Finite Group Databasebeta

← #33004683

Subgroups — #33004683

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
1C2×#17988825C211yesmaximal, characteristic, complemented
2C2×#17988820C211yesmaximal, characteristic
3C2×#17988391C211yesmaximal, characteristic, complemented
4C2×#17988388C211yesmaximal, characteristic, complemented
5C2×#17988366C211yesmaximal, characteristic, complemented
6#22608023C211yesmaximal, characteristic, complemented
7C22×SD32×C73C211yesmaximal, characteristic, complemented
8C22×C8×D73C2211yescharacteristic
9C2×#14556567—21no—
10C2×#14556562—21no—
11C22×#11924003C2211yescharacteristic
12#17988825—22no—
13#17988825—22no—
14#17988820—22no—
15#17988820—22no—
16C2×#14556525C2211yescharacteristic
17C2×#14556525C2211yescharacteristic
18C2×#14556515—21no—
19C2×#14556512—21no—
20#17988391—22no—
21#17988388—22no—
22#17988366—22no—
23C22×Q16×C73C2211yescharacteristic
24C22×D8×C73C2211yescharacteristic
25C2×SD32×C73—22no—
26C2×SD32×C73—22no—
27C2×SD32×C73—22no—
28C22×C16×C73C2211yescharacteristic
29C22×C22⋊D73—41no—
30C22×C4×D73D411yescharacteristic
31C22×#10596831D411yescharacteristic
32C2×C8×D73—21no—
33C2×C8×D73—21no—
34C2×C8×D73—21no—
35C2×C8×D73—21no—
36C2×C8×D73—21no—
37C2×C8×D73—21no—
38#14556567—24no—
39#14556567—42no—
40#14556562—24no—
41#14556562—42no—
42C2×C4⋊Dic73—21no—
43C2×C4⋊Dic73—21no—
44C2×#11924007—41no—
45C2×C22⋊Dic73—41no—
46C2×#11924005—41no—
47C2×#11924003—21no—
48C2×#11924003—21no—
49C2×#11924003—21no—
50#14556525—21no—
51#14556525—21no—
52#14556525D412yes—
53#14556525D412yescomplemented
54#14556515—24no—
55#14556512—24no—
56C22×Q8×C73—21no—
57C22×D4×C73—21no—
58C2×Q16×C73—21no—
59C2×Q16×C73—21no—
60C2×Q16×C73—21no—
61C2×Q16×C73—21no—
62C2×Q16×C73—21no—
63C2×Q16×C73D412yescomplemented
64C2×D8×C73—21no—
65C2×D8×C73—21no—
66C2×D8×C73—21no—
67C2×D8×C73—21no—
68C2×D8×C73—21no—
69C2×D8×C73D412yes—
70C22×C8×C73C2311yescharacteristic
71SD32×C73—28no—
72C2×C16×C73—21no—
73C2×C16×C73—21no—
74C2×C16×C73D412yescomplemented
75C23×D73D811yescharacteristic
76C2×C22⋊D73—44no—
77C2×C22⋊D73—82no—
78C2×C22⋊D73—82no—
79C22×Dic73—41no—
80C22×Dic73—41no—
81C22×Dic73D811yescharacteristic
82C2×C4×D73—21no—
83C2×C4×D73—21no—
84C2×C4×D73—21no—
85C2×C4×D73—21no—
86C2×C4×D73—21no—
87C2×C4×D73—21no—
88C2×#10596831—21no—
89C2×#10596831—21no—
90C2×#10596831—21no—
91C8×D73—41no—
92C8×D73—41no—
93C8×D73—41no—
94C8×D73—41no—
95C4⋊Dic73—22no—
96C4⋊Dic73—22no—
97C4⋊Dic73—41no—
98C4⋊Dic73—41no—
99#11924007—44no—
100C22⋊Dic73—44no—
101#11924005—44no—
102#11924003—41no—
103C24×C73—41no—
104C2×Q8×C73—22no—
105C2×Q8×C73—41no—
106C2×Q8×C73—41no—
107C2×Q8×C73—41no—
108C2×Q8×C73—41no—
109C2×Q8×C73—41no—
110C2×D4×C73—22no—
111C2×D4×C73—41no—
112C2×D4×C73—41no—
113C2×D4×C73—41no—
114C2×D4×C73—41no—
115C2×D4×C73—41no—
116C22×C4×C73—41no—
117C22×C4×C73C2×D411yescharacteristic
118Q16×C73—22no—
119Q16×C73—22no—
120Q16×C73—41no—
121Q16×C73—41no—
122D8×C73—22no—
123D8×C73—22no—
124D8×C73—41no—
125D8×C73—41no—
126C2×C8×C73C2×D411yescharacteristic
127C2×C8×C73C2×D411yescharacteristic
128C2×C8×C73C2×D411yescharacteristic
129C2×C8×C73C2×D411yescharacteristic
130C2×C8×C73C2×D411yescharacteristic
131C2×C8×C73#4111yescharacteristic
132C16×C73—22no—
133C23×C73—42no—
134C23×C73—81no—
135C23×C73—81no—
136C23×C73—81no—
137C23×C73—81no—
138C23×C73—81no—
139C23×C73—81no—
140C23×C73C2×D811yescharacteristic
141C22×D73—21no—
142C22×D73—21no—
143C22×D73—21no—
144C22×D73—21no—
145C22×D73—21no—
146C22×D73—21no—
147C22×D73SD3212yescomplemented
148Q8×C73—42no—
149Q8×C73—42no—
150Q8×C73—81no—
151Q8×C73—81no—
152D4×C73—42no—
153D4×C73—42no—
154D4×C73—81no—
155D4×C73—81no—
156C2×C4×C73—42no—
157C2×C4×C73—81no—
158C2×C4×C73—81no—
159C2×C4×C73#12011yescharacteristic
160C2×C4×C73#12011yescharacteristic
161C2×C4×C73#12111yescharacteristic
162C2×C4×C73#13511yescharacteristic
163C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
165C22⋊D73—88no—
166C2×Dic73—21no—
167C2×Dic73—21no—
168C2×Dic73—21no—
169C2×Dic73—42no—
170C2×Dic73—42no—
171C2×Dic73—81no—
172C2×Dic73—81no—
173C2×Dic73—81no—
174C2×Dic73—81no—
175C4×D73—41no—
176C4×D73—41no—
177C4×D73—41no—
178C4×D73—41no—
179C8×C73#12011yescharacteristic
180C8×C73#12111yescharacteristic
181C8×C73#12111yescharacteristic
182C8×C73#12711yescharacteristic
183#10596831—41no—
184C22×C73—82no—
185C22×C73—82no—
186C22×C73—82no—
187C22×C73—161no—
188C22×C73—161no—
189C22×C73—161no—
190C22×C73—161no—
191C22×C73#44711yescharacteristic
192C22×C73#44711yescharacteristic
193C22×C73#46611yescharacteristic
194C22×C73C2×SD3211yescharacteristic
195C22×C73#50811yescharacteristic
196C22×C73#50911yescharacteristic
197C22×C73#51011yescharacteristic
198C2×D73—22no—
199C2×D73—22no—
200C2×D73—22no—
201C2×D73—41no—
202C2×D73—41no—
203C2×D73—41no—
204C2×D73—41no—
205C4×C73—82no—
206C4×C73#39211yescharacteristic
207C4×C73#44911yescharacteristic
208C4×C73#44911yescharacteristic
209C4×C73#46911yescharacteristic
210Dic73—41no—
211Dic73—82no—
212Dic73—82no—
213#13305—731nomaximal, Sylow
214C2×C73—162no—
215C2×C73#199911yescharacteristic
216C2×C73#218511yescharacteristic
217C2×C73#218611yescharacteristic
218C2×C73#218711yescharacteristic
219C2×C73#218911yescharacteristic
220C2×C73#221711yescharacteristic
221C2×C73#222011yescharacteristic
222D73—42no—
223C22×SD32D7311yescharacteristic, complemented
224C2×#467—731no—
225C2×#466—731no—
226C2×#358—731no—
227C2×#357—731no—
228C2×C22⋊C16—731no—
229#1380—731no—
230C73#1330511yescharacteristic, complemented, Sylow
231C22×Q16C2×D7311yescharacteristic
232C22×D8C2×D7311yescharacteristic
233C23×C8—731no—
234C2×C22⋊Q8—1461no—
235C2×#121—1461no—
236C2×SD32—22no—
237C2×SD32—22no—
238C2×SD32—22no—
239C22×C16—731no—
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—1462no—
252C22×Q8—21no—
253C22×D4—21no—
254C22×D4—2921no—
255C23×C4—731no—
256C2×Q16—21no—
257C2×Q16—21no—
258C2×Q16—21no—
259C2×Q16—21no—
260C2×Q16—21no—
261C2×Q16C22⋊D7312yescomplemented
262C2×D8—21no—
263C2×D8—21no—
264C2×D8—21no—
265C2×D8—21no—
266C2×D8—21no—
267C2×D8C22⋊D7312yes—
268C22×C8—731no—
269C22×C8—1461no—
270C22×C8—1461no—
271C22×C8—1461no—
272C22×C8—1461no—
273C22×C8—1461no—
274C22×C8—1461no—
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—28no—
286C2×C16—21no—
287C2×C16—21no—
288C2×C16—1461no—
289C2×C16—1461no—
290C2×C16—1461no—
291C2×C16C22⋊D7312yescomplemented
292#107—732no—
293#107—732no—
294#107—1461no—
295#107—1461no—
296#103—1464no—
297#102—1464no—
298C24—41no—
299C24—731no—
300C2×Q8—22no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×D4—22no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—2924no—
313C2×D4—5842no—
314C2×D4—5842no—
315C22×C4—41no—
316C22×C4—2921no—
317C22×C4—2921no—
318C22×C4—731no—
319C22×C4—1461no—
320C22×C4—1461no—
321C22×C4—1461no—
322C22×C4—1461no—
323C22×C4—1461no—
324C22×C4—1461no—
325C22×C4D4×D7311yescharacteristic
326Q16—22no—
327Q16—22no—
328Q16—41no—
329Q16—41no—
330D8—22no—
331D8—22no—
332D8—41no—
333D8—41no—
334C2×C8—2921no—
335C2×C8—2921no—
336C2×C8—2921no—
337C2×C8—2921no—
338C2×C8—1461no—
339C2×C8—1461no—
340C2×C8—1461no—
341C2×C8D4×D7311yescharacteristic
342C2×C8D4×D7311yescharacteristic
343C2×C8C2×C22⋊D7311yescharacteristic
344C2×C8C2×C22⋊D7311yescharacteristic
345C2×C8C2×C22⋊D7311yescharacteristic
346C2×C8#1192402711yescharacteristic
347C4⋊C4—2924no—
348C4⋊C4—2921no—
349C4⋊C4—2921no—
350C4⋊C4—1462no—
351C4⋊C4—1462no—
352C22⋊C4—2924no—
353C22⋊C4—2924no—
354C16—22no—
355C16—2921no—
356C23—42no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—732no—
364C23—1461no—
365C23—1461no—
366C23—1461no—
367C23—1461no—
368C23—1461no—
369C23—1461no—
370C23D8×D7311yescharacteristic
371Q8—42no—
372Q8—42no—
373Q8—81no—
374Q8—81no—
375D4—42no—
376D4—42no—
377D4—81no—
378D4—81no—
379D4—5848no—
380C2×C4—42no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—2921no—
384C2×C4—2921no—
385C2×C4—2921no—
386C2×C4—2921no—
387C2×C4—2922no—
388C2×C4—2922no—
389C2×C4—5841no—
390C2×C4—5841no—
391C2×C4—5841no—
392C2×C4—5841no—
393C2×C4—1461no—
394C2×C4—1461no—
395C2×C4—1461no—
396C2×C4#1455655911yescharacteristic
397C2×C4#1455655911yescharacteristic
398C2×C4#1455656211yescharacteristic
399C2×C4#1455661011yescharacteristic
400C2×C4#1455661511yescharacteristic
401C2×C4#1455661511yescharacteristic
402C8—2921no—
403C8#1455656011yescharacteristic
404C8#1455656311yescharacteristic
405C8#1455656311yescharacteristic
406C8#1455658711yescharacteristic
407C22—82no—
408C22—82no—
409C22—82no—
410C22—161no—
411C22—161no—
412C22—161no—
413C22—161no—
414C22—2921no—
415C22—2921no—
416C22—2921no—
417C22—2921no—
418C22—1462no—
419C22—1462no—
420C22—1462no—
421C22#1798871911yescharacteristic
422C22#1798871911yescharacteristic
423C22#1798882011yescharacteristic
424C22SD32×D7311yescharacteristic
425C22#1798901811yescharacteristic
426C22#1798902411yescharacteristic
427C22#1798902611yescharacteristic
428C4—82no—
429C4—2921no—
430C4—5842no—
431C4—5842no—
432C4#1798849511yescharacteristic
433C4#1798872911yescharacteristic
434C4#1798872911yescharacteristic
435C4#1798883711yescharacteristic
436C2—162no—
437C2—2922no—
438C2#2261092611yescharacteristic
439C2#2261184611yescharacteristic
440C2#2261185311yescharacteristic
441C2#2261185811yescharacteristic
442C2#2261186811yescharacteristic
443C2#2261202411yescharacteristic
444C2#2261204111yescharacteristic
subgroupG/H
centerC23D8×D73
commutator, cosocleC22×C8×C73C23
FrattiniC22×C8C22×D73
FittingC22×SD32×C73C2
socleC23×C73C2×D8
prime radicalC2×#17988820C2
hypercenterC22×SD32D73
hypocenterC73#13305
LCSG ▷ C22×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C22×SD32

Complements

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

NH ≅ G/Ntype
1. C2×#17988825C2semidirect
3. C2×#17988391C2semidirect
4. C2×#17988388C2semidirect
5. C2×#17988366C2semidirect
6. #22608023C2semidirect
7. C22×SD32×C73C2semidirect
53. #14556525D4semidirect
63. C2×Q16×C73D4semidirect
74. C2×C16×C73D4semidirect
147. C22×D73SD32semidirect
223. C22×SD32D73semidirect
230. C73#13305semidirect
261. C2×Q16C22⋊D73semidirect
291. C2×C16C22⋊D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179888252
C2×#179888202
C2×#179883912
C2×#179883882
C2×#179883662
#226080232
C22×SD32×C732
#1330573