Finite Group Databasebeta

← #29575337

Subgroups — #29575337

Order 15616.

426 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 103 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#21456144C211yesmaximal, characteristic
2#21454921C211yesmaximal, characteristic, complemented
3#21444237C212yesmaximal
4#21443657C212yesmaximal, complemented
5C61×#1526C211yesmaximal, characteristic, complemented
6#16840312C2211yescharacteristic
7#16840119—22no—
8#16840114C412yes—
9#16840110C412yes—
10#16839893—22no—
11Q8×#92418—22no—
12#16839823—22no—
13#16839815C2212yes—
14#16839756—22no—
15C2×C4×#92418C2211yescharacteristic
16#16839491—22no—
17#16839491—22no—
18C61×#533C2211yescharacteristic
19C4×M16×C61C2212yes—
20C61×#330—22no—
21C2×#11758222—21no—
22C22×#92418—21no—
23#14519777C2×C411yescharacteristic
24#14519774C2×C411yescharacteristic
25#14519765—21no—
26#14519751—22no—
27C4×D244—22no—
28C4⋊#92418—24no—
29C4⋊#92418D412yes—
30C4⋊#92418D412yes—
31#14519700—24no—
32#14519697—24no—
33C22⋊#92418—22no—
34#14519688—22no—
35C4×#92418—22no—
36C4×#92418—21no—
37C4×#92418—22no—
38C4×#92418C2×C412yes—
39C2×M16×C61—22no—
40C4⋊Q8×C61C2×C412yes—
41#125×C61D412yes—
42C22⋊Q8×C61—22no—
43C4×Q8×C61—22no—
44#117×C61D412yes—
45C2×C42×C61C2311yescharacteristic
46#97×C61—22no—
47C4×C8×C61—22no—
48C2×Q8×C61—22no—
49C22×C4×C61C2×D411yescharacteristic
50C22×C4×C61#4112yes—
51C2×C22⋊D61—41no—
52#11758222—44no—
53C2×D244—21no—
54C2×D244—22no—
55C2×C4×D61—41no—
56C2×Dic122—21no—
57M16×C61—24no—
58M16×C61—42no—
59C2×C8×C61—42no—
60C4⋊C4×C61—22no—
61C4⋊C4×C61—24no—
62C4⋊C4×C61C22⋊C414yes—
63C22⋊C4×C61—22no—
64C42×C61—22no—
65C42×C61C22×C411yescharacteristic
66C42×C61C2×D411yescharacteristic
67C42×C61C2×D412yes—
68#11758198—42no—
69C4⋊Dic61—22no—
70C2×#92418—22no—
71C2×#92418—22no—
72C2×#92418—22no—
73C2×#92418—42no—
74C23×C61#12311yescharacteristic
75C22×D61—41no—
76Q8×C61—42no—
77Q8×C61—42no—
78C2×C4×C61—22no—
79C2×C4×C61—22no—
80C2×C4×C61—22no—
81C2×C4×C61—24no—
82C2×C4×C61C2×C22⋊C411yescharacteristic
83C2×C4×C61C4×D412yes—
84C2×C4×C61#12011yescharacteristic
85C2×C4×C61#12112yes—
86C22⋊D61—82no—
87C2×Dic61—41no—
88D244—42no—
89D244—42no—
90D244—42no—
91C4×D61—82no—
92Dic122—42no—
93C8×C61—44no—
94#92418—42no—
95#92418—42no—
96#92418—42no—
97#9070—611nomaximal, Sylow
98C22×C61—22no—
99C22×C61—21no—
100C22×C61#38611yescharacteristic
101C2×D61—82no—
102C2×D61—81no—
103C4×C61—22no—
104C4×C61—21no—
105C4×C61—44no—
106C4×C61#44312yes—
107C4×C61#44412yes—
108C4×C61#45412yes—
109Dic61—81no—
110#3131—611no—
111#2986—611no—
112#1526D6111yescharacteristic, complemented
113#1484—612no—
114#1278—612no—
115C2×C61—41no—
116C2×C61#187612yes—
117C2×C61#188811yescharacteristic
118D61—161no—
119#533C2×D6111yescharacteristic
120#532—611no—
121#488—1222no—
122#487—612no—
123#486—612no—
124#446—1222no—
125C8×Q8—1222no—
126#433—1222no—
127#431—612no—
128#416—1222no—
129C4×M16C2×D6112yes—
130C2×C4×C8—611no—
131#330—22no—
132D4⋊C8—1222no—
133D4⋊C8—1222no—
134C61#907011yescharacteristic, complemented, Sylow
135C2×#41—1221no—
136C2×M16—22no—
137C22×C8—1221no—
138C4⋊Q8—611no—
139C4⋊Q8C4×D6112yes—
140#127—611no—
141#125C22⋊D6112yes—
142#124—1221no—
143C22⋊Q8—22no—
144#121—1222no—
145C4×Q8—22no—
146C4×D4—1222no—
147#117D24412yes—
148C2×C42C22×D6111yescharacteristic
149C4⋊C8—612no—
150C4⋊C8—612no—
151C4⋊C8—1224no—
152#103—1224no—
153#102—1224no—
154C22⋊C8—1222no—
155#97—22no—
156#97—1222no—
157C4×C8—22no—
158C4×C8—612no—
159C4×C8—1222no—
160C4×C8—1222no—
161C4×C8—1221no—
162#41—2444no—
163C2×Q8—22no—
164C2×Q8—1221no—
165C2×D4—1221no—
166C2×D4—1222no—
167C2×D4—2441no—
168C22×C4—2441no—
169C22×C4C2×D24411yescharacteristic
170C22×C4#1175822412yes—
171M16—24no—
172M16—42no—
173C2×C8—42no—
174C2×C8—1222no—
175C2×C8—1222no—
176C2×C8—1222no—
177C2×C8—2442no—
178C4⋊C4—22no—
179C4⋊C4—24no—
180C4⋊C4—1222no—
181C4⋊C4#1175819814yes—
182C22⋊C4—22no—
183C22⋊C4—2442no—
184C42—22no—
185C42C2×C4×D6111yescharacteristic
186C42D4×D6112yes—
187C42C2×C22⋊D6111yescharacteristic
188C23—2441no—
189C23#1451976111yescharacteristic
190Q8—42no—
191Q8—42no—
192Q8—2442no—
193D4—4882no—
194D4—2442no—
195D4—2442no—
196D4—2442no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—4882no—
202C2×C4—2441no—
203C2×C4C2×#1175819811yescharacteristic
204C2×C4#1451973712yes—
205C2×C4#1451974711yescharacteristic
206C2×C4#1451974912yes—
207C8—44no—
208C8—2442no—
209C8—2442no—
210C8—2442no—
211C22—22no—
212C22—21no—
213C22—4882no—
214C22—4881no—
215C22#1683963911yescharacteristic
216C4—22no—
217C4—21no—
218C4—44no—
219C4—4881no—
220C4#1683987612yes—
221C4#1683988112yes—
222C4#1683993712yes—
223C2—41no—
224C2—9761no—
225C2#2144614412yes—
226C2#2144622011yescharacteristic
subgroupG/H
centerC2×C4#14519747
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1526C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#1526D61
hypocenterC61#9070
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1526

Complements

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

NH ≅ G/Ntype
2. #21454921C2semidirect
4. #21443657C2semidirect
5. C61×#1526C2semidirect
112. #1526D61semidirect
134. C61#9070semidirect

Maximal subgroups

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

subgroupindex
#214561442
#214549212
#214442372
#214436572
C61×#15262
#907061