Finite Group Databasebeta

← #32993261

Subgroups — #32993261

Order 18688.

868 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618521C211yesmaximal, characteristic, complemented
2C2×#17988561C212yesmaximal
3#22609642C212yesmaximal, complemented
4C2×#17988344C211yesmaximal, characteristic, complemented
5C73×#2879C211yesmaximal, characteristic
6C2×#14556550C2212yes—
7C2×#14556550C2212yes—
8#17988629—24no—
9#17988625—22no—
10#17988561—24no—
11#17988561—24no—
12C2×#14556506—24no—
13#17988348—24no—
14#17988344—22no—
15C22×#41×C73C2211yescharacteristic
16C22×M16×C73C2211yescharacteristic
17C22×M16×C73C2211yescharacteristic
18#408×C73—22no—
19#408×C73—22no—
20#407×C73C414yescomplemented
21#14556550—22no—
22#14556550—22no—
23#14556550—22no—
24#14556550—22no—
25#14556550—22no—
26#14556550—22no—
27#14556550D414yes—
28#14556550D414yes—
29C2×C4⋊Dic73—22no—
30C2×C4⋊Dic73—22no—
31C2×C22⋊Dic73—22no—
32C2×C22⋊Dic73—22no—
33C2×C4×Dic73—22no—
34C2×C4×Dic73—22no—
35#14556525—24no—
36#14556523—24no—
37#14556506—216no—
38C2×#41×C73—21no—
39C2×#41×C73—21no—
40C2×#41×C73—22no—
41C2×#41×C73—22no—
42C2×#41×C73—21no—
43C2×#41×C73—21no—
44C2×#41×C73—22no—
45C2×#41×C73—22no—
46C22×Q8×C73Q811yescharacteristic
47C22×D4×C73D412yes—
48C22×D4×C73Q811yescharacteristic
49C23×C4×C73C2×C412yes—
50C23×C4×C73C2311yescharacteristic
51C2×M16×C73—22no—
52C2×M16×C73—22no—
53C2×M16×C73—21no—
54C2×M16×C73—22no—
55C2×M16×C73—22no—
56C2×M16×C73C2×C412yes—
57C2×M16×C73C2×C412yes—
58C2×M16×C73D412yes—
59C22×C8×C73—21no—
60C22×C8×C73D412yes—
61C22⋊C8×C73—28no—
62C22×Dic73—42no—
63C22×Dic73—42no—
64C4⋊Dic73—24no—
65C4⋊Dic73—24no—
66C4⋊Dic73—42no—
67C4⋊Dic73—42no—
68C22⋊Dic73—24no—
69C22⋊Dic73—24no—
70C22⋊Dic73—42no—
71C22⋊Dic73—42no—
72C4×Dic73—24no—
73C4×Dic73—24no—
74C4×Dic73—42no—
75C4×Dic73—42no—
76C24×C73C4⋊C412yes—
77C24×C73C2×D411yescharacteristic
78#41×C73—41no—
79#41×C73—42no—
80#41×C73—42no—
81#41×C73—42no—
82#41×C73—41no—
83#41×C73—42no—
84#41×C73—42no—
85#41×C73—41no—
86#41×C73—42no—
87#41×C73—41no—
88C2×Q8×C73—22no—
89C2×Q8×C73—22no—
90C2×Q8×C73—21no—
91C2×Q8×C73—21no—
92C2×D4×C73—24no—
93C2×D4×C73—22no—
94C2×D4×C73—22no—
95C2×D4×C73—24no—
96C2×D4×C73—22no—
97C2×D4×C73—22no—
98C2×D4×C73—21no—
99C2×D4×C73—21no—
100C22×C4×C73—22no—
101C22×C4×C73—22no—
102C22×C4×C73—24no—
103C22×C4×C73—24no—
104C22×C4×C73C4⋊C412yes—
105C22×C4×C73C22×C411yescharacteristic
106C22×C4×C73C2×D412yes—
107C22×C4×C73C2×D412yes—
108C22×C4×C73C2×D411yescharacteristic
109C22×C4×C73C2×D411yescharacteristic
110C22×C4×C73C2×Q811yescharacteristic
111C22×C4×C73#4112yes—
112C22×C4×C73#4112yes—
113C22×C4×C73#4111yescharacteristic
114C22×C4×C73#4111yescharacteristic
115M16×C73—24no—
116M16×C73—24no—
117M16×C73—24no—
118M16×C73—41no—
119M16×C73—41no—
120C2×C8×C73—22no—
121C2×C8×C73—22no—
122C2×C8×C73—22no—
123C2×C8×C73—42no—
124C2×C8×C73C22⋊C414yes—
125C23×C73—22no—
126C23×C73—24no—
127C23×C73—81no—
128C23×C73—82no—
129C23×C73C2×C4⋊C411yescharacteristic
130C23×C73#12012yes—
131C23×C73#12111yescharacteristic
132C23×C73#12111yescharacteristic
133C23×C73#12312yes—
134Q8×C73—41no—
135Q8×C73—42no—
136Q8×C73—41no—
137D4×C73—42no—
138D4×C73—44no—
139D4×C73—42no—
140D4×C73—41no—
141D4×C73—42no—
142D4×C73—41no—
143C2×C4×C73—22no—
144C2×C4×C73—24no—
145C2×C4×C73—42no—
146C2×C4×C73—44no—
147C2×C4×C73—42no—
148C2×C4×C73C2×C22⋊C411yescharacteristic
149C2×C4×C73#11711yescharacteristic
150C2×C4×C73C4×D412yes—
151C2×C4×C73C4×D412yes—
152C2×C4×C73#12011yescharacteristic
153C2×C4×C73#12112yes—
154C2×C4×C73#12112yes—
155C2×C4×C73#12112yes—
156C2×C4×C73C22⋊Q812yes—
157C2×C4×C73C22⋊Q812yes—
158C2×C4×C73C22⋊Q811yescharacteristic
159C2×C4×C73C22⋊Q811yescharacteristic
160C2×C4×C73#12311yescharacteristic
161C2×C4×C73#12312yes—
162C2×C4×C73#12412yes—
163C2×C4×C73#12411yescharacteristic
164C2×C4×C73#12612yes—
165C2×C4×C73#12711yescharacteristic
166C2×Dic73—44no—
167C2×Dic73—44no—
168C2×Dic73—84no—
169C2×Dic73—84no—
170C8×C73—24no—
171C8×C73—42no—
172C22×C73—42no—
173C22×C73—42no—
174C22×C73—41no—
175C22×C73—41no—
176C22×C73—41no—
177C22×C73—44no—
178C22×C73—42no—
179C22×C73—84no—
180C22×C73#38011yescharacteristic
181C22×C73#38512yes—
182C22×C73#39211yescharacteristic
183C22×C73#39412yes—
184C22×C73#39711yescharacteristic
185C4×C73—42no—
186C4×C73#38611yescharacteristic
187C4×C73#38812yes—
188C4×C73#39011yescharacteristic
189C4×C73#39311yescharacteristic
190C4×C73#39612yes—
191C4×C73#39911yescharacteristic
192Dic73—84no—
193Dic73—84no—
194#11102—731nomaximal, Sylow
195C2×C73—21no—
196C2×C73—22no—
197C2×C73—21no—
198C2×C73—22no—
199C2×C73—21no—
200C2×C73—82no—
201C2×C73#143911yescharacteristic
202#2911—731no—
203C2×#410—732no—
204#2879D7311yescharacteristic
205#1733—732no—
206C2×H(3,ℤ/4)—731no—
207C73#1110211yescharacteristic, complemented, Sylow
208C22×#41C2×D7311yescharacteristic
209C22×M16C2×D7311yescharacteristic
210C22×M16C2×D7311yescharacteristic
211C2×#117—732no—
212C2×#117—732no—
213#428—1464no—
214#427—1462no—
215#410—1464no—
216#410—1464no—
217C2×#99—1464no—
218#408—22no—
219#408—22no—
220#407Dic7314yescomplemented
221#339—1464no—
222H(3,ℤ/4)—1462no—
223C2×#41—22no—
224C2×#41—22no—
225C2×#41—21no—
226C2×#41—21no—
227C2×#41—22no—
228C2×#41—22no—
229C2×#41—21no—
230C2×#41—21no—
231C22×Q8Dic14611yescharacteristic
232C22×D4Dic14611yescharacteristic
233C22×D4D29212yes—
234C23×C4C2×Dic7312yes—
235C23×C4C22×D7311yescharacteristic
236C2×M16—22no—
237C2×M16—22no—
238C2×M16—21no—
239C2×M16—22no—
240C2×M16—22no—
241C2×M16C2×Dic7312yes—
242C2×M16C2×Dic7312yes—
243C2×M16C22⋊D7312yes—
244C22×C8—21no—
245C22×C8C22⋊D7312yes—
246#117—734no—
247#117—734no—
248#117—1462no—
249#117—1462no—
250#117—1462no—
251#117—1462no—
252#117—1462no—
253#117—1462no—
254C2×C4⋊C4—1462no—
255C2×C4⋊C4—1462no—
256C2×C22⋊C4—1462no—
257C2×C22⋊C4—1462no—
258C2×C42—1462no—
259C2×C42—1462no—
260#107—1464no—
261#106—1464no—
262#99—14616no—
263C22⋊C8—28no—
264C24C4⋊Dic7312yes—
265C24C2×D29211yescharacteristic
266#41—41no—
267#41—42no—
268#41—42no—
269#41—42no—
270#41—41no—
271#41—42no—
272#41—42no—
273#41—41no—
274#41—42no—
275#41—41no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×D4—24no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—24no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—2922no—
293C22×C4—2922no—
294C22×C4C4⋊Dic7312yes—
295C22×C4C22×Dic7311yescharacteristic
296C22×C4D4×D7312yes—
297C22×C4D4×D7312yes—
298C22×C4C2×C22⋊D7311yescharacteristic
299C22×C4C2×C22⋊D7311yescharacteristic
300C22×C4C2×Dic14611yescharacteristic
301C22×C4#1192402712yes—
302C22×C4#1192402712yes—
303C22×C4#1192402811yescharacteristic
304C22×C4#1192402811yescharacteristic
305M16—24no—
306M16—24no—
307M16—24no—
308M16—41no—
309M16—41no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—42no—
314C2×C8C22⋊Dic7314yes—
315C4⋊C4—2922no—
316C4⋊C4—2922no—
317C4⋊C4—1464no—
318C4⋊C4—1464no—
319C22⋊C4—2922no—
320C22⋊C4—2922no—
321C22⋊C4—1464no—
322C22⋊C4—1464no—
323C42—2922no—
324C42—2922no—
325C42—1464no—
326C42—1464no—
327C23—22no—
328C23—24no—
329C23—81no—
330C23—82no—
331C23C2×C4⋊Dic7311yescharacteristic
332C23#1455656112yes—
333C23#1455656511yescharacteristic
334C23#1455656511yescharacteristic
335C23#1455657512yes—
336Q8—41no—
337Q8—42no—
338Q8—41no—
339D4—42no—
340D4—44no—
341D4—42no—
342D4—41no—
343D4—42no—
344D4—41no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—2924no—
351C2×C4—2924no—
352C2×C4—5844no—
353C2×C4—5844no—
354C2×C4C2×C22⋊Dic7311yescharacteristic
355C2×C4#1455655011yescharacteristic
356C2×C4D4×Dic7312yes—
357C2×C4D4×Dic7312yes—
358C2×C4#1455655911yescharacteristic
359C2×C4#1455656212yes—
360C2×C4#1455656312yes—
361C2×C4#1455656412yes—
362C2×C4#1455656812yes—
363C2×C4#1455656812yes—
364C2×C4#1455657011yescharacteristic
365C2×C4#1455657011yescharacteristic
366C2×C4#1455657211yescharacteristic
367C2×C4#1455657312yes—
368C2×C4#1455657712yes—
369C2×C4#1455657811yescharacteristic
370C2×C4#1455658412yes—
371C2×C4#1455658711yescharacteristic
372C8—24no—
373C8—42no—
374C22—42no—
375C22—42no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—44no—
380C22—42no—
381C22—84no—
382C22#1798844711yescharacteristic
383C22#1798846212yes—
384C22#1798849611yescharacteristic
385C22#1798850312yes—
386C22#1798851211yescharacteristic
387C4—42no—
388C4—5844no—
389C4—5844no—
390C4#1798846711yescharacteristic
391C4#1798848012yes—
392C4#1798849011yescharacteristic
393C4#1798849911yescharacteristic
394C4#1798851012yes—
395C4#1798852311yescharacteristic
396C2—21no—
397C2—22no—
398C2—21no—
399C2—22no—
400C2—21no—
401C2—82no—
402C2#2260817611yescharacteristic
subgroupG/H
centerC4#17988490
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#2879C2
socleC2×C73#1439
prime radicalC22×#41×C73C22
cosocleC23×C4×C73C23
hypercenter#2879D73
hypocenterC73#11102
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C23×C4 ◁ #2879

Complements

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

NH ≅ G/Ntype
1. #22618521C2semidirect
3. #22609642C2semidirect
4. C2×#17988344C2semidirect
20. #407×C73C4semidirect
207. C73#11102semidirect
220. #407Dic73semidirect

Maximal subgroups

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

subgroupindex
#226185212
C2×#179885612
#226096422
C2×#179883442
C73×#28792
#1110273