Finite Group Databasebeta

← #29596863

Subgroups — #29596863

Order 15616.

416 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 85 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#21454883C211yesmaximal, characteristic
2#21447502C211yesmaximal, characteristic
3#21446200C211yesmaximal, characteristic
4#21446041C211yesmaximal, characteristic, complemented
5#21446038C211yesmaximal, characteristic, complemented
6#21443947C211yesmaximal, characteristic, complemented
7C61×#1395C211yesmaximal, characteristic, complemented
8#16840529C2211yescharacteristic
9#16840308C2211yescharacteristic
10#16839838—21no—
11#16839837—21no—
12C8×C22⋊D61—21no—
13#16839825—21no—
14#16839812—21no—
15#16839811—21no—
16#16839810—21no—
17#16839780—21no—
18#16839746C2211yescharacteristic
19#16839729C2211yescharacteristic
20#16839727—21no—
21#16839560C412yescomplemented
22#16839558C412yescomplemented
23#16839527—21no—
24#16839526—21no—
25#16839521—21no—
26C61×#413C2211yescharacteristic
27C61×#410C2211yescharacteristic
28C61×#408C2211yescharacteristic
29C2×#11758222D411yescharacteristic
30#14519784—42no—
31C2×#11758194—21no—
32C2×#11758194—21no—
33C2×#11758194—21no—
34C2×#11758194—21no—
35C2×#11758194D411yescharacteristic
36C2×#11758189—21no—
37C22×#92418D411yescharacteristic
38C2×C8×D61—21no—
39#14519775C2×C411yescharacteristic
40#14519773Q811yescharacteristic
41#14519766C2×C411yescharacteristic
42#14519764Q811yescharacteristic
43#14519749—21no—
44#14519749—21no—
45C4×C22⋊D61—21no—
46C4×C22⋊D61—21no—
47C2×C4×Dic61D411yescharacteristic
48#14519713—42no—
49#14519712—42no—
50#14519706—21no—
51#14519706—21no—
52#14519705—42no—
53#14519696C2×C412yes—
54#14519694—21no—
55C22⋊#92418—21no—
56C22⋊#92418—21no—
57#14519689—21no—
58#14519689—21no—
59C8×Dic61—21no—
60C8×Dic61—21no—
61C2×#41×C61C2311yescharacteristic
62C2×M16×C61—21no—
63C2×M16×C61D411yescharacteristic
64C22×C8×C61D411yescharacteristic
65#117×C61—21no—
66#101×C61C2×C412yes—
67#100×C61—21no—
68#99×C61—21no—
69#99×C61—21no—
70C22⋊C8×C61—21no—
71C22⋊C8×C61—21no—
72#41×C61—41no—
73#41×C61—41no—
74C2×Q8×C61C22×C411yescharacteristic
75C2×D4×C61C2×Q811yescharacteristic
76C2×D4×C61#4111yescharacteristic
77C2×D4×C61#4111yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C2×C22⋊D61—21no—
82C2×C22⋊D61—21no—
83C2×C22⋊D61—21no—
84C22×Dic61—21no—
85#11758222—41no—
86#11758222—41no—
87C2×D244C4⋊C411yescharacteristic
88C2×C4×D61—21no—
89C2×Dic122C4⋊C411yescharacteristic
90M16×C61—22no—
91M16×C61—41no—
92M16×C61—41no—
93C2×C8×C61—21no—
94C2×C8×C61—21no—
95C2×C8×C61—41no—
96C2×C8×C61—41no—
97C4⋊C4×C61—41no—
98C22⋊C4×C61—41no—
99C42×C61—41no—
100C22⋊Dic61—21no—
101C22⋊Dic61—21no—
102#11758198—21no—
103#11758198—21no—
104#11758196—21no—
105#11758196—21no—
106C4×Dic61—21no—
107C4×Dic61C4⋊C411yescharacteristic
108C4×Dic61C4⋊C411yescharacteristic
109#11758194—22no—
110#11758194—22no—
111#11758194—22no—
112#11758194—22no—
113#11758194—41no—
114#11758194—41no—
115#11758194—41no—
116C2×#92418—21no—
117C2×#92418—21no—
118C2×#92418—41no—
119C2×#92418—41no—
120C2×#92418—41no—
121C2×#92418—41no—
122C2×#92418—41no—
123#11758189—81no—
124C8×D61—81no—
125C23×C61C22⋊Q811yescharacteristic
126C23×C61C22⋊Q811yescharacteristic
127C23×C61#12311yescharacteristic
128C22×D61—21no—
129Q8×C61—41no—
130D4×C61—41no—
131D4×C61—41no—
132D4×C61—41no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—81no—
140C2×C4×C61C2×C4⋊C411yescharacteristic
141C2×C4×C61C4×D411yescharacteristic
142C2×C4×C61C4×D411yescharacteristic
143C2×C4×C61#12011yescharacteristic
144C22⋊D61—42no—
145C22⋊D61—42no—
146C22⋊D61—81no—
147C2×Dic61—21no—
148C2×Dic61—21no—
149C2×Dic61—21no—
150C2×Dic61—42no—
151D244—41no—
152C4×D61—41no—
153C4×D61—41no—
154Dic122—41no—
155C8×C61—42no—
156C8×C61—41no—
157C8×C61—41no—
158#92418—42no—
159#92418—41no—
160#92418—42no—
161#92418—41no—
162#12975—611nomaximal, Sylow
163C22×C61—21no—
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—41no—
167C22×C61—41no—
168C22×C61—41no—
169C22×C61#38511yescharacteristic
170C2×D61—41no—
171C2×D61—41no—
172C2×D61—41no—
173C4×C61—21no—
174C4×C61—21no—
175C4×C61—21no—
176C4×C61—161no—
177C4×C61B(4,𝔽2)11yescharacteristic
178C4×C61#45811yescharacteristic
179Dic61—22no—
180Dic61—41no—
181Dic61—41no—
182#2979—611no—
183#2132—611no—
184#1884—611no—
185#1858—611no—
186#1858—611no—
187#1395—611no—
188#1395D6111yescharacteristic, complemented
189C2×C61—21no—
190C2×C61—41no—
191C2×C61—41no—
192C2×C61—41no—
193C2×C61#186711yescharacteristic
194D61—81no—
195#568—611no—
196#532—611no—
197#436—1221no—
198#436—1221no—
199C8×D4—1221no—
200C8×D4—1221no—
201#431—1221no—
202#430—1221no—
203#430—1221no—
204#421—1221no—
205#413—611no—
206#413C2×D6111yescharacteristic
207#410C2×D6111yescharacteristic
208#408—611no—
209#408—1221no—
210#408C2×D6111yescharacteristic
211#356—612no—
212#355—612no—
213#341—1221no—
214#341—1221no—
215#339—1221no—
216C61#1297511yescharacteristic, complemented, Sylow
217C2×#41—611no—
218C2×#41C22×D6111yescharacteristic
219#131—2442no—
220C2×M16—21no—
221C2×M16—611no—
222C2×M16—1221no—
223C2×M16—1221no—
224C2×M16—1221no—
225C2×M16—1221no—
226C2×M16—1221no—
227C2×M16D24411yescharacteristic
228C22×C8—611no—
229C22×C8—1221no—
230C22×C8D24411yescharacteristic
231C4⋊Q8—611no—
232#127—611no—
233#124—611no—
234#124—611no—
235#121—1221no—
236#121—1221no—
237C4×D4—1221no—
238C4×D4—1221no—
239#117—21no—
240C2×C42—611no—
241#108—2442no—
242#108—2442no—
243C4⋊C8—1221no—
244C4⋊C8—1221no—
245#104—2442no—
246#101—612no—
247#101C4×D6112yes—
248#100—21no—
249#100—1221no—
250#99—21no—
251#99—21no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—1221no—
255C22⋊C8—1221no—
256C22⋊C8—1221no—
257C22⋊C8—1221no—
258C4×C8—1221no—
259C4×C8—1221no—
260#41—41no—
261#41—41no—
262#41—2441no—
263#41—2441no—
264C2×Q8—611no—
265C2×Q8C2×C4×D6111yescharacteristic
266C2×D4—611no—
267C2×D4—1221no—
268C2×D4—1221no—
269C2×D4—1221no—
270C2×D4#1175822211yescharacteristic
271C2×D4#1175822411yescharacteristic
272C2×D4Q8×D6111yescharacteristic
273C22×C4—1221no—
274C22×C4—1221no—
275C22×C4C2×D24411yescharacteristic
276C22×C4D4×D6111yescharacteristic
277C22×C4D4×D6111yescharacteristic
278M16—22no—
279M16—41no—
280M16—41no—
281M16—1222no—
282M16—1222no—
283M16—1222no—
284M16—1222no—
285M16—4881no—
286M16—2441no—
287M16—2441no—
288M16—2441no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—1221no—
294C2×C8—1221no—
295C2×C8—4881no—
296C2×C8—2441no—
297C2×C8—2441no—
298C2×C8—2441no—
299C2×C8—2441no—
300C2×C8—2441no—
301C4⋊C4—41no—
302C4⋊C4—1221no—
303C4⋊C4—1221no—
304C22⋊C4—41no—
305C22⋊C4—1221no—
306C22⋊C4—1221no—
307C22⋊C4—1221no—
308C22⋊C4—1221no—
309C42—41no—
310C42—611no—
311C42—611no—
312C42—1221no—
313C23—1221no—
314C23#1451975511yescharacteristic
315C23#1451975711yescharacteristic
316C23#1451975911yescharacteristic
317Q8—41no—
318Q8—2441no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—4881no—
323D4—2441no—
324D4—2442no—
325D4—2442no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—81no—
333C2×C4—1221no—
334C2×C4—1221no—
335C2×C4—1221no—
336C2×C4—2442no—
337C2×C4—2441no—
338C2×C4—2441no—
339C2×C4C4⋊C4×D6111yescharacteristic
340C2×C4#1451973711yescharacteristic
341C2×C4C4×D24411yescharacteristic
342C2×C4#1451974711yescharacteristic
343C8—42no—
344C8—41no—
345C8—41no—
346C8—2442no—
347C8—2441no—
348C8—2442no—
349C8—2441no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—2441no—
357C22—2441no—
358C22—2441no—
359C22#1683963611yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—161no—
364C4—1222no—
365C4—2441no—
366C4—2441no—
367C4#1683995011yescharacteristic
368C4#1683995311yescharacteristic
369C2—21no—
370C2—41no—
371C2—41no—
372C2—41no—
373C2—4881no—
374C2#2144610111yescharacteristic
subgroupG/H
centerC4#16839950
commutatorC2×Q8×C61C22×C4
FrattiniC2×#41C22×D61
FittingC61×#1395C2
socleC2×C61#1867
prime radical#16840308C22
cosocleC2×#41×C61C23
hypercenter#1395D61
hypocenterC61#12975
LCSG ▷ C2×Q8×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1395

Complements

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

NH ≅ G/Ntype
4. #21446041C2semidirect
5. #21446038C2semidirect
6. #21443947C2semidirect
7. C61×#1395C2semidirect
21. #16839560C4semidirect
22. #16839558C4semidirect
188. #1395D61semidirect
216. C61#12975semidirect

Maximal subgroups

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

subgroupindex
#214548832
#214475022
#214462002
#214460412
#214460382
#214439472
C61×#13952
#1297561