Finite Group Databasebeta

← #29599161

Subgroups — #29599161

Order 15616.

540 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21453805C211yesmaximal, characteristic, complemented
2#21445778C211yesmaximal, characteristic
3#21445778C211yesmaximal, characteristic
4#21443946C211yesmaximal, characteristic, complemented
5#21443945C211yesmaximal, characteristic, complemented
6#21443945C211yesmaximal, characteristic, complemented
7C61×#1793C211yesmaximal, characteristic
8C2×#14519704—21no—
9C2×#14519704—21no—
10#16839763—21no—
11#16839763—21no—
12C2×#14519696C2211yescharacteristic
13C2×#14519696C2211yescharacteristic
14#16839704—21no—
15#16839696—21no—
16#16839695—21no—
17#16839669C2211yescharacteristic
18#16839669C2211yescharacteristic
19#16839659—21no—
20#16839646—21no—
21#16839645—21no—
22C2×2−1+4×C61C2211yescharacteristic
23C2×#104×C61—21no—
24C61×#417—21no—
25C2×#101×C61C2211yescharacteristic
26C61×#391C2211yescharacteristic
27C2×#11758194—21no—
28C2×#11758194—21no—
29C2×C4⋊Dic61—21no—
30C2×C4⋊Dic61D411yescharacteristic
31C2×C4⋊Dic61D411yescharacteristic
32C2×#11758196—21no—
33C2×#11758196—21no—
34C2×#11758196—21no—
35C2×#11758196—21no—
36C2×C4×Dic61D411yescharacteristic
37C2×C4×Dic61D411yescharacteristic
38#14519704—42no—
39#14519704—42no—
40#14519701—41no—
41#14519701—41no—
42#14519698—41no—
43#14519698—41no—
44#14519696—21no—
45#14519696—21no—
46#14519696—21no—
47#14519696—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
552−1+4×C61—21no—
562−1+4×C61—21no—
572−1+4×C61—21no—
582−1+4×C61—42no—
592−1+4×C61Q812yes—
60C2×#41×C61—21no—
61C2×#41×C61—21no—
62C2×#41×C61—21no—
63C2×#41×C61—41no—
64C22×Q8×C61—41no—
65C22×Q8×C61C2311yescharacteristic
66C2×M16×C61—21no—
67C2×C4⋊C4×C61D411yescharacteristic
68C2×C42×C61D411yescharacteristic
69#104×C61—42no—
70#103×C61—41no—
71#102×C61—41no—
72#101×C61—21no—
73#101×C61—21no—
74#95×C61—21no—
75#95×C61—21no—
76#41×C61—22no—
77#41×C61—22no—
78#41×C61—22no—
79#41×C61—22no—
80#41×C61—22no—
81#41×C61—22no—
82#41×C61—41no—
83#41×C61—41no—
84#41×C61—41no—
85#41×C61—42no—
86#41×C61—42no—
87#41×C61—42no—
88#41×C61—42no—
89#41×C61—41no—
90#41×C61—41no—
91#41×C61—41no—
92C2×Q8×C61—21no—
93C2×Q8×C61—21no—
94C2×Q8×C61—21no—
95C2×Q8×C61—41no—
96C2×Q8×C61—41no—
97C2×Q8×C61—42no—
98C2×Q8×C61—42no—
99C2×Q8×C61—42no—
100C2×Q8×C61—42no—
101C2×Q8×C61C2×Q811yescharacteristic
102C2×Q8×C61#4111yescharacteristic
103C2×Q8×C61#4111yescharacteristic
104C2×Q8×C61#4111yescharacteristic
105C2×D4×C61—21no—
106C2×D4×C61—21no—
107C2×D4×C61—21no—
108C2×D4×C61—41no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—41no—
112C22×C4×C61—41no—
113C22×C4×C61—41no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×Dic61—21no—
118C22×Dic61—21no—
119C22×Dic61—21no—
120C22×Dic61—21no—
121M16×C61—41no—
122M16×C61—41no—
123C2×C8×C61—41no—
124C4⋊C4×C61—41no—
125C42×C61—41no—
126C4⋊Dic61—42no—
127C4⋊Dic61—41no—
128C4⋊Dic61—41no—
129#11758196—42no—
130#11758196—42no—
131#11758196—81no—
132#11758196—81no—
133C4×Dic61—41no—
134C4×Dic61—41no—
135#11758194—41no—
136#11758194—41no—
137#11758194—41no—
138#11758194—41no—
139C2×#92418—41no—
140C2×#92418—41no—
141C23×C61—41no—
142C23×C61#12011yescharacteristic
143Q8×C61—22no—
144Q8×C61—22no—
145Q8×C61—22no—
146Q8×C61—41no—
147Q8×C61—41no—
148Q8×C61—41no—
149Q8×C61—41no—
150Q8×C61—41no—
151Q8×C61—41no—
152Q8×C61—41no—
153D4×C61—22no—
154D4×C61—22no—
155D4×C61—22no—
156D4×C61—41no—
157D4×C61—41no—
158D4×C61—41no—
159D4×C61—42no—
160D4×C61—42no—
161C2×C4×C61—21no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—42no—
168C2×C4×C61—42no—
169C2×C4×C61—42no—
170C2×C4×C61—42no—
171C2×C4×C61—42no—
172C2×C4×C61—42no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61C22⋊Q811yescharacteristic
181C2×C4×C61C22⋊Q811yescharacteristic
182C2×C4×C61C22⋊Q811yescharacteristic
183C2×C4×C61#12411yescharacteristic
184C2×C4×C61#12411yescharacteristic
185C2×C4×C61#12411yescharacteristic
186C2×Dic61—22no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—42no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C8×C61—81no—
198#92418—81no—
199#92418—81no—
200#13379—611nomaximal, Sylow
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—42no—
204C22×C61—81no—
205C22×C61—81no—
206C22×C61#39311yescharacteristic
207C22×C61B(4,𝔽2)11yescharacteristic
208C22×C61B(4,𝔽2)11yescharacteristic
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—21no—
215C4×C61—42no—
216C4×C61—81no—
217C4×C61—81no—
218Dic61—42no—
219Dic61—81no—
220Dic61—81no—
221Dic61—81no—
222#2809—611no—
223#1793—611no—
224#1793—611no—
225#1793D6111yescharacteristic
226#1394—611no—
227#1394—611no—
228#1394—611no—
229C2×C61—21no—
230C2×C61—21no—
231C2×C61—82no—
232C2×C61#202911yescharacteristic
233C2×C61#220311yescharacteristic
234C2×C61#220411yescharacteristic
235C2×2−1+4C2×D6111yescharacteristic
236C2×#104—21no—
237C2×#104—1221no—
238C2×#104—1221no—
239#417—21no—
240#417—1221no—
241#417—1221no—
242C2×#101—611no—
243C2×#101—611no—
244C2×#101C2×D6111yescharacteristic
245#400—1221no—
246#398—1221no—
247#398—1221no—
248#391—611no—
249#391—611no—
250#391C2×D6111yescharacteristic
251#389—1221no—
252#387—1221no—
253#387—1221no—
254C61#1337911yescharacteristic, complemented, Sylow
2552−1+4—21no—
2562−1+4—21no—
2572−1+4—21no—
2582−1+4—42no—
2592−1+4Dic12212yes—
260C2×#41—21no—
261C2×#41—21no—
262C2×#41—21no—
263C2×#41—41no—
264C22×Q8—41no—
265C22×Q8C22×D6111yescharacteristic
266C2×M16—21no—
267C2×M16—1221no—
268C2×M16—1221no—
269C2×C4⋊C4—611no—
270C2×C4⋊C4—611no—
271C2×C4⋊C4—1221no—
272C2×C4⋊C4—1221no—
273C2×C4⋊C4—1221no—
274C2×C4⋊C4—1221no—
275C2×C4⋊C4—1221no—
276C2×C4⋊C4C22⋊D6111yescharacteristic
277C2×C42—611no—
278C2×C42—611no—
279C2×C42C22⋊D6111yescharacteristic
280#104—42no—
281#104—2442no—
282#104—2442no—
283#103—41no—
284#103—2441no—
285#103—2441no—
286#102—41no—
287#102—2441no—
288#102—2441no—
289#101—21no—
290#101—21no—
291#101—1221no—
292#101—1221no—
293#101—1221no—
294#101—1221no—
295#95—21no—
296#95—21no—
297#95—1221no—
298#95—1221no—
299#95—1221no—
300#95—1221no—
301#95—1221no—
302#95—1221no—
303#95—1221no—
304#41—22no—
305#41—22no—
306#41—22no—
307#41—22no—
308#41—22no—
309#41—22no—
310#41—41no—
311#41—41no—
312#41—41no—
313#41—42no—
314#41—42no—
315#41—42no—
316#41—42no—
317#41—41no—
318#41—41no—
319#41—41no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8C2×Dic12211yescharacteristic
330C2×Q8#1175822211yescharacteristic
331C2×Q8#1175822411yescharacteristic
332C2×Q8#1175822411yescharacteristic
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—41no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—1221no—
343C22×C4—1221no—
344C22×C4—1221no—
345C22×C4—1221no—
346C22×C4D4×D6111yescharacteristic
347C22×C4D4×D6111yescharacteristic
348C22×C4C2×C22⋊D6111yescharacteristic
349M16—41no—
350M16—41no—
351M16—2441no—
352M16—2441no—
353M16—2441no—
354M16—2441no—
355C2×C8—41no—
356C2×C8—2441no—
357C2×C8—2441no—
358C4⋊C4—41no—
359C4⋊C4—4881no—
360C4⋊C4—4881no—
361C4⋊C4—2441no—
362C4⋊C4—2441no—
363C4⋊C4—2442no—
364C4⋊C4—2442no—
365C4⋊C4—2442no—
366C42—41no—
367C42—2441no—
368C42—2441no—
369C23—41no—
370C23#1451974511yescharacteristic
371Q8—22no—
372Q8—22no—
373Q8—22no—
374Q8—41no—
375Q8—41no—
376Q8—41no—
377Q8—41no—
378Q8—41no—
379Q8—41no—
380Q8—41no—
381D4—22no—
382D4—22no—
383D4—22no—
384D4—41no—
385D4—41no—
386D4—41no—
387D4—42no—
388D4—42no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—1222no—
409C2×C4—2441no—
410C2×C4—2441no—
411C2×C4—2441no—
412C2×C4—2441no—
413C2×C4—2442no—
414C2×C4—2441no—
415C2×C4—2441no—
416C2×C4—2441no—
417C2×C4—2441no—
418C2×C4—2441no—
419C2×C4#1451975411yescharacteristic
420C2×C4#1451975411yescharacteristic
421C2×C4#1451975611yescharacteristic
422C2×C4#1451976311yescharacteristic
423C2×C4#1451976311yescharacteristic
424C2×C4#1451976411yescharacteristic
425C8—81no—
426C8—4881no—
427C8—4881no—
428C22—21no—
429C22—21no—
430C22—42no—
431C22—81no—
432C22—81no—
433C22#1683967411yescharacteristic
434C22#1683994811yescharacteristic
435C22#1683994811yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—42no—
443C4—81no—
444C4—81no—
445C4—4881no—
446C4—4881no—
447C4—4881no—
448C4—2442no—
449C2—21no—
450C2—21no—
451C2—82no—
452C2#2144698311yescharacteristic
453C2#2144783811yescharacteristic
454C2#2144784311yescharacteristic
subgroupG/H
centerC22#16839948
commutator, cosocleC22×Q8×C61C23
FrattiniC22×Q8C22×D61
FittingC61×#1793C2
socleC22×C61B(4,𝔽2)
prime radicalC2×2−1+4×C61C22
hypercenter#1793D61
hypocenterC61#13379
LCSG ▷ C22×Q8×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1793

Complements

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

NH ≅ G/Ntype
1. #21453805C2semidirect
4. #21443946C2semidirect
5. #21443945C2semidirect
6. #21443945C2semidirect
254. C61#13379semidirect

Maximal subgroups

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

subgroupindex
#214538052
#214457782
#214439462
#214439452
C61×#17932
#1337961