Finite Group Databasebeta

← #14528012

Subgroups — #14528012

Order 2176.

212 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 57 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#11763159C211yesmaximal, characteristic, complemented
2#11763089C211yesmaximal, characteristic, complemented
3#11763014C211yesmaximal, characteristic, complemented
4#11762558C211yesmaximal, characteristic, complemented
5D4⋊#3622C211yesmaximal, characteristic, complemented
6D4⋊#3622C211yesmaximal, characteristic, complemented
7C17×#555C211yesmaximal, characteristic
8C17×C4⋊Q8C2211yescharacteristic
9C17×#124—22no—
10C17×#121—21no—
11C17×#121—21no—
12C4×D4×C17C2211yescharacteristic
13C4×D4×C17C2211yescharacteristic
14C17×#117—21no—
15C17×#117—21no—
16C2×#63229—21no—
17#10587555C2211yescharacteristic
18#10587506—22no—
19#10587503—21no—
20#10587481—21no—
21#10587478—21no—
22C4⋊#3622C2211yescharacteristic
23C4⋊#3622C2211yescharacteristic
24C4×#3622C2211yescharacteristic
25C2×Q8×C17D412yes—
26C2×D4×C17—22no—
27C2×D4×C17D411yescharacteristic
28C2×D4×C17D411yescharacteristic
29C22×C4×C17—21no—
30C22×C4×C17—21no—
31C2×Dic34—21no—
32C4⋊C4×C17—21no—
33C4⋊C4×C17—21no—
34C4⋊C4×C17D411yescharacteristic
35C4⋊C4×C17D411yescharacteristic
36C22⋊C4×C17—22no—
37C22⋊C4×C17—22no—
38C22⋊C4×C17—21no—
39C22⋊C4×C17—21no—
40C42×C17—22no—
41C42×C17C2311yescharacteristic
42#63229—42no—
43C4⋊Dic17—21no—
44C4⋊Dic17—21no—
45C2×#3622—21no—
46C2×#3622—21no—
47C2×#3622—21no—
48C23×C17—21no—
49C23×C17—21no—
50Q8×C17—42no—
51D4×C17—21no—
52D4×C17—21no—
53D4×C17—21no—
54D4×C17—21no—
55D4×C17—82no—
56C2×C4×C17—21no—
57C2×C4×C17—21no—
58C2×C4×C17—22no—
59C2×C4×C17—41no—
60C2×C4×C17—41no—
61C2×C4×C17—41no—
62C2×C4×C17—41no—
63C2×C4×C17C2×D411yescharacteristic
64C2×C4×C17C2×D411yescharacteristic
65C2×C4×C17C2×D411yescharacteristic
66C2×Dic17—41no—
67Dic34—41no—
68Dic34—41no—
69#3622—41no—
70#3622—41no—
71#3622—41no—
72#1646—171nomaximal, Sylow
73C22×C17—41no—
74C22×C17—41no—
75C22×C17—41no—
76C22×C17—41no—
77C22×C17—41no—
78C22×C17—41no—
79C22×C17#12011yescharacteristic
80C4×C17—21no—
81C4×C17—41no—
82C4×C17—41no—
83C4×C17—42no—
84C4×C17#13511yescharacteristic
85C4×C17#13511yescharacteristic
86C4×C17AGL(1,ℤ/8)11yescharacteristic
87C4×C17#13711yescharacteristic
88Dic17—81no—
89#555D1711yescharacteristic
90#487—171no—
91#474—171no—
92#461—171no—
93#333—171no—
94D4⋊C8—171no—
95D4⋊C8—171no—
96C2×C17—81no—
97C2×C17—81no—
98C2×C17#44911yescharacteristic
99C2×C17#45211yescharacteristic
100C2×C17#45511yescharacteristic
101C2×SD16—341no—
102C4⋊Q8—171no—
103C4⋊Q8C2×D1711yescharacteristic
104#124—22no—
105#121—21no—
106#121—21no—
107C4×D4C2×D1711yescharacteristic
108C4×D4C2×D1711yescharacteristic
109#117—21no—
110#117—21no—
111#107—341no—
112C4⋊C8—171no—
113C4⋊C8—171no—
114#103—342no—
115#103—341no—
116#102—341no—
117C4×C8—171no—
118C17#164611yescharacteristic, complemented, Sylow
119C2×Q8—341no—
120C2×Q8C22⋊D1712yes—
121C2×D4—22no—
122C2×D4C22⋊D1711yescharacteristic
123C2×D4C22⋊D1711yescharacteristic
124C22×C4—21no—
125C22×C4—21no—
126SD16—682no—
127C2×C8—341no—
128C2×C8—341no—
129C2×C8—341no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—341no—
133C4⋊C4—341no—
134C4⋊C4C22⋊D1711yescharacteristic
135C4⋊C4C22⋊D1711yescharacteristic
136C22⋊C4—22no—
137C22⋊C4—22no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C42—22no—
141C42C22×D1711yescharacteristic
142C23—21no—
143C23—21no—
144Q8—42no—
145Q8—681no—
146Q8—681no—
147D4—21no—
148D4—21no—
149D4—21no—
150D4—21no—
151D4—82no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—681no—
160C2×C4C2×C22⋊D1711yescharacteristic
161C2×C4C2×C22⋊D1711yescharacteristic
162C2×C4C2×C22⋊D1711yescharacteristic
163C8—681no—
164C8—681no—
165C8—681no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22#1058763911yescharacteristic
173C4—21no—
174C4—41no—
175C4—41no—
176C4—42no—
177C4—1361no—
178C4#1058763511yescharacteristic
179C4#1058763611yescharacteristic
180C4#1058763611yescharacteristic
181C4#1058763711yescharacteristic
182C2—81no—
183C2—81no—
184C2#1176295411yescharacteristic
185C2#1176297111yescharacteristic
186C2#1176298711yescharacteristic
subgroupG/H
centerC22#10587639
commutator, cosocleC42×C17C23
FrattiniC42C22×D17
Fitting, prime radicalC17×#555C2
socleC22×C17#120
hypercenter#555D17
hypocenterC17#1646
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #11763159C2semidirect
2. #11763089C2semidirect
3. #11763014C2semidirect
4. #11762558C2semidirect
5. D4⋊#3622C2semidirect
6. D4⋊#3622C2semidirect
118. C17#1646semidirect

Maximal subgroups

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

subgroupindex
#117631592
#117630892
#117630142
#117625582
D4⋊#36222
C17×#5552
#164617