Finite Group Databasebeta

← #29593739

Subgroups — #29593739

Order 15616.

296 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 67 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#21458688C211yesmaximal, characteristic
2#21448048C211yesmaximal, characteristic, complemented
3#21447865C211yesmaximal, characteristic
4#21445190C211yesmaximal, characteristic, complemented
5#21443797C211yesmaximal, characteristic, complemented
6#21443779C211yesmaximal, characteristic
7C61×#1365C211yesmaximal, characteristic, complemented
8#16840473—21no—
9C2×#14519722—21no—
10#16840163—21no—
11#16840142C2211yescharacteristic
12#16840065—22no—
13#16840049—21no—
14#16840048—21no—
15C8×Dic122C2211yescharacteristic
16C4×D488C2211yescharacteristic
17#16839844—21no—
18#16839567—21no—
19#16839565—22no—
20C4×#11758185C2211yescharacteristic
21#16839507—21no—
22C61×#500C2211yescharacteristic
23C61×C4⋊C16C2211yescharacteristic
24C61×#335C2211yescharacteristic
25C2×Dic244—21no—
26C2×Dic244—21no—
27C2×D488D411yescharacteristic
28#14519777—21no—
29#14519768—22no—
30#14519757—41no—
31#14519753—42no—
32C4×Dic122D411yescharacteristic
33C4×D244—21no—
34#14519739—22no—
35C4⋊C4×D61—41no—
36#14519722—42no—
37C2×#11758185—21no—
38#14519711D411yescharacteristic
39#14519709—41no—
40#14519708—22no—
41C4⋊#92418D411yescharacteristic
42#14519706—21no—
43#14519702—41no—
44#14519699—21no—
45#14519697—22no—
46C8×Dic61—21no—
47C4⋊Q8×C61—21no—
48C2×C16×C61—21no—
49#107×C61—21no—
50#107×C61D412yes—
51C4⋊C8×C61—21no—
52C4×C8×C61C2311yescharacteristic
53C2×Q8×C61—41no—
54C2×D244—21no—
55C2×C4×D61—41no—
56C2×C4×D61—42no—
57C2×Dic122—41no—
58C2×Dic122D811yescharacteristic
59C2×C8×C61—41no—
60C2×C8×C61C2×D411yescharacteristic
61C2×C8×C61C2×D411yescharacteristic
62C4⋊C4×C61—22no—
63C4⋊C4×C61—41no—
64C42×C61C2×D411yescharacteristic
65#11758198—42no—
66#11758198—41no—
67C4⋊Dic61—21no—
68C4⋊Dic61—42no—
69C4⋊Dic61—41no—
70C4⋊Dic61—41no—
71C4⋊Dic61D811yescharacteristic
72#11758196—21no—
73#11758196—42no—
74C4×Dic61—21no—
75C2×#92418—21no—
76Dic244—42no—
77Dic244—41no—
78Dic244—41no—
79D488—21no—
80D488SD1612yes—
81C16×C61—41no—
82#11758185—41no—
83C22×D61—41no—
84Q8×C61—161no—
85C2×C4×C61—42no—
86C2×C4×C61#12011yescharacteristic
87C2×C4×C61C2×D811yescharacteristic
88C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
89C2×Dic61—21no—
90C2×Dic61—21no—
91C2×Dic61—41no—
92C2×Dic61—81no—
93D244—22no—
94D244—41no—
95C4×D61—81no—
96C4×D61—81no—
97C4×D61—162no—
98Dic122—21no—
99Dic122—21no—
100Dic122—81no—
101Dic122—81no—
102C8×C61—21no—
103C8×C61—81no—
104C8×C61C2×SD1611yescharacteristic
105C8×C61#13711yescharacteristic
106#92418—41no—
107#12359—611nomaximal, Sylow
108C22×C61#44711yescharacteristic
109C2×D61—42no—
110C2×D61—81no—
111C2×D61—81no—
112C4×C61—21no—
113C4×C61—82no—
114C4×C61#44811yescharacteristic
115C4×C61#45411yescharacteristic
116C4×C61#50811yescharacteristic
117C4×C61#51011yescharacteristic
118Dic61—41no—
119Dic61—41no—
120Dic61—81no—
121Dic61—161no—
122#3380—611no—
123#2241—611no—
124#2208—611no—
125#1670—611no—
126#1365D6111yescharacteristic, complemented
127Q8⋊C16—611no—
128#1331—611no—
129C2×C61#162711yescharacteristic
130C2×C61#218911yescharacteristic
131C2×C61#219411yescharacteristic
132D61—82no—
133#554—1221no—
134C2×SD32—1221no—
135#500C2×D6111yescharacteristic
136#499—1221no—
137#494—611no—
138#478—1222no—
139#476—1221no—
140#475—1221no—
141C8×Q8—611no—
142C4×D8—611no—
143C4×D8—1221no—
144C4⋊C16C2×D6111yescharacteristic
145#358—1221no—
146#358—1222no—
147C4×C16—611no—
148#335C2×D6111yescharacteristic
149#332—1221no—
150C61#1235911yescharacteristic, complemented, Sylow
151C2×Q16—1221no—
152C2×Q16—1221no—
153C2×D8—611no—
154C4⋊Q8—21no—
155C4⋊Q8—1221no—
156#125—1222no—
157C22⋊Q8—2441no—
158C22⋊Q8—2442no—
159C4×Q8—611no—
160C4×D4—1222no—
161C4×D4—1221no—
162C2×C4⋊C4—2441no—
163SD32—2442no—
164C2×C16—21no—
165C2×C16—1221no—
166#107—21no—
167#107—611no—
168#107D24412yes—
169#106—1222no—
170#106—2441no—
171C4⋊C8—21no—
172C4⋊C8—611no—
173C4⋊C8—1221no—
174#103—2441no—
175#102—1222no—
176#102—1221no—
177C4×C8—1221no—
178C4×C8C22×D6111yescharacteristic
179C2×Q8—41no—
180C2×Q8—611no—
181C2×Q8—2441no—
182C2×D4—1221no—
183C22×C4—2442no—
184C22×C4—2441no—
185Q16—2442no—
186Q16—2441no—
187Q16—2441no—
188D8—612no—
189D8—1221no—
190C2×C8—41no—
191C2×C8—1221no—
192C2×C8C2×D24411yescharacteristic
193C2×C8D4×D6111yescharacteristic
194C4⋊C4—22no—
195C4⋊C4—41no—
196C4⋊C4—611no—
197C4⋊C4—1221no—
198C4⋊C4—1221no—
199C4⋊C4—2442no—
200C4⋊C4—2442no—
201C4⋊C4—2441no—
202C4⋊C4—2441no—
203C22⋊C4—2442no—
204C22⋊C4—2441no—
205C42—1221no—
206C42D4×D6111yescharacteristic
207C16—41no—
208C16—2441no—
209C23—2441no—
210Q8—161no—
211Q8—1221no—
212Q8—1221no—
213Q8—4881no—
214Q8—4881no—
215D4—1222no—
216D4—2441no—
217C2×C4—42no—
218C2×C4—1221no—
219C2×C4—1221no—
220C2×C4—9762no—
221C2×C4—4881no—
222C2×C4—4881no—
223C2×C4—4881no—
224C2×C4—2441no—
225C2×C4#1451974711yescharacteristic
226C2×C4D8×D6111yescharacteristic
227C2×C4#1451980411yescharacteristic
228C8—21no—
229C8—81no—
230C8—2441no—
231C8SD16×D6111yescharacteristic
232C8#1451980911yescharacteristic
233C22—4881no—
234C22—4881no—
235C22—2442no—
236C22#1683989711yescharacteristic
237C4—21no—
238C4—82no—
239C4—9761no—
240C4—4881no—
241C4—2441no—
242C4—2441no—
243C4#1683990211yescharacteristic
244C4#1683993711yescharacteristic
245C4#1684019311yescharacteristic
246C4#1684020411yescharacteristic
247C2—4882no—
248C2#2144492311yescharacteristic
249C2#2144776211yescharacteristic
250C2#2144779311yescharacteristic
subgroupG/H
centerC22#16839897
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#1365C2
socleC22×C61#447
prime radicalC2×D488D4
hypercenter#1365D61
hypocenterC61#12359
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1365

Complements

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

NH ≅ G/Ntype
2. #21448048C2semidirect
4. #21445190C2semidirect
5. #21443797C2semidirect
7. C61×#1365C2semidirect
126. #1365D61semidirect
150. C61#12359semidirect

Maximal subgroups

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

subgroupindex
#214586882
#214480482
#214478652
#214451902
#214437972
#214437792
C61×#13652
#1235961