Finite Group Databasebeta

← #14544692

Subgroups — #14544692

Order 2176.

306 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 68 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#11763780C211yesmaximal, characteristic
2C4×C22⋊#595C211yesmaximal, characteristic
3#11763679C212yesmaximal, complemented
4#11763679C212yesmaximal, complemented
5#117×D17C211yesmaximal, characteristic
6C22×C4×D17C2211yescharacteristic
7C2×C22⋊#595—21no—
8C2×C4×#595C2212yes—
9C2×#63243C2212yes—
10C17×#117C411yescharacteristic, complemented
11#10587612C411yescharacteristic, complemented
12#10587578C2211yescharacteristic
13#10587578C2211yescharacteristic
14C4⋊C4×D17C411yescharacteristic, complemented
15C4⋊C4×D17C411yescharacteristic, complemented
16C22⋊C4×D17—21no—
17#10587563—21no—
18#10587558—21no—
19C42×D17—21no—
20C22⋊#3624—21no—
21(C2×#595)⋊C4—21no—
22#10587543—21no—
23C23×D17D411yescharacteristic
24C22×#595—22no—
25C22×C4×C17C2×C411yescharacteristic
26C22×Dic17D411yescharacteristic
27C2×C4×D17—24no—
28C2×C4×D17—21no—
29C2×C4×D17—21no—
30C2×C4×D17C2×C411yescharacteristic
31C2×C4×D17D411yescharacteristic
32C2×C4×D17D411yescharacteristic
33C2×C4×D17C2311yescharacteristic
34C22⋊#595—24no—
35C2×#3624—22no—
36C4×#595—24no—
37#63243—24no—
38C4⋊C4×C17C2×C411yescharacteristic
39C4⋊C4×C17C2×C411yescharacteristic
40C22⋊C4×C17—21no—
41C42×C17—21no—
42C22⋊Dic17—21no—
43#63227—21no—
44#63227—21no—
45C4⋊Dic17C2×C411yescharacteristic
46C4⋊Dic17C2×C411yescharacteristic
47#63225—21no—
48#63225—21no—
49C4×Dic17—21no—
50C4×Dic17—21no—
51C4×Dic17—21no—
52C23×C17C22⋊C411yescharacteristic
53C22×D17—22no—
54C22×D17—22no—
55C22×D17—41no—
56C22×D17C22⋊C411yescharacteristic
57C22×D17C2×D411yescharacteristic
58C2×#595—24no—
59C2×#595—42no—
60C2×#595—42no—
61C2×C4×C17—22no—
62C2×C4×C17—21no—
63C2×C4×C17—21no—
64C2×C4×C17C22⋊C411yescharacteristic
65C2×C4×C17C22×C411yescharacteristic
66C2×Dic17—22no—
67C2×Dic17—21no—
68C2×Dic17—21no—
69C2×Dic17C22⋊C411yescharacteristic
70C2×Dic17C2×D411yescharacteristic
71C4×D17—22no—
72C4×D17—22no—
73C4×D17—22no—
74C4×D17—42no—
75C4×D17—42no—
76C4×D17#4114yes—
77#3624—42no—
78#1826—171nomaximal, Sylow
79C22×C17—21no—
80C22×C17—21no—
81C22×C17—21no—
82C22×C17C2×C22⋊C411yescharacteristic
83C2×D17—21no—
84C2×D17—21no—
85C2×D17—21no—
86C2×D17—21no—
87C2×D17—44no—
88C2×D17#10412yes—
89C2×D17#12312yes—
90#595—44no—
91C4×C17—21no—
92C4×C17—41no—
93C4×C17—41no—
94C4×C17#11712yes—
95Dic17—21no—
96Dic17—41no—
97Dic17—41no—
98Dic17#12312yes—
99C2×#117—171no—
100#407—171no—
101C4×C22⋊C4—171no—
102#339—172no—
103#339—172no—
104C2×C17—41no—
105C2×C17#38111yescharacteristic
106C2×C17C2×#10411yescharacteristic
107C2×C17#42111yescharacteristic
108D17—22no—
109D17—22no—
110C23×C4—171no—
111C2×M16—172no—
112#117—171no—
113#117—171no—
114#117—171no—
115#117—341no—
116#117—341no—
117#117#59511yescharacteristic, complemented
118C2×C4⋊C4—171no—
119C2×C4⋊C4—171no—
120C2×C22⋊C4—341no—
121C2×C22⋊C4—341no—
122C2×C42—172no—
123C2×C42—341no—
124C22⋊C8—341no—
125C22⋊C8—341no—
126#95—341no—
127C17#182611yescharacteristic, complemented, Sylow
128C24—171no—
129C22×C4—171no—
130C22×C4—171no—
131C22×C4—171no—
132C22×C4—171no—
133C22×C4—171no—
134C22×C4—344no—
135C22×C4—341no—
136C22×C4—341no—
137C22×C4—342no—
138C22×C4C2×#59511yescharacteristic
139M16—344no—
140C2×C8—342no—
141C4⋊C4—171no—
142C4⋊C4—171no—
143C4⋊C4—341no—
144C4⋊C4—341no—
145C4⋊C4C2×#59511yescharacteristic
146C4⋊C4C2×#59511yescharacteristic
147C22⋊C4—21no—
148C22⋊C4—341no—
149C22⋊C4—341no—
150C22⋊C4—344no—
151C22⋊C4—341no—
152C42—21no—
153C42—341no—
154C42—341no—
155C42—344no—
156C42—341no—
157C23—171no—
158C23—171no—
159C23—342no—
160C23—342no—
161C23—681no—
162C23C22⋊#59511yescharacteristic
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—171no—
167C2×C4—174no—
168C2×C4—171no—
169C2×C4—342no—
170C2×C4—342no—
171C2×C4—342no—
172C2×C4—342no—
173C2×C4—344no—
174C2×C4—341no—
175C2×C4—341no—
176C2×C4—682no—
177C2×C4—682no—
178C2×C4—682no—
179C2×C4—682no—
180C2×C4C22⋊#59511yescharacteristic
181C2×C4C22×#59511yescharacteristic
182C8—682no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—172no—
187C22—172no—
188C22—341no—
189C22—341no—
190C22—341no—
191C22—341no—
192C22—684no—
193C22C2×C22⋊#59511yescharacteristic
194C4—21no—
195C4—41no—
196C4—41no—
197C4—172no—
198C4—341no—
199C4—684no—
200C4—681no—
201C4—681no—
202C4#1058767712yes—
203C2—41no—
204C2—342no—
205C2—342no—
206C2#1176375211yescharacteristic
207C2D17⋊#10411yescharacteristic
208C2#1176380611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#117C4
socleC22×C17C2×C22⋊C4
prime radicalC23×D17D4
cosocleC2×C4×D17C23
hypercenter#117#595
hypocenterC17#1826
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
3. #11763679C2semidirect
4. #11763679C2semidirect
10. C17×#117C4semidirect
11. #10587612C4semidirect
14. C4⋊C4×D17C4semidirect
15. C4⋊C4×D17C4semidirect
117. #117#595semidirect
127. C17#1826semidirect

Maximal subgroups

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

subgroupindex
#117637802
C4×C22⋊#5952
#117636792
#117×D172
#182617