Finite Group Databasebeta

← #32988757

Subgroups — #32988757

Order 18688.

802 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988959C211yesmaximal, characteristic, complemented
2C2×#17988933C211yesmaximal, characteristic, complemented
3#22610892C211yesmaximal, characteristic, complemented
4#22610738C211yesmaximal, characteristic, complemented
5#22610738C211yesmaximal, characteristic, complemented
6#22607758C211yesmaximal, characteristic, complemented
7C2×#493×C73C211yesmaximal, characteristic
8C22×#11924016—22no—
9C2×#14556591C2211yescharacteristic
10#17988959—22no—
11#17988959—22no—
12#17988933—22no—
13#17988933—22no—
14C2×#14556512—22no—
15C2×#14556512—22no—
16C2×#14556512—22no—
17C2×C4×#10596831C2211yescharacteristic
18C2×C4×#10596831C2211yescharacteristic
19#17988457C2211yescharacteristic
20#17988350—22no—
21C22×D8×C73—22no—
22C2×#127×C73C2211yescharacteristic
23C2×#127×C73C2211yescharacteristic
24#493×C73—22no—
25#493×C73—22no—
26C2×C4×C8×C73C2211yescharacteristic
27C22×Dic146—21no—
28C2×#11924016—28no—
29C2×#11924016—44no—
30C2×#11924016—44no—
31C22×#10596831D412yes—
32C22×#10596831D412yes—
33#14556591—21no—
34#14556591—21no—
35#14556591—21no—
36#14556591D412yes—
37C2×C4⋊Dic73—21no—
38C2×C4⋊Dic73—21no—
39C2×C4⋊Dic73—21no—
40C2×C4⋊Dic73—21no—
41C2×C4⋊Dic73—21no—
42#14556512—28no—
43#14556512—44no—
44#14556512—44no—
45C4×#10596831—21no—
46C4×#10596831—21no—
47C4×#10596831—21no—
48C4×#10596831—21no—
49C22×D4×C73—21no—
50C22×D4×C73—21no—
51C22×D4×C73—21no—
52C22×D4×C73—21no—
53C22×D4×C73—21no—
54C22×D4×C73—21no—
55C2×D8×C73—28no—
56C2×D8×C73—44no—
57C2×D8×C73—44no—
58C22×C8×C73D412yes—
59#127×C73—21no—
60#127×C73—21no—
61#127×C73—21no—
62#127×C73—21no—
63#127×C73—21no—
64#127×C73—21no—
65#127×C73D412yes—
66#127×C73D412yes—
67C2×C42×C73C2311yescharacteristic
68C4×C8×C73—21no—
69C4×C8×C73—21no—
70C2×Dic146—22no—
71C2×Dic146—42no—
72C2×Dic146—42no—
73C2×Dic146—41no—
74C22×Dic73—41no—
75C22×Dic73—41no—
76#11924016—416no—
77C2×#10596831—22no—
78C2×#10596831—22no—
79C2×#10596831—22no—
80C2×#10596831—22no—
81C2×#10596831—22no—
82C2×#10596831D814yescomplemented
83C4⋊Dic73—22no—
84C4⋊Dic73—22no—
85C4⋊Dic73—41no—
86C4⋊Dic73—41no—
87C4⋊Dic73—42no—
88C4⋊Dic73—42no—
89C4⋊Dic73—42no—
90C24×C73—41no—
91C24×C73—41no—
92C2×D4×C73—22no—
93C2×D4×C73—22no—
94C2×D4×C73—22no—
95C2×D4×C73—22no—
96C2×D4×C73—22no—
97C2×D4×C73—22no—
98C2×D4×C73—41no—
99C2×D4×C73—41no—
100C2×D4×C73—41no—
101C2×D4×C73—41no—
102C2×D4×C73—41no—
103C2×D4×C73—41no—
104C2×D4×C73—42no—
105C2×D4×C73—42no—
106C2×D4×C73—42no—
107C2×D4×C73—42no—
108C2×D4×C73—42no—
109C2×D4×C73—42no—
110C2×D4×C73—42no—
111C2×D4×C73—42no—
112C2×D4×C73—42no—
113C2×D4×C73—42no—
114C2×D4×C73—42no—
115C2×D4×C73—42no—
116C22×C4×C73C2×D411yescharacteristic
117C22×C4×C73C2×D411yescharacteristic
118C22×C4×C73C2×D411yescharacteristic
119D8×C73—416no—
120C2×C8×C73—22no—
121C2×C8×C73—22no—
122C2×C8×C73SD1614yescomplemented
123C42×C73C2×D411yescharacteristic
124C42×C73C2×D411yescharacteristic
125C42×C73C2×D411yescharacteristic
126C42×C73#4111yescharacteristic
127C23×C73—42no—
128C23×C73—42no—
129C23×C73—81no—
130C23×C73—81no—
131C23×C73—81no—
132C23×C73—81no—
133C23×C73—81no—
134C23×C73—81no—
135C23×C73—81no—
136C23×C73—81no—
137C23×C73—81no—
138C23×C73—81no—
139C23×C73—81no—
140C23×C73—81no—
141C23×C73#12711yescharacteristic
142D4×C73—44no—
143D4×C73—44no—
144D4×C73—44no—
145D4×C73—44no—
146D4×C73—44no—
147D4×C73—44no—
148D4×C73—82no—
149D4×C73—82no—
150D4×C73—82no—
151D4×C73—82no—
152D4×C73—82no—
153D4×C73—82no—
154C2×C4×C73#12011yescharacteristic
155C2×C4×C73#12011yescharacteristic
156C2×C4×C73#12011yescharacteristic
157C2×C4×C73#12111yescharacteristic
158C2×C4×C73#12111yescharacteristic
159C2×C4×C73#12111yescharacteristic
160C2×C4×C73C2×D812yes—
161C2×C4×C73C2×SD1612yes—
162C2×C4×C73#13512yes—
163C2×C4×C73AGL(1,ℤ/8)12yes—
164C2×C4×C73AGL(1,ℤ/8)12yes—
165C2×C4×C73AGL(1,ℤ/8)12yes—
166C2×Dic73—42no—
167C2×Dic73—81no—
168C2×Dic73—81no—
169C2×Dic73—81no—
170C2×Dic73—81no—
171C2×Dic73—81no—
172Dic146—44no—
173Dic146—82no—
174C8×C73—24no—
175#10596831—24no—
176#10596831—42no—
177C22×C73—82no—
178C22×C73—82no—
179C22×C73—82no—
180C22×C73—82no—
181C22×C73—82no—
182C22×C73—82no—
183C22×C73—161no—
184C22×C73—161no—
185C22×C73—161no—
186C22×C73—161no—
187C22×C73—161no—
188C22×C73—161no—
189C22×C73—161no—
190C22×C73—161no—
191C22×C73#39211yescharacteristic
192C22×C73#49211yescharacteristic
193C22×C73#49311yescharacteristic
194C22×C73#49511yescharacteristic
195C22×C73#49611yescharacteristic
196C22×C73#49611yescharacteristic
197C22×C73#49611yescharacteristic
198C4×C73#44712yes—
199C4×C73#44912yes—
200C4×C73#45012yes—
201C4×C73#45912yes—
202C4×C73#46012yes—
203C4×C73#46312yes—
204Dic73—82no—
205Dic73—161no—
206#10305—731nomaximal, Sylow
207C2×C73—162no—
208C2×C73—162no—
209C2×C73#170611yescharacteristic
210C2×C73#171511yescharacteristic
211C2×C73#171711yescharacteristic
212C2×C73#172111yescharacteristic
213C2×C73#201111yescharacteristic
214C2×C73#201211yescharacteristic
215C2×C73#201411yescharacteristic
216C2×#493D7311yescharacteristic
217C2×#492—731no—
218C2×#486—731no—
219#1990—731no—
220#1957—731no—
221#1957—731no—
222#1273—731no—
223C73#1030511yescharacteristic, complemented, Sylow
224C22×SD16—1462no—
225C22×D8—22no—
226C2×C4⋊Q8—731no—
227C2×#127C2×D7311yescharacteristic
228C2×#127C2×D7311yescharacteristic
229#493—22no—
230#493—22no—
231#492—1462no—
232#492—1462no—
233#486—1462no—
234#486—1462no—
235C2×#102—1462no—
236C2×#102—1462no—
237C2×#102—1462no—
238C2×C4×C8—731no—
239C2×C4×C8—731no—
240C2×C4×C8C2×D7311yescharacteristic
241#384—731no—
242#340—1462no—
243C22×Q8—1461no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4—21no—
250C2×SD16—2924no—
251C2×SD16—2924no—
252C2×SD16—1468no—
253C2×D8—28no—
254C2×D8—44no—
255C2×D8—44no—
256C22×C8—732no—
257C22×C8—732no—
258C22×C8C22⋊D7312yes—
259C4⋊Q8—732no—
260C4⋊Q8—1461no—
261C4⋊Q8—1461no—
262C4⋊Q8—1461no—
263#127—21no—
264#127—21no—
265#127—21no—
266#127—21no—
267#127—21no—
268#127—21no—
269#127C22⋊D7312yes—
270#127C22⋊D7312yes—
271C2×C4⋊C4—1461no—
272C2×C4⋊C4—1461no—
273C2×C4⋊C4—1461no—
274C2×C4⋊C4—1461no—
275C2×C4⋊C4—1461no—
276C2×C42C22×D7311yescharacteristic
277#102—2924no—
278#102—2924no—
279#102—1468no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—1461no—
283C4×C8—1461no—
284C4×C8—1461no—
285C4×C8—1461no—
286C24—41no—
287C24—41no—
288C2×Q8—2921no—
289C2×Q8—2922no—
290C2×Q8—2922no—
291C2×Q8—1462no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C22×C4—2921no—
317C22×C4—2921no—
318C22×C4D4×D7311yescharacteristic
319C22×C4D4×D7311yescharacteristic
320C22×C4C2×C22⋊D7311yescharacteristic
321SD16—29216no—
322D8—416no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—734no—
326C2×C8—1462no—
327C2×C8—1462no—
328C2×C8—1462no—
329C2×C8—1462no—
330C2×C8—1462no—
331C2×C8#1192401614yescomplemented
332C4⋊C4—2921no—
333C4⋊C4—2921no—
334C4⋊C4—2922no—
335C4⋊C4—2922no—
336C4⋊C4—2922no—
337C4⋊C4—1462no—
338C4⋊C4—1462no—
339C42D4×D7311yescharacteristic
340C42C2×C22⋊D7311yescharacteristic
341C42C2×C22⋊D7311yescharacteristic
342C42#1192402711yescharacteristic
343C23—42no—
344C23—42no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23#1455658711yescharacteristic
358Q8—2924no—
359Q8—5842no—
360D4—44no—
361D4—44no—
362D4—44no—
363D4—44no—
364D4—44no—
365D4—44no—
366D4—82no—
367D4—82no—
368D4—82no—
369D4—82no—
370D4—82no—
371D4—82no—
372C2×C4—2922no—
373C2×C4—5841no—
374C2×C4—5841no—
375C2×C4—5841no—
376C2×C4—5841no—
377C2×C4—5841no—
378C2×C4#1455655911yescharacteristic
379C2×C4#1455655911yescharacteristic
380C2×C4#1455656011yescharacteristic
381C2×C4#1455656211yescharacteristic
382C2×C4#1455656311yescharacteristic
383C2×C4#1455656311yescharacteristic
384C2×C4D8×D7312yes—
385C2×C4C2×#1192401612yes—
386C2×C4#1455661012yes—
387C2×C4#1455661512yes—
388C2×C4#1455661512yes—
389C2×C4#1455661712yes—
390C8—24no—
391C8—2922no—
392C8—1464no—
393C22—82no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22—82no—
398C22—82no—
399C22—161no—
400C22—161no—
401C22—161no—
402C22—161no—
403C22—161no—
404C22—161no—
405C22—161no—
406C22—161no—
407C22#1798849511yescharacteristic
408C22#1798895911yescharacteristic
409C22#1798896211yescharacteristic
410C22#1798897011yescharacteristic
411C22#1798897411yescharacteristic
412C22#1798897511yescharacteristic
413C22#1798897511yescharacteristic
414C4—5842no—
415C4—11681no—
416C4#1798871912yes—
417C4#1798872912yes—
418C4#1798873812yes—
419C4#1798878012yes—
420C4#1798878412yes—
421C4#1798880012yes—
422C2—162no—
423C2—162no—
424C2#2260952011yescharacteristic
425C2#2260955711yescharacteristic
426C2#2260956911yescharacteristic
427C2#2260959611yescharacteristic
428C2#2261100011yescharacteristic
429C2#2261100611yescharacteristic
430C2#2261101911yescharacteristic
subgroupG/H
centerC23#14556587
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
Fitting, prime radicalC2×#493×C73C2
socleC23×C73#127
hypercenterC2×#493D73
hypocenterC73#10305
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#493

Complements

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

NH ≅ G/Ntype
1. C2×#17988959C2semidirect
2. C2×#17988933C2semidirect
3. #22610892C2semidirect
4. #22610738C2semidirect
5. #22610738C2semidirect
6. #22607758C2semidirect
82. C2×#10596831D8semidirect
122. C2×C8×C73SD16semidirect
223. C73#10305semidirect
331. C2×C8#11924016semidirect

Maximal subgroups

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

subgroupindex
C2×#179889592
C2×#179889332
#226108922
#226107382
#226077582
C2×#493×C732
#1030573