Finite Group Databasebeta

← #30608303

Subgroups — #30608303

Order 15616.

498 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 60 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#21462090C211yesmaximal, characteristic, complemented
2#21461273C211yesmaximal, characteristic, complemented
3#21460942C214yesmaximal, complemented
4#21455168C211yesmaximal, characteristic
5#16840741—21no—
6C4×#11758217—21no—
7#16840668C2212yescomplemented
8#16840666—21no—
9#16840618C2212yescomplemented
10#16840617—21no—
11#16840560—21no—
12#16840308C2211yescharacteristic
13#16839945C412yescomplemented
14#16839940—21no—
15#16839935C2212yes—
16#16839930—21no—
17C2×#14519704—21no—
18#16839746C411yescharacteristic, complemented
19C61×#585C411yescharacteristic, complemented
20C2×#11758217—41no—
21#14519856—41no—
22#14519836—22no—
23#14519835—41no—
24#14519832—22no—
25C4×#92420—41no—
26C2×#11758222D411yescharacteristic
27#14519808—41no—
28#14519807—22no—
29#14519803—41no—
30#14519802—22no—
31#14519797—42no—
32C2×#11758194—21no—
33#14519775C2×C411yescharacteristic
34#14519773D411yescharacteristic
35#14519766C2311yescharacteristic
36#14519764D411yescharacteristic
37#14519749—41no—
38C4×C22⋊D61—41no—
39C2×C4×Dic61D411yescharacteristic
40#14519704—22no—
41#14519704—22no—
42#14519696C2×C412yes—
43#14519694—21no—
442−1+4×C61—41no—
452−1+4×C61C2×C412yes—
46H(5,𝔽2)×C61—21no—
47H(5,𝔽2)×C61—41no—
48H(5,𝔽2)×C61—41no—
49C2×#41×C61—21no—
50C2×#41×C61—41no—
51C2×#41×C61—42no—
52C2×#41×C61C2×C411yescharacteristic
53#41×C61—22no—
54#41×C61—22no—
55#41×C61—41no—
56#41×C61—42no—
57#41×C61—41no—
58#41×C61—41no—
59#41×C61—42no—
60#41×C61—42no—
61#41×C61—44no—
62#41×C61—42no—
63#41×C61—41no—
64#41×C61—42no—
65C2×Q8×C61—21no—
66C2×Q8×C61—42no—
67C2×Q8×C61—41no—
68C2×Q8×C61C22×C411yescharacteristic
69C2×D4×C61—21no—
70C2×D4×C61—21no—
71C2×D4×C61—42no—
72C2×D4×C61—42no—
73C2×D4×C61—44no—
74C2×D4×C61—41no—
75C2×D4×C61—41no—
76C2×D4×C61C22⋊C411yescharacteristic
77C22×C4×C61—21no—
78C22×C4×C61—41no—
79C22×C4×C61—42no—
80C22×C4×C61C22⋊C411yescharacteristic
81C2×C22⋊D61—21no—
82C2×C22⋊D61—41no—
83C22×Dic61—21no—
84#11758222—41no—
85#11758222—41no—
86C2×D244C2×D411yescharacteristic
87C2×C4×D61—21no—
88C2×Dic122C22⋊C411yescharacteristic
89C22⋊#6923—42no—
90#11758217—44no—
91C2×#92420—42no—
92C4⋊#6923—81no—
93C4×#6923—81no—
94C22⋊Dic61—41no—
95#11758202—42no—
96#11758201—42no—
97#11758200—42no—
98#11758199—42no—
99#11758198—41no—
100#11758196—41no—
101C4×Dic61—21no—
102C4×Dic61C22⋊C411yescharacteristic
103C4×Dic61C2×D411yescharacteristic
104#11758194—22no—
105#11758194—41no—
106C2×#92418—41no—
107C23×C61—21no—
108C23×C61—41no—
109C23×C61—42no—
110C23×C61#9911yescharacteristic
111C22×D61—21no—
112C2×#6923—82no—
113Q8×C61—22no—
114Q8×C61—41no—
115Q8×C61—41no—
116Q8×C61—42no—
117D4×C61—22no—
118D4×C61—41no—
119D4×C61—42no—
120D4×C61—42no—
121D4×C61—42no—
122D4×C61—44no—
123D4×C61—42no—
124D4×C61—41no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—42no—
129C2×C4×C61—41no—
130C2×C4×C61—44no—
131C2×C4×C61—42no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—42no—
135C2×C4×C61#9911yescharacteristic
136C2×C4×C61C2×C22⋊C411yescharacteristic
137C22⋊D61—82no—
138C22⋊D61—81no—
139C2×Dic61—21no—
140C2×Dic61—21no—
141C2×Dic61—21no—
142C2×Dic61—42no—
143D244—41no—
144C4×D61—41no—
145C4×D61—41no—
146Dic122—41no—
147#92420—44no—
148#92418—42no—
149#13568—611nomaximal, Sylow
150C22×C61—21no—
151C22×C61—41no—
152C22×C61—42no—
153C22×C61—41no—
154C22×C61—81no—
155C22×C61—161no—
156C22×C61—162no—
157C22×C61C2×#9911yescharacteristic
158C2×D61—41no—
159C2×D61—81no—
160#6923—162no—
161C4×C61—21no—
162C4×C61—41no—
163C4×C61—42no—
164C4×C61AΣL(1,𝔽16)11yescharacteristic
165C4×C61AΣL(1,𝔽16)11yescharacteristic
166Dic61—22no—
167Dic61—41no—
168Dic61—41no—
169#3018—611no—
170#2126—611no—
171#1525—611no—
172#1404—614no—
173C2×C61—21no—
174C2×C61—41no—
175C2×C61—81no—
176C2×C61—82no—
177C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
178D61—81no—
179#585#692311yescharacteristic, complemented
180#575—1221no—
181#532—611no—
182#456—612no—
183#455—1221no—
184#454—612no—
185#453—1221no—
186C2×#104—1221no—
187#413—611no—
188#410—1221no—
189C4×M16—1221no—
190#354—612no—
191#353—1221no—
192#329—612no—
193#329—1221no—
194C61#1356811yescharacteristic, complemented, Sylow
1952−1+4—41no—
1962−1+4C2×#692312yes—
197H(5,𝔽2)—21no—
198H(5,𝔽2)—41no—
199H(5,𝔽2)—41no—
200C2×#41—21no—
201C2×#41—41no—
202C2×#41—42no—
203C2×#41—611no—
204C2×#41C2×#692311yescharacteristic
205#137—1222no—
206#137—2441no—
207AGL(1,ℤ/8)—1222no—
208AGL(1,ℤ/8)—2441no—
209#135—2442no—
210C2×M16—1221no—
211C2×M16—2441no—
212C4⋊Q8—611no—
213#127—611no—
214#124—611no—
215#124—611no—
216#121—2441no—
217C4×D4—2441no—
218#117—2441no—
219C2×C42—611no—
220#104—1222no—
221#104—1222no—
222#101—612no—
223#100—1221no—
224#99—1222no—
225#99—2441no—
226#97—1222no—
227C4×C8—2441no—
228#41—22no—
229#41—22no—
230#41—41no—
231#41—42no—
232#41—41no—
233#41—41no—
234#41—42no—
235#41—42no—
236#41—44no—
237#41—42no—
238#41—41no—
239#41—42no—
240#41—2441no—
241#41—2441no—
242C2×Q8—21no—
243C2×Q8—42no—
244C2×Q8—41no—
245C2×Q8—611no—
246C2×Q8C22×#692311yescharacteristic
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—44no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—611no—
255C2×D4—1221no—
256C2×D4—2441no—
257C2×D4C22⋊#692311yescharacteristic
258C22×C4—21no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—1221no—
262C22×C4—1221no—
263C22×C4C22⋊#692311yescharacteristic
264Q16—2442no—
265SD16—2442no—
266SD16—2442no—
267D8—2442no—
268M16—1222no—
269M16—2441no—
270M16—2444no—
271C2×C8—2441no—
272C2×C8—2442no—
273C4⋊C4—4881no—
274C4⋊C4—2441no—
275C22⋊C4—2441no—
276C22⋊C4—2441no—
277C22⋊C4—2442no—
278C42—611no—
279C42—611no—
280C42—1221no—
281C42—4881no—
282C23—21no—
283C23—41no—
284C23—42no—
285C23—1221no—
286C23#1451983511yescharacteristic
287Q8—22no—
288Q8—41no—
289Q8—42no—
290Q8—41no—
291Q8—2441no—
292D4—22no—
293D4—41no—
294D4—42no—
295D4—42no—
296D4—42no—
297D4—44no—
298D4—42no—
299D4—41no—
300D4—4882no—
301D4—4881no—
302D4—2441no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—1221no—
314C2×C4—1221no—
315C2×C4—1221no—
316C2×C4—4882no—
317C2×C4—2442no—
318C2×C4—2441no—
319C2×C4—2441no—
320C2×C4#1451983511yescharacteristic
321C2×C4C2×C22⋊#692311yescharacteristic
322C8—2442no—
323C8—2444no—
324C22—21no—
325C22—41no—
326C22—42no—
327C22—41no—
328C22—81no—
329C22—161no—
330C22—162no—
331C22—4881no—
332C22—2441no—
333C22C2×#1451983511yescharacteristic
334C4—21no—
335C4—41no—
336C4—42no—
337C4—1222no—
338C4—9762no—
339C4—2441no—
340C4—2441no—
341C4#1684066211yescharacteristic
342C4#1684066211yescharacteristic
343C2—21no—
344C2—41no—
345C2—81no—
346C2—82no—
347C2—4881no—
348C2C2×#1684066211yescharacteristic
subgroupG/H
centerC4#16840662
commutatorC2×Q8×C61C22×C4
FrattiniC2×Q8C22×#6923
FittingC61×#585C4
socleC2×C61C2×AΣL(1,𝔽16)
prime radical#21455168C2
cosocle#14519766C23
hypercenter#585#6923
hypocenterC61#13568
LCSG ▷ C2×Q8×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×#41 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #21462090C2semidirect
2. #21461273C2semidirect
3. #21460942C2semidirect
7. #16840668C22semidirect
9. #16840618C22semidirect
13. #16839945C4semidirect
18. #16839746C4semidirect
19. C61×#585C4semidirect
179. #585#6923semidirect
194. C61#13568semidirect

Maximal subgroups

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

subgroupindex
#214620902
#214612732
#214609422
#214551682
#1356861