Finite Group Databasebeta

← #29584152

Subgroups — #29584152

Order 15616.

386 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 73 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#21457200C211yesmaximal, characteristic
2#21445993C211yesmaximal, characteristic, complemented
3#21444542C212yesmaximal, complemented
4#21444431C212yesmaximal, complemented
5C61×#1307C211yesmaximal, characteristic, complemented
6#16840312C2212yes—
7#16840053—22no—
8#16840015—22no—
9#16839958—22no—
10C2×#14519712—22no—
11#16839786C2211yescharacteristic
12C2×#14519688C2211yescharacteristic
13#16839509—22no—
14#16839505—22no—
15#16839501—24no—
16C61×#432C2212yes—
17C61×#423C2211yescharacteristic
18C2×#11758222—22no—
19C2×D488—22no—
20C2×#11758194—21no—
21C22×#92418—21no—
22#14519777D412yes—
23#14519774D412yes—
24#14519765—22no—
25#14519751—22no—
26#14519751—22no—
27C4×D244—22no—
28C4×D244—22no—
29#14519712—44no—
30#14519709—22no—
31C4⋊#92418—22no—
32#14519699—24no—
33#14519688—21no—
34#14519688D412yes—
35C2×M16×C61D412yes—
36C2×C42×C61C2311yescharacteristic
37C4⋊C8×C61—22no—
38C4⋊C8×C61—22no—
39C22⋊C8×C61—22no—
40#97×C61—22no—
41C22×C4×C61C2×D411yescharacteristic
42C22×C4×C61#4112yes—
43C2×C22⋊D61—42no—
44#11758222—84no—
45C2×D244—22no—
46C2×D244—22no—
47C2×D244—22no—
48C2×C4×D61—42no—
49C2×Dic122—22no—
50M16×C61—42no—
51C2×C8×C61—22no—
52C2×C8×C61—42no—
53C42×C61C2×D412yes—
54C42×C61C2×D411yescharacteristic
55C42×C61#4111yescharacteristic
56#11758198—42no—
57#11758198—42no—
58C4⋊Dic61—22no—
59C4⋊Dic61—22no—
60#11758194—42no—
61C2×#92418—22no—
62C2×#92418—22no—
63C2×#92418—22no—
64C2×#92418—41no—
65D488—82no—
66C23×C61#12411yescharacteristic
67C22×D61—42no—
68C2×C4×C61—21no—
69C2×C4×C61—21no—
70C2×C4×C61—42no—
71C2×C4×C61#12011yescharacteristic
72C2×C4×C61#12112yes—
73C2×C4×C61#12111yescharacteristic
74C2×C4×C61#12312yes—
75C22⋊D61—84no—
76C2×Dic61—42no—
77D244—42no—
78D244—42no—
79D244—82no—
80D244—82no—
81C4×D61—84no—
82Dic122—82no—
83C8×C61—42no—
84C8×C61—82no—
85#92418—24no—
86#92418—81no—
87#10524—611nomaximal, Sylow
88C22×C61—22no—
89C22×C61—21no—
90C22×C61#39411yescharacteristic
91C2×D61—82no—
92C2×D61—82no—
93C2×D61—82no—
94C4×C61—22no—
95C4×C61—22no—
96C4×C61—21no—
97C4×C61#45311yescharacteristic
98C4×C61#45611yescharacteristic
99Dic61—82no—
100#3253—611no—
101#1849—611no—
102#1557—612no—
103#1530—612no—
104#1307D6111yescharacteristic, complemented
105C2×C61—41no—
106C2×C61#203911yescharacteristic
107C2×C61#204812yes—
108D61—162no—
109#532—612no—
110#476—1222no—
111#469—1222no—
112#459—1222no—
113#432C2×D6112yes—
114C2×#108—1222no—
115#423—611no—
116#423C2×D6111yescharacteristic
117C2×#97—611no—
118#332—1222no—
119#331—1222no—
120#329—1224no—
121C61#1052411yescharacteristic, complemented, Sylow
122C2×#41—1222no—
123C2×D8—1222no—
124C2×M16—1221no—
125C2×M16D24412yes—
126C22×C8—1221no—
127C4⋊Q8—612no—
128#127—612no—
129#124—1222no—
130#121—1222no—
131#121—1222no—
132C4×D4—1222no—
133C4×D4—1222no—
134C2×C42C22×D6111yescharacteristic
135#108—2444no—
136#106—1222no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—1222no—
140#102—1224no—
141C22⋊C8—22no—
142#97—22no—
143#97—612no—
144#97—1221no—
145#41—4884no—
146C2×Q8—1222no—
147C2×D4—1222no—
148C2×D4—1222no—
149C2×D4—1222no—
150C2×D4—2442no—
151C22×C4—2442no—
152C22×C4C2×D24411yescharacteristic
153C22×C4#1175822212yes—
154D8—4882no—
155M16—42no—
156M16—2442no—
157C2×C8—22no—
158C2×C8—42no—
159C2×C8—1222no—
160C2×C8—1222no—
161C2×C8—1222no—
162C2×C8—2441no—
163C4⋊C4—1222no—
164C4⋊C4—1222no—
165C22⋊C4—2442no—
166C22⋊C4—2442no—
167C42D4×D6112yes—
168C42D4×D6111yescharacteristic
169C42#1175822611yescharacteristic
170C23—2442no—
171C23#1451976511yescharacteristic
172Q8—4882no—
173D4—4882no—
174D4—4882no—
175D4—4884no—
176D4—2442no—
177D4—2442no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—4884no—
182C2×C4—2442no—
183C2×C4#1451974711yescharacteristic
184C2×C4#1451975012yes—
185C2×C4#1451975211yescharacteristic
186C2×C4#1451976212yes—
187C8—42no—
188C8—82no—
189C8—1224no—
190C8—4881no—
191C22—21no—
192C22—22no—
193C22—4882no—
194C22—4882no—
195C22—4882no—
196C22#1683968011yescharacteristic
197C4—21no—
198C4—22no—
199C4—22no—
200C4—4882no—
201C4#1683993211yescharacteristic
202C4#1683994711yescharacteristic
203C2—41no—
204C2—9762no—
205C2#2144704411yescharacteristic
206C2#2144709512yes—
subgroupG/H
centerC22#16839680
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1307C2
socleC22×C61#394
prime radical#21457200C2
hypercenter#1307D61
hypocenterC61#10524
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1307

Complements

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

NH ≅ G/Ntype
2. #21445993C2semidirect
3. #21444542C2semidirect
4. #21444431C2semidirect
5. C61×#1307C2semidirect
104. #1307D61semidirect
121. C61#10524semidirect

Maximal subgroups

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

subgroupindex
#214572002
#214459932
#214445422
#214444312
C61×#13072
#1052461