Finite Group Databasebeta

← #14527804

Subgroups — #14527804

Order 2176.

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

normal subgroups only

This group has 288 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#11763125C212yesmaximal, complemented
2#11762745C211yesmaximal, characteristic, complemented
3#11762543C212yesmaximal, complemented
4#11762533C211yesmaximal, characteristic, complemented
5C17×#534C211yesmaximal, characteristic
6C22×D4×C17—21no—
7C17×#123—22no—
8C17×#121—21no—
9C17×#121C2212yes—
10C17×#120—22no—
11C2×C22⋊C4×C17C2211yescharacteristic
12C2×C22⋊Dic17—21no—
13C2×C4⋊Dic17C2211yescharacteristic, complemented
14C2×#63225—21no—
15#10587503—22no—
16#10587502C2211yescharacteristic, complemented
17C22⋊#3622C2212yescomplemented
18#10587479—22no—
19C24×C17—21no—
20C24×C17D411yescharacteristic
21C2×D4×C17—22no—
22C2×D4×C17—22no—
23C2×D4×C17—21no—
24C2×D4×C17—42no—
25C2×D4×C17—42no—
26C2×D4×C17D412yes—
27C22×C4×C17—21no—
28C22×C4×C17D411yescharacteristic
29C22×C4×C17C2311yescharacteristic
30C22×Dic17—21no—
31C22×Dic17—21no—
32C4⋊C4×C17—21no—
33C4⋊C4×C17D412yes—
34C22⋊C4×C17—22no—
35C22⋊C4×C17—22no—
36C22⋊C4×C17—22no—
37C22⋊Dic17—42no—
38C4⋊Dic17—22no—
39#63225—42no—
40C2×#3622—22no—
41C23×C17—21no—
42C23×C17—21no—
43C23×C17—22no—
44C23×C17—41no—
45C23×C17—41no—
46C23×C17—42no—
47C23×C17—81no—
48C23×C17—81no—
49C23×C17SD1612yes—
50C23×C17C2×D411yescharacteristic
51D4×C17—42no—
52D4×C17—44no—
53D4×C17—42no—
54D4×C17—42no—
55D4×C17—81no—
56C2×C4×C17—21no—
57C2×C4×C17—22no—
58C2×C4×C17—41no—
59C2×C4×C17—42no—
60C2×C4×C17—41no—
61C2×C4×C17C2×D412yes—
62C2×Dic17—22no—
63C2×Dic17—41no—
64C2×Dic17—41no—
65C2×Dic17—41no—
66C2×Dic17—41no—
67C2×Dic17—41no—
68#3622—42no—
69#1613—171nomaximal, Sylow
70C22×C17—21no—
71C22×C17—21no—
72C22×C17—22no—
73C22×C17—42no—
74C22×C17—41no—
75C22×C17—41no—
76C22×C17—42no—
77C22×C17—42no—
78C22×C17—42no—
79C22×C17—81no—
80C22×C17—84no—
81C22×C17#12011yescharacteristic
82C22×C17C2×SD1611yescharacteristic
83C22×C17AGL(1,ℤ/8)11yescharacteristic
84C4×C17—22no—
85C4×C17—42no—
86C4×C17—41no—
87Dic17—42no—
88Dic17—81no—
89#534D1711yescharacteristic
90#481—172no—
91#399—171no—
92#327—172no—
93#323—171no—
94C2×C17—21no—
95C2×C17—21no—
96C2×C17—42no—
97C2×C17—82no—
98C2×C17#45011yescharacteristic
99C2×C17#45311yescharacteristic
100C2×C17#45811yescharacteristic
101C22×D4—21no—
102#123—22no—
103#121—21no—
104#121C2×D1712yes—
105#120—22no—
106C2×C4⋊C4—171no—
107C2×C4⋊C4—341no—
108C2×C22⋊C4—341no—
109C2×C22⋊C4C2×D1711yescharacteristic
110#106—342no—
111#102—342no—
112C22⋊C8—172no—
113#95—171no—
114C17#161311yescharacteristic, complemented, Sylow
115C24—21no—
116C24C22⋊D1711yescharacteristic
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—21no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4C22⋊D1712yes—
123C22×C4—21no—
124C22×C4—341no—
125C22×C4—341no—
126C22×C4C22⋊D1711yescharacteristic
127C22×C4C22×D1711yescharacteristic
128C2×C8—342no—
129C4⋊C4—21no—
130C4⋊C4—342no—
131C4⋊C4—682no—
132C4⋊C4C22⋊D1712yes—
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—682no—
137C23—21no—
138C23—21no—
139C23—22no—
140C23—41no—
141C23—41no—
142C23—42no—
143C23—81no—
144C23—81no—
145C23#6322912yes—
146C23C2×C22⋊D1711yescharacteristic
147D4—42no—
148D4—44no—
149D4—42no—
150D4—42no—
151D4—81no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—342no—
158C2×C4—681no—
159C2×C4—681no—
160C2×C4—681no—
161C2×C4—681no—
162C2×C4—681no—
163C2×C4C2×C22⋊D1712yes—
164C8—682no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—42no—
172C22—42no—
173C22—42no—
174C22—81no—
175C22—84no—
176C22#1058761811yescharacteristic
177C22C2×#6322911yescharacteristic
178C22#1058763911yescharacteristic
179C4—22no—
180C4—42no—
181C4—41no—
182C4—682no—
183C4—1361no—
184C2—21no—
185C2—21no—
186C2—42no—
187C2—82no—
188C2#1176296011yescharacteristic
189C2#1176297711yescharacteristic
190C2#1176299911yescharacteristic
subgroupG/H
centerC22#10587639
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
Fitting, prime radicalC17×#534C2
socleC22×C17#120
hypercenter#534D17
hypocenterC17#1613
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #11763125C2semidirect
2. #11762745C2semidirect
3. #11762543C2semidirect
4. #11762533C2semidirect
13. C2×C4⋊Dic17C22semidirect
16. #10587502C22semidirect
17. C22⋊#3622C22semidirect
114. C17#1613semidirect

Maximal subgroups

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

subgroupindex
#117631252
#117627452
#117625432
#117625332
C17×#5342
#161317