#161311yescharacteristic
155#3197D7311yescharacteristic
156#2239—731no—
157#2238—731no—
158#2091—731no—
159#1342—731no—
160#1341—731no—
161#1320—731no—
162C73#1230011yescharacteristic, complemented, Sylow
163#525—21no—
164#485—21no—
165#482—21no—
166#470C2×D7311yescharacteristic
167#469C2×D7311yescharacteristic
168#468—21no—
169#452—21no—
170#449—21no—
171#430—731no—
172#430—1461no—
173#407—1461no—
174#407C2×D7311yescharacteristic
175#365—1461no—
176#365—1461no—
177#362—1461no—
178#361—1461no—
179#349—731no—
180#349—731no—
181#338—731no—
182C2×#41—21no—
183C23×C4C22⋊D7311yescharacteristic
184C2×Q16C22⋊D7311yescharacteristic
185C2×SD16—21no—
186C2×D8C22⋊D7311yescharacteristic
187C2×M16—1461no—
188C2×M16—1461no—
189C2×M16C22⋊D7311yescharacteristic
190C2×M16C22×D7311yescharacteristic
191#123—41no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194#121—21no—
195#121—21no—
196#120—41no—
197C4×D4—41no—
198C4×D4—41no—
199#117—21no—
200M32—1461no—
201M32—1461no—
202#108—2922no—
203#108—1461no—
204#108—1461no—
205#107—21no—
206#106C22⋊D7311yescharacteristic
207#106C22⋊D7311yescharacteristic
208#103—21no—
209#103—21no—
210#102—21no—
211#102—21no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C22⋊C8—2922no—
215C24#1192401611yescharacteristic
216#41—81no—
217#41—81no—
218C2×Q8—21no—
219C2×D4—21no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4#1192401611yescharacteristic
229C22×C4C2×C22⋊D7311yescharacteristic
230Q16—41no—
231SD16—81no—
232D8—41no—
233M16—41no—
234M16—41no—
235M16—2921no—
236M16—2921no—
237M16—2921no—
238M16—2921no—
239C2×C8—21no—
240C2×C8—2921no—
241C2×C8—1462no—
242C2×C8C2×C22⋊D7311yescharacteristic
243C2×C8C2×C22⋊D7311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C42—41no—
254C16—2921no—
255C16—2921no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—81no—
260C23C2×#1192401611yescharacteristic
261Q8—41no—
262Q8—41no—
263D4—41no—
264D4—41no—
265D4—82no—
266D4—82no—
267D4—82no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—81no—
278C2×C4—82no—
279C2×C4#1455656011yescharacteristic
280C2×C4#1455661711yescharacteristic
281C8—21no—
282C8—21no—
283C8—41no—
284C8—2922no—
285C8—5841no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—42no—
291C22—41no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22#1798873811yescharacteristic
297C4—21no—
298C4—41no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4#1798875511yescharacteristic
303C4#1798877711yescharacteristic
304C2—21no—
305C2—41no—
306C2—81no—
307C2—161no—
308C2#2260895111yescharacteristic
subgroupG/H
centerC2#22608951
commutator, cosocleC2×M16×C73C23
FrattiniC2×M16C22×D73
Fitting, prime radicalC73×#3197C2
socleC2×C73#1613
hypercenter#3197D73
hypocenterC73#12300
LCSG ▷ C2×M16×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3197

Complements

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

NH ≅ G/Ntype
1. #22612148C2semidirect
2. #22612143C2semidirect
3. #22611447C2semidirect
4. #22607925C2semidirect
5. #22607922C2semidirect
6. #22607867C2semidirect
162. C73#12300semidirect

Maximal subgroups

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

subgroupindex
#226121482
#226121432
#226114472
#226079252
#226079222
#226078672
C73×#31972
#1230073