Finite Group Databasebeta

← #14527921

Subgroups — #14527921

Order 2176.

214 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 55 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#11763519C211yesmaximal, characteristic
2#11763189C211yesmaximal, characteristic, complemented
3#11763104C211yesmaximal, characteristic, complemented
4#11763100C211yesmaximal, characteristic
5#11762557C211yesmaximal, characteristic, complemented
6#11762538C211yesmaximal, characteristic
7C17×Q8⋊C8C211yesmaximal, characteristic, complemented
8C4×Q8×C17C2211yescharacteristic
9#10587632—21no—
10C2×Dic68—22no—
11#10587582—22no—
12C4⋊C4×D17—21no—
13C4×D68—21no—
14C4×D68C2211yescharacteristic
15#10587556—21no—
16#10587555C2211yescharacteristic
17#10587555C2211yescharacteristic
18C17×C4⋊C8C2211yescharacteristic
19C4×C8×C17C2211yescharacteristic
20#10587506—21no—
21#10587492—21no—
22#10587488—21no—
23#10587478—21no—
24C4⋊#3622C2211yescharacteristic
25C2×Q8×C17D411yescharacteristic
26C2×D68D411yescharacteristic
27C2×C4×D17—21no—
28C2×C4×D17—22no—
29C2×Dic34—21no—
30C2×Dic34D412yes—
31C2×C8×C17—21no—
32C2×C8×C17—21no—
33C4⋊C4×C17—21no—
34C4⋊C4×C17D411yescharacteristic
35C42×C17—21no—
36C42×C17C2311yescharacteristic
37#63227—22no—
38#63227—21no—
39C4⋊Dic17—21no—
40C4⋊Dic17—21no—
41C4⋊Dic17—21no—
42C4⋊Dic17—21no—
43C4⋊Dic17D411yescharacteristic
44#63225—22no—
45#63225—22no—
46C2×#3622—21no—
47Dic68—44no—
48C22×D17—21no—
49Q8×C17—21no—
50Q8×C17—21no—
51C2×C4×C17—21no—
52C2×C4×C17—21no—
53C2×C4×C17C2×D411yescharacteristic
54C2×C4×C17C2×D411yescharacteristic
55C2×C4×C17C2×D411yescharacteristic
56C2×Dic17—22no—
57C2×Dic17—21no—
58C2×Dic17—41no—
59D68—21no—
60D68Q1612yescomplemented
61C4×D17—41no—
62C4×D17—41no—
63C4×D17—82no—
64Dic34—42no—
65Dic34—42no—
66C8×C17—22no—
67C8×C17—41no—
68#3622—41no—
69#1632—171nomaximal, Sylow
70C22×C17#12011yescharacteristic
71C2×D17—22no—
72C2×D17—41no—
73C2×D17—41no—
74C4×C17—21no—
75C4×C17—41no—
76C4×C17—41no—
77C4×C17C2×Q1611yescharacteristic
78C4×C17#13511yescharacteristic
79C4×C17#13711yescharacteristic
80C4×C17#13711yescharacteristic
81Dic17—41no—
82Dic17—42no—
83Dic17—81no—
84#554—171no—
85#494—171no—
86#477—171no—
87#476—171no—
88#333—171no—
89Q8⋊C8D1711yescharacteristic, complemented
90D4⋊C8—171no—
91C2×C17#45111yescharacteristic
92C2×C17#45211yescharacteristic
93C2×C17#45311yescharacteristic
94D17—42no—
95C2×Q16—342no—
96C4⋊Q8—171no—
97C4⋊Q8—171no—
98#125—341no—
99C22⋊Q8—342no—
100C22⋊Q8—341no—
101C4×Q8C2×D1711yescharacteristic
102C4×D4—171no—
103C4×D4—341no—
104C2×C4⋊C4—341no—
105#107—341no—
106#106—341no—
107C4⋊C8—171no—
108C4⋊C8C2×D1711yescharacteristic
109#103—341no—
110#102—341no—
111C4×C8C2×D1711yescharacteristic
112C17#163211yescharacteristic, complemented, Sylow
113C2×Q8—172no—
114C2×Q8—341no—
115C2×Q8D6811yescharacteristic
116C2×D4—171no—
117C22×C4—342no—
118C22×C4—341no—
119Q16—684no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—341no—
123C4⋊C4—21no—
124C4⋊C4—171no—
125C4⋊C4—342no—
126C4⋊C4—342no—
127C4⋊C4—341no—
128C4⋊C4—341no—
129C4⋊C4—341no—
130C4⋊C4—341no—
131C4⋊C4D6811yescharacteristic
132C22⋊C4—342no—
133C22⋊C4—341no—
134C42—21no—
135C42C22×D1711yescharacteristic
136C23—341no—
137Q8—21no—
138Q8—21no—
139Q8—682no—
140Q8—682no—
141D4—172no—
142D4—341no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—341no—
146C2×C4—342no—
147C2×C4—681no—
148C2×C4—681no—
149C2×C4—681no—
150C2×C4—1362no—
151C2×C4C2×D6811yescharacteristic
152C2×C4D4×D1711yescharacteristic
153C2×C4D4×D1711yescharacteristic
154C8—22no—
155C8—41no—
156C8—681no—
157C22—342no—
158C22—681no—
159C22—681no—
160C22#1058756811yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—681no—
165C4—682no—
166C4—1361no—
167C4#1058759011yescharacteristic
168C4#1058759511yescharacteristic
169C4#1058760111yescharacteristic
170C4Q16×D1711yescharacteristic
171C2—682no—
172C2#1176296611yescharacteristic
173C2#1176297311yescharacteristic
174C2#1176297811yescharacteristic
subgroupG/H
centerC22#10587568
commutator, cosocleC42×C17C23
FrattiniC42C22×D17
FittingC17×Q8⋊C8C2
socleC22×C17#120
prime radicalC2×D68D4
hypercenterQ8⋊C8D17
hypocenterC17#1632
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #11763189C2semidirect
3. #11763104C2semidirect
5. #11762557C2semidirect
7. C17×Q8⋊C8C2semidirect
60. D68Q16semidirect
89. Q8⋊C8D17semidirect
112. C17#1632semidirect

Maximal subgroups

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

subgroupindex
#117635192
#117631892
#117631042
#117631002
#117625572
#117625382
C17×Q8⋊C82
#163217