Finite Group Databasebeta

← #29577615

Subgroups — #29577615

Order 15616.

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

normal subgroups only

This group has 400 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#21456201C211yesmaximal, characteristic
2#21454921C211yesmaximal, characteristic
3#21444505C212yesmaximal
4#21443708C212yesmaximal
5C61×#1561C211yesmaximal, characteristic
6C2×#14519777C2211yescharacteristic
7#16840132C414yes—
8#16840114—22no—
9#16839893—22no—
10Q8×#92418—22no—
11#16839823—22no—
12#16839815C2212yes—
13#16839770—22no—
14C4×#11758194C2211yescharacteristic
15#16839507—22no—
16#16839507—22no—
17C61×#533C2211yescharacteristic
18C61×#423C2212yes—
19C61×#332—22no—
20C22×Dic122—21no—
21C2×#11758194—21no—
22#14519777—22no—
23#14519777—21no—
24#14519777C2×C412yes—
25C2×C4⋊Dic61—22no—
26C4⋊#92418—24no—
27C4⋊#92418D412yes—
28C4⋊#92418D412yes—
29#14519700—28no—
30C22⋊#92418—22no—
31#14519688—22no—
32#14519688C2×C412yes—
33C4×#92418—22no—
34C4×#92418—21no—
35C4×#92418—22no—
36C2×M16×C61—22no—
37C4⋊Q8×C61D412yes—
38#125×C61C2×C412yes—
39C22⋊Q8×C61—22no—
40C4×Q8×C61—22no—
41#117×C61D412yes—
42C2×C42×C61C2311yescharacteristic
43C4⋊C8×C61—22no—
44C4⋊C8×C61—22no—
45C2×Q8×C61—22no—
46C22×C4×C61C2×D411yescharacteristic
47C22×C4×C61#4112yes—
48C22×Dic61—41no—
49C2×Dic122—22no—
50C2×Dic122—44no—
51C2×Dic122—41no—
52M16×C61—82no—
53C2×C8×C61—42no—
54C4⋊C4×C61—24no—
55C4⋊C4×C61—22no—
56C4⋊C4×C61C22⋊C414yes—
57C22⋊C4×C61—22no—
58C42×C61—22no—
59C42×C61C22×C411yescharacteristic
60C42×C61C2×D411yescharacteristic
61C42×C61C2×D412yes—
62C4⋊Dic61—24no—
63C4⋊Dic61—42no—
64#11758194—42no—
65C2×#92418—22no—
66C2×#92418—22no—
67C2×#92418—22no—
68C23×C61#12311yescharacteristic
69Q8×C61—42no—
70Q8×C61—42no—
71C2×C4×C61—22no—
72C2×C4×C61—22no—
73C2×C4×C61—22no—
74C2×C4×C61—24no—
75C2×C4×C61C2×C22⋊C411yescharacteristic
76C2×C4×C61C4×D412yes—
77C2×C4×C61#12011yescharacteristic
78C2×C4×C61#12112yes—
79C2×Dic61—42no—
80C2×Dic61—82no—
81Dic122—44no—
82Dic122—82no—
83C8×C61—82no—
84#92418—42no—
85#92418—42no—
86#92418—42no—
87#9436—611nomaximal, Sylow
88C22×C61—21no—
89C22×C61—22no—
90C22×C61#38611yescharacteristic
91C4×C61—21no—
92C4×C61—22no—
93C4×C61—44no—
94C4×C61#45412yes—
95C4×C61#47112yes—
96C4×C61#47312yes—
97Dic61—82no—
98#3139—611no—
99#2986—611no—
100#1561D6111yescharacteristic
101#1549—612no—
102#1299—612no—
103C2×C61—41no—
104C2×C61#188811yescharacteristic
105C2×C61#190812yes—
106#533C2×D6111yescharacteristic
107C2×C4⋊Q8—611no—
108#491—614no—
109#487—1222no—
110#446—1222no—
111C8×Q8—1222no—
112#433—1222no—
113#431—612no—
114#423C2×D6112yes—
115#419—1222no—
116C4×M16—611no—
117#332—22no—
118#332—1222no—
119#332—1222no—
120C61#943611yescharacteristic, complemented, Sylow
121C22×Q8—1221no—
122C2×M16—22no—
123C2×M16—1221no—
124C4⋊Q8—612no—
125C4⋊Q8—1222no—
126C4⋊Q8—1221no—
127C4⋊Q8C22⋊D6112yes—
128#125C4×D6112yes—
129C22⋊Q8—22no—
130C4×Q8—22no—
131#117D24412yes—
132C2×C4⋊C4—1222no—
133C2×C42C22×D6111yescharacteristic
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—612no—
137C4⋊C8—612no—
138C4⋊C8—1224no—
139#103—1228no—
140C22⋊C8—1222no—
141#97—612no—
142#97—1222no—
143C4×C8—1222no—
144C4×C8—1222no—
145C4×C8—1221no—
146C2×Q8—22no—
147C2×Q8—1222no—
148C2×Q8—2444no—
149C2×Q8—2441no—
150C22×C4—2441no—
151C22×C4C2×D24411yescharacteristic
152C22×C4#1175822412yes—
153M16—82no—
154M16—2442no—
155C2×C8—42no—
156C2×C8—1222no—
157C2×C8—1222no—
158C2×C8—1222no—
159C4⋊C4—24no—
160C4⋊C4—22no—
161C4⋊C4—1224no—
162C4⋊C4—2442no—
163C4⋊C4#1175819814yes—
164C22⋊C4—22no—
165C42—22no—
166C42C2×C4×D6111yescharacteristic
167C42D4×D6112yes—
168C42C2×C22⋊D6111yescharacteristic
169C23#1451976111yescharacteristic
170Q8—42no—
171Q8—42no—
172Q8—4882no—
173Q8—2444no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—4882no—
179C2×C4—2442no—
180C2×C4C2×#1175819811yescharacteristic
181C2×C4#1451973712yes—
182C2×C4#1451974711yescharacteristic
183C2×C4#1451974912yes—
184C8—82no—
185C8—2442no—
186C8—2442no—
187C8—2442no—
188C22—21no—
189C22—22no—
190C22#1683963911yescharacteristic
191C4—21no—
192C4—22no—
193C4—44no—
194C4—4882no—
195C4#1683993712yes—
196C4#1684002312yes—
197C4#1684003512yes—
198C2—41no—
199C2#2144622011yescharacteristic
200C2#2144633212yes—
subgroupG/H
centerC22#16839639
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1561C2
socleC22×C61#386
prime radicalC23×C61#123
cosocleC2×C42×C61C23
hypercenter#1561D61
hypocenterC61#9436
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1561

Complements

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

NH ≅ G/Ntype
120. C61#9436semidirect

Maximal subgroups

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

subgroupindex
#214562012
#214549212
#214445052
#214437082
C61×#15612
#943661