Finite Group Databasebeta

← #14538713

Subgroups — #14538713

Order 2176.

414 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 133 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×#10587612C211yesmaximal, characteristic, complemented
2#11763147C214yesmaximal, complemented
3#11763130C214yesmaximal, complemented
4#11762852C212yesmaximal, complemented
5#11762810C212yesmaximal, complemented
6#11762773C211yesmaximal, characteristic, complemented
7C17×#525C211yesmaximal, characteristic
8C2×#41×C17C2211yescharacteristic
9C23×C4×C17C2211yescharacteristic
10C17×#123—21no—
11C17×C22⋊Q8C2212yes—
12C17×#121C2212yes—
13C17×#120—21no—
14C4×D4×C17—22no—
15C17×#117C2211yescharacteristic
16C2×C22⋊Dic17—21no—
17#10587612—22no—
18#10587612C2214yes—
19C2×C4⋊Dic17C2212yes—
20C2×C4×Dic17—21no—
21C2×#63223C2212yes—
22#10587506C2214yescomplemented
23#10587503C2214yescomplemented
24C22⋊#3622C2214yes—
25#10587479C2214yescomplemented
26#10587478C2214yescomplemented
27C24×C17D411yescharacteristic
28#41×C17—24no—
29C2×Q8×C17C2311yescharacteristic
30C2×D4×C17—21no—
31C2×D4×C17—22no—
32C2×D4×C17C2311yescharacteristic
33C22×C4×C17—24no—
34C22×C4×C17—21no—
35C22×C4×C17D411yescharacteristic
36C22×C4×C17D412yes—
37C22×C4×C17C2312yes—
38C22×C4×C17C2311yescharacteristic
39C22×Dic17—22no—
40C4⋊C4×C17—22no—
41C4⋊C4×C17C2312yes—
42C22⋊C4×C17—22no—
43C22⋊C4×C17—21no—
44C22⋊C4×C17—22no—
45C42×C17—21no—
46C22⋊Dic17—24no—
47C4⋊Dic17—22no—
48C4⋊Dic17C2314yes—
49C4×Dic17—24no—
50#63223—24no—
51C2×#3622C2314yes—
52C23×C17—22no—
53C23×C17—21no—
54C23×C17—42no—
55C23×C17C2×D412yes—
56C23×C17C2×D411yescharacteristic
57Q8×C17—22no—
58D4×C17—22no—
59D4×C17—44no—
60D4×C17—42no—
61C2×C4×C17—22no—
62C2×C4×C17—24no—
63C2×C4×C17—24no—
64C2×C4×C17—22no—
65C2×C4×C17—21no—
66C2×C4×C17—42no—
67C2×C4×C17C2×D412yes—
68C2×C4×C17C2×D411yescharacteristic
69C2×C4×C17#4114yes—
70C2×C4×C17C2411yescharacteristic
71C2×Dic17—24no—
72C2×Dic17—44no—
73#3622—24no—
74#3093—171nomaximal, Sylow
75C22×C17—22no—
76C22×C17—22no—
77C22×C17—22no—
78C22×C17—22no—
79C22×C17—44no—
80C22×C17—41no—
81C22×C17—42no—
82C22×C17#12312yes—
83C22×C17C22×D411yescharacteristic
84C4×C17—22no—
85C4×C17—41no—
86C4×C17—42no—
87C4×C17#12312yes—
88C4×C17C2×#4112yes—
89Dic17—44no—
90#525D1711yescharacteristic
91C2×#117—171no—
92#485—174no—
93#482—174no—
94#428—172no—
95#417—172no—
96#407—171no—
97C2×C17—22no—
98C2×C17—42no—
99C2×C17—81no—
100C2×C17C2×#12311yescharacteristic
101C2×C17#57512yes—
102C2×#41C2×D1711yescharacteristic
103C23×C4C2×D1711yescharacteristic
104C2×M16—172no—
105#123—21no—
106C22⋊Q8C2×D1712yes—
107#121C2×D1712yes—
108#120—21no—
109C4×D4—22no—
110#117—174no—
111#117—342no—
112#117C2×D1711yescharacteristic
113C2×C4⋊C4—172no—
114C2×C22⋊C4—341no—
115C2×C42—341no—
116#107—174no—
117#106—174no—
118#103—174no—
119#102—174no—
120C22⋊C8—174no—
121C17#309311yescharacteristic, complemented, Sylow
122C24C22⋊D1711yescharacteristic
123#41—24no—
124C2×Q8C22×D1711yescharacteristic
125C2×D4—22no—
126C2×D4—21no—
127C2×D4C22×D1711yescharacteristic
128C22×C4—24no—
129C22×C4—21no—
130C22×C4—342no—
131C22×C4C22⋊D1711yescharacteristic
132C22×C4C22⋊D1712yes—
133C22×C4C22×D1712yes—
134C22×C4C22×D1711yescharacteristic
135M16—344no—
136C2×C8—174no—
137C4⋊C4—22no—
138C4⋊C4—174no—
139C4⋊C4—342no—
140C4⋊C4C22×D1712yes—
141C22⋊C4—21no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—344no—
145C42—21no—
146C42—344no—
147C23—22no—
148C23—21no—
149C23—42no—
150C23C2×C22⋊D1712yes—
151C23C2×C22⋊D1711yescharacteristic
152Q8—22no—
153D4—22no—
154D4—44no—
155D4—42no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—344no—
163C2×C4—684no—
164C2×C4#6325414yes—
165C2×C4C2×C22⋊D1712yes—
166C2×C4C2×C22⋊D1711yescharacteristic
167C2×C4C23×D1711yescharacteristic
168C8—344no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—41no—
175C22—42no—
176C22#1058762112yes—
177C22C22×C22⋊D1711yescharacteristic
178C4—22no—
179C4—41no—
180C4—42no—
181C4—684no—
182C4#1058762112yes—
183C4C2×#6325412yes—
184C2—22no—
185C2—42no—
186C2—81no—
187C2C2×#1058762111yescharacteristic
188C2#1176360712yes—
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
Fitting, prime radicalC17×#525C2
socleC22×C17C22×D4
hypercenter#525D17
hypocenterC17#3093
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#10587612C2semidirect
2. #11763147C2semidirect
3. #11763130C2semidirect
4. #11762852C2semidirect
5. #11762810C2semidirect
6. #11762773C2semidirect
22. #10587506C22semidirect
23. #10587503C22semidirect
25. #10587479C22semidirect
26. #10587478C22semidirect
121. C17#3093semidirect

Maximal subgroups

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

subgroupindex
C2×#105876122
#117631472
#117631302
#117628522
#117628102
#117627732
C17×#5252
#309317