Finite Group Databasebeta

← #33003038

Subgroups — #33003038

Order 18688.

590 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
1C22×#14556532C211yesmaximal, characteristic
2C2×#17988669C211yesmaximal, characteristic, complemented
3C2×#17988389C211yesmaximal, characteristic, complemented
4#22610798C211yesmaximal, characteristic, complemented
5#22608022C211yesmaximal, characteristic, complemented
6#22608008C211yesmaximal, characteristic, complemented
7C2×#357×C73C211yesmaximal, characteristic, complemented
8C22×D584C2211yescharacteristic
9C2×#14556553—21no—
10C2×#14556532—22no—
11C2×#14556532—22no—
12C2×#14556532C414yescomplemented
13C22×#11924003C2211yescharacteristic
14#17988669—22no—
15#17988669—22no—
16C2×#14556525C2211yescharacteristic
17C2×#14556512—21no—
18C2×#14556511—21no—
19C2×C8×Dic73C2211yescharacteristic
20#17988490—21no—
21#17988389—22no—
22C22×D8×C73C2211yescharacteristic
23C22×C16×C73C2211yescharacteristic
24C2×#107×C73C2211yescharacteristic
25#357×C73—22no—
26C22×D292—21no—
27C22×C4×D73—41no—
28C2×D584—21no—
29C2×D584—21no—
30C2×D584—21no—
31C2×D584C2×C412yes—
32C2×D584D412yes—
33C2×D584D412yes—
34C22×#10596831D411yescharacteristic
35#14556553—24no—
36#14556553—42no—
37C2×C4⋊Dic73—21no—
38C2×C22⋊Dic73—41no—
39C2×C22⋊Dic73—41no—
40C2×#11924005—41no—
41C2×C4×Dic73D411yescharacteristic
42#14556532—28no—
43C2×#11924003—21no—
44C2×#11924003—21no—
45C2×#11924003C2×C412yescomplemented
46#14556525—21no—
47#14556525—21no—
48#14556512—42no—
49#14556511—24no—
50C8×Dic73—21no—
51C8×Dic73—21no—
52C22×D4×C73—21no—
53C2×D8×C73—21no—
54C2×D8×C73—21no—
55C2×D8×C73—21no—
56C2×D8×C73—21no—
57C2×D8×C73—21no—
58C2×D8×C73C2×C412yescomplemented
59C22×C8×C73C2311yescharacteristic
60C2×C4⋊C4×C73—21no—
61C2×C16×C73—21no—
62C2×C16×C73—21no—
63C2×C16×C73—21no—
64#107×C73—21no—
65#107×C73D412yes—
66C23×D73—41no—
67C2×D292—22no—
68C2×D292—22no—
69C2×D292—22no—
70C2×D292—41no—
71C2×D292—41no—
72C2×D292—41no—
73C22×Dic73—41no—
74C22×Dic73D811yescharacteristic
75C22×Dic73D811yescharacteristic
76C2×C4×D73—44no—
77C2×C4×D73—82no—
78C2×C4×D73—82no—
79D584—22no—
80D584—22no—
81D584—22no—
82D584C22⋊C414yescomplemented
83C2×#10596831—21no—
84C2×#10596831—21no—
85C2×#10596831—21no—
86C4⋊Dic73—41no—
87C4⋊Dic73—41no—
88C22⋊Dic73—82no—
89C22⋊Dic73—82no—
90#11924005—44no—
91C4×Dic73—21no—
92C4×Dic73—21no—
93#11924003—22no—
94C24×C73—41no—
95C2×D4×C73—22no—
96C2×D4×C73—41no—
97C2×D4×C73—41no—
98C2×D4×C73—41no—
99C2×D4×C73—41no—
100C2×D4×C73—41no—
101C22×C4×C73—41no—
102C22×C4×C73C2×D411yescharacteristic
103D8×C73—22no—
104D8×C73—22no—
105D8×C73—41no—
106D8×C73—41no—
107C2×C8×C73C22×C411yescharacteristic
108C2×C8×C73C2×D411yescharacteristic
109C2×C8×C73C2×D411yescharacteristic
110C2×C8×C73C2×D411yescharacteristic
111C2×C8×C73#4111yescharacteristic
112C2×C8×C73#4111yescharacteristic
113C4⋊C4×C73—22no—
114C4⋊C4×C73—41no—
115C16×C73—41no—
116C23×C73—42no—
117C23×C73—81no—
118C23×C73—81no—
119C23×C73—81no—
120C23×C73—81no—
121C23×C73—81no—
122C23×C73—81no—
123C23×C73C2×D811yescharacteristic
124C22×D73—42no—
125C22×D73—42no—
126C22×D73—42no—
127C22×D73—81no—
128C22×D73—81no—
129C22×D73—81no—
130C22×D73—81no—
131D4×C73—42no—
132D4×C73—42no—
133D4×C73—81no—
134D4×C73—81no—
135C2×C4×C73—42no—
136C2×C4×C73—81no—
137C2×C4×C73—81no—
138C2×C4×C73C4×D411yescharacteristic
139C2×C4×C73#12011yescharacteristic
140C2×C4×C73#12111yescharacteristic
141C2×C4×C73#13511yescharacteristic
142C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
144C2×Dic73—21no—
145C2×Dic73—21no—
146C2×Dic73—21no—
147C2×Dic73—21no—
148C2×Dic73—21no—
149C2×Dic73—81no—
150C2×Dic73—81no—
151C2×Dic73—81no—
152C2×Dic73D1612yescomplemented
153D292—24no—
154D292—42no—
155D292—42no—
156D292—42no—
157C4×D73—88no—
158C8×C73C2×C22⋊C411yescharacteristic
159C8×C73C4×D411yescharacteristic
160C8×C73#12111yescharacteristic
161C8×C73#12311yescharacteristic
162#10596831—41no—
163C22×C73—82no—
164C22×C73—82no—
165C22×C73—82no—
166C22×C73—161no—
167C22×C73—161no—
168C22×C73—161no—
169C22×C73—161no—
170C22×C73C4×D811yescharacteristic
171C22×C73#44711yescharacteristic
172C22×C73#45911yescharacteristic
173C22×C73C2×D1611yescharacteristic
174C22×C73#50811yescharacteristic
175C22×C73#50911yescharacteristic
176C22×C73#50911yescharacteristic
177C2×D73—44no—
178C2×D73—82no—
179C2×D73—82no—
180C2×D73—82no—
181C2×D73—82no—
182C2×D73—82no—
183C2×D73—82no—
184C4×C73—82no—
185C4×C73#38611yescharacteristic
186C4×C73#44211yescharacteristic
187C4×C73#44911yescharacteristic
188C4×C73#46311yescharacteristic
189Dic73—22no—
190Dic73—41no—
191Dic73—161no—
192#13012—731nomaximal, Sylow
193C2×C73—162no—
194C2×C73#187911yescharacteristic
195C2×C73C4×D1611yescharacteristic
196C2×C73#217711yescharacteristic
197C2×C73#218411yescharacteristic
198C2×C73#218611yescharacteristic
199C2×C73#220611yescharacteristic
200C2×C73#221011yescharacteristic
201D73—84no—
202C22×D16—731no—
203C2×C4×D8—731no—
204C2×#357—731no—
205C2×#357D7311yescharacteristic, complemented
206#1970—731no—
207#1380—731no—
208#1374—731no—
209C73#1301211yescharacteristic, complemented, Sylow
210C22×D8—731no—
211C22×D8C2×D7311yescharacteristic
212C2×C4×D4—1461no—
213C2×D16—734no—
214C2×D16—1462no—
215C2×D16—1462no—
216C22×C16—731no—
217C22×C16C2×D7311yescharacteristic
218C4×D8—1462no—
219C4×D8—1462no—
220C2×#107—731no—
221C2×#107C2×D7311yescharacteristic
222C2×#102—1461no—
223C2×#102—1461no—
224C2×C4×C8—731no—
225#390—1461no—
226#357—22no—
227#357—1462no—
228C22×D4—21no—
229C22×D4—1461no—
230C23×C4—2921no—
231C2×D8—21no—
232C2×D8—21no—
233C2×D8—21no—
234C2×D8—21no—
235C2×D8—21no—
236C2×D8—732no—
237C2×D8—732no—
238C2×D8—732no—
239C2×D8—1461no—
240C2×D8—1461no—
241C2×D8—1461no—
242C2×D8C4×D7312yescomplemented
243C22×C8—731no—
244C22×C8C22×D7311yescharacteristic
245C4×D4—2922no—
246C4×D4—1464no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—1461no—
249C2×C22⋊C4—2921no—
250C2×C22⋊C4—2921no—
251C2×C22⋊C4—2921no—
252C2×C42—731no—
253D16—1468no—
254C2×C16—21no—
255C2×C16—21no—
256C2×C16—21no—
257C2×C16—732no—
258C2×C16—1461no—
259C2×C16—1461no—
260#107—21no—
261#107—1461no—
262#107—1461no—
263#107C22⋊D7312yes—
264#102—2922no—
265#102—1464no—
266C4×C8—1461no—
267C4×C8—1461no—
268C24—41no—
269C24—2921no—
270C2×D4—22no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—2921no—
277C2×D4—2921no—
278C2×D4—2921no—
279C2×D4—1462no—
280C2×D4—1462no—
281C2×D4—1462no—
282C22×C4—41no—
283C22×C4—2924no—
284C22×C4—2921no—
285C22×C4—5842no—
286C22×C4—5842no—
287C22×C4—731no—
288C22×C4—731no—
289C22×C4D4×D7311yescharacteristic
290D8—22no—
291D8—22no—
292D8—41no—
293D8—41no—
294D8—734no—
295D8—1462no—
296D8—1462no—
297D8—1462no—
298C2×C8—1461no—
299C2×C8—1461no—
300C2×C8—1461no—
301C2×C8C2×C4×D7311yescharacteristic
302C2×C8D4×D7311yescharacteristic
303C2×C8D4×D7311yescharacteristic
304C2×C8C2×C22⋊D7311yescharacteristic
305C2×C8#1192402711yescharacteristic
306C2×C8#1192402811yescharacteristic
307C4⋊C4—22no—
308C4⋊C4—41no—
309C4⋊C4—2921no—
310C4⋊C4—2921no—
311C22⋊C4—2924no—
312C22⋊C4—5842no—
313C22⋊C4—5842no—
314C42—1461no—
315C42—1461no—
316C16—41no—
317C16—1462no—
318C23—42no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—2922no—
326C23—2922no—
327C23—2922no—
328C23—5841no—
329C23—5841no—
330C23—5841no—
331C23—5841no—
332C23D8×D7311yescharacteristic
333D4—42no—
334D4—42no—
335D4—81no—
336D4—81no—
337D4—2922no—
338D4—2922no—
339D4—2922no—
340D4—1464no—
341C2×C4—42no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—5848no—
345C2×C4—5841no—
346C2×C4—5841no—
347C2×C4—5841no—
348C2×C4—732no—
349C2×C4—1461no—
350C2×C4—1461no—
351C2×C4—1461no—
352C2×C4—1461no—
353C2×C4—1461no—
354C2×C4#1455655111yescharacteristic
355C2×C4#1455655911yescharacteristic
356C2×C4#1455656411yescharacteristic
357C2×C4#1455661211yescharacteristic
358C2×C4#1455661511yescharacteristic
359C2×C4#1455661811yescharacteristic
360C8—2921no—
361C8C22⋊C4×D7311yescharacteristic
362C8C4×C22⋊D7311yescharacteristic
363C8#1455656211yescharacteristic
364C8#1455657311yescharacteristic
365C22—82no—
366C22—82no—
367C22—82no—
368C22—161no—
369C22—161no—
370C22—161no—
371C22—161no—
372C22—2924no—
373C22—5842no—
374C22—5842no—
375C22—5842no—
376C22—5842no—
377C22—5842no—
378C22—5842no—
379C22#1798866711yescharacteristic
380C22#1798871911yescharacteristic
381C22#1798878111yescharacteristic
382C22D16×D7311yescharacteristic
383C22#1798901811yescharacteristic
384C22#1798902111yescharacteristic
385C22#1798902411yescharacteristic
386C4—82no—
387C4—2921no—
388C4—11681no—
389C4—1462no—
390C4#1798846311yescharacteristic
391C4#1798869611yescharacteristic
392C4#1798873111yescharacteristic
393C4#1798880111yescharacteristic
394C2—162no—
395C2—5844no—
396C2#2261027611yescharacteristic
397C2#2261177511yescharacteristic
398C2#2261180411yescharacteristic
399C2#2261184111yescharacteristic
400C2#2261185311yescharacteristic
401C2#2261196511yescharacteristic
402C2#2261198511yescharacteristic
subgroupG/H
centerC23D8×D73
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC2×#357×C73C2
socleC23×C73C2×D8
prime radicalC22×#14556532C2
cosocleC22×C8×C73C23
hypercenterC2×#357D73
hypocenterC73#13012
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×#17988669C2semidirect
3. C2×#17988389C2semidirect
4. #22610798C2semidirect
5. #22608022C2semidirect
6. #22608008C2semidirect
7. C2×#357×C73C2semidirect
12. C2×#14556532C4semidirect
45. C2×#11924003C2×C4semidirect
58. C2×D8×C73C2×C4semidirect
82. D584C22⋊C4semidirect
152. C2×Dic73D16semidirect
205. C2×#357D73semidirect
209. C73#13012semidirect
242. C2×D8C4×D73semidirect

Maximal subgroups

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

subgroupindex
C22×#145565322
C2×#179886692
C2×#179883892
#226107982
#226080222
#226080082
C2×#357×C732
#1301273