Finite Group Databasebeta

← #29712830

Subgroups — #29712830

Order 15616.

1302 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1302 conjugacy classes of subgroups — too many to draw. Only its 187 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#21459477C211yesmaximal, characteristic
2#21455750C212yesmaximal, complemented
3#21447029C214yesmaximal, complemented
4#21446262C212yesmaximal, complemented
5#21446033C212yesmaximal, complemented
6#21445777C212yesmaximal, complemented
7#21445747C211yesmaximal, characteristic, complemented
8C61×#2297C211yesmaximal, characteristic, complemented
9C2×#14519825C2211yescharacteristic
10#16840431—24no—
11#16840419—24no—
12#16840329—22no—
13#16840329C2212yes—
14#16840273—22no—
15#16840273C2212yes—
16#16840240C2211yescharacteristic
17#16840103—24no—
18#16840086—24no—
19#16839802C2214yes—
20C2×#14519705C2214yes—
21#16839761C2214yes—
22C2×#14519696C2212yes—
23C2×#14519694C2212yes—
24#16839726C2212yes—
25#16839522C2214yes—
26C22×#41×C61C2211yescharacteristic
27C2×#117×C61C2211yescharacteristic
28C22×C42×C61C2211yescharacteristic
29C61×#391C2212yes—
30C61×#390C2212yes—
31C61×#386—22no—
32C4×C22⋊C4×C61—22no—
33#14519825—24no—
34#14519825—42no—
35C2×#11758226—22no—
36C2×#11758222—22no—
37C2×#11758222C2312yes—
38C2×Q8×D61—22no—
39C2×#11758194C2314yes—
40#14519773—24no—
41#14519773—22no—
42#14519764—24no—
43#14519764—22no—
44#14519760—22no—
45#14519756—24no—
46#14519756D414yes—
47#14519751—24no—
48#14519751D414yes—
49#14519748—28no—
50#14519748—24no—
51#14519746—24no—
52#14519746—22no—
53Q8×Dic61—24no—
54C4×C22⋊D61—28no—
55C4×C22⋊D61—24no—
56#14519736—22no—
57#14519736C2312yes—
58#14519710—24no—
59#14519708—24no—
60#14519705—44no—
61#14519700—24no—
62#14519697—24no—
63#14519696—24no—
64#14519694—42no—
65C22⋊#92418—24no—
66C2×#41×C61—24no—
67C2×#41×C61—24no—
68C2×#41×C61—42no—
69C22×Q8×C61C2311yescharacteristic
70C22×D4×C61D412yes—
71C22×D4×C61C2311yescharacteristic
72C23×C4×C61—21no—
73C23×C4×C61D412yes—
74C23×C4×C61C2311yescharacteristic
75#117×C61—24no—
76C2×C4⋊C4×C61C2312yes—
77C2×C22⋊C4×C61—24no—
78C2×C22⋊C4×C61—21no—
79C2×C42×C61—21no—
80C2×C42×C61—21no—
81C2×C42×C61—42no—
82C2×C42×C61C2312yes—
83#95×C61—24no—
84C24×C61C2×D412yes—
85C24×C61C2×D411yescharacteristic
86#41×C61—416no—
87C2×Q8×C61—22no—
88C2×Q8×C61—22no—
89C2×Q8×C61#4114yes—
90C2×D4×C61—24no—
91C2×D4×C61—24no—
92C2×D4×C61—22no—
93C2×D4×C61—22no—
94C2×D4×C61—22no—
95C2×D4×C61—42no—
96C22×C4×C61—22no—
97C22×C4×C61—22no—
98C22×C4×C61—22no—
99C22×C4×C61—24no—
100C22×C4×C61—24no—
101C22×C4×C61—22no—
102C22×C4×C61—42no—
103C22×C4×C61—44no—
104C22×C4×C61—42no—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D414yes—
107C22×C4×C61C2×D412yes—
108C22×C4×C61C2411yescharacteristic
109C2×C22⋊D61—22no—
110C2×C22⋊D61—28no—
111C2×C22⋊D61—24no—
112#11758226—48no—
113Q8×D61—48no—
114#11758222—24no—
115#11758222—24no—
116#11758222—44no—
117#11758222—44no—
118C2×D244—22no—
119C2×D244C2×D412yes—
120C2×C4×D61—24no—
121C2×C4×D61—22no—
122C2×Dic122—22no—
123C2×Dic122C2×D412yes—
124C4⋊C4×C61—24no—
125C22⋊C4×C61—48no—
126C22⋊C4×C61—42no—
127C42×C61—24no—
128C42×C61—42no—
129C42×C61—42no—
130C22⋊Dic61—28no—
131C22⋊Dic61—24no—
132C22⋊Dic61—22no—
133#11758198—24no—
134C4⋊Dic61—24no—
135C4⋊Dic61C2×D414yes—
136#11758196—24no—
137C4×Dic61—24no—
138C4×Dic61—22no—
139#11758194—24no—
140#11758194—24no—
141C2×#92418—24no—
142C23×C61—21no—
143C23×C61—21no—
144C23×C61—22no—
145C23×C61—22no—
146C23×C61—22no—
147C23×C61—42no—
148C23×C61—44no—
149C23×C61#12012yes—
150C23×C61C22×D411yescharacteristic
151C22×D61—22no—
152Q8×C61—28no—
153D4×C61—48no—
154D4×C61—44no—
155C2×C4×C61—22no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—22no—
159C2×C4×C61—28no—
160C2×C4×C61—24no—
161C2×C4×C61—48no—
162C2×C4×C61—42no—
163C2×C4×C61—48no—
164C2×C4×C61—42no—
165C2×C4×C61—44no—
166C2×C4×C61—42no—
167C2×C4×C61#12012yes—
168C2×C4×C61#12314yes—
169C2×C4×C61C22×D412yes—
170C2×C4×C61C2×#4112yes—
171C2×C4×C61H(5,𝔽2)12yes—
172C22⋊D61—42no—
173C22⋊D61—42no—
174C22⋊D61—84no—
175C22⋊D61—88no—
176C2×Dic61—24no—
177C2×Dic61—22no—
178D244—24no—
179D244—44no—
180C4×D61—48no—
181C4×D61—44no—
182Dic122—24no—
183Dic122—44no—
184#92418—44no—
185#24020—611nomaximal, Sylow
186C22×C61—22no—
187C22×C61—22no—
188C22×C61—24no—
189C22×C61—42no—
190C22×C61—42no—
191C22×C61—44no—
192C22×C61—42no—
193C22×C61—44no—
194C22×C61—82no—
195C22×C61C2×#12011yescharacteristic
196C22×C61C2×#12311yescharacteristic
197C22×C61#53411yescharacteristic
198C2×D61—42no—
199C2×D61—44no—
200C4×C61—24no—
201C4×C61—22no—
202C4×C61—44no—
203C4×C61—82no—
204C4×C61#54614yes—
205Dic61—42no—
206Dic61—44no—
207#3487—611no—
208#3093—612no—
209#2297D6111yescharacteristic, complemented
210#2037—614no—
211#1896—612no—
212#1857—612no—
213#1793—612no—
214#1785—611no—
215C2×C61—21no—
216C2×C61—21no—
217C2×C61—42no—
218C2×C61—82no—
219C2×C61#240511yescharacteristic
220C2×C61#301812yes—
221D61—82no—
222C2×2−1+4—611no—
223C22×#41C2×D6111yescharacteristic
224#550—1224no—
225#548—1224no—
226#535—612no—
227#535—1222no—
228#525—612no—
229#525—1222no—
230#519—611no—
231C2×#117C2×D6111yescharacteristic
232C22×C42C2×D6111yescharacteristic
233#485—1224no—
234#482—1224no—
235#428—614no—
236C2×#104—614no—
237#417—614no—
238C2×#101—612no—
239C2×#100—612no—
240#407—612no—
241#391C2×D6112yes—
242#390C2×D6112yes—
243#386—22no—
244C4×C22⋊C4—22no—
245#339—614no—
246C61#2402011yescharacteristic, complemented, Sylow
2472−1+4—1224no—
2482−1+4—2442no—
249C2×#41—24no—
250C2×#41—24no—
251C2×#41—42no—
252C2×#41—612no—
253C2×#41—1222no—
254C2×#41—1222no—
255C22×Q8—1222no—
256C22×Q8C22×D6111yescharacteristic
257C22×D4C22⋊D6112yes—
258C22×D4C22×D6111yescharacteristic
259C23×C4—21no—
260C23×C4C22⋊D6112yes—
261C23×C4C22×D6111yescharacteristic
262C2×M16—614no—
263#127—1222no—
264#127—1224no—
265#124—1222no—
266#124—1224no—
267#123—1222no—
268C22⋊Q8—614no—
269C22⋊Q8—1224no—
270#121—614no—
271#121—1224no—
272#121—1224no—
273#121—1228no—
274#120—1222no—
275#120—1224no—
276C4×Q8—1224no—
277C4×D4—1228no—
278C4×D4—1224no—
279#117—24no—
280#117—612no—
281#117—1222no—
282C2×C4⋊C4C22×D6112yes—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—24no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—42no—
288C2×C42C22×D6112yes—
289#107—1224no—
290#106—1224no—
291#104—2444no—
292#103—1224no—
293#102—1224no—
294#101—1224no—
295#100—2442no—
296C22⋊C8—1224no—
297#95—24no—
298C24C2×C22⋊D6112yes—
299C24C2×C22⋊D6111yescharacteristic
300#41—416no—
301#41—1224no—
302#41—1224no—
303#41—2444no—
304#41—2448no—
305#41—2444no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—612no—
309C2×Q8—1222no—
310C2×Q8—2448no—
311C2×Q8#1175822214yes—
312C2×D4—24no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—24no—
317C2×D4—42no—
318C2×D4—612no—
319C2×D4—1222no—
320C2×D4—1228no—
321C2×D4—1224no—
322C2×D4—1222no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—42no—
330C22×C4—44no—
331C22×C4—42no—
332C22×C4—1224no—
333C22×C4—1222no—
334C22×C4D4×D6114yes—
335C22×C4C2×C22⋊D6111yescharacteristic
336C22×C4C2×C22⋊D6112yes—
337C22×C4C23×D6111yescharacteristic
338M16—1224no—
339M16—1224no—
340C2×C8—1224no—
341C4⋊C4—24no—
342C4⋊C4—614no—
343C4⋊C4—1224no—
344C4⋊C4—1224no—
345C22⋊C4—48no—
346C22⋊C4—42no—
347C22⋊C4—1228no—
348C22⋊C4—1222no—
349C22⋊C4—1224no—
350C22⋊C4—1224no—
351C42—24no—
352C42—42no—
353C42—42no—
354C42—1224no—
355C42—1222no—
356C23—21no—
357C23—21no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—42no—
362C23—44no—
363C23—1222no—
364C23#1451974512yes—
365C23C22×C22⋊D6111yescharacteristic
366Q8—28no—
367Q8—1224no—
368Q8—2444no—
369D4—48no—
370D4—44no—
371D4—1224no—
372D4—4884no—
373D4—4888no—
374D4—2444no—
375D4—2442no—
376D4—2442no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—28no—
382C2×C4—24no—
383C2×C4—48no—
384C2×C4—42no—
385C2×C4—48no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—1224no—
390C2×C4—1222no—
391C2×C4—2448no—
392C2×C4—2444no—
393C2×C4#1451974512yes—
394C2×C4#1451976014yes—
395C2×C4C2×D4×D6112yes—
396C2×C4C2×#1175822212yes—
397C2×C4#1451982212yes—
398C8—2444no—
399C22—22no—
400C22—24no—
401C22—22no—
402C22—42no—
403C22—42no—
404C22—44no—
405C22—44no—
406C22—42no—
407C22—82no—
408C22—2444no—
409C22—2442no—
410C22C2×#1451974511yescharacteristic
411C22C2×#1451976011yescharacteristic
412C22#1684032111yescharacteristic
413C4—22no—
414C4—24no—
415C4—44no—
416C4—82no—
417C4—2444no—
418C4—2442no—
419C4#1684040214yes—
420C2—21no—
421C2—21no—
422C2—42no—
423C2—82no—
424C2—4882no—
425C2#2144914611yescharacteristic
426C2#2145516712yes—
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2297C2
socleC22×C61C2×#120
prime radical#21459477C2
hypercenter#2297D61
hypocenterC61#24020
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2297

Complements

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

NH ≅ G/Ntype
2. #21455750C2semidirect
3. #21447029C2semidirect
4. #21446262C2semidirect
5. #21446033C2semidirect
6. #21445777C2semidirect
7. #21445747C2semidirect
8. C61×#2297C2semidirect
209. #2297D61semidirect
246. C61#24020semidirect

Maximal subgroups

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

subgroupindex
#214594772
#214557502
#214470292
#214462622
#214460332
#214457772
#214457472
C61×#22972
#2402061