Finite Group Databasebeta

← #14544204

Subgroups — #14544204

Order 2176.

252 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 56 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, complemented
2#11763754C211yesmaximal, characteristic, complemented
3#11763660C214yesmaximal, complemented
4#11763357C211yesmaximal, characteristic
5C2×#63254—21no—
6C17×#127C411yescharacteristic, complemented
7#10587624—21no—
8#10587624C2212yes—
9#10587622—21no—
10D4×Dic17—21no—
11D4×Dic17C412yescomplemented
12C42×D17C2211yescharacteristic
13#10587555C411yescharacteristic, complemented
14(C2×#595)⋊C4—21no—
15(C2×#595)⋊C4—21no—
16(C2×#595)⋊C4C2212yes—
17#10587544—21no—
18#10587543C2212yes—
19#10587542—21no—
20C22×#595—22no—
21C2×D4×C17—21no—
22C2×D4×C17—21no—
23C2×D4×C17C2×C412yes—
24C2×C22⋊D17—22no—
25C22×Dic17—22no—
26#63254—44no—
27C2×C4×D17D411yescharacteristic
28C2×C4×D17D411yescharacteristic
29C2×C4×D17C2311yescharacteristic
30C2×Dic34—21no—
31C2×#3624—22no—
32C42×C17C2×C411yescharacteristic
33C22⋊Dic17—22no—
34C22⋊Dic17—22no—
35C4⋊Dic17—21no—
36C4⋊Dic17C2×C412yes—
37C4×Dic17C2×C411yescharacteristic
38C4×Dic17D411yescharacteristic
39C4×Dic17D411yescharacteristic
40C23×C17—22no—
41C22×D17C2×D411yescharacteristic
42C2×#595—42no—
43C2×#595—42no—
44C2×#595—42no—
45D4×C17—42no—
46D4×C17—42no—
47D4×C17—42no—
48C2×C4×C17C22⋊C411yescharacteristic
49C2×C4×C17C22⋊C411yescharacteristic
50C2×C4×C17C22×C411yescharacteristic
51C22⋊D17—44no—
52C2×Dic17—22no—
53C2×Dic17—44no—
54C2×Dic17C22⋊C411yescharacteristic
55C2×Dic17C22⋊C411yescharacteristic
56C2×Dic17C2×D411yescharacteristic
57C4×D17—22no—
58C4×D17—41no—
59C4×D17—41no—
60Dic34—42no—
61#3624—42no—
62#1514—171nomaximal, Sylow
63C22×C17—42no—
64C22×C17—42no—
65C22×C17—42no—
66C22×C17C2×C22⋊C411yescharacteristic
67C2×D17—21no—
68C2×D17—21no—
69C2×D17AGL(1,ℤ/8)12yes—
70#595—82no—
71C4×C17—21no—
72C4×C17—21no—
73C4×C17—21no—
74Dic17—21no—
75Dic17—21no—
76Dic17—21no—
77Dic17—42no—
78#532—171no—
79#432—171no—
80#383—171no—
81#327—174no—
82C2×C17—82no—
83C2×C17#41011yescharacteristic
84C2×C17#41811yescharacteristic
85C2×C17#42111yescharacteristic
86D17—22no—
87C2×#41—341no—
88C4⋊Q8—171no—
89#127#59511yescharacteristic, complemented
90#124—341no—
91#121—172no—
92#121—341no—
93C4×D4—172no—
94C4×D4—341no—
95C2×C42—171no—
96C4⋊C8—341no—
97C22⋊C8—172no—
98#97—341no—
99#95—172no—
100#95—341no—
101#95—341no—
102C17#151411yescharacteristic, complemented, Sylow
103#41—684no—
104C2×Q8—341no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—342no—
108C2×D4C2×#59512yes—
109C22×C4—171no—
110C22×C4—171no—
111C22×C4—171no—
112C22×C4—342no—
113C22×C4—342no—
114C2×C8—342no—
115C4⋊C4—172no—
116C4⋊C4—341no—
117C22⋊C4—342no—
118C22⋊C4—342no—
119C42—171no—
120C42—171no—
121C42—171no—
122C42C2×#59511yescharacteristic
123C23—22no—
124C23—171no—
125Q8—682no—
126D4—42no—
127D4—42no—
128D4—42no—
129D4—684no—
130C2×C4—171no—
131C2×C4—171no—
132C2×C4—171no—
133C2×C4—342no—
134C2×C4—342no—
135C2×C4—681no—
136C2×C4—681no—
137C2×C4—684no—
138C2×C4—682no—
139C2×C4—682no—
140C2×C4—682no—
141C2×C4C22⋊#59511yescharacteristic
142C2×C4C22⋊#59511yescharacteristic
143C2×C4C22×#59511yescharacteristic
144C8—682no—
145C22—42no—
146C22—42no—
147C22—42no—
148C22—172no—
149C22—341no—
150C22—341no—
151C22C2×C22⋊#59511yescharacteristic
152C4—21no—
153C4—21no—
154C4—21no—
155C4—341no—
156C4—341no—
157C4—341no—
158C4—682no—
159C4—1362no—
160C2—82no—
161C2—342no—
162C2#1176378911yescharacteristic
163C2#1176380111yescharacteristic
164C2#1176380711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#127C4
socleC22×C17C2×C22⋊C4
prime radical#11763357C2
cosocleC2×C4×D17C23
hypercenter#127#595
hypocenterC17#1514
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #11763832C2semidirect
2. #11763754C2semidirect
3. #11763660C2semidirect
6. C17×#127C4semidirect
11. D4×Dic17C4semidirect
13. #10587555C4semidirect
89. #127#595semidirect
102. C17#1514semidirect

Maximal subgroups

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

subgroupindex
#117638322
#117637542
#117636602
#117633572
#151417