Finite Group Databasebeta

← #29596041

Subgroups — #29596041

Order 15616.

420 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 89 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#21454534C211yesmaximal, characteristic
2#21447482C212yesmaximal
3#21445863C212yesmaximal, complemented
4#21443940C211yesmaximal, characteristic, complemented
5C61×#1391C211yesmaximal, characteristic, complemented
6C2×#14519785C411yescharacteristic, complemented
7C2×#14519784C2211yescharacteristic
8#16840308C2211yescharacteristic
9#16839836—22no—
10#16839830—22no—
11#16839798—22no—
12#16839785C411yescharacteristic, complemented
13#16839736C2212yes—
14#16839728C411yescharacteristic, complemented
15C2×#14519687C411yescharacteristic, complemented
16#16839556—22no—
17#16839554—22no—
18#16839518—22no—
19C61×#410C2212yes—
20C61×#408C2211yescharacteristic
21C2×#11758222C2×C411yescharacteristic
22#14519785—42no—
23#14519784—42no—
24C2×#11758194C2×C411yescharacteristic
25C2×#11758194D412yes—
26C2×#11758189—21no—
27C22×#92418C2×C411yescharacteristic
28C22×#92418D412yes—
29C2×C8×D61—21no—
30#14519775Q811yescharacteristic
31#14519773D411yescharacteristic
32#14519766Q811yescharacteristic
33#14519764D411yescharacteristic
34#14519749—22no—
35C4×C22⋊D61—22no—
36#14519736—22no—
37C2×C4×Dic61C2×C411yescharacteristic
38#14519710—42no—
39#14519708—42no—
40#14519706—22no—
41#14519692—22no—
42#14519692—22no—
43#14519689—22no—
44#14519687—21no—
45#14519687—21no—
46C2×#41×C61C2311yescharacteristic
47C2×M16×C61C2×C411yescharacteristic
48C22×C8×C61C2×C411yescharacteristic
49#117×C61—22no—
50#99×C61—22no—
51#99×C61—22no—
52C22⋊C8×C61—22no—
53#41×C61—41no—
54#41×C61—41no—
55C2×Q8×C61C2×Q811yescharacteristic
56C2×D4×C61C2×D411yescharacteristic
57C2×D4×C61#4112yes—
58C22×C4×C61C22×C411yescharacteristic
59C22×C4×C61C2×D412yes—
60C2×C22⋊D61—21no—
61C2×C22⋊D61—22no—
62C22×Dic61—21no—
63#11758222—41no—
64#11758222—41no—
65C2×D244C4⋊C411yescharacteristic
66C2×C4×D61—21no—
67C2×Dic122C4⋊C411yescharacteristic
68M16×C61—41no—
69C2×C8×C61—21no—
70C2×C8×C61—21no—
71C2×C8×C61—41no—
72C4⋊C4×C61—42no—
73C22⋊C4×C61—42no—
74C42×C61—42no—
75C22⋊Dic61—22no—
76C22⋊Dic61—42no—
77#11758198—22no—
78C4⋊Dic61—42no—
79#11758196—22no—
80C4×Dic61—21no—
81C4×Dic61—42no—
82C4×Dic61C4⋊C411yescharacteristic
83C4×Dic61C4⋊C411yescharacteristic
84#11758194—41no—
85#11758194—42no—
86C2×#92418—22no—
87C2×#92418—22no—
88C2×#92418—22no—
89C2×#92418C22⋊C411yescharacteristic
90C2×#92418C22⋊C412yes—
91C2×#92418C22⋊C411yescharacteristic
92#11758189—42no—
93C8×D61—42no—
94C23×C61#11711yescharacteristic
95C23×C61#12112yes—
96C22×D61—21no—
97Q8×C61—41no—
98D4×C61—41no—
99D4×C61—42no—
100C2×C4×C61—22no—
101C2×C4×C61—21no—
102C2×C4×C61—22no—
103C2×C4×C61—21no—
104C2×C4×C61—82no—
105C2×C4×C61C2×C22⋊C411yescharacteristic
106C2×C4×C61C2×C4⋊C411yescharacteristic
107C2×C4×C61C22⋊Q812yes—
108C22⋊D61—42no—
109C22⋊D61—82no—
110C2×Dic61—21no—
111C2×Dic61—21no—
112C2×Dic61—21no—
113C2×Dic61—81no—
114C2×Dic61—82no—
115D244—41no—
116C4×D61—41no—
117C4×D61—41no—
118Dic122—41no—
119C8×C61—41no—
120C8×C61—41no—
121#92418—21no—
122#92418—22no—
123#92418#9912yes—
124#12808—611nomaximal, Sylow
125C22×C61—22no—
126C22×C61—21no—
127C22×C61—41no—
128C22×C61—42no—
129C22×C61#38011yescharacteristic
130C2×D61—42no—
131C2×D61—41no—
132C4×C61—21no—
133C4×C61—22no—
134C4×C61—162no—
135C4×C61C2×#9911yescharacteristic
136C4×C61#41011yescharacteristic
137Dic61—41no—
138Dic61—41no—
139Dic61—41no—
140Dic61—162no—
141#2932—611no—
142#2126—612no—
143#1812—612no—
144#1391—611no—
145#1391D6111yescharacteristic, complemented
146C2×C61—21no—
147C2×C61—42no—
148C2×C61—41no—
149C2×C61#179911yescharacteristic
150D61—81no—
151C2×#131—611no—
152C2×#131—611no—
153#532—611no—
154#435—1222no—
155#435—1222no—
156#427—1222no—
157#423—611no—
158#410—612no—
159#410C2×D6112yes—
160#408—611no—
161#408C2×D6111yescharacteristic
162C2×#97—611no—
163#354—1222no—
164#353—1222no—
165H(3,ℤ/4)—1222no—
166C61#1280811yescharacteristic, complemented, Sylow
167C2×#41—611no—
168C2×#41C22×D6111yescharacteristic
169#131—2442no—
170#131—2442no—
171C2×M16—611no—
172C2×M16—612no—
173C2×M16—1221no—
174C2×M16C4×D6111yescharacteristic
175C22×C8—611no—
176C22×C8—612no—
177C22×C8—1221no—
178C22×C8C4×D6111yescharacteristic
179C4⋊Q8—611no—
180#127—611no—
181#124—611no—
182#124—611no—
183#121—1222no—
184C4×D4—1222no—
185#117—22no—
186#117—1222no—
187C2×C42—611no—
188#107—2442no—
189#106—2442no—
190C4⋊C8—1222no—
191#99—22no—
192#99—22no—
193#99—1222no—
194#99—1222no—
195C22⋊C8—22no—
196C22⋊C8—1222no—
197#97—1221no—
198#97—1221no—
199#41—41no—
200#41—41no—
201#41—2441no—
202#41—2441no—
203C2×Q8—611no—
204C2×Q8Q8×D6111yescharacteristic
205C2×D4—611no—
206C2×D4—1222no—
207C2×D4—1221no—
208C2×D4#1175822212yes—
209C2×D4D4×D6111yescharacteristic
210C22×C4—1221no—
211C22×C4—1221no—
212C22×C4C2×C4×D6111yescharacteristic
213C22×C4D4×D6112yes—
214M16—41no—
215M16—2442no—
216M16—2442no—
217M16—2441no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—41no—
221C2×C8—611no—
222C2×C8—611no—
223C2×C8—612no—
224C2×C8—1222no—
225C2×C8—1222no—
226C2×C8—1222no—
227C2×C8—2442no—
228C4⋊C4—42no—
229C4⋊C4—1222no—
230C4⋊C4—2442no—
231C22⋊C4—42no—
232C22⋊C4—1222no—
233C22⋊C4—1222no—
234C22⋊C4—2442no—
235C42—42no—
236C42—611no—
237C42—611no—
238C42—1221no—
239C42—2442no—
240C23—1221no—
241C23#1451973411yescharacteristic
242C23#1451975012yes—
243Q8—41no—
244Q8—2441no—
245D4—42no—
246D4—41no—
247D4—4882no—
248D4—2441no—
249D4—2442no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—82no—
255C2×C4—1221no—
256C2×C4—1221no—
257C2×C4—1221no—
258C2×C4—4882no—
259C2×C4—4881no—
260C2×C4—2441no—
261C2×C4—2441no—
262C2×C4C22⋊C4×D6111yescharacteristic
263C2×C4C4⋊C4×D6111yescharacteristic
264C2×C4#1451975512yes—
265C8—41no—
266C8—41no—
267C8—612no—
268C8—1221no—
269C8—1222no—
270C22—22no—
271C22—21no—
272C22—41no—
273C22—42no—
274C22—2442no—
275C22—2441no—
276C22#1683962111yescharacteristic
277C4—21no—
278C4—22no—
279C4—162no—
280C4—9762no—
281C4—2441no—
282C4—2441no—
283C4—2441no—
284C4#99×D6111yescharacteristic
285C4#1683973411yescharacteristic
286C2—21no—
287C2—42no—
288C2—41no—
289C2—4881no—
290C2#2144580611yescharacteristic
subgroupG/H
centerC4#16839734
commutatorC22×C4×C61C22×C4
FrattiniC2×#41C22×D61
FittingC61×#1391C2
socleC2×C61#1799
prime radical#16840308C22
cosocleC2×#41×C61C23
hypercenter#1391D61
hypocenterC61#12808
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1391

Complements

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

NH ≅ G/Ntype
3. #21445863C2semidirect
4. #21443940C2semidirect
5. C61×#1391C2semidirect
6. C2×#14519785C4semidirect
12. #16839785C4semidirect
14. #16839728C4semidirect
15. C2×#14519687C4semidirect
145. #1391D61semidirect
166. C61#12808semidirect

Maximal subgroups

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

subgroupindex
#214545342
#214474822
#214458632
#214439402
C61×#13912
#1280861