Finite Group Databasebeta

← #33071324

Subgroups — #33071324

Order 18688.

644 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22621059C211yesmaximal, characteristic
2#22619403C211yesmaximal, characteristic
3#22619012C211yesmaximal, characteristic, complemented
4#22609349C214yesmaximal, complemented
5#22609315C214yesmaximal, complemented
6#22608701C211yesmaximal, characteristic, complemented
7#22608612C211yesmaximal, characteristic, complemented
8#22608581C211yesmaximal, characteristic
9C2×#477×C73C211yesmaximal, characteristic, complemented
10C2×#14556622—21no—
11#17989137C2211yescharacteristic
12#17988983C2212yes—
13#17988935C2212yes—
14#17988905—22no—
15#17988874—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#17988337C2214yes—
27#17988336C2214yes—
28#17988334C2212yes—
29#17988330C2212yes—
30C2×C4⋊Q8×C73C2211yescharacteristic
31C2×C4×Q8×C73C2211yescharacteristic
32#477×C73—22no—
33#477×C73C2214yes—
34C2×#107×C73—21no—
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#14556522—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
65C22×C8×C73—21no—
66C4⋊Q8×C73—21no—
67C4⋊Q8×C73—21no—
68C4⋊Q8×C73—21no—
69C4⋊Q8×C73C2312yes—
70C4×Q8×C73—21no—
71C4×Q8×C73—21no—
72C4×Q8×C73D412yes—
73C4×Q8×C73C2312yes—
74C2×C4⋊C4×C73—21no—
75C2×C4⋊C4×C73—21no—
76C2×C4⋊C4×C73—21no—
77C2×C4⋊C4×C73D411yescharacteristic
78C2×C42×C73—21no—
79C2×C42×C73C2311yescharacteristic
80#107×C73—24no—
81C4⋊C8×C73—21no—
82C4⋊C8×C73C2312yes—
83#103×C73—24no—
84#11924028—42no—
85#11924028—42no—
86C2×Dic146—21no—
87C2×D292—21no—
88C2×D292—21no—
89C2×D292—21no—
90C2×C22⋊D73—41no—
91C2×C4×D73—41no—
92#11924018—44no—
93#11924015—44no—
94#11924012—42no—
95C2×#10596831—22no—
96C2×#10596831—22no—
97C2×#10596831C2×D414yes—
98C4⋊Dic73—21no—
99C4⋊Dic73—21no—
100#11924005—41no—
101#11924005—41no—
102C2×Q8×C73—21no—
103C2×Q8×C73—22no—
104C2×Q8×C73—22no—
105C2×Q8×C73—41no—
106C2×Q8×C73—82no—
107C2×Q8×C73SD1614yes—
108C2×Q8×C73C2×D412yes—
109C22×C4×C73—21no—
110C22×C4×C73—21no—
111C22×C4×C73—41no—
112C22×C4×C73C2×D411yescharacteristic
113C22×C4×C73C2×D411yescharacteristic
114C22×C4×C73#4111yescharacteristic
115C2×C8×C73—22no—
116C2×C8×C73—42no—
117C4⋊C4×C73—22no—
118C4⋊C4×C73—21no—
119C4⋊C4×C73—22no—
120C4⋊C4×C73—22no—
121C4⋊C4×C73—22no—
122C4⋊C4×C73—41no—
123C4⋊C4×C73—41no—
124C4⋊C4×C73C2×D412yes—
125C42×C73—22no—
126C42×C73—22no—
127C42×C73C2×D411yescharacteristic
128C42×C73C2×D411yescharacteristic
129C42×C73#4111yescharacteristic
130C42×C73C2411yescharacteristic
131C23×C73#12111yescharacteristic
132C22×D73—41no—
133Q8×C73—22no—
134Q8×C73—24no—
135Q8×C73—82no—
136Q8×C73—82no—
137Q8×C73C2×SD1614yes—
138C2×C4×C73—22no—
139C2×C4×C73—24no—
140C2×C4×C73—22no—
141C2×C4×C73—22no—
142C2×C4×C73—41no—
143C2×C4×C73—42no—
144C2×C4×C73—81no—
145C2×C4×C73—82no—
146C2×C4×C73#12111yescharacteristic
147C2×C4×C73#12111yescharacteristic
148C2×C4×C73#12111yescharacteristic
149C2×C4×C73C2×SD1612yes—
150C2×C4×C73#13512yes—
151C2×C4×C73C22×D411yescharacteristic
152C2×C4×C73C22×D411yescharacteristic
153C2×C4×C73C2×#4111yescharacteristic
154C22⋊D73—82no—
155C2×Dic73—41no—
156D292—41no—
157D292—41no—
158D292—42no—
159D292—81no—
160C4×D73—82no—
161Dic146—41no—
162Dic146—41no—
163C8×C73—42no—
164#10596831—24no—
165#10596831—42no—
166#10596831—42no—
167C22×C73—21no—
168C22×C73—21no—
169C22×C73#46512yes—
170C22×C73C2×#12111yescharacteristic
171C2×D73—81no—
172C2×D73—81no—
173C2×D73—81no—
174C4×C73—21no—
175C4×C73—24no—
176C4×C73—21no—
177C4×C73—21no—
178C4×C73—42no—
179C4×C73—82no—
180C4×C73#46512yes—
181C4×C73C22×SD1611yescharacteristic
182C4×C73C2×#13511yescharacteristic
183C4×C73#57812yes—
184Dic73—81no—
185#19403—731nomaximal, Sylow
186C2×C73—22no—
187C2×C73#304411yescharacteristic
188C2×C73C2×#46511yescharacteristic
189C2×C73#306811yescharacteristic
190D73—161no—
191#3227—731no—
192C2×#477D7311yescharacteristic, complemented
193#3037—731no—
194#2982—731no—
195#1678—734no—
196#1672—734no—
197#1570—731no—
198#1548—731no—
199#1540—731no—
200C73#1940311yescharacteristic, complemented, Sylow
201C2×#137—1461no—
202#532—731no—
203C2×C4⋊Q8C2×D7311yescharacteristic
204C2×C4×Q8C2×D7311yescharacteristic
205#498—732no—
206#486—732no—
207#481—1462no—
208#477—22no—
209#477C2×D7314yes—
210#476—1462no—
211#464—1462no—
212#462—732no—
213#460—732no—
214C8×Q8—734no—
215C8×Q8—1462no—
216#431—731no—
217#431—731no—
218#431—1462no—
219C2×#107—21no—
220#423—731no—
221C2×C4⋊C8C2×D7311yescharacteristic
222#417—1461no—
223C2×#103—21no—
224#334—734no—
225#333—734no—
226#332—732no—
227#331—732no—
228C2×#41—1461no—
229C22×Q8—21no—
230C22×Q8C22⋊D7311yescharacteristic
231#137—2924no—
232C2×Q16—1462no—
233C2×SD16—1462no—
234C2×M16—1461no—
235C22×C8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—731no—
240C4⋊Q8C22×D7312yes—
241#127—731no—
242#124—1461no—
243#121—1461no—
244#121—1461no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8C22⋊D7312yes—
248C4×Q8C22×D7312yes—
249C4×D4—1461no—
250C4×D4—1461no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4C22⋊D7311yescharacteristic
255C2×C42—21no—
256C2×C42C22×D7311yescharacteristic
257#107—24no—
258#106—1464no—
259C4⋊C8—21no—
260C4⋊C8—732no—
261C4⋊C8—732no—
262C4⋊C8—732no—
263C4⋊C8—1464no—
264C4⋊C8—1461no—
265C4⋊C8C22×D7312yes—
266#103—24no—
267#103—1462no—
268#102—1464no—
269#102—1462no—
270C22⋊C8—732no—
271C22⋊C8—1461no—
272C4×C8—732no—
273C4×C8—1464no—
274C4×C8—1461no—
275#41—2922no—
276#41—2922no—
277C2×Q8—21no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—41no—
281C2×Q8—82no—
282C2×Q8—1461no—
283C2×Q8#1192401514yes—
284C2×Q8C2×C22⋊D7312yes—
285C2×D4—2921no—
286C2×D4—1461no—
287C2×D4—1461no—
288C2×D4—1461no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—41no—
292C22×C4—2921no—
293C22×C4D4×D7311yescharacteristic
294C22×C4C2×C22⋊D7311yescharacteristic
295C22×C4#1192402711yescharacteristic
296Q16—2924no—
297SD16—2924no—
298M16—2922no—
299C2×C8—22no—
300C2×C8—42no—
301C2×C8—734no—
302C2×C8—1462no—
303C2×C8—1462no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—1461no—
312C4⋊C4—1461no—
313C4⋊C4C2×C22⋊D7312yes—
314C22⋊C4—2921no—
315C22⋊C4—2921no—
316C42—22no—
317C42—22no—
318C42D4×D7311yescharacteristic
319C42C2×C22⋊D7311yescharacteristic
320C42#1192402711yescharacteristic
321C42C23×D7311yescharacteristic
322C23—2921no—
323C23#1455656311yescharacteristic
324Q8—24no—
325Q8—22no—
326Q8—82no—
327Q8—82no—
328Q8—2921no—
329Q8—2921no—
330Q8C2×#1192401514yes—
331D4—2922no—
332D4—2921no—
333D4—2921no—
334D4—5841no—
335D4—5842no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—81no—
343C2×C4—82no—
344C2×C4—2921no—
345C2×C4—5842no—
346C2×C4#1455656311yescharacteristic
347C2×C4#1455656311yescharacteristic
348C2×C4#1455656311yescharacteristic
349C2×C4C2×#1192401512yes—
350C2×C4#1455661112yes—
351C2×C4C2×D4×D7311yescharacteristic
352C2×C4C22×C22⋊D7311yescharacteristic
353C2×C4C2×#1192402711yescharacteristic
354C8—42no—
355C8—2922no—
356C8—2922no—
357C8—1464no—
358C22—21no—
359C22—21no—
360C22—5841no—
361C22—5841no—
362C22—5841no—
363C22#1798881812yes—
364C22C2×#1455656311yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—24no—
369C4—42no—
370C4—82no—
371C4—5841no—
372C4#1798881812yes—
373C4C22×#1192401511yescharacteristic
374C4C2×#1455661111yescharacteristic
375C4#1798940012yes—
376C2—22no—
377C2—11681no—
378C2#2261945811yescharacteristic
379C2C2×#1798881811yescharacteristic
380C2#2261965611yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC2×#477×C73C2
socleC22×C73C2×#121
prime radical#17989137C22
hypercenterC2×#477D73
hypocenterC73#19403
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
3. #22619012C2semidirect
4. #22609349C2semidirect
5. #22609315C2semidirect
6. #22608701C2semidirect
7. #22608612C2semidirect
9. C2×#477×C73C2semidirect
192. C2×#477D73semidirect
200. C73#19403semidirect

Maximal subgroups

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

subgroupindex
#226210592
#226194032
#226190122
#226093492
#226093152
#226087012
#226086122
#226085812
C2×#477×C732
#1940373