Finite Group Databasebeta

← #29567564

Subgroups — #29567564

Order 15616.

228 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 41 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#21447043C211yesmaximal, characteristic
2#21443765C211yesmaximal, characteristic, complemented
3C61×#1296C211yesmaximal, characteristic, complemented
4#16840343C411yescharacteristic
5C2×#14519705—21no—
6#16839761—21no—
7#16839531C411yescharacteristic
8#16839488—21no—
9C2×C4⋊C8×C61—21no—
10C61×#384C2211yescharacteristic
11C2×#11758222—21no—
12C2×#11758194—21no—
13#14519757—41no—
14#14519752—41no—
15#14519751—41no—
16#14519705—41no—
17#14519705—41no—
18#14519700—41no—
19#14519697—41no—
20C22⋊#92418—41no—
21C22×C8×C61—41no—
22C2×C4⋊C4×C61—41no—
23C2×C4⋊C4×C61C2×C411yescharacteristic
24C2×C4⋊C4×C61D411yescharacteristic
25C2×C42×C61D411yescharacteristic
26C4⋊C8×C61—22no—
27C4⋊C8×C61—41no—
28C22×C4×C61—21no—
29C22×C4×C61—21no—
30C22×C4×C61—21no—
31C22×C4×C61C22⋊C411yescharacteristic
32C2×C22⋊D61—41no—
33#11758222—41no—
34#11758222—41no—
35#11758222—41no—
36#11758222—41no—
37C2×D244—21no—
38C2×D244—41no—
39C2×C4×D61—41no—
40C2×Dic122—21no—
41C2×C8×C61—42no—
42C2×C8×C61—81no—
43C2×C8×C61—81no—
44C4⋊C4×C61—41no—
45C4⋊C4×C61—41no—
46C4⋊C4×C61—81no—
47C4⋊C4×C61—81no—
48C42×C61—21no—
49C42×C61D811yescharacteristic
50C42×C61SD1611yescharacteristic
51#11758198—42no—
52C4⋊Dic61—41no—
53#11758194—41no—
54#11758194—41no—
55C2×#92418—41no—
56C2×#92418—81no—
57C23×C61#9911yescharacteristic
58C22×D61—41no—
59C2×C4×C61—21no—
60C2×C4×C61—21no—
61C2×C4×C61—21no—
62C2×C4×C61—21no—
63C2×C4×C61—21no—
64C2×C4×C61—21no—
65C2×C4×C61—21no—
66C2×C4×C61—21no—
67C2×C4×C61—41no—
68C2×C4×C61—41no—
69C2×C4×C61—81no—
70C2×C4×C61—81no—
71C2×C4×C61#10211yescharacteristic
72C2×C4×C61#10411yescharacteristic
73C22⋊D61—81no—
74C22⋊D61—81no—
75C2×Dic61—41no—
76D244—41no—
77D244—41no—
78D244—161no—
79C4×D61—81no—
80C4×D61—81no—
81Dic122—41no—
82Dic122—41no—
83C8×C61—82no—
84#92418—81no—
85#92418—161no—
86#7417—611nomaximal, Sylow
87C22×C61—21no—
88C22×C61—21no—
89C22×C61#32711yescharacteristic
90C22×C61AΣL(1,𝔽16)11yescharacteristic
91C22×C61#35511yescharacteristic
92C2×D61—81no—
93C2×D61—81no—
94C2×D61—81no—
95C4×C61—21no—
96C4×C61—21no—
97C4×C61—21no—
98C4×C61—21no—
99C4×C61—21no—
100C4×C61—21no—
101C4×C61—81no—
102C4×C61—81no—
103Dic61—81no—
104#2039—611no—
105#1325—611no—
106#1296D6111yescharacteristic, complemented
107C2×C61—21no—
108C2×C61—21no—
109C2×C61#134511yescharacteristic
110C2×C61#140211yescharacteristic
111C2×C61#140511yescharacteristic
112D61—161no—
113#537—611no—
114C2×C4⋊C8—21no—
115C2×#104—1221no—
116#417—1221no—
117#384C2×D6111yescharacteristic
118#343—611no—
119#324—1221no—
120C61#741711yescharacteristic, complemented, Sylow
121C2×#41—1221no—
122C2×M16—1221no—
123C22×C8—41no—
124C22⋊Q8—2441no—
125#121—2441no—
126#121—2441no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4C4×D6111yescharacteristic
129C2×C4⋊C4D24411yescharacteristic
130C2×C42C22⋊D6111yescharacteristic
131C4⋊C8—22no—
132C4⋊C8—41no—
133#104—2441no—
134#104—2441no—
135#103—2441no—
136#102—2441no—
137C22⋊C8—2441no—
138#41—2441no—
139#41—2441no—
140#41—2441no—
141#41—2441no—
142C2×Q8—1221no—
143C2×D4—1221no—
144C2×D4—2441no—
145C2×D4—2441no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—2441no—
150C22×C4#1175819811yescharacteristic
151M16—2441no—
152M16—2441no—
153C2×C8—42no—
154C2×C8—81no—
155C2×C8—81no—
156C2×C8—4881no—
157C2×C8—2441no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—81no—
161C4⋊C4—81no—
162C4⋊C4—2441no—
163C22⋊C4—2442no—
164C42—21no—
165C42#1175819911yescharacteristic
166C42#1175820111yescharacteristic
167C23—2441no—
168C23#1451969111yescharacteristic
169Q8—2441no—
170Q8—2441no—
171D4—9761no—
172D4—4881no—
173D4—4881no—
174D4—2441no—
175D4—2441no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—4881no—
189C2×C4—4881no—
190C2×C4—2441no—
191C2×C4#1451969711yescharacteristic
192C2×C4#1451970511yescharacteristic
193C8—82no—
194C8—9761no—
195C8—4881no—
196C22—21no—
197C22—21no—
198C22—4881no—
199C22—4881no—
200C22—4881no—
201C22#1683949511yescharacteristic
202C22#1683955011yescharacteristic
203C22#1683955811yescharacteristic
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—21no—
210C4—81no—
211C4—81no—
212C4—4881no—
213C2—21no—
214C2—21no—
215C2—9761no—
216C2#2144382111yescharacteristic
217C2#2144396511yescharacteristic
218C2#2144397411yescharacteristic
subgroupG/H
centerC22#16839550
commutatorC2×C4⋊C4×C61C2×C4
Frattini#384C2×D61
FittingC61×#1296C2
socleC22×C61AΣL(1,𝔽16)
prime radical#16840343C4
cosocleC61×#384C22
hypercenter#1296D61
hypocenterC61#7417
LCSG ▷ C2×C4⋊C4×C61 ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #384 ◁ #1296

Complements

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

NH ≅ G/Ntype
2. #21443765C2semidirect
3. C61×#1296C2semidirect
106. #1296D61semidirect
120. C61#7417semidirect

Maximal subgroups

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

subgroupindex
#214470432
#214437652
C61×#12962
#741761