Finite Group Databasebeta

← #33004688

Subgroups — #33004688

Order 18688.

618 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556536C211yesmaximal, characteristic, complemented
2C2×#17988828C211yesmaximal, characteristic, complemented
3C2×#17988823C211yesmaximal, characteristic
4C2×#17988392C211yesmaximal, characteristic, complemented
5C2×#17988389C211yesmaximal, characteristic, complemented
6#22608023C211yesmaximal, characteristic, complemented
7C2×C22⋊C16×C73C211yesmaximal, characteristic, complemented
8C22×Dic292C2211yescharacteristic
9C22×D584C2211yescharacteristic
10C2×#14556570—21no—
11C2×#14556565—21no—
12C2×#14556536—22no—
13C2×#14556536—22no—
14C2×#14556536—22no—
15#17988828—22no—
16#17988828—22no—
17#17988823—22no—
18#17988823—22no—
19C2×#14556525C2211yescharacteristic
20C2×#14556525C2211yescharacteristic
21C2×#14556516—21no—
22C2×#14556513—21no—
23#17988392—22no—
24#17988389—22no—
25C23×C8×C73C2211yescharacteristic
26C22×C16×C73C2211yescharacteristic
27C22×C16×C73C2211yescharacteristic
28C22⋊C16×C73—22no—
29C22×Dic146—21no—
30C22×D292—21no—
31C22×C22⋊D73—41no—
32C2×Dic292—21no—
33C2×Dic292—21no—
34C2×Dic292—21no—
35C2×Dic292—21no—
36C2×Dic292—21no—
37C2×Dic292D412yescomplemented
38C2×D584—21no—
39C2×D584—21no—
40C2×D584—21no—
41C2×D584—21no—
42C2×D584—21no—
43C2×D584D412yes—
44#14556570—24no—
45#14556570—42no—
46#14556565—24no—
47#14556565—42no—
48C2×C4⋊Dic73—21no—
49C2×C4⋊Dic73—21no—
50C2×#11924007—41no—
51C2×C22⋊Dic73—41no—
52C2×#11924005—41no—
53#14556536—28no—
54#14556525—21no—
55#14556525—21no—
56#14556525D412yes—
57#14556525D412yescomplemented
58#14556516—24no—
59#14556513—24no—
60C23×C4×C73D411yescharacteristic
61C22×C8×C73—21no—
62C22×C8×C73—21no—
63C22×C8×C73—21no—
64C22×C8×C73—21no—
65C22×C8×C73—21no—
66C22×C8×C73—21no—
67C22×C8×C73D411yescharacteristic
68C22×C8×C73C2311yescharacteristic
69C2×C16×C73—21no—
70C2×C16×C73—21no—
71C2×C16×C73—21no—
72C2×C16×C73—21no—
73C2×C16×C73—21no—
74C2×C16×C73D412yescomplemented
75C23×D73—41no—
76C2×Dic146—22no—
77C2×Dic146—41no—
78C2×Dic146—41no—
79C2×Dic146—41no—
80C2×Dic146—41no—
81C2×Dic146—41no—
82C2×D292—22no—
83C2×D292—41no—
84C2×D292—41no—
85C2×D292—41no—
86C2×D292—41no—
87C2×D292—41no—
88C2×C22⋊D73—44no—
89C2×C22⋊D73—82no—
90C2×C22⋊D73—82no—
91C22×Dic73—41no—
92C22×Dic73—41no—
93C22×Dic73—41no—
94Dic292—22no—
95Dic292—22no—
96Dic292—41no—
97Dic292—41no—
98D584—22no—
99D584—22no—
100D584—41no—
101D584—41no—
102C4⋊Dic73—22no—
103C4⋊Dic73—22no—
104C4⋊Dic73—41no—
105C4⋊Dic73—41no—
106#11924007—44no—
107C22⋊Dic73—44no—
108#11924005—44no—
109C24×C73D811yescharacteristic
110C22×C4×C73—21no—
111C22×C4×C73—21no—
112C22×C4×C73—21no—
113C22×C4×C73—21no—
114C22×C4×C73—21no—
115C22×C4×C73—21no—
116C22×C4×C73D811yescharacteristic
117C22×C4×C73C2×D411yescharacteristic
118C2×C8×C73—21no—
119C2×C8×C73—21no—
120C2×C8×C73—21no—
121C2×C8×C73—41no—
122C2×C8×C73—41no—
123C2×C8×C73—41no—
124C2×C8×C73—41no—
125C2×C8×C73C2×D411yescharacteristic
126C2×C8×C73C2×D411yescharacteristic
127C2×C8×C73C2×D411yescharacteristic
128C2×C8×C73C2×D411yescharacteristic
129C2×C8×C73C2×D411yescharacteristic
130C2×C8×C73#4111yescharacteristic
131C16×C73—22no—
132C16×C73—41no—
133C23×C73—21no—
134C23×C73—21no—
135C23×C73—21no—
136C23×C73—21no—
137C23×C73—21no—
138C23×C73—21no—
139C23×C73SD3212yescomplemented
140C23×C73C2×D811yescharacteristic
141C22×D73—42no—
142C22×D73—81no—
143C22×D73—81no—
144C22×D73—81no—
145C22×D73—81no—
146C22×D73—81no—
147C22×D73—81no—
148C2×C4×C73—21no—
149C2×C4×C73—21no—
150C2×C4×C73—21no—
151C2×C4×C73—41no—
152C2×C4×C73—41no—
153C2×C4×C73—41no—
154C2×C4×C73—41no—
155C2×C4×C73#12011yescharacteristic
156C2×C4×C73#12011yescharacteristic
157C2×C4×C73#12111yescharacteristic
158C2×C4×C73#13511yescharacteristic
159C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
161C22⋊D73—88no—
162C2×Dic73—42no—
163C2×Dic73—42no—
164C2×Dic73—42no—
165C2×Dic73—81no—
166C2×Dic73—81no—
167C2×Dic73—81no—
168C2×Dic73—81no—
169C2×Dic73—81no—
170C2×Dic73—81no—
171D292—42no—
172D292—42no—
173D292—81no—
174D292—81no—
175Dic146—42no—
176Dic146—42no—
177Dic146—81no—
178Dic146—81no—
179C8×C73—41no—
180C8×C73#12011yescharacteristic
181C8×C73#12111yescharacteristic
182C8×C73#12111yescharacteristic
183C8×C73#12711yescharacteristic
184C22×C73—22no—
185C22×C73—22no—
186C22×C73—22no—
187C22×C73—41no—
188C22×C73—41no—
189C22×C73—41no—
190C22×C73—41no—
191C22×C73#44711yescharacteristic
192C22×C73#44711yescharacteristic
193C22×C73#46611yescharacteristic
194C22×C73C2×SD3211yescharacteristic
195C22×C73#50811yescharacteristic
196C22×C73#50911yescharacteristic
197C22×C73#51011yescharacteristic
198C2×D73—82no—
199C2×D73—82no—
200C2×D73—82no—
201C2×D73—161no—
202C2×D73—161no—
203C2×D73—161no—
204C2×D73—161no—
205C4×C73—41no—
206C4×C73#39211yescharacteristic
207C4×C73#44911yescharacteristic
208C4×C73#44911yescharacteristic
209C4×C73#46911yescharacteristic
210Dic73—82no—
211Dic73—82no—
212Dic73—82no—
213#13305—731nomaximal, Sylow
214C2×C73—42no—
215C2×C73#199911yescharacteristic
216C2×C73#218511yescharacteristic
217C2×C73#218611yescharacteristic
218C2×C73#218711yescharacteristic
219C2×C73#218911yescharacteristic
220C2×C73#221711yescharacteristic
221C2×C73#222011yescharacteristic
222D73—162no—
223C22×SD32—731no—
224C2×#467—731no—
225C2×#466—731no—
226C2×#358—731no—
227C2×#357—731no—
228C2×C22⋊C16D7311yescharacteristic, complemented
229#1380—731no—
230C73#1330511yescharacteristic, complemented, Sylow
231C22×Q16—731no—
232C22×D8—731no—
233C23×C8C2×D7311yescharacteristic
234C2×C22⋊Q8—1461no—
235C2×#121—1461no—
236C2×SD32—1462no—
237C2×SD32—1462no—
238C2×SD32—1462no—
239C22×C16C2×D7311yescharacteristic
240C22×C16C2×D7311yescharacteristic
241#467—1462no—
242#467—1462no—
243#466—1462no—
244#466—1462no—
245C2×#107—731no—
246C2×#107—731no—
247C2×#103—1461no—
248C2×#102—1461no—
249#358—1462no—
250#357—1462no—
251C22⋊C16—22no—
252C22×Q8—1461no—
253C22×D4—2921no—
254C22×D4—1461no—
255C23×C4D29211yescharacteristic
256C2×Q16—732no—
257C2×Q16—1461no—
258C2×Q16—1461no—
259C2×Q16—1461no—
260C2×Q16—1461no—
261C2×Q16—1461no—
262C2×D8—732no—
263C2×D8—1461no—
264C2×D8—1461no—
265C2×D8—1461no—
266C2×D8—1461no—
267C2×D8—1461no—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—21no—
273C22×C8—21no—
274C22×C8D29211yescharacteristic
275C22×C8C22×D7311yescharacteristic
276C22⋊Q8—2922no—
277C22⋊Q8—1464no—
278#121—2922no—
279#121—1464no—
280C2×C4⋊C4—2921no—
281C2×C4⋊C4—1461no—
282C2×C4⋊C4—1461no—
283C2×C22⋊C4—2921no—
284C2×C22⋊C4—2921no—
285SD32—1468no—
286C2×C16—21no—
287C2×C16—21no—
288C2×C16—21no—
289C2×C16—21no—
290C2×C16—21no—
291C2×C16C22⋊D7312yescomplemented
292#107—732no—
293#107—732no—
294#107—1461no—
295#107—1461no—
296#103—1464no—
297#102—1464no—
298C24—2921no—
299C24D58411yescharacteristic
300C2×Q8—2921no—
301C2×Q8—2921no—
302C2×Q8—2921no—
303C2×Q8—2921no—
304C2×Q8—2921no—
305C2×Q8—1462no—
306C2×D4—2924no—
307C2×D4—2921no—
308C2×D4—2921no—
309C2×D4—2921no—
310C2×D4—2921no—
311C2×D4—2921no—
312C2×D4—5842no—
313C2×D4—5842no—
314C2×D4—1462no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—2921no—
322C22×C4—2921no—
323C22×C4—2921no—
324C22×C4D58411yescharacteristic
325C22×C4C2×D29211yescharacteristic
326Q16—2921no—
327Q16—2921no—
328Q16—1462no—
329Q16—1462no—
330D8—2921no—
331D8—2921no—
332D8—1462no—
333D8—1462no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8D4×D7311yescharacteristic
342C2×C8D4×D7311yescharacteristic
343C2×C8D4×D7311yescharacteristic
344C2×C8D4×D7311yescharacteristic
345C2×C8C2×C22⋊D7311yescharacteristic
346C2×C8#1192402811yescharacteristic
347C4⋊C4—2924no—
348C4⋊C4—2921no—
349C4⋊C4—2921no—
350C4⋊C4—1462no—
351C4⋊C4—1462no—
352C22⋊C4—2924no—
353C22⋊C4—2924no—
354C16—22no—
355C16—41no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—2922no—
363C23—5841no—
364C23—5841no—
365C23—5841no—
366C23—5841no—
367C23—5841no—
368C23—5841no—
369C23#1455653612yescomplemented
370C23C2×D58411yescharacteristic
371Q8—2922no—
372Q8—2922no—
373Q8—5841no—
374Q8—5841no—
375D4—2922no—
376D4—2922no—
377D4—5848no—
378D4—5841no—
379D4—5841no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—2922no—
388C2×C4—2922no—
389C2×C4—2922no—
390C2×C4—5841no—
391C2×C4—5841no—
392C2×C4—5841no—
393C2×C4—5841no—
394C2×C4—5841no—
395C2×C4—5841no—
396C2×C4#1455656111yescharacteristic
397C2×C4#1455656111yescharacteristic
398C2×C4#1455656511yescharacteristic
399C2×C4#1455661311yescharacteristic
400C2×C4#1455661911yescharacteristic
401C2×C4#1455661911yescharacteristic
402C8—41no—
403C8#1455655911yescharacteristic
404C8#1455656411yescharacteristic
405C8#1455656411yescharacteristic
406C8#1455658711yescharacteristic
407C22—22no—
408C22—22no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—5842no—
415C22—5842no—
416C22—5842no—
417C22—11681no—
418C22—11681no—
419C22—11681no—
420C22—11681no—
421C22#1798872311yescharacteristic
422C22#1798872311yescharacteristic
423C22#1798882311yescharacteristic
424C22C2×#1455653611yescharacteristic
425C22#1798901911yescharacteristic
426C22#1798902511yescharacteristic
427C22#1798903011yescharacteristic
428C4—41no—
429C4—5842no—
430C4—5842no—
431C4—5842no—
432C4#1798849611yescharacteristic
433C4#1798873411yescharacteristic
434C4#1798873411yescharacteristic
435C4#1798884011yescharacteristic
436C2—42no—
437C2—11682no—
438C2#2261092911yescharacteristic
439C2#2261185011yescharacteristic
440C2#2261185611yescharacteristic
441C2#2261186211yescharacteristic
442C2#2261187311yescharacteristic
443C2#2261202711yescharacteristic
444C2#2261204411yescharacteristic
subgroupG/H
centerC23C2×D584
commutator, cosocleC22×C8×C73C23
FrattiniC22×C8C22×D73
FittingC2×C22⋊C16×C73C2
socleC23×C73C2×D8
prime radicalC2×#17988823C2
hypercenterC2×C22⋊C16D73
hypocenterC73#13305
LCSG ▷ C22×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×C22⋊C16

Complements

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

NH ≅ G/Ntype
1. C22×#14556536C2semidirect
2. C2×#17988828C2semidirect
4. C2×#17988392C2semidirect
5. C2×#17988389C2semidirect
6. #22608023C2semidirect
7. C2×C22⋊C16×C73C2semidirect
37. C2×Dic292D4semidirect
57. #14556525D4semidirect
74. C2×C16×C73D4semidirect
139. C23×C73SD32semidirect
228. C2×C22⋊C16D73semidirect
230. C73#13305semidirect
291. C2×C16C22⋊D73semidirect
369. C23#14556536semidirect

Maximal subgroups

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

subgroupindex
C22×#145565362
C2×#179888282
C2×#179888232
C2×#179883922
C2×#179883892
#226080232
C2×C22⋊C16×C732
#1330573