Finite Group Databasebeta

← #14529913

Subgroups — #14529913

Order 2176.

430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#10587621C212yesmaximal, complemented
2C2×#10587505C212yesmaximal, complemented
3#11762670C211yesmaximal, characteristic, complemented
4#11762576C211yesmaximal, characteristic, complemented
5C17×#534C211yesmaximal, characteristic
6C22×D4×C17C2212yes—
7C23×Dic17C2211yescharacteristic
8C17×#123—22no—
9C17×#121—22no—
10C17×#120—22no—
11C2×C22⋊C4×C17C2211yescharacteristic
12C2×C22⋊Dic17—22no—
13C2×C22⋊Dic17C2211yescharacteristic
14C2×C22⋊Dic17C2212yes—
15#10587621—24no—
16#10587621—24no—
17C2×#63225—22no—
18#10587505—24no—
19#10587502—22no—
20C24×C17D412yes—
21C24×C17C2311yescharacteristic
22C2×D4×C17—22no—
23C2×D4×C17—22no—
24C2×D4×C17—22no—
25C2×D4×C17—22no—
26C2×D4×C17—22no—
27C2×D4×C17—22no—
28C2×D4×C17—22no—
29C22×C4×C17D412yes—
30C22×C4×C17D412yes—
31C22×Dic17—21no—
32C22×Dic17—22no—
33C22×Dic17—22no—
34C22×Dic17—22no—
35C22×Dic17—21no—
36C22×Dic17—21no—
37C22×Dic17D412yescomplemented
38C4⋊C4×C17—22no—
39C22⋊C4×C17—24no—
40C22⋊C4×C17—22no—
41C22⋊Dic17—28no—
42C22⋊Dic17—22no—
43C22⋊Dic17—22no—
44C22⋊Dic17D414yescomplemented
45#63225—28no—
46C23×C17—22no—
47C23×C17—22no—
48C23×C17—22no—
49C23×C17—42no—
50C23×C17—44no—
51C23×C17C2×D411yescharacteristic
52C23×C17C2×D412yes—
53C23×C17C2×D412yes—
54C23×C17C2×D411yescharacteristic
55C23×C17#4111yescharacteristic
56D4×C17—42no—
57D4×C17—42no—
58D4×C17—42no—
59D4×C17—42no—
60D4×C17—82no—
61C2×C4×C17—22no—
62C2×C4×C17—22no—
63C2×C4×C17—22no—
64C2×C4×C17—22no—
65C2×C4×C17—42no—
66C2×Dic17—24no—
67C2×Dic17—24no—
68C2×Dic17—22no—
69C2×Dic17—44no—
70C2×Dic17—41no—
71C2×Dic17—42no—
72C2×Dic17—41no—
73C2×Dic17—42no—
74C2×Dic17—41no—
75#2022—171nomaximal, Sylow
76C22×C17—22no—
77C22×C17—24no—
78C22×C17—41no—
79C22×C17—42no—
80C22×C17—41no—
81C22×C17—44no—
82C22×C17—42no—
83C22×C17—44no—
84C22×C17—82no—
85C22×C17#12012yes—
86C22×C17#12011yescharacteristic
87C22×C17#12111yescharacteristic
88C22×C17#12112yes—
89C22×C17#12711yescharacteristic
90C4×C17—42no—
91C4×C17—42no—
92Dic17—44no—
93Dic17—42no—
94Dic17—81no—
95#534D1711yescharacteristic
96C2×#123—172no—
97C2×#99—172no—
98#381—171no—
99#342—171no—
100C2×C17—22no—
101C2×C17—21no—
102C2×C17—22no—
103C2×C17—21no—
104C2×C17—82no—
105C2×C17#39211yescharacteristic
106C2×C17#45812yes—
107C22×D4C2×D1712yes—
108C23×C4—171no—
109#123—22no—
110#123—344no—
111#123—344no—
112#121—22no—
113#120—22no—
114C2×C4⋊C4—342no—
115C2×C22⋊C4—172no—
116C2×C22⋊C4—171no—
117C2×C22⋊C4—342no—
118C2×C22⋊C4C2×D1711yescharacteristic
119#99—344no—
120#95—342no—
121C17#202211yescharacteristic, complemented, Sylow
122C24C22⋊D1712yes—
123C24C22×D1711yescharacteristic
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—22no—
131C22×C4—172no—
132C22×C4—342no—
133C22×C4—341no—
134C22×C4—341no—
135C22×C4—342no—
136C22×C4—342no—
137C22×C4—341no—
138C22×C4C22⋊D1712yes—
139C22×C4C22⋊D1712yes—
140C4⋊C4—22no—
141C4⋊C4—348no—
142C22⋊C4—24no—
143C22⋊C4—22no—
144C22⋊C4—174no—
145C22⋊C4—348no—
146C22⋊C4—342no—
147C22⋊C4—342no—
148C23—22no—
149C23—22no—
150C23—22no—
151C23—42no—
152C23—44no—
153C23D4×D1711yescharacteristic
154C23D4×D1712yes—
155C23#6325411yescharacteristic
156C23C2×C22⋊D1712yes—
157C23C2×C22⋊D1711yescharacteristic
158D4—42no—
159D4—42no—
160D4—42no—
161D4—42no—
162D4—82no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—42no—
168C2×C4—344no—
169C2×C4—344no—
170C2×C4—342no—
171C2×C4—684no—
172C2×C4—682no—
173C2×C4—681no—
174C2×C4—681no—
175C2×C4—682no—
176C2×C4—681no—
177C22—22no—
178C22—24no—
179C22—41no—
180C22—42no—
181C22—41no—
182C22—44no—
183C22—42no—
184C22—44no—
185C22—82no—
186C22#1058762312yes—
187C22#1058762411yescharacteristic
188C22#1058762512yes—
189C22#1058762611yescharacteristic
190C22#1058763911yescharacteristic
191C4—42no—
192C4—42no—
193C4—684no—
194C4—682no—
195C4—1361no—
196C2—22no—
197C2—21no—
198C2—22no—
199C2—21no—
200C2—82no—
201C2#1176271711yescharacteristic
202C2#1176299912yes—
subgroupG/H
centerC22#10587639
commutator, cosocleC24×C17C23
FrattiniC24C22×D17
Fitting, prime radicalC17×#534C2
socleC22×C17#120
hypercenter#534D17
hypocenterC17#2022
LCSG ▷ C24×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#10587621C2semidirect
2. C2×#10587505C2semidirect
3. #11762670C2semidirect
4. #11762576C2semidirect
37. C22×Dic17D4semidirect
44. C22⋊Dic17D4semidirect
121. C17#2022semidirect

Maximal subgroups

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

subgroupindex
C2×#105876212
C2×#105875052
#117626702
#117625762
C17×#5342
#202217