Finite Group Databasebeta

← #29738936

Subgroups — #29738936

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#21456355C211yesmaximal, characteristic, complemented
2#21455675C212yesmaximal
3#21454415C211yesmaximal, characteristic
4#21447325C212yesmaximal, complemented
5#21447164C212yesmaximal, complemented
6#21446602C211yesmaximal, characteristic, complemented
7#21446567C211yesmaximal, characteristic, complemented
8#21446044C212yesmaximal
9#21445884C211yesmaximal, characteristic
10#21445656C211yesmaximal, characteristic
11C61×#1845C211yesmaximal, characteristic, complemented
12C2×#14519783C2211yescharacteristic
13#16840393C2212yes—
14#16840243C2212yes—
15#16840170—22no—
16#16840166—21no—
17#16840163—21no—
18#16840068—22no—
19#16840065—22no—
20#16840056—22no—
21#16840041—22no—
22C2×#14519713C2211yescharacteristic
23#16839803—22no—
24#16839802C2212yes—
25#16839802C2212yes—
26#16839802C2212yes—
27#16839800C2211yescharacteristic
28#16839799—21no—
29#107×D61—21no—
30#16839788C2212yes—
31C2×#14519705C2211yescharacteristic
32#16839761C2212yes—
33#16839759C2211yescharacteristic
34#16839717C2211yescharacteristic
35#16839531C2212yes—
36#16839531C2212yes—
37#16839527C2211yescharacteristic
38#16839522C2211yescharacteristic
39#16839522C2212yes—
40C2×#107×C61C2212yes—
41C2×#107×C61C2211yescharacteristic
42C2×#107×C61C2211yescharacteristic
43C2×C4×C8×C61C2211yescharacteristic
44C61×#384C2212yes—
45C2×#11758222C2311yescharacteristic
46#14519783—22no—
47#14519783Q814yescomplemented
48C2×#11758194C2312yes—
49C2×#11758194C2311yescharacteristic
50C2×#11758194C2311yescharacteristic
51C2×#11758189—21no—
52C2×C8×D61—21no—
53#14519776—22no—
54#14519776—22no—
55#14519768—22no—
56#14519768—22no—
57#14519756—42no—
58#14519742—22no—
59#14519739—22no—
60C4×C22⋊D61—42no—
61#14519736C2311yescharacteristic
62#14519736C2312yes—
63#14519735—22no—
64C4⋊C4×D61—22no—
65#14519713—22no—
66#14519711D411yescharacteristic
67#14519711D411yescharacteristic
68#14519710—22no—
69#14519710—22no—
70#14519710—22no—
71#14519709D412yes—
72#14519708—22no—
73#14519708—22no—
74#14519708—22no—
75#14519706—22no—
76#14519706—22no—
77#14519705Q814yescomplemented
78#14519702D411yescharacteristic
79#14519702D411yescharacteristic
80#14519700—22no—
81#14519699D411yescharacteristic
82#14519699D411yescharacteristic
83#14519697—22no—
84#14519687—21no—
85C8×Dic61—21no—
86C22×C8×C61C2311yescharacteristic
87C22×C8×C61C2311yescharacteristic
88C2×C4⋊C4×C61—42no—
89C2×C4⋊C4×C61C2312yes—
90C2×C4⋊C4×C61C2312yes—
91C2×C42×C61C2311yescharacteristic
92#107×C61—22no—
93#107×C61—21no—
94#107×C61—22no—
95#107×C61—21no—
96#107×C61—21no—
97#107×C61—21no—
98C4×C8×C61—22no—
99C22×C4×C61—21no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61C2411yescharacteristic
103C2×C22⋊D61—21no—
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
113C2×C8×C61—21no—
114C2×C8×C61—21no—
115C2×C8×C61C2×D411yescharacteristic
116C2×C8×C61C2×D411yescharacteristic
117C2×C8×C61C2×D411yescharacteristic
118C2×C8×C61C2×D411yescharacteristic
119C2×C8×C61C2×Q812yes—
120C2×C8×C61#4112yes—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—84no—
126C42×C61—21no—
127C42×C61C2×Q812yes—
128C22⋊Dic61—22no—
129C22⋊Dic61—21no—
130#11758198—42no—
131C4⋊Dic61—42no—
132C4⋊Dic61C2×D411yescharacteristic
133C4⋊Dic61C2×D411yescharacteristic
134C4⋊Dic61C2×D412yes—
135C4⋊Dic61C2×D412yes—
136#11758196—42no—
137#11758196—42no—
138#11758196—42no—
139C4×Dic61—21no—
140C4×Dic61—22no—
141C4×Dic61—42no—
142#11758194—21no—
143#11758194—21no—
144#11758194—22no—
145#11758194—22no—
146#11758194C2×Q812yes—
147#11758194C2×Q812yes—
148C2×#92418—21no—
149C2×#92418—21no—
150C2×#92418—22no—
151#11758189—24no—
152C8×D61—24no—
153C23×C61H(5,𝔽2)11yescharacteristic
154C22×D61—21no—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—42no—
159C2×C4×C61—42no—
160C2×C4×C61—82no—
161C2×C4×C61—82no—
162C2×C4×C61C22×D411yescharacteristic
163C2×C4×C61C22×D411yescharacteristic
164C2×C4×C61C22×Q811yescharacteristic
165C2×C4×C61C22×Q811yescharacteristic
166C2×C4×C61C2×#4111yescharacteristic
167C2×C4×C612−1+411yescharacteristic
168C22⋊D61—22no—
169C22⋊D61—22no—
170C2×Dic61—21no—
171C2×Dic61—22no—
172C2×Dic61—21no—
173C2×Dic61—21no—
174C2×Dic61—22no—
175D244—21no—
176D244C4⋊Q812yes—
177C4×D61—22no—
178C4×D61—22no—
179C4×D61—162no—
180Dic122—21no—
181Dic122—162no—
182Dic122C4⋊Q812yes—
183C8×C61—21no—
184C8×C61—21no—
185C8×C61C22⋊Q812yes—
186C8×C61C22⋊Q812yes—
187#92418—22no—
188#92418—22no—
189#92418—42no—
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—24no—
200C4×C61—82no—
201C4×C61—82no—
202C4×C61C2×C22⋊Q811yescharacteristic
203C4×C61C2×C4⋊Q811yescharacteristic
204C4×C61#54711yescharacteristic
205C4×C61D4×Q811yescharacteristic
206Dic61—22no—
207Dic61—41no—
208Dic61—42no—
209Dic61—42no—
210Dic61—41no—
211#3164—611no—
212#3083—612no—
213#2913—611no—
214#2089—612no—
215#2060—612no—
216#1952—611no—
217#1945—611no—
218#1859—612no—
219#1845D6111yescharacteristic, complemented
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×#108—611no—
239#428—612no—
240#428—612no—
241#428—612no—
242#428—1222no—
243#427—611no—
244#427—1221no—
245C2×#107—1221no—
246C2×#107C2×D6112yes—
247C2×#107C2×D6111yescharacteristic
248C2×#107C2×D6111yescharacteristic
249#424—612no—
250C2×#104—611no—
251#417—612no—
252#416—611no—
253#405—611no—
254C2×C4×C8C2×D6111yescharacteristic
255#384C2×D6112yes—
256#343—612no—
257#343—612no—
258#341—611no—
259#339—611no—
260#339—612no—
261C61#2608311yescharacteristic, complemented, Sylow
262C2×#41—611no—
263#131—614no—
264#131—1222no—
265C2×M16—611no—
266C2×M16—611no—
267C2×M16—612no—
268C2×M16—1221no—
269C22×C8—1221no—
270C22×C8C22×D6111yescharacteristic
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—611no—
282#117—1222no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—1222no—
285C2×C4⋊C4C22×D6112yes—
286C2×C4⋊C4C22×D6112yes—
287C2×C42C22×D6111yescharacteristic
288#108—1222no—
289#107—21no—
290#107—21no—
291#107—22no—
292#107—22no—
293#107—21no—
294#107—21no—
295#107—611no—
296#107—611no—
297#107—1222no—
298#107—1222no—
299#107—1222no—
300#106—612no—
301#106—1222no—
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—1221no—
314C4×C8—22no—
315C4×C8—1221no—
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—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—1221no—
328C22×C4—2442no—
329C22×C4C23×D6111yescharacteristic
330M16—612no—
331M16—612no—
332M16—1221no—
333M16—1221no—
334M16—1224no—
335M16—1222no—
336M16—1222no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—1221no—
340C2×C8—1221no—
341C2×C8—1224no—
342C2×C8—1222no—
343C2×C8#1175822212yes—
344C2×C8D4×D6111yescharacteristic
345C2×C8D4×D6111yescharacteristic
346C2×C8D4×D6111yescharacteristic
347C2×C8D4×D6111yescharacteristic
348C2×C8Q8×D6112yes—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—84no—
354C4⋊C4—611no—
355C4⋊C4—611no—
356C4⋊C4—612no—
357C4⋊C4—612no—
358C4⋊C4—2442no—
359C4⋊C4—2442no—
360C4⋊C4—2442no—
361C4⋊C4—2442no—
362C22⋊C4—1222no—
363C22⋊C4—1221no—
364C22⋊C4—2442no—
365C42—21no—
366C42—1221no—
367C42—1222no—
368C42—2442no—
369C42Q8×D6112yes—
370C23—1221no—
371C23#1451982211yescharacteristic
372Q8—612no—
373Q8—1221no—
374Q8—9762no—
375D4—612no—
376D4—1222no—
377D4—1222no—
378D4—1221no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—1221no—
387C2×C4—1222no—
388C2×C4—1222no—
389C2×C4—1222no—
390C2×C4—1221no—
391C2×C4—1222no—
392C2×C4—1221no—
393C2×C4—9762no—
394C2×C4C2×D4×D6111yescharacteristic
395C2×C4C2×D4×D6111yescharacteristic
396C2×C4C2×Q8×D6111yescharacteristic
397C2×C4C2×Q8×D6111yescharacteristic
398C2×C4C2×#1175822211yescharacteristic
399C2×C4#1451982511yescharacteristic
400C8—21no—
401C8—21no—
402C8—1222no—
403C8—1222no—
404C8—2442no—
405C8#1451975512yes—
406C8#1451975512yes—
407C22—21no—
408C22—21no—
409C22—1222no—
410C22—2441no—
411C22—2441no—
412C22#1684038911yescharacteristic
413C22#1684039711yescharacteristic
414C22#1684046711yescharacteristic
415C4—24no—
416C4—82no—
417C4—82no—
418C4—1222no—
419C4—2442no—
420C4—2441no—
421C4—2442no—
422C4—2441no—
423C4C2×#1451975511yescharacteristic
424C4C4⋊Q8×D6111yescharacteristic
425C4#1684041311yescharacteristic
426C4#1684042711yescharacteristic
427C2—21no—
428C2—21no—
429C2—2442no—
430C2#2144952411yescharacteristic
431C2#2145761811yescharacteristic
432C2#2145762311yescharacteristic
subgroupG/H
centerC22#16840397
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1845C2
socleC22×C61D42
prime radicalC2×#11758222C23
hypercenter#1845D61
hypocenterC61#26083
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1845

Complements

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

NH ≅ G/Ntype
1. #21456355C2semidirect
4. #21447325C2semidirect
5. #21447164C2semidirect
6. #21446602C2semidirect
7. #21446567C2semidirect
11. C61×#1845C2semidirect
47. #14519783Q8semidirect
77. #14519705Q8semidirect
219. #1845D61semidirect
261. C61#26083semidirect

Maximal subgroups

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

subgroupindex
#214563552
#214556752
#214544152
#214473252
#214471642
#214466022
#214465672
#214460442
#214458842
#214456562
C61×#18452
#2608361