Finite Group Databasebeta

← #14545067

Subgroups — #14545067

Order 2176.

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

normal subgroups only

This group has 238 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
2#11763716C212yesmaximal, complemented
3#11763715C212yesmaximal, complemented
4#11763384C211yesmaximal, characteristic
5C2×#63256C2211yescharacteristic
6C2×Q8×D17C411yescharacteristic, complemented
7#10587677—22no—
8C17×C4⋊Q8C411yescharacteristic, complemented
9#10587633C2211yescharacteristic
10#10587630C411yescharacteristic, complemented
11#10587582—22no—
12#10587580—22no—
13#10587561C2211yescharacteristic
14#10587558C411yescharacteristic, complemented
15#10587539—22no—
16#10587539C2214yes—
17C2×Q8×C17C2×C411yescharacteristic
18C2×Q8×C17C2×C411yescharacteristic
19#63256—42no—
20Q8×D17—42no—
21C2×D68D412yes—
22C2×D68C2311yescharacteristic
23C2×C4×D17—21no—
24C2×C4×D17C2×C411yescharacteristic
25C2×C4×D17D412yes—
26C2×Dic34—21no—
27C2×Dic34C2×C411yescharacteristic
28C22⋊#595—24no—
29C4⋊#595—42no—
30C4×#595—42no—
31C4⋊C4×C17—22no—
32C42×C17C2×C411yescharacteristic
33#63227—21no—
34#63227—22no—
35#63227—22no—
36#63225—22no—
37#63225—21no—
38C4×Dic17C2×C411yescharacteristic
39C22×D17C22⋊C411yescharacteristic
40C22×D17C2×D412yes—
41C2×#595—44no—
42Q8×C17—41no—
43Q8×C17—41no—
44C2×C4×C17—21no—
45C2×C4×C17—21no—
46C2×C4×C17C22⋊C412yes—
47C2×C4×C17C22×C411yescharacteristic
48C2×Dic17—21no—
49C2×Dic17—21no—
50C2×Dic17C22⋊C411yescharacteristic
51D68—42no—
52D68—41no—
53C4×D17—22no—
54C4×D17—22no—
55C4×D17—22no—
56C4×D17—42no—
57Dic34—42no—
58Dic34—41no—
59#2129—171nomaximal, Sylow
60C22×C17C2×C22⋊C411yescharacteristic
61C2×D17—21no—
62C2×D17—22no—
63C2×D17—41no—
64C2×D17—42no—
65#595—84no—
66C4×C17—21no—
67C4×C17—22no—
68C4×C17—41no—
69C4×C17—41no—
70Dic17—41no—
71Dic17—41no—
72Dic17#9912yes—
73#536—171no—
74#410—172no—
75#354—172no—
76#354—172no—
77C2×C17—21no—
78C2×C17C2×#9911yescharacteristic
79D17—42no—
80D17—41no—
81C2×#41—171no—
82C22×Q8—171no—
83C4⋊Q8—171no—
84C4⋊Q8#59511yescharacteristic, complemented
85#124—171no—
86#124—171no—
87#123—342no—
88C22⋊Q8—342no—
89#117—171no—
90#117—342no—
91#99—174no—
92#99—342no—
93C17#212911yescharacteristic, complemented, Sylow
94#41—682no—
95C2×Q8—171no—
96C2×Q8—341no—
97C2×Q8—682no—
98C2×Q8C2×#59511yescharacteristic
99C2×Q8C2×#59511yescharacteristic
100C2×D4—172no—
101C2×D4—171no—
102C22×C4—172no—
103C22×C4—171no—
104C22×C4—341no—
105C4⋊C4—22no—
106C4⋊C4—341no—
107C4⋊C4—342no—
108C4⋊C4—682no—
109C22⋊C4—341no—
110C22⋊C4—342no—
111C22⋊C4—342no—
112C22⋊C4—344no—
113C42—171no—
114C42—682no—
115C42C2×#59511yescharacteristic
116C23—172no—
117C23—171no—
118Q8—41no—
119Q8—41no—
120Q8—681no—
121Q8—682no—
122D4—682no—
123D4—681no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—171no—
127C2×C4—342no—
128C2×C4—342no—
129C2×C4—342no—
130C2×C4—341no—
131C2×C4—341no—
132C2×C4—682no—
133C2×C4—684no—
134C2×C4C22⋊#59512yes—
135C2×C4C22×#59511yescharacteristic
136C22—342no—
137C22—341no—
138C22—682no—
139C22—681no—
140C22C2×C22⋊#59511yescharacteristic
141C4—22no—
142C4—21no—
143C4—41no—
144C4—41no—
145C4—172no—
146C4—681no—
147C4—681no—
148C4—1364no—
149C2—21no—
150C2—682no—
151C2—681no—
152C2D17⋊#9911yescharacteristic
subgroupG/H
centerC2D17⋊#99
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×C4⋊Q8C4
socleC2×C17C2×#99
prime radicalC2×#63256C22
cosocleC2×D68C23
hypercenterC4⋊Q8#595
hypocenterC17#2129
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #11763716C2semidirect
3. #11763715C2semidirect
6. C2×Q8×D17C4semidirect
8. C17×C4⋊Q8C4semidirect
10. #10587630C4semidirect
14. #10587558C4semidirect
84. C4⋊Q8#595semidirect
93. C17#2129semidirect

Maximal subgroups

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

subgroupindex
#117637902
#117637162
#117637152
#117633842
#212917