Finite Group Databasebeta

← #33068704

Subgroups — #33068704

Order 18688.

596 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620517C211yesmaximal, characteristic, complemented
2#22619719C212yesmaximal
3#22609513C212yesmaximal, complemented
4#22609506C212yesmaximal, complemented
5#22609459C211yesmaximal, characteristic, complemented
6#22609454C211yesmaximal, characteristic, complemented
7#22609425C211yesmaximal, characteristic, complemented
8#22609420C211yesmaximal, characteristic, complemented
9C2×#17988338C211yesmaximal, characteristic, complemented
10#22608586C212yesmaximal, complemented
11C73×#2965C211yesmaximal, characteristic
12#17989144C2212yes—
13#17988992C2211yescharacteristic
14#17988967C2211yescharacteristic
15#17988938C2212yes—
16#17988894—22no—
17#17988887—22no—
18#17988882C2212yes—
19#17988877C2212yes—
20#17988872C2212yes—
21#17988867C2212yes—
22#17988828—21no—
23#17988828—21no—
24C4×Dic292—21no—
25C4×Dic292—21no—
26#17988623—22no—
27C2×C4⋊#10596831C2212yes—
28#17988583—22no—
29#17988338C2211yescharacteristic
30#17988338C2211yescharacteristic
31#17988338C2212yes—
32#17988335C2212yes—
33#17988335C2212yes—
34#17988334C2212yes—
35#17988334C2212yes—
36C4×#41×C73C2211yescharacteristic
37#446×C73C2211yescharacteristic
38#446×C73C2211yescharacteristic
39#436×C73C2211yescharacteristic
40#436×C73C2211yescharacteristic
41#435×C73—21no—
42#435×C73—21no—
43#431×C73—21no—
44#431×C73C2211yescharacteristic
45C4×M16×C73—21no—
46C2×C4×C8×C73C2211yescharacteristic
47C2×Dic292—21no—
48C2×Dic292—21no—
49C22×#10596831—22no—
50#14556591C2312yes—
51#14556591C2312yes—
52#14556583—22no—
53#14556570—22no—
54#14556570—22no—
55C4×Dic146—22no—
56C4×Dic146—22no—
57#14556550—22no—
58#14556525—21no—
59#14556525—21no—
60#14556524—22no—
61#14556524—22no—
62#14556522—22no—
63#14556522—22no—
64C4⋊#10596831—22no—
65C4⋊#10596831C2312yes—
66C4⋊#10596831C2312yes—
67#14556516—22no—
68#14556516—22no—
69#14556515—22no—
70#14556515—22no—
71#14556512—22no—
72#14556512—22no—
73C2×#41×C73D411yescharacteristic
74C2×M16×C73—21no—
75C2×M16×C73—21no—
76C22×C8×C73D411yescharacteristic
77C22×C8×C73D411yescharacteristic
78C4×Q8×C73D411yescharacteristic
79C4×Q8×C73C2311yescharacteristic
80C4×D4×C73—21no—
81C4×D4×C73—21no—
82C4×D4×C73D411yescharacteristic
83C4×D4×C73C2311yescharacteristic
84#117×C73—21no—
85#117×C73D411yescharacteristic
86C2×C42×C73—21no—
87C2×C42×C73C2311yescharacteristic
88C4⋊C8×C73—21no—
89C4⋊C8×C73—21no—
90C4⋊C8×C73C2311yescharacteristic
91C4⋊C8×C73C2311yescharacteristic
92C22⋊C8×C73—21no—
93C22⋊C8×C73—21no—
94C22⋊C8×C73—21no—
95#97×C73—21no—
96#97×C73—21no—
97C4×C8×C73—21no—
98C4×C8×C73D411yescharacteristic
99C4×C8×C73D411yescharacteristic
100C4×C8×C73C2311yescharacteristic
101C4×C8×C73C2311yescharacteristic
102C2×Dic146—22no—
103C2×Dic146—22no—
104Dic292—42no—
105Dic292—42no—
106C2×#10596831—22no—
107C2×#10596831—22no—
108C2×#10596831—44no—
109C4⋊Dic73—22no—
110C4⋊Dic73—22no—
111C4⋊Dic73—22no—
112C4⋊Dic73—22no—
113#11924007—42no—
114#11924007—42no—
115C22⋊Dic73—42no—
116C4×Dic73—42no—
117#41×C73—24no—
118C2×Q8×C73C2×D411yescharacteristic
119C2×D4×C73—21no—
120C2×D4×C73C2×D411yescharacteristic
121C22×C4×C73—21no—
122C22×C4×C73—21no—
123C22×C4×C73—21no—
124C22×C4×C73C2×D411yescharacteristic
125C22×C4×C73C2×D411yescharacteristic
126C22×C4×C73#4111yescharacteristic
127M16×C73—42no—
128M16×C73—42no—
129C2×C8×C73—22no—
130C2×C8×C73—21no—
131C2×C8×C73—21no—
132C2×C8×C73—41no—
133C2×C8×C73C2×D411yescharacteristic
134C2×C8×C73C2×D411yescharacteristic
135C2×C8×C73C2×D411yescharacteristic
136C2×C8×C73C2×D411yescharacteristic
137C4⋊C4×C73—21no—
138C4⋊C4×C73—21no—
139C4⋊C4×C73C2×D411yescharacteristic
140C4⋊C4×C73C2×D411yescharacteristic
141C22⋊C4×C73—21no—
142C22⋊C4×C73—21no—
143C22⋊C4×C73—21no—
144C42×C73—21no—
145C42×C73—21no—
146C42×C73—21no—
147C42×C73C2×D411yescharacteristic
148C42×C73C2×D411yescharacteristic
149C42×C73#4111yescharacteristic
150C42×C73C2411yescharacteristic
151C23×C73—21no—
152C23×C73#12111yescharacteristic
153Q8×C73—22no—
154D4×C73—22no—
155D4×C73—42no—
156C2×C4×C73—21no—
157C2×C4×C73—21no—
158C2×C4×C73—21no—
159C2×C4×C73—21no—
160C2×C4×C73—21no—
161C2×C4×C73—41no—
162C2×C4×C73—42no—
163C2×C4×C73—41no—
164C2×C4×C73—41no—
165C2×C4×C73—42no—
166C2×C4×C73#12111yescharacteristic
167C2×C4×C73#12111yescharacteristic
168C2×C4×C73#12111yescharacteristic
169C2×C4×C73#13714yes—
170C2×C4×C73C22×D411yescharacteristic
171C2×C4×C73C22×D411yescharacteristic
172C2×C4×C73C2×#4111yescharacteristic
173C2×Dic73—42no—
174C2×Dic73—42no—
175Dic146—42no—
176Dic146—42no—
177Dic146—82no—
178C8×C73—21no—
179C8×C73—21no—
180C8×C73—21no—
181C8×C73—41no—
182C8×C73—41no—
183C8×C73#13512yes—
184#10596831—42no—
185#10596831—42no—
186C22×C73—21no—
187C22×C73—21no—
188C22×C73—41no—
189C22×C73—41no—
190C22×C73—41no—
191C22×C73#47012yes—
192C22×C73C2×#12111yescharacteristic
193C4×C73—21no—
194C4×C73—21no—
195C4×C73—21no—
196C4×C73—41no—
197C4×C73—41no—
198C4×C73—41no—
199C4×C73#47012yes—
200C4×C73C2×#13511yescharacteristic
201C4×C73C2×#13712yes—
202C4×C73#57511yescharacteristic
203Dic73—82no—
204Dic73—82no—
205#19158—731nomaximal, Sylow
206C2×C73—22no—
207C2×C73—81no—
208C2×C73#303511yescharacteristic
209C2×C73C2×#47011yescharacteristic
210C2×C73#306711yescharacteristic
211#3170—731no—
212#3076—732no—
213#2965D7311yescharacteristic
214#1704—732no—
215#1703—732no—
216#1696—731no—
217#1695—731no—
218#1690—731no—
219#1689—731no—
220C2×#334—731no—
221#1542—732no—
222C73#1915811yescharacteristic, complemented, Sylow
223#533—732no—
224C4×#41C2×D7311yescharacteristic
225#500—731no—
226#494—731no—
227#487—732no—
228#479—1462no—
229#478—1462no—
230#477—732no—
231#476—732no—
232#475—732no—
233#474—732no—
234#467—1461no—
235#467—1461no—
236#446C2×D7311yescharacteristic
237#446C2×D7311yescharacteristic
238C4×Q16—1461no—
239C4×Q16—1461no—
240#436C2×D7311yescharacteristic
241#436C2×D7311yescharacteristic
242#435—21no—
243#435—21no—
244#431—21no—
245#431C2×D7311yescharacteristic
246#427—1462no—
247C2×C4⋊C8—732no—
248#416—1462no—
249C4×M16—21no—
250C2×C4×C8C2×D7311yescharacteristic
251#334—731no—
252#334—731no—
253#334—732no—
254#333—732no—
255#333—732no—
256#332—732no—
257#332—732no—
258C2×#41D29211yescharacteristic
259C2×Q16—1461no—
260C2×Q16—1461no—
261C2×M16—21no—
262C2×M16—21no—
263C22×C8—1462no—
264C22×C8C22⋊D7311yescharacteristic
265C22×C8C22⋊D7311yescharacteristic
266C4⋊Q8—732no—
267C4⋊Q8—732no—
268#125—1462no—
269C22⋊Q8—1462no—
270C22⋊Q8—1462no—
271C4×Q8—1462no—
272C4×Q8—1462no—
273C4×Q8D29211yescharacteristic
274C4×Q8C22×D7311yescharacteristic
275C4×D4—21no—
276C4×D4—21no—
277C4×D4D29211yescharacteristic
278C4×D4C22×D7311yescharacteristic
279#117—21no—
280#117—1462no—
281#117D29211yescharacteristic
282C2×C42—21no—
283C2×C42C22×D7311yescharacteristic
284#107—1461no—
285#107—1461no—
286#107—1462no—
287#107—1462no—
288#106—1462no—
289#106—1462no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—732no—
293C4⋊C8—732no—
294C4⋊C8—1462no—
295C4⋊C8C22×D7311yescharacteristic
296C4⋊C8C22×D7311yescharacteristic
297#103—1462no—
298#103—1462no—
299#103—1462no—
300#103—1462no—
301#102—1462no—
302#102—1462no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306#97—21no—
307#97—21no—
308C4×C8—21no—
309C4×C8C22⋊D7311yescharacteristic
310C4×C8C22⋊D7311yescharacteristic
311C4×C8C22×D7311yescharacteristic
312C4×C8C22×D7311yescharacteristic
313#41—24no—
314C2×Q8—1462no—
315C2×Q8—1462no—
316C2×Q8C2×D29211yescharacteristic
317C2×D4—21no—
318C2×D4C2×D29211yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4C2×C22⋊D7311yescharacteristic
323C22×C4C2×D29211yescharacteristic
324C22×C4#1192402811yescharacteristic
325Q16—2922no—
326Q16—2922no—
327M16—42no—
328M16—42no—
329C2×C8—22no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—41no—
333C2×C8—2924no—
334C2×C8—1462no—
335C2×C8—1462no—
336C2×C8C2×C22⋊D7311yescharacteristic
337C2×C8C2×C22⋊D7311yescharacteristic
338C2×C8C2×C22⋊D7311yescharacteristic
339C2×C8C2×C22⋊D7311yescharacteristic
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—2922no—
343C4⋊C4—2922no—
344C4⋊C4—1462no—
345C4⋊C4—1462no—
346C4⋊C4—1462no—
347C4⋊C4—1462no—
348C4⋊C4C2×D29211yescharacteristic
349C4⋊C4C2×D29211yescharacteristic
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—2922no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—2922no—
358C42C2×C22⋊D7311yescharacteristic
359C42C2×D29211yescharacteristic
360C42#1192402811yescharacteristic
361C42C23×D7311yescharacteristic
362C23—21no—
363C23#1455656511yescharacteristic
364Q8—22no—
365Q8—2922no—
366Q8—2922no—
367Q8—5842no—
368D4—22no—
369D4—42no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—2922no—
381C2×C4—2922no—
382C2×C4#1455656511yescharacteristic
383C2×C4#1455656511yescharacteristic
384C2×C4#1455656511yescharacteristic
385C2×C4#1455662414yes—
386C2×C4C22×C22⋊D7311yescharacteristic
387C2×C4C22×D29211yescharacteristic
388C2×C4C2×#1192402811yescharacteristic
389C8—21no—
390C8—21no—
391C8—21no—
392C8—41no—
393C8—41no—
394C8—2922no—
395C8—2922no—
396C8#1455661112yes—
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22#1798884512yes—
403C22C2×#1455656511yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—5842no—
411C4—5842no—
412C4#1798884512yes—
413C4C2×#1455661111yescharacteristic
414C4C2×#1455662412yes—
415C4#1798938511yescharacteristic
416C2—22no—
417C2—81no—
418C2#2261938611yescharacteristic
419C2C2×#1798884511yescharacteristic
420C2#2261965111yescharacteristic
subgroupG/H
centerC2×C4#14556565
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#2965C2
socleC22×C73C2×#121
prime radicalC2×#41×C73D4
hypercenter#2965D73
hypocenterC73#19158
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2965

Complements

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

NH ≅ G/Ntype
1. #22620517C2semidirect
3. #22609513C2semidirect
4. #22609506C2semidirect
5. #22609459C2semidirect
6. #22609454C2semidirect
7. #22609425C2semidirect
8. #22609420C2semidirect
9. C2×#17988338C2semidirect
10. #22608586C2semidirect
222. C73#19158semidirect

Maximal subgroups

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

subgroupindex
#226205172
#226197192
#226095132
#226095062
#226094592
#226094542
#226094252
#226094202
C2×#179883382
#226085862
C73×#29652
#1915873