Finite Group Databasebeta

← #14544177

Subgroups — #14544177

Order 2176.

228 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 58 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#11763832C211yesmaximal, characteristic
2#11763745C211yesmaximal, characteristic
3#11763663C212yesmaximal
4#11763662C212yesmaximal
5#11763363C211yesmaximal, characteristic
6C2×C4×#595—21no—
7C4×Q8×C17C411yescharacteristic, complemented
8#10587632C2211yescharacteristic
9#10587630C411yescharacteristic, complemented
10#10587583C2211yescharacteristic
11#10587582—21no—
12#10587575—21no—
13#10587574C411yescharacteristic, complemented
14#10587558—21no—
15C42×D17C2211yescharacteristic
16C4×Dic34—21no—
17C4×Dic34C411yescharacteristic, complemented
18(C2×#595)⋊C4—21no—
19(C2×#595)⋊C4C2212yes—
20#10587544—21no—
21#10587543C2212yes—
22#10587542—21no—
23C22×#595—22no—
24C2×Q8×C17C2×C411yescharacteristic
25C2×C4×D17D411yescharacteristic
26C2×C4×D17D411yescharacteristic
27C2×C4×D17C2311yescharacteristic
28C2×Dic34—21no—
29C2×Dic34C2×C411yescharacteristic
30C2×#3624—22no—
31C4×#595—24no—
32C4⋊C4×C17—21no—
33C4⋊C4×C17C2×C411yescharacteristic
34C42×C17—21no—
35C42×C17C2×C411yescharacteristic
36#63227—21no—
37#63227—21no—
38C4⋊Dic17—21no—
39C4⋊Dic17C2×C411yescharacteristic
40#63225—21no—
41#63225—21no—
42#63225—21no—
43#63225—21no—
44#63225—21no—
45C4×Dic17—21no—
46C4×Dic17C2×C411yescharacteristic
47C4×Dic17D411yescharacteristic
48C4×Dic17D411yescharacteristic
49C22×D17C2×D411yescharacteristic
50C2×#595—24no—
51C2×#595—42no—
52C2×#595—42no—
53Q8×C17—41no—
54C2×C4×C17—21no—
55C2×C4×C17—21no—
56C2×C4×C17C22⋊C411yescharacteristic
57C2×C4×C17C22⋊C411yescharacteristic
58C2×C4×C17C22×C411yescharacteristic
59C2×Dic17—21no—
60C2×Dic17—21no—
61C2×Dic17C22⋊C411yescharacteristic
62C2×Dic17C22⋊C411yescharacteristic
63C2×Dic17C2×D411yescharacteristic
64C4×D17—22no—
65C4×D17—22no—
66C4×D17—41no—
67Dic34—42no—
68Dic34—41no—
69#3624—42no—
70#1506—171nomaximal, Sylow
71C22×C17C2×C22⋊C411yescharacteristic
72C2×D17—21no—
73C2×D17—21no—
74C2×D17#13712yes—
75#595—44no—
76C4×C17—21no—
77C4×C17—21no—
78C4×C17—21no—
79C4×C17—41no—
80C4×C17—41no—
81Dic17—21no—
82Dic17—21no—
83Dic17—41no—
84Dic17—41no—
85Dic17#10412yes—
86#533—171no—
87#432—171no—
88#376—171no—
89#328—172no—
90#328—172no—
91C2×C17#41011yescharacteristic
92C2×C17#41911yescharacteristic
93C2×C17C2×#10411yescharacteristic
94D17—22no—
95C4⋊Q8—171no—
96C4⋊Q8—171no—
97#125—341no—
98C22⋊Q8—171no—
99C22⋊Q8—171no—
100C22⋊Q8—341no—
101C4×Q8—171no—
102C4×Q8—341no—
103C4×Q8#59511yescharacteristic, complemented
104#117—341no—
105C2×C42—171no—
106C2×C42—341no—
107C4⋊C8—341no—
108C22⋊C8—172no—
109#97—341no—
110#95—172no—
111#95—341no—
112C17#150611yescharacteristic, complemented, Sylow
113C2×Q8—171no—
114C2×Q8—341no—
115C2×Q8C2×#59511yescharacteristic
116C22×C4—171no—
117C22×C4—171no—
118C22×C4—171no—
119C22×C4—342no—
120C2×C8—342no—
121C4⋊C4—21no—
122C4⋊C4—171no—
123C4⋊C4—341no—
124C4⋊C4—341no—
125C4⋊C4—341no—
126C4⋊C4—341no—
127C4⋊C4—341no—
128C4⋊C4—341no—
129C4⋊C4C2×#59511yescharacteristic
130C22⋊C4—341no—
131C22⋊C4—341no—
132C42—21no—
133C42—171no—
134C42—171no—
135C42—171no—
136C42—344no—
137C42—341no—
138C42C2×#59511yescharacteristic
139C23—171no—
140Q8—41no—
141Q8—681no—
142Q8—682no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—171no—
146C2×C4—171no—
147C2×C4—171no—
148C2×C4—342no—
149C2×C4—342no—
150C2×C4—344no—
151C2×C4—341no—
152C2×C4—341no—
153C2×C4—681no—
154C2×C4—682no—
155C2×C4—682no—
156C2×C4C22⋊#59511yescharacteristic
157C2×C4C22⋊#59511yescharacteristic
158C2×C4C22×#59511yescharacteristic
159C8—682no—
160C22—172no—
161C22—341no—
162C22—341no—
163C22C2×C22⋊#59511yescharacteristic
164C4—21no—
165C4—21no—
166C4—21no—
167C4—41no—
168C4—41no—
169C4—172no—
170C4—341no—
171C4—341no—
172C4—684no—
173C4—681no—
174C4—681no—
175C2—342no—
176C2#1176378811yescharacteristic
177C2#1176380211yescharacteristic
178C2D17⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC4×Q8×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC22×D17C2×D4
cosocleC2×C4×D17C23
hypercenterC4×Q8#595
hypocenterC17#1506
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
7. C4×Q8×C17C4semidirect
9. #10587630C4semidirect
13. #10587574C4semidirect
17. C4×Dic34C4semidirect
103. C4×Q8#595semidirect
112. C17#1506semidirect

Maximal subgroups

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

subgroupindex
#117638322
#117637452
#117636632
#117636622
#117633632
#150617