Finite Group Databasebeta

← #14545064

Subgroups — #14545064

Order 2176.

270 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 52 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#11763790C212yesmaximal, complemented
2#11763714C212yesmaximal, complemented
3#11763713C212yesmaximal, complemented
4#11763377C211yesmaximal, characteristic
5C2×#63256C2211yescharacteristic
6C2×D4×D17C411yescharacteristic, complemented
7#10587677—22no—
8C17×#124C411yescharacteristic, complemented
9#10587633C411yescharacteristic, complemented
10#10587626C2211yescharacteristic
11#10587570—22no—
12#10587568—22no—
13#10587560C2211yescharacteristic
14#10587558C411yescharacteristic, complemented
15#10587539—22no—
16#10587539C2214yescomplemented
17C23×D17—21no—
18C2×Q8×C17C2×C411yescharacteristic
19C2×D4×C17C2×C411yescharacteristic
20C2×C22⋊D17—22no—
21C2×C22⋊D17—21no—
22#63256—42no—
23D4×D17—42no—
24C2×D68—22no—
25C2×D68C2×C411yescharacteristic
26C2×D68D412yes—
27C2×D68C2311yescharacteristic
28C2×C4×D17C2×C411yescharacteristic
29C2×C4×D17D412yes—
30C22⋊#595—24no—
31C4⋊#595—42no—
32C4×#595—42no—
33C22⋊C4×C17—22no—
34C42×C17C2×C411yescharacteristic
35#63227—21no—
36#63227—22no—
37#63225—21no—
38C4×Dic17C2×C411yescharacteristic
39C23×C17—21no—
40C22×D17—21no—
41C22×D17—42no—
42C22×D17—82no—
43C22×D17C22⋊C411yescharacteristic
44C22×D17C2×D412yes—
45C2×#595—44no—
46Q8×C17—41no—
47D4×C17—41no—
48C2×C4×C17—21no—
49C2×C4×C17C22⋊C412yes—
50C2×C4×C17C22×C411yescharacteristic
51C22⋊D17—42no—
52C22⋊D17—82no—
53C2×Dic17—21no—
54C2×Dic17C22⋊C411yescharacteristic
55D68—42no—
56D68—41no—
57D68—41no—
58D68—82no—
59C4×D17—22no—
60C4×D17—22no—
61C4×D17—22no—
62#2128—171nomaximal, Sylow
63C22×C17—42no—
64C22×C17—41no—
65C22×C17C2×C22⋊C411yescharacteristic
66C2×D17—22no—
67C2×D17—21no—
68C2×D17—42no—
69C2×D17—41no—
70C2×D17—42no—
71C2×D17—41no—
72C2×D17—84no—
73#595—84no—
74C4×C17—21no—
75C4×C17—22no—
76C4×C17—41no—
77Dic17—41no—
78Dic17#9912yes—
79#535—171no—
80#410—172no—
81#353—172no—
82#353—172no—
83C2×C17—21no—
84C2×C17—81no—
85C2×C17C2×#9911yescharacteristic
86D17—42no—
87D17—41no—
88D17—81no—
89C2×#41—171no—
90C22×D4—171no—
91#127—171no—
92#127—171no—
93#124—171no—
94#124#59511yescharacteristic, complemented
95#121—342no—
96#120—342no—
97#117—171no—
98#117—342no—
99#99—174no—
100#99—342no—
101C17#212811yescharacteristic, complemented, Sylow
102C24—341no—
103#41—682no—
104C2×Q8C2×#59511yescharacteristic
105C2×D4—172no—
106C2×D4—171no—
107C2×D4—171no—
108C2×D4—342no—
109C2×D4—341no—
110C2×D4—342no—
111C2×D4—682no—
112C2×D4C2×#59511yescharacteristic
113C22×C4—172no—
114C22×C4—171no—
115C4⋊C4—341no—
116C4⋊C4—682no—
117C22⋊C4—22no—
118C22⋊C4—341no—
119C22⋊C4—342no—
120C22⋊C4—344no—
121C42—171no—
122C42—682no—
123C42C2×#59511yescharacteristic
124C23—21no—
125C23—172no—
126C23—171no—
127C23—341no—
128C23—682no—
129C23—1362no—
130Q8—41no—
131D4—41no—
132D4—682no—
133D4—681no—
134D4—681no—
135D4—682no—
136D4—1362no—
137D4—1362no—
138C2×C4—21no—
139C2×C4—171no—
140C2×C4—342no—
141C2×C4—342no—
142C2×C4—342no—
143C2×C4—341no—
144C2×C4—684no—
145C2×C4C22⋊#59512yes—
146C2×C4C22×#59511yescharacteristic
147C22—42no—
148C22—41no—
149C22—342no—
150C22—341no—
151C22—682no—
152C22—681no—
153C22—682no—
154C22—681no—
155C22—1364no—
156C22C2×C22⋊#59511yescharacteristic
157C4—22no—
158C4—21no—
159C4—41no—
160C4—172no—
161C4—681no—
162C4—1364no—
163C2—21no—
164C2—81no—
165C2—682no—
166C2—681no—
167C2—1361no—
168C2D17⋊#9911yescharacteristic
subgroupG/H
centerC2D17⋊#99
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#124C4
socleC2×C17C2×#99
prime radical#11763377C2
cosocleC2×D68C23
hypercenter#124#595
hypocenterC17#2128
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #11763790C2semidirect
2. #11763714C2semidirect
3. #11763713C2semidirect
6. C2×D4×D17C4semidirect
8. C17×#124C4semidirect
9. #10587633C4semidirect
14. #10587558C4semidirect
16. #10587539C22semidirect
94. #124#595semidirect
101. C17#2128semidirect

Maximal subgroups

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

subgroupindex
#117637902
#117637142
#117637132
#117633772
#212817