Finite Group Databasebeta

← #14544789

Subgroups — #14544789

Order 2176.

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

normal subgroups only

This group has 360 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#11763876C211yesmaximal, characteristic
2#11763802C212yesmaximal, complemented
3D17⋊#100C211yesmaximal, characteristic
4#11763680C212yesmaximal, complemented
5#124×D17C211yesmaximal, characteristic
6C2×Q8×D17C2211yescharacteristic
7C2×D4×D17C2211yescharacteristic
8C2×C22⋊#595—22no—
9D4×#595—24no—
10#10587677C2212yes—
11D17⋊M16C2212yes—
12C17×#124C411yescharacteristic, complemented
13#10587633C411yescharacteristic, complemented
14#10587622C411yescharacteristic, complemented
15#10587571—22no—
16C22⋊C4×D17—22no—
17#10587561C411yescharacteristic, complemented
18C42×D17C2211yescharacteristic
19#10587553—22no—
20#10587549—24no—
21C23×D17D412yes—
22C22×#595—24no—
23C2×Q8×C17C2×C411yescharacteristic, complemented
24C2×D4×C17C2×C411yescharacteristic
25C2×C22⋊D17D412yes—
26Q8×D17—22no—
27Q8×D17—22no—
28D4×D17—22no—
29D4×D17—22no—
30C2×D68C2×C411yescharacteristic
31C2×C4×D17—21no—
32C2×C4×D17—21no—
33C2×C4×D17C2311yescharacteristic
34C2×Dic34—21no—
35C2×Dic34C2×C411yescharacteristic, complemented
36C22⋊#595—24no—
37C22⋊#595—22no—
38#63247—22no—
39C4⋊#595D414yes—
40C4×#595—22no—
41#63243—22no—
42#63242—22no—
43C22⋊C4×C17—22no—
44C42×C17C2×C411yescharacteristic, complemented
45C22⋊Dic17—22no—
46#63227—22no—
47#63227—22no—
48C4×Dic17—21no—
49C4×Dic17C2×C411yescharacteristic, complemented
50C23×C17C22⋊C412yes—
51C22×D17—22no—
52C22×D17—22no—
53C22×D17—44no—
54C22×D17C22⋊C412yes—
55C22×D17C2×D411yescharacteristic
56C2×#595—24no—
57C2×#595—84no—
58Q8×C17—22no—
59D4×C17—22no—
60C2×C4×C17—21no—
61C2×C4×C17—21no—
62C2×C4×C17C22×C411yescharacteristic
63C22⋊D17—24no—
64C2×Dic17—21no—
65C2×Dic17—21no—
66C2×Dic17C2×D411yescharacteristic
67D68—22no—
68C4×D17—42no—
69C4×D17—41no—
70C4×D17—41no—
71C4×D17C2×D412yes—
72C4×D17#4112yes—
73Dic34—22no—
74Dic34—42no—
75#3624—42no—
76#1885—171nomaximal, Sylow
77C22×C17—22no—
78C22×C17—42no—
79C22×C17C2×C22⋊C411yescharacteristic
80C2×D17—22no—
81C2×D17—22no—
82C2×D17—44no—
83C2×D17—42no—
84C2×D17—82no—
85C2×D17#12011yescharacteristic
86C2×D17#12311yescharacteristic
87#595—44no—
88C4×C17—41no—
89C4×C17—41no—
90C4×C17C4×D412yes—
91Dic17—41no—
92Dic17—41no—
93Dic17#12112yes—
94C2×#124—171no—
95#518—171no—
96#419—172no—
97C2×#100—171no—
98#339—172no—
99C2×C17—21no—
100C2×C17—42no—
101C2×C17#38611yescharacteristic
102D17—21no—
103D17—42no—
104D17#45512yes—
105C22×Q8—171no—
106C22×D4—171no—
107C2×M16—172no—
108#124—171no—
109#124—171no—
110#124—171no—
111#124—342no—
112#124#59511yescharacteristic, complemented
113C4×D4—344no—
114#117—172no—
115C2×C22⋊C4—342no—
116C2×C22⋊C4—342no—
117C2×C42—171no—
118#103—344no—
119#100—342no—
120C17#188511yescharacteristic, complemented, Sylow
121C24—172no—
122C2×Q8—171no—
123C2×Q8—341no—
124C2×Q8—342no—
125C2×Q8—342no—
126C2×Q8C2×#59511yescharacteristic, complemented
127C2×D4—172no—
128C2×D4—171no—
129C2×D4—342no—
130C2×D4—342no—
131C2×D4C2×#59511yescharacteristic
132C22×C4—171no—
133C22×C4—341no—
134C22×C4—341no—
135C22×C4—344no—
136M16—342no—
137M16—342no—
138C2×C8—342no—
139C4⋊C4—174no—
140C22⋊C4—22no—
141C22⋊C4—342no—
142C22⋊C4—342no—
143C22⋊C4—342no—
144C22⋊C4—344no—
145C22⋊C4—342no—
146C42—171no—
147C42—341no—
148C42—342no—
149C42C2×#59511yescharacteristic, complemented
150C23—171no—
151C23—172no—
152C23—342no—
153C23—342no—
154C23—684no—
155C23C22⋊#59512yes—
156Q8—22no—
157Q8—342no—
158Q8—682no—
159D4—22no—
160D4—344no—
161D4—342no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—171no—
165C2×C4—172no—
166C2×C4—172no—
167C2×C4—344no—
168C2×C4—341no—
169C2×C4—341no—
170C2×C4—681no—
171C2×C4—681no—
172C2×C4—682no—
173C2×C4—1364no—
174C2×C4C22×#59511yescharacteristic
175C8—682no—
176C22—22no—
177C22—42no—
178C22—171no—
179C22—171no—
180C22—342no—
181C22—342no—
182C22—682no—
183C22—684no—
184C22—1362no—
185C22C2×C22⋊#59511yescharacteristic
186C4—41no—
187C4—41no—
188C4—172no—
189C4—681no—
190C4—684no—
191C4—681no—
192C4D4×#59512yes—
193C2—21no—
194C2—42no—
195C2—172no—
196C2—341no—
197C2—682no—
198C2#1176375711yescharacteristic
subgroupG/H
centerC2#11763757
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#124C4
socleC2×C17#386
prime radicalC2×D4×D17C22
cosocleC2×C4×D17C23
hypercenter#124#595
hypocenterC17#1885
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #11763802C2semidirect
4. #11763680C2semidirect
12. C17×#124C4semidirect
13. #10587633C4semidirect
14. #10587622C4semidirect
17. #10587561C4semidirect
23. C2×Q8×C17C2×C4semidirect
35. C2×Dic34C2×C4semidirect
44. C42×C17C2×C4semidirect
49. C4×Dic17C2×C4semidirect
112. #124#595semidirect
120. C17#1885semidirect
126. C2×Q8C2×#595semidirect
149. C42C2×#595semidirect

Maximal subgroups

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

subgroupindex
#117638762
#117638022
D17⋊#1002
#117636802
#124×D172
#188517