Finite Group Databasebeta

← #33071333

Subgroups — #33071333

Order 18688.

676 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22621052C211yesmaximal, characteristic
2#22619403C211yesmaximal, characteristic
3#22619012C211yesmaximal, characteristic, complemented
4#22609377C214yesmaximal, complemented
5#22609323C214yesmaximal, complemented
6#22608705C211yesmaximal, characteristic, complemented
7#22608603C211yesmaximal, characteristic, complemented
8#22608581C211yesmaximal, characteristic
9C2×#462×C73C211yesmaximal, characteristic, complemented
10C2×#14556622—21no—
11#17989137C2211yescharacteristic
12#17988990C2212yes—
13#17988935C2212yes—
14#17988900—22no—
15#17988864—22no—
16#17988809—22no—
17#17988797C2212yes—
18#17988787C2212yes—
19Q8×#10596831—22no—
20Q8×#10596831C2214yes—
21#17988640—22no—
22#17988640C2211yescharacteristic
23#17988640C2211yescharacteristic
24#17988611C2211yescharacteristic
25#17988586—21no—
26#17988339C2214yes—
27#17988334C2212yes—
28#17988333C2214yes—
29#17988330C2212yes—
30C22×Q16×C73—21no—
31C2×C4⋊Q8×C73C2211yescharacteristic
32C2×C4×Q8×C73C2211yescharacteristic
33#462×C73—22no—
34#462×C73C2214yes—
35C2×C4⋊C8×C73C2211yescharacteristic
36C2×#103×C73—21no—
37C2×#11924028—21no—
38#14556622—44no—
39C2×#11924018—22no—
40C2×#11924015—22no—
41C2×#11924012—21no—
42#14556591C2311yescharacteristic
43#14556588C2311yescharacteristic
44#14556579—21no—
45#14556565—21no—
46#14556565—21no—
47C4×D292—21no—
48C4×D292—21no—
49#14556524—24no—
50C4⋊#10596831—24no—
51C4⋊#10596831—21no—
52C4⋊#10596831D412yes—
53C4⋊#10596831C2312yes—
54C4⋊#10596831C2312yes—
55#14556514—22no—
56#14556511—24no—
57#14556511—22no—
58C22⋊#10596831—21no—
59C22⋊#10596831D412yes—
60C4×#10596831—24no—
61C4×#10596831—21no—
62C4×#10596831C2312yes—
63C22×Q8×C73—21no—
64C22×Q8×C73D411yescharacteristic
65C2×Q16×C73—24no—
66C2×Q16×C73—44no—
67C22×C8×C73—21no—
68C4⋊Q8×C73—21no—
69C4⋊Q8×C73—21no—
70C4⋊Q8×C73—21no—
71C4⋊Q8×C73C2312yes—
72C4×Q8×C73—21no—
73C4×Q8×C73—21no—
74C4×Q8×C73D412yes—
75C4×Q8×C73C2312yes—
76C2×C4⋊C4×C73—21no—
77C2×C4⋊C4×C73—21no—
78C2×C4⋊C4×C73—21no—
79C2×C4⋊C4×C73D411yescharacteristic
80C2×C42×C73—21no—
81C2×C42×C73C2311yescharacteristic
82C4⋊C8×C73—21no—
83C4⋊C8×C73C2312yes—
84#103×C73—24no—
85#11924028—42no—
86#11924028—42no—
87C2×Dic146—21no—
88C2×D292—21no—
89C2×D292—21no—
90C2×D292—21no—
91C2×C22⋊D73—41no—
92C2×C4×D73—41no—
93#11924018—44no—
94#11924015—44no—
95#11924012—42no—
96C2×#10596831—22no—
97C2×#10596831—22no—
98C2×#10596831C2×D414yes—
99C4⋊Dic73—21no—
100C4⋊Dic73—21no—
101#11924005—41no—
102#11924005—41no—
103C2×Q8×C73—21no—
104C2×Q8×C73—22no—
105C2×Q8×C73—22no—
106C2×Q8×C73—41no—
107C2×Q8×C73—42no—
108C2×Q8×C73—42no—
109C2×Q8×C73D814yes—
110C2×Q8×C73C2×D412yes—
111C22×C4×C73—21no—
112C22×C4×C73—21no—
113C22×C4×C73—41no—
114C22×C4×C73C2×D411yescharacteristic
115C22×C4×C73C2×D411yescharacteristic
116C22×C4×C73#4111yescharacteristic
117Q16×C73—48no—
118C2×C8×C73—22no—
119C2×C8×C73—42no—
120C4⋊C4×C73—22no—
121C4⋊C4×C73—21no—
122C4⋊C4×C73—22no—
123C4⋊C4×C73—22no—
124C4⋊C4×C73—22no—
125C4⋊C4×C73—41no—
126C4⋊C4×C73—41no—
127C4⋊C4×C73C2×D412yes—
128C42×C73—22no—
129C42×C73—22no—
130C42×C73C2×D411yescharacteristic
131C42×C73C2×D411yescharacteristic
132C42×C73#4111yescharacteristic
133C42×C73C2411yescharacteristic
134C23×C73#12111yescharacteristic
135C22×D73—41no—
136Q8×C73—22no—
137Q8×C73—24no—
138Q8×C73—42no—
139Q8×C73—42no—
140Q8×C73—82no—
141Q8×C73C2×D814yes—
142C2×C4×C73—22no—
143C2×C4×C73—24no—
144C2×C4×C73—22no—
145C2×C4×C73—22no—
146C2×C4×C73—41no—
147C2×C4×C73—42no—
148C2×C4×C73—81no—
149C2×C4×C73—82no—
150C2×C4×C73#12111yescharacteristic
151C2×C4×C73#12111yescharacteristic
152C2×C4×C73#12111yescharacteristic
153C2×C4×C73C2×D812yes—
154C2×C4×C73#13512yes—
155C2×C4×C73C22×D411yescharacteristic
156C2×C4×C73C22×D411yescharacteristic
157C2×C4×C73C2×#4111yescharacteristic
158C22⋊D73—82no—
159C2×Dic73—41no—
160D292—41no—
161D292—41no—
162D292—42no—
163D292—81no—
164C4×D73—82no—
165Dic146—41no—
166Dic146—41no—
167C8×C73—42no—
168#10596831—24no—
169#10596831—42no—
170#10596831—42no—
171C22×C73—21no—
172C22×C73—21no—
173C22×C73#46612yes—
174C22×C73C2×#12111yescharacteristic
175C2×D73—81no—
176C2×D73—81no—
177C2×D73—81no—
178C4×C73—21no—
179C4×C73—24no—
180C4×C73—21no—
181C4×C73—21no—
182C4×C73—42no—
183C4×C73—82no—
184C4×C73#46612yes—
185C4×C73C22×D811yescharacteristic
186C4×C73C2×#13511yescharacteristic
187C4×C73#57812yes—
188Dic73—81no—
189#19404—731nomaximal, Sylow
190C2×C73—22no—
191C2×C73#304411yescharacteristic
192C2×C73C2×#46611yescharacteristic
193C2×C73#306911yescharacteristic
194D73—161no—
195#3226—731no—
196#3037—731no—
197C2×#462D7311yescharacteristic, complemented
198#2982—731no—
199#1682—734no—
200#1674—734no—
201#1571—731no—
202#1546—731no—
203#1540—731no—
204C73#1940411yescharacteristic, complemented, Sylow
205C2×#137—1461no—
206C22×Q16—21no—
207#532—731no—
208C2×C4⋊Q8C2×D7311yescharacteristic
209C2×C4×Q8C2×D7311yescharacteristic
210#500—732no—
211#486—732no—
212#480—1462no—
213#474—1462no—
214#464—1462no—
215#462—22no—
216#462—732no—
217#462C2×D7314yes—
218#460—732no—
219C8×Q8—734no—
220C8×Q8—1462no—
221#431—731no—
222#431—731no—
223#431—1462no—
224#423—731no—
225C2×C4⋊C8C2×D7311yescharacteristic
226#417—1461no—
227C2×#103—21no—
228#335—734no—
229#332—734no—
230#332—732no—
231#331—732no—
232C2×#41—1461no—
233C22×Q8—21no—
234C22×Q8C22⋊D7311yescharacteristic
235#137—2924no—
236C2×Q16—24no—
237C2×Q16—44no—
238C2×Q16—1462no—
239C2×SD16—1462no—
240C2×M16—1461no—
241C22×C8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—731no—
246C4⋊Q8C22×D7312yes—
247#127—731no—
248#124—1461no—
249#121—1461no—
250#121—1461no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8C22⋊D7312yes—
254C4×Q8C22×D7312yes—
255C4×D4—1461no—
256C4×D4—1461no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4C22⋊D7311yescharacteristic
261C2×C42—21no—
262C2×C42C22×D7311yescharacteristic
263#107—1464no—
264C4⋊C8—21no—
265C4⋊C8—732no—
266C4⋊C8—732no—
267C4⋊C8—732no—
268C4⋊C8—1464no—
269C4⋊C8—1461no—
270C4⋊C8C22×D7312yes—
271#103—24no—
272#103—1462no—
273#102—1464no—
274#102—1462no—
275C22⋊C8—732no—
276C22⋊C8—1461no—
277C4×C8—732no—
278C4×C8—1464no—
279C4×C8—1461no—
280#41—2922no—
281#41—2922no—
282C2×Q8—21no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—41no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—1461no—
289C2×Q8#1192401314yes—
290C2×Q8C2×C22⋊D7312yes—
291C2×D4—2921no—
292C2×D4—1461no—
293C2×D4—1461no—
294C2×D4—1461no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—41no—
298C22×C4—2921no—
299C22×C4D4×D7311yescharacteristic
300C22×C4C2×C22⋊D7311yescharacteristic
301C22×C4#1192402711yescharacteristic
302Q16—48no—
303Q16—2924no—
304SD16—2924no—
305M16—2922no—
306C2×C8—22no—
307C2×C8—42no—
308C2×C8—734no—
309C2×C8—1462no—
310C2×C8—1462no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—1461no—
319C4⋊C4—1461no—
320C4⋊C4C2×C22⋊D7312yes—
321C22⋊C4—2921no—
322C22⋊C4—2921no—
323C42—22no—
324C42—22no—
325C42D4×D7311yescharacteristic
326C42C2×C22⋊D7311yescharacteristic
327C42#1192402711yescharacteristic
328C42C23×D7311yescharacteristic
329C23—2921no—
330C23#1455656311yescharacteristic
331Q8—24no—
332Q8—22no—
333Q8—42no—
334Q8—42no—
335Q8—82no—
336Q8—2921no—
337Q8—2921no—
338Q8C2×#1192401314yes—
339D4—2922no—
340D4—2921no—
341D4—2921no—
342D4—5841no—
343D4—5842no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—81no—
351C2×C4—82no—
352C2×C4—2921no—
353C2×C4—5842no—
354C2×C4#1455656311yescharacteristic
355C2×C4#1455656311yescharacteristic
356C2×C4#1455656311yescharacteristic
357C2×C4C2×#1192401312yes—
358C2×C4#1455661112yes—
359C2×C4C2×D4×D7311yescharacteristic
360C2×C4C22×C22⋊D7311yescharacteristic
361C2×C4C2×#1192402711yescharacteristic
362C8—42no—
363C8—2922no—
364C8—2922no—
365C8—1464no—
366C22—21no—
367C22—21no—
368C22—5841no—
369C22—5841no—
370C22—5841no—
371C22#1798882112yes—
372C22C2×#1455656311yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—24no—
377C4—42no—
378C4—82no—
379C4—5841no—
380C4#1798882112yes—
381C4C22×#1192401311yescharacteristic
382C4C2×#1455661111yescharacteristic
383C4#1798940012yes—
384C2—22no—
385C2—11681no—
386C2#2261945811yescharacteristic
387C2C2×#1798882111yescharacteristic
388C2#2261966511yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC2×#462×C73C2
socleC22×C73C2×#121
prime radical#17989137C22
hypercenterC2×#462D73
hypocenterC73#19404
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
3. #22619012C2semidirect
4. #22609377C2semidirect
5. #22609323C2semidirect
6. #22608705C2semidirect
7. #22608603C2semidirect
9. C2×#462×C73C2semidirect
197. C2×#462D73semidirect
204. C73#19404semidirect

Maximal subgroups

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

subgroupindex
#226210522
#226194032
#226190122
#226093772
#226093232
#226087052
#226086032
#226085812
C2×#462×C732
#1940473