Finite Group Databasebeta

← #33059907

Subgroups — #33059907

Order 18688.

940 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620703C212yesmaximal
2#22615744C211yesmaximal, characteristic
3#22608973C214yesmaximal, complemented
4#22608905C214yesmaximal, complemented
5#22608503C212yesmaximal
6C2×#17988313C211yesmaximal, characteristic
7C2×#544×C73C211yesmaximal, characteristic
8C2×#14556570C2211yescharacteristic
9#17988922—22no—
10#17988920C2214yes—
11#17988826—24no—
12#17988743—22no—
13#17988742C2214yes—
14C2×C22⋊#10596831C2212yes—
15#17988526C2212yes—
16#17988520—22no—
17#17988507C2212yes—
18#17988460—21no—
19#17988450—21no—
20#17988447C2211yescharacteristic
21#17988324C2218yes—
22#17988313C2214yes—
23#544×C73—24no—
24#544×C73—22no—
25#544×C73C2214yes—
26C2×C4⋊Q8×C73—22no—
27C2×#125×C73—22no—
28C2×C22⋊Q8×C73—21no—
29C2×C22⋊Q8×C73C2212yes—
30C2×#117×C73—21no—
31C22×C4⋊C4×C73C2211yescharacteristic
32C22×Dic146C2311yescharacteristic
33C2×#11924018—24no—
34C22×#10596831—22no—
35#14556570—22no—
36#14556570—22no—
37C2×C4⋊Dic73—21no—
38C2×C4⋊Dic73C2311yescharacteristic
39C2×#11924007—21no—
40C2×#11924007—22no—
41C2×C22⋊Dic73—21no—
42C2×C22⋊Dic73—21no—
43#14556524—24no—
44#14556515—24no—
45#14556514—24no—
46C22⋊#10596831—22no—
47C22⋊#10596831C2314yes—
48#14556498—22no—
49#14556498—22no—
50#14556498—22no—
51#14556498C2312yes—
52C22×Q8×C73—21no—
53C22×Q8×C73D412yes—
54C23×C4×C73D411yescharacteristic
55C23×C4×C73D411yescharacteristic
56C23×C4×C73C2311yescharacteristic
57C4⋊Q8×C73—28no—
58C4⋊Q8×C73—28no—
59#125×C73—28no—
60#125×C73—28no—
61C22⋊Q8×C73—24no—
62C22⋊Q8×C73—24no—
63C22⋊Q8×C73—22no—
64C22⋊Q8×C73—22no—
65C22⋊Q8×C73D414yes—
66C22⋊Q8×C73C2314yes—
67#117×C73—24no—
68#117×C73—24no—
69C2×C4⋊C4×C73—21no—
70C2×C4⋊C4×C73—21no—
71C2×C4⋊C4×C73—21no—
72C2×C4⋊C4×C73—22no—
73C2×C4⋊C4×C73—22no—
74C2×C4⋊C4×C73—22no—
75C2×C4⋊C4×C73—22no—
76C2×C4⋊C4×C73D412yes—
77C2×C4⋊C4×C73D412yes—
78C2×C4⋊C4×C73C2312yes—
79C2×C22⋊C4×C73—22no—
80C2×C42×C73—22no—
81C2×Dic146—22no—
82C2×Dic146—81no—
83C22×Dic73—21no—
84C22×Dic73—22no—
85C22×Dic73—21no—
86#11924018—84no—
87C2×#10596831—24no—
88C2×#10596831—82no—
89C4⋊Dic73—22no—
90C4⋊Dic73—81no—
91#11924007—42no—
92#11924007—82no—
93C22⋊Dic73—41no—
94C22⋊Dic73—41no—
95C22⋊Dic73—42no—
96C24×C73C2×D411yescharacteristic
97C2×Q8×C73—24no—
98C2×Q8×C73—22no—
99C2×Q8×C73—82no—
100C2×Q8×C73—82no—
101C2×Q8×C73C2×D414yes—
102C22×C4×C73—21no—
103C22×C4×C73—21no—
104C22×C4×C73—22no—
105C22×C4×C73—22no—
106C22×C4×C73—22no—
107C22×C4×C73—21no—
108C22×C4×C73—21no—
109C22×C4×C73—81no—
110C22×C4×C73—81no—
111C22×C4×C73Q1614yescomplemented
112C22×C4×C73C2×D412yes—
113C22×C4×C73C2×D412yes—
114C22×C4×C73C2×D411yescharacteristic
115C22×C4×C73C2×D412yes—
116C22×C4×C73C2×D412yes—
117C22×C4×C73C2411yescharacteristic
118C4⋊C4×C73—24no—
119C4⋊C4×C73—24no—
120C4⋊C4×C73—24no—
121C4⋊C4×C73—28no—
122C4⋊C4×C73—24no—
123C4⋊C4×C73—24no—
124C4⋊C4×C73—22no—
125C4⋊C4×C73—24no—
126C4⋊C4×C73—24no—
127C4⋊C4×C73C2×D414yes—
128C22⋊C4×C73—24no—
129C22⋊C4×C73—24no—
130C42×C73—28no—
131C23×C73—41no—
132C23×C73—41no—
133C23×C73#12011yescharacteristic
134C23×C73#12011yescharacteristic
135C23×C73C2×Q1612yes—
136C23×C73#13712yes—
137C23×C73C22×D411yescharacteristic
138Q8×C73—44no—
139Q8×C73—82no—
140Q8×C73—84no—
141C2×C4×C73—22no—
142C2×C4×C73—22no—
143C2×C4×C73—24no—
144C2×C4×C73—24no—
145C2×C4×C73—24no—
146C2×C4×C73—24no—
147C2×C4×C73—22no—
148C2×C4×C73—42no—
149C2×C4×C73—41no—
150C2×C4×C73—42no—
151C2×C4×C73—41no—
152C2×C4×C73—44no—
153C2×C4×C73—42no—
154C2×C4×C73—44no—
155C2×C4×C73—42no—
156C2×C4×C73—81no—
157C2×C4×C73#12012yes—
158C2×C4×C73C2×Q1614yes—
159C2×C4×C73C22×D412yes—
160C2×Dic73—42no—
161C2×Dic73—42no—
162C2×Dic73—42no—
163C2×Dic73—42no—
164C2×Dic73—41no—
165C2×Dic73—41no—
166C2×Dic73—42no—
167Dic146—44no—
168#10596831—44no—
169C22×C73—21no—
170C22×C73—21no—
171C22×C73—21no—
172C22×C73—21no—
173C22×C73—21no—
174C22×C73—21no—
175C22×C73—42no—
176C22×C73—42no—
177C22×C73#45112yes—
178C22×C73#45112yes—
179C22×C73C2×#12011yescharacteristic
180C22×C73C22×Q1611yescharacteristic
181C22×C73C2×#13711yescharacteristic
182C4×C73—24no—
183C4×C73—22no—
184C4×C73—22no—
185C4×C73—44no—
186C4×C73—42no—
187C4×C73—44no—
188Dic73—82no—
189Dic73—81no—
190Dic73—81no—
191#18295—731nomaximal, Sylow
192C2×C73—21no—
193C2×C73—21no—
194C2×C73—22no—
195C2×C73—22no—
196C2×C73C2×#45111yescharacteristic
197C2×C73#302011yescharacteristic
198C2×C73#302611yescharacteristic
199C2×#544D7311yescharacteristic
200#3189—732no—
201#2627—731no—
202#1617—734no—
203#1605—734no—
204#1520—732no—
205C2×#324—731no—
206C73#1829511yescharacteristic, complemented, Sylow
207#544—24no—
208#544—22no—
209#544C2×D7314yes—
210C2×C4⋊Q8—22no—
211C2×#125—22no—
212C2×C22⋊Q8—21no—
213C2×C22⋊Q8—731no—
214C2×C22⋊Q8C2×D7312yes—
215C2×#117—21no—
216C22×C4⋊C4C2×D7311yescharacteristic
217#484—734no—
218#484—1462no—
219#467—1464no—
220#451—734no—
221#451—1462no—
222C2×C22⋊C8—732no—
223#400—732no—
224#398—1462no—
225#395—732no—
226#385—1461no—
227#381—1461no—
228#380—731no—
229#328—738no—
230#324—734no—
231C22×Q8—21no—
232C22×Q8—731no—
233C22×Q8C22⋊D7312yes—
234C23×C4C22⋊D7311yescharacteristic
235C23×C4C22⋊D7311yescharacteristic
236C23×C4C22×D7311yescharacteristic
237C2×Q16—1464no—
238C22×C8—1462no—
239C4⋊Q8—28no—
240C4⋊Q8—28no—
241#125—28no—
242#125—28no—
243C22⋊Q8—24no—
244C22⋊Q8—24no—
245C22⋊Q8—22no—
246C22⋊Q8—22no—
247C22⋊Q8—1462no—
248C22⋊Q8—1462no—
249C22⋊Q8C22⋊D7314yes—
250C22⋊Q8C22×D7314yes—
251#117—24no—
252#117—24no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—731no—
261C2×C4⋊C4—1461no—
262C2×C4⋊C4—1461no—
263C2×C4⋊C4—1462no—
264C2×C4⋊C4C22⋊D7312yes—
265C2×C4⋊C4C22⋊D7312yes—
266C2×C4⋊C4C22×D7312yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—1461no—
269C2×C22⋊C4—1461no—
270C2×C42—22no—
271#107—1464no—
272#103—1464no—
273#103—1464no—
274C22⋊C8—734no—
275C22⋊C8—1462no—
276#95—732no—
277#95—1462no—
278#95—1462no—
279#95—1462no—
280C24C2×C22⋊D7311yescharacteristic
281C2×Q8—24no—
282C2×Q8—22no—
283C2×Q8—82no—
284C2×Q8—82no—
285C2×Q8—5841no—
286C2×Q8—1462no—
287C2×Q8C2×C22⋊D7314yes—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4—1462no—
298C22×C4—1461no—
299C22×C4—1461no—
300C22×C4#1192401814yescomplemented
301C22×C4C2×C22⋊D7312yes—
302C22×C4C2×C22⋊D7312yes—
303C22×C4C2×C22⋊D7311yescharacteristic
304C22×C4C2×C22⋊D7312yes—
305C22×C4C2×C22⋊D7312yes—
306C22×C4C23×D7311yescharacteristic
307Q16—5844no—
308C2×C8—5842no—
309C2×C8—1464no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—28no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—22no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—2922no—
320C4⋊C4—5842no—
321C4⋊C4—5841no—
322C4⋊C4—1462no—
323C4⋊C4C2×C22⋊D7314yes—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—2921no—
327C22⋊C4—2921no—
328C22⋊C4—2922no—
329C42—28no—
330C23—41no—
331C23—41no—
332C23#1455656011yescharacteristic
333C23#1455656011yescharacteristic
334C23C2×#1192401812yes—
335C23#1455662212yes—
336C23C22×C22⋊D7311yescharacteristic
337Q8—44no—
338Q8—82no—
339Q8—84no—
340Q8—2924no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—81no—
357C2×C4—2922no—
358C2×C4—2922no—
359C2×C4—2922no—
360C2×C4—2922no—
361C2×C4—2922no—
362C2×C4—2921no—
363C2×C4—2921no—
364C2×C4#1455656012yes—
365C2×C4C2×#1192401814yes—
366C2×C4C22×C22⋊D7312yes—
367C8—2924no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—42no—
375C22—42no—
376C22#1798874312yes—
377C22#1798874312yes—
378C22C2×#1455656011yescharacteristic
379C22C22×#1192401811yescharacteristic
380C22C2×#1455662211yescharacteristic
381C4—22no—
382C4—22no—
383C4—24no—
384C4—44no—
385C4—42no—
386C4—44no—
387C4—5842no—
388C4—5841no—
389C4—5841no—
390C2—21no—
391C2—22no—
392C2—22no—
393C2—21no—
394C2C2×#1798874311yescharacteristic
395C2#2261929511yescharacteristic
396C2#2261932811yescharacteristic
subgroupG/H
centerC22C2×#14556560
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#544×C73C2
socleC22×C73C2×#120
prime radicalC24×C73C2×D4
hypercenterC2×#544D73
hypocenterC73#18295
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #22608973C2semidirect
4. #22608905C2semidirect
111. C22×C4×C73Q16semidirect
206. C73#18295semidirect
300. C22×C4#11924018semidirect

Maximal subgroups

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

subgroupindex
#226207032
#226157442
#226089732
#226089052
#226085032
C2×#179883132
C2×#544×C732
#1829573