Finite Group Databasebeta

← #29574396

Subgroups — #29574396

Order 15616.

396 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 111 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#21449005C211yesmaximal, characteristic
2#21445954C212yesmaximal
3C2×#16839504C211yesmaximal, characteristic
4#21443701C212yesmaximal
5C61×#1992C211yesmaximal, characteristic
6C2×#14519688C2212yes—
7#16839661—22no—
8#16839648—22no—
9#16839648—22no—
10#16839628C2212yes—
11#16839524—22no—
12#16839524—22no—
13#16839504C414yescomplemented
14C2×#125×C61C2211yescharacteristic
15C2×#97×C61C2211yescharacteristic
16C61×#384C2211yescharacteristic
17C61×#340—22no—
18C22×#92418—22no—
19C2×C4⋊Dic61—22no—
20C2×C4⋊Dic61D412yes—
21C2×C4⋊Dic61D412yes—
22C2×C4×Dic61—22no—
23#14519688—22no—
24#14519688C2×C414yes—
25#14519684—22no—
26#14519684—22no—
27#14519684—22no—
28#14519684—22no—
29C22×C8×C61—21no—
30#125×C61—22no—
31#125×C61C2×C412yes—
32#125×C61Q812yes—
33C2×C4⋊C4×C61—22no—
34C2×C4⋊C4×C61—21no—
35C2×C4⋊C4×C61—22no—
36C2×C4⋊C4×C61D412yes—
37C2×C42×C61C2311yescharacteristic
38#97×C61—22no—
39C22×C4×C61—22no—
40C22×C4×C61—41no—
41C22×C4×C61C2×D411yescharacteristic
42C22×C4×C61C2×D412yes—
43C22×Dic61—22no—
44C22×Dic61—22no—
45C2×C8×C61—22no—
46C2×C8×C61—41no—
47C2×C8×C61—41no—
48C4⋊C4×C61—24no—
49C4⋊C4×C61—24no—
50C4⋊C4×C61—22no—
51C4⋊C4×C61—44no—
52C4⋊C4×C61—42no—
53C4⋊C4×C61C4⋊C414yes—
54C42×C61C22×C411yescharacteristic
55C42×C61C2×Q811yescharacteristic
56C42×C61#4112yes—
57C4⋊Dic61—42no—
58C4⋊Dic61—42no—
59C4⋊Dic61—82no—
60C4×Dic61—82no—
61C2×#92418—24no—
62C2×#92418—42no—
63C2×#92418—42no—
64C23×C61#12011yescharacteristic
65C2×C4×C61—22no—
66C2×C4×C61—21no—
67C2×C4×C61—22no—
68C2×C4×C61—21no—
69C2×C4×C61—24no—
70C2×C4×C61—42no—
71C2×C4×C61—42no—
72C2×C4×C61—82no—
73C2×C4×C61—81no—
74C2×C4×C61C2×C4⋊C411yescharacteristic
75C2×C4×C61C4×D412yes—
76C2×C4×C61C22⋊Q812yes—
77C2×C4×C61#12311yescharacteristic
78C2×Dic61—42no—
79C2×Dic61—42no—
80C2×Dic61—42no—
81C2×Dic61—42no—
82C2×Dic61—42no—
83C2×Dic61—42no—
84C8×C61—42no—
85#92418—44no—
86#8915—611nomaximal, Sylow
87C22×C61#38511yescharacteristic
88C22×C61#45412yes—
89C22×C61#45411yescharacteristic
90C22×C61#45512yes—
91C22×C61#45611yescharacteristic
92C4×C61—21no—
93C4×C61—21no—
94C4×C61—22no—
95C4×C61—22no—
96C4×C61—44no—
97C4×C61—161no—
98Dic61—82no—
99Dic61—82no—
100#2382—611no—
101#1992D6111yescharacteristic
102#1836—612no—
103C2×#330—611no—
104#1295—612no—
105C2×C61#165811yescharacteristic
106C2×C61#166211yescharacteristic
107C2×C61#166412yes—
108C2×C61#185711yescharacteristic
109C2×C61#186012yes—
110C2×#125C2×D6111yescharacteristic
111C2×#97—612no—
112C2×#97C2×D6111yescharacteristic
113#389—1222no—
114#387—1222no—
115#387—1222no—
116#384C2×D6111yescharacteristic
117#382—612no—
118#340—22no—
119#340—1222no—
120#340—1222no—
121#330—614no—
122C61#891511yescharacteristic, complemented, Sylow
123C22×C8—21no—
124C22×C8—1222no—
125#125—22no—
126#125Dic12212yes—
127#125C4×D6112yes—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—612no—
132C2×C4⋊C4—612no—
133C2×C4⋊C4—1222no—
134C2×C4⋊C4C22⋊D6112yes—
135C2×C42—1222no—
136C2×C42C22×D6111yescharacteristic
137#97—22no—
138#97—614no—
139#97—1222no—
140#95—1222no—
141#95—1222no—
142#95—1222no—
143#95—1222no—
144C22×C4—22no—
145C22×C4—41no—
146C22×C4—1222no—
147C22×C4—1222no—
148C22×C4D4×D6112yes—
149C22×C4C2×C22⋊D6111yescharacteristic
150C2×C8—22no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—1224no—
154C2×C8—2442no—
155C2×C8—2442no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—22no—
159C4⋊C4—44no—
160C4⋊C4—42no—
161C4⋊C4—4882no—
162C4⋊C4—2442no—
163C4⋊C4—2442no—
164C4⋊C4#1175819614yes—
165C42—4882no—
166C42C2×Dic12211yescharacteristic
167C42C2×C4×D6111yescharacteristic
168C42#1175822412yes—
169C23#1451974511yescharacteristic
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—82no—
178C2×C4—81no—
179C2×C4—2442no—
180C2×C4—2442no—
181C2×C4—2442no—
182C2×C4—2442no—
183C2×C4—2442no—
184C2×C4—2442no—
185C2×C4C2×#1175819611yescharacteristic
186C2×C4#1451973712yes—
187C2×C4#1451975412yes—
188C2×C4#1451975811yescharacteristic
189C8—42no—
190C8—2444no—
191C22#1683963411yescharacteristic
192C22#1683993312yes—
193C22#1683993411yescharacteristic
194C22#1683993812yes—
195C22#1683994411yescharacteristic
196C4—21no—
197C4—21no—
198C4—22no—
199C4—22no—
200C4—44no—
201C4—161no—
202C4—4882no—
203C4—4882no—
204C2#2144512011yescharacteristic
205C2#2144514011yescharacteristic
206C2#2144515112yes—
207C2#2144603311yescharacteristic
208C2#2144605012yes—
subgroupG/H
centerC23#14519745
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1992C2
socle, prime radicalC23×C61#120
cosocleC2×C42×C61C23
hypercenter#1992D61
hypocenterC61#8915
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1992

Complements

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

NH ≅ G/Ntype
13. #16839504C4semidirect
122. C61#8915semidirect

Maximal subgroups

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

subgroupindex
#214490052
#214459542
C2×#168395042
#214437012
C61×#19922
#891561