Finite Group Databasebeta

← #29738935

Subgroups — #29738935

Order 15616.

690 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456358C211yesmaximal, characteristic, complemented
2#21455669C212yesmaximal
3#21454415C211yesmaximal, characteristic
4#21447328C212yesmaximal, complemented
5#21447162C212yesmaximal, complemented
6#21446565C211yesmaximal, characteristic, complemented
7#21446048C212yesmaximal
8#21445977C211yesmaximal, characteristic, complemented
9#21445880C211yesmaximal, characteristic
10#21445658C211yesmaximal, characteristic
11C61×#1952C211yesmaximal, characteristic, complemented
12C2×#14519783C2211yescharacteristic
13#16840390C2212yes—
14#16840243C2212yes—
15#16840171—22no—
16#16840167—21no—
17#16840164—21no—
18#16840072—22no—
19#16840061—22no—
20#16840056—22no—
21#16840041—22no—
22#16839804C2212yes—
23#16839803—22no—
24#16839802C2212yes—
25#16839800C2211yescharacteristic
26#16839799—21no—
27C2×#14519711C2211yescharacteristic
28C2×#14519710C2212yes—
29#107×D61—21no—
30#16839790C2212yes—
31C2×#14519703C2211yescharacteristic
32#16839764C2212yes—
33#16839756C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35#16839631C2212yes—
36#16839530C2212yes—
37#16839526C2211yescharacteristic
38#16839523C2212yes—
39#16839521C2211yescharacteristic
40C2×#108×C61C2211yescharacteristic
41C61×#428C2212yes—
42C2×#107×C61C2211yescharacteristic
43C61×#405C2211yescharacteristic
44C61×#343C2212yes—
45C2×#11758222C2311yescharacteristic
46#14519783—22no—
47#14519783Q814yescomplemented
48C2×#11758194C2311yescharacteristic
49C2×#11758189—21no—
50C22×#92418C2311yescharacteristic
51C2×C8×D61—21no—
52#14519777—22no—
53#14519776—22no—
54#14519769—22no—
55#14519768—22no—
56#14519754—42no—
57#14519742—22no—
58#14519739—22no—
59C4×C22⋊D61—42no—
60#14519736C2312yes—
61#14519735—22no—
62C2×C4⋊Dic61C2312yes—
63C2×#11758196—42no—
64C4⋊C4×D61—22no—
65C2×C4×Dic61C2311yescharacteristic
66#14519711—21no—
67#14519711—21no—
68#14519711—22no—
69#14519711D411yescharacteristic
70#14519711D411yescharacteristic
71#14519710—22no—
72#14519710—22no—
73#14519710—22no—
74#14519709—22no—
75#14519709D412yes—
76#14519708—22no—
77#14519706—22no—
78#14519706—22no—
79#14519703Q814yescomplemented
80#14519702—22no—
81#14519700D411yescharacteristic
82#14519700D411yescharacteristic
83#14519699—22no—
84#14519697D411yescharacteristic
85#14519697D411yescharacteristic
86C8×Dic61—22no—
87C2×M16×C61C2312yes—
88C2×M16×C61C2311yescharacteristic
89C22×C8×C61C2311yescharacteristic
90#117×C61C2311yescharacteristic
91C2×C4⋊C4×C61C2312yes—
92#108×C61—22no—
93#107×C61—21no—
94#107×C61—21no—
95#107×C61—22no—
96#106×C61—22no—
97#97×C61—21no—
98C4×C8×C61—21no—
99C22×C4×C61—22no—
100C22×C4×C61C2411yescharacteristic
101C2×C22⋊D61—21no—
102C22×Dic61—21no—
103C22×Dic61—22no—
104#11758222—21no—
105#11758222—21no—
106#11758222C2×Q812yes—
107#11758222C2×Q812yes—
108C2×D244C2×D411yescharacteristic
109C2×C4×D61—21no—
110C2×C4×D61—42no—
111C2×Dic122—42no—
112C2×Dic122C2×D411yescharacteristic
113M16×C61—21no—
114M16×C61—22no—
115M16×C61—22no—
116M16×C61C2×Q812yes—
117C2×C8×C61—21no—
118C2×C8×C61—22no—
119C2×C8×C61C2×D411yescharacteristic
120C2×C8×C61C2×D411yescharacteristic
121C2×C8×C61C2×Q812yes—
122C2×C8×C61#4112yes—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61C2×D411yescharacteristic
126C4⋊C4×C61C2×D411yescharacteristic
127C22⋊C4×C61—21no—
128C42×C61—21no—
129C22⋊Dic61—22no—
130#11758198—42no—
131C4⋊Dic61—22no—
132C4⋊Dic61—22no—
133C4⋊Dic61—42no—
134C4⋊Dic61C2×D412yes—
135C4⋊Dic61C2×D412yes—
136#11758196—42no—
137#11758196—42no—
138#11758196—42no—
139#11758196—84no—
140C4×Dic61—22no—
141C4×Dic61—21no—
142C4×Dic61—42no—
143C4×Dic61C2×Q812yes—
144#11758194—21no—
145#11758194C2×Q812yes—
146C2×#92418—21no—
147C2×#92418—21no—
148C2×#92418—21no—
149C2×#92418C2×D411yescharacteristic
150C2×#92418C2×D411yescharacteristic
151#11758189—24no—
152C8×D61—24no—
153C23×C61H(5,𝔽2)11yescharacteristic
154C22×D61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—42no—
158C2×C4×C61—82no—
159C2×C4×C61C22×D411yescharacteristic
160C2×C4×C61C22×D411yescharacteristic
161C2×C4×C61C22×Q811yescharacteristic
162C2×C4×C61C22×Q811yescharacteristic
163C2×C4×C61C2×#4111yescharacteristic
164C2×C4×C612−1+411yescharacteristic
165C22⋊D61—22no—
166C22⋊D61—22no—
167C2×Dic61—22no—
168C2×Dic61—22no—
169C2×Dic61—21no—
170C2×Dic61—22no—
171C2×Dic61—22no—
172C2×Dic61—22no—
173C2×Dic61—42no—
174C2×Dic61—82no—
175D244—21no—
176D244C4⋊Q812yes—
177C4×D61—22no—
178C4×D61—22no—
179C4×D61—162no—
180Dic122—21no—
181Dic122—162no—
182Dic122C4⋊Q812yes—
183C8×C61—22no—
184C8×C61—42no—
185C8×C61C22⋊Q812yes—
186C8×C61C22⋊Q812yes—
187#92418—22no—
188#92418—21no—
189#92418—21no—
190#26083—611nomaximal, Sylow
191C22×C61—21no—
192C22×C61—21no—
193C22×C61#54411yescharacteristic
194C22×C61D4211yescharacteristic
195C22×C61#55411yescharacteristic
196C2×D61—22no—
197C2×D61—41no—
198C2×D61—41no—
199C4×C61—41no—
200C4×C61—41no—
201C4×C61—82no—
202C4×C61C2×C22⋊Q811yescharacteristic
203C4×C61C2×C4⋊Q811yescharacteristic
204C4×C61#54711yescharacteristic
205C4×C61D4×Q811yescharacteristic
206Dic61—24no—
207Dic61—22no—
208Dic61—42no—
209Dic61—42no—
210Dic61—82no—
211#3164—611no—
212#3083—612no—
213#2913—611no—
214#2089—612no—
215#2060—612no—
216#1952D6111yescharacteristic, complemented
217#1945—611no—
218#1859—612no—
219#1845—611no—
220#1816—611no—
221#1764—611no—
222C2×C61—21no—
223C2×C61—21no—
224C2×C61#244011yescharacteristic
225C2×C61#329211yescharacteristic
226C2×C61#329311yescharacteristic
227D61—42no—
228C2×#131—611no—
229#544—612no—
230#520—612no—
231#501—1222no—
232#500—1221no—
233#499—1221no—
234#479—1222no—
235#478—1222no—
236#477—1222no—
237#474—1222no—
238C2×#108C2×D6111yescharacteristic
239#428—612no—
240#428—612no—
241#428—1222no—
242#428C2×D6112yes—
243#427—611no—
244#427—1221no—
245C2×#107—611no—
246C2×#107—612no—
247C2×#107—1221no—
248C2×#107C2×D6111yescharacteristic
249#424—612no—
250C2×#104—611no—
251#417—612no—
252#416—611no—
253#405C2×D6111yescharacteristic
254C2×C4×C8—611no—
255#384—612no—
256#343—612no—
257#343C2×D6112yes—
258#341—611no—
259#339—612no—
260#339—611no—
261C61#2608311yescharacteristic, complemented, Sylow
262C2×#41—611no—
263#131—614no—
264#131—1222no—
265C2×M16—611no—
266C2×M16—1221no—
267C2×M16C22×D6112yes—
268C2×M16C22×D6111yescharacteristic
269C22×C8—611no—
270C22×C8—1221no—
271C22×C8C22×D6111yescharacteristic
272C4⋊Q8—1222no—
273C4⋊Q8—1222no—
274#125—1222no—
275#125—1222no—
276C22⋊Q8—2442no—
277C4×Q8—1222no—
278C4×D4—1222no—
279C4×D4—2442no—
280#117—612no—
281#117—1222no—
282#117C22×D6111yescharacteristic
283C2×C4⋊C4—612no—
284C2×C4⋊C4—1222no—
285C2×C4⋊C4—2442no—
286C2×C4⋊C4C22×D6112yes—
287C2×C42—611no—
288#108—22no—
289#107—21no—
290#107—21no—
291#107—22no—
292#107—611no—
293#107—611no—
294#107—1222no—
295#107—1222no—
296#107—1222no—
297#107—1222no—
298#107—1221no—
299#107—1221no—
300#106—22no—
301#106—612no—
302#106—1222no—
303#106—1222no—
304C4⋊C8—1222no—
305C4⋊C8—1222no—
306#104—614no—
307#103—611no—
308#103—611no—
309#103—1222no—
310#102—611no—
311#102—611no—
312#102—1222no—
313#97—21no—
314C4×C8—21no—
315C4×C8—1222no—
316#41—612no—
317#41—612no—
318#41—1221no—
319#41—1221no—
320C2×Q8—611no—
321C2×Q8—2442no—
322C2×D4—611no—
323C2×D4—1221no—
324C22×C4—22no—
325C22×C4—1221no—
326C22×C4—1221no—
327C22×C4—1222no—
328C22×C4—2442no—
329C22×C4C23×D6111yescharacteristic
330M16—21no—
331M16—22no—
332M16—22no—
333M16—612no—
334M16—1221no—
335M16—1224no—
336M16Q8×D6112yes—
337C2×C8—21no—
338C2×C8—22no—
339C2×C8—611no—
340C2×C8—611no—
341C2×C8—1221no—
342C2×C8—1221no—
343C2×C8—1221no—
344C2×C8—1224no—
345C2×C8#1175822212yes—
346C2×C8D4×D6111yescharacteristic
347C2×C8D4×D6111yescharacteristic
348C2×C8Q8×D6112yes—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—612no—
352C4⋊C4—612no—
353C4⋊C4—1222no—
354C4⋊C4—1222no—
355C4⋊C4—4884no—
356C4⋊C4—2442no—
357C4⋊C4—2442no—
358C4⋊C4—2442no—
359C4⋊C4—2442no—
360C4⋊C4D4×D6111yescharacteristic
361C4⋊C4D4×D6111yescharacteristic
362C22⋊C4—21no—
363C22⋊C4—1222no—
364C22⋊C4—2442no—
365C42—21no—
366C42—612no—
367C42—1222no—
368C42—1221no—
369C42—2442no—
370C23—1221no—
371C23#1451982311yescharacteristic
372Q8—612no—
373Q8—1221no—
374Q8—9762no—
375D4—612no—
376D4—1222no—
377D4—1222no—
378D4—1221no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—1222no—
384C2×C4—1222no—
385C2×C4—1222no—
386C2×C4—1221no—
387C2×C4—1222no—
388C2×C4—1222no—
389C2×C4—1222no—
390C2×C4—1222no—
391C2×C4—9762no—
392C2×C4—4882no—
393C2×C4—2442no—
394C2×C4C2×D4×D6111yescharacteristic
395C2×C4C2×D4×D6111yescharacteristic
396C2×C4C2×Q8×D6111yescharacteristic
397C2×C4C2×Q8×D6111yescharacteristic
398C2×C4C2×#1175822211yescharacteristic
399C2×C4#1451982411yescharacteristic
400C8—22no—
401C8—42no—
402C8—1222no—
403C8—1221no—
404C8—1221no—
405C8#1451975512yes—
406C8#1451975512yes—
407C22—21no—
408C22—21no—
409C22—1222no—
410C22—2441no—
411C22—2441no—
412C22#1684039211yescharacteristic
413C22#1684039611yescharacteristic
414C22#1684047211yescharacteristic
415C4—41no—
416C4—41no—
417C4—82no—
418C4—1224no—
419C4—1222no—
420C4—4882no—
421C4—2442no—
422C4—2442no—
423C4C2×#1451975511yescharacteristic
424C4C4⋊Q8×D6111yescharacteristic
425C4#1684041711yescharacteristic
426C4#1684042511yescharacteristic
427C2—21no—
428C2—21no—
429C2—2442no—
430C2#2144952311yescharacteristic
431C2#2145761711yescharacteristic
432C2#2145762211yescharacteristic
subgroupG/H
centerC22#16840396
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1952C2
socleC22×C61D42
prime radicalC2×#11758222C23
hypercenter#1952D61
hypocenterC61#26083
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1952

Complements

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

NH ≅ G/Ntype
1. #21456358C2semidirect
4. #21447328C2semidirect
5. #21447162C2semidirect
6. #21446565C2semidirect
8. #21445977C2semidirect
11. C61×#1952C2semidirect
47. #14519783Q8semidirect
79. #14519703Q8semidirect
216. #1952D61semidirect
261. C61#26083semidirect

Maximal subgroups

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

subgroupindex
#214563582
#214556692
#214544152
#214473282
#214471622
#214465652
#214460482
#214459772
#214458802
#214456582
C61×#19522
#2608361