Finite Group Databasebeta

← #14544221

Subgroups — #14544221

Order 2176.

340 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 70 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#11763777C211yesmaximal, characteristic, complemented
2#11763751C211yesmaximal, characteristic, complemented
3#11763661C214yesmaximal, complemented
4C2×#10587581C211yesmaximal, characteristic
5C22×D68—21no—
6C22×D68C411yescharacteristic, complemented
7C22×C4×D17C2211yescharacteristic
8C2×C22⋊#595—21no—
9C2×C4⋊#595—21no—
10C2×#63242—21no—
11C2×C4⋊C4×C17C411yescharacteristic, complemented
12C2×#63227—21no—
13#10587581—21no—
14#10587581—21no—
15#10587581C412yescomplemented
16#10587581C2212yes—
17C22⋊#3624—21no—
18(C2×#595)⋊C4C2212yescomplemented
19#10587543C2212yescomplemented
20C23×D17—21no—
21C23×D17D411yescharacteristic
22C22×#595—22no—
23C22×C4×C17—21no—
24C22×C4×C17C2×C411yescharacteristic
25C22×Dic17D411yescharacteristic
26C2×D68—22no—
27C2×D68—22no—
28C2×D68—21no—
29C2×D68—42no—
30C2×D68—42no—
31C2×D68—42no—
32C2×D68C2×C412yes—
33C2×C4×D17—42no—
34C2×C4×D17C2×C411yescharacteristic
35C2×C4×D17D411yescharacteristic
36C2×C4×D17D411yescharacteristic
37C2×C4×D17C2311yescharacteristic
38C22⋊#595—42no—
39C2×#3624—22no—
40C4⋊#595—42no—
41#63242—42no—
42C4⋊C4×C17—21no—
43C4⋊C4×C17C2×C412yes—
44#63227—22no—
45#63227—22no—
46C23×C17C22⋊C411yescharacteristic
47C22×D17—22no—
48C22×D17—41no—
49C22×D17—41no—
50C22×D17—42no—
51C22×D17—42no—
52C22×D17—42no—
53C22×D17C22⋊C411yescharacteristic
54C22×D17D812yes—
55C22×D17SD1612yes—
56C22×D17C2×D411yescharacteristic
57C2×#595—42no—
58C2×#595—42no—
59C2×#595—42no—
60C2×C4×C17—22no—
61C2×C4×C17—22no—
62C2×C4×C17—42no—
63C2×C4×C17C22⋊C411yescharacteristic
64C2×C4×C17C22×C411yescharacteristic
65C2×Dic17—41no—
66C2×Dic17C22⋊C411yescharacteristic
67C2×Dic17C2×D411yescharacteristic
68D68—42no—
69D68—44no—
70D68—42no—
71D68—42no—
72D68—42no—
73C4×D17—22no—
74C4×D17—22no—
75C4×D17—41no—
76C4×D17—41no—
77#3624—42no—
78#1519—171nomaximal, Sylow
79C22×C17—21no—
80C22×C17—21no—
81C22×C17#10212yes—
82C22×C17C2×C22⋊C411yescharacteristic
83C2×D17—21no—
84C2×D17—21no—
85C2×D17—21no—
86C2×D17—21no—
87C2×D17—42no—
88C2×D17—42no—
89C2×D17—42no—
90C2×D17—42no—
91C2×D17—42no—
92C2×D17—84no—
93C2×D17#10212yes—
94C2×D17C2×D811yescharacteristic
95C2×D17C2×SD1611yescharacteristic
96#595—82no—
97C4×C17—21no—
98C4×C17—42no—
99C4×C17#9912yes—
100Dic17—21no—
101Dic17—21no—
102C2×#121—171no—
103C2×C22⋊C8—171no—
104#380—171no—
105#327—174no—
106C2×C17—22no—
107C2×C17C2×#9911yescharacteristic
108C2×C17C2×#10211yescharacteristic
109C2×C17#42111yescharacteristic
110D17—21no—
111D17—22no—
112D17—21no—
113D17—82no—
114C22×D4—171no—
115C22×D4—341no—
116C23×C4—171no—
117C22×C8—341no—
118#121—172no—
119#121—172no—
120#121—341no—
121#121—341no—
122C2×C4⋊C4—341no—
123C2×C4⋊C4#59511yescharacteristic, complemented
124C2×C22⋊C4—341no—
125C2×C22⋊C4—341no—
126C22⋊C8—172no—
127C22⋊C8—341no—
128#95—172no—
129C17#151911yescharacteristic, complemented, Sylow
130C24—171no—
131C24—341no—
132C2×D4—172no—
133C2×D4—342no—
134C2×D4—342no—
135C2×D4—341no—
136C2×D4—682no—
137C2×D4—682no—
138C2×D4—682no—
139C22×C4—21no—
140C22×C4—171no—
141C22×C4—171no—
142C22×C4—171no—
143C22×C4—171no—
144C22×C4—171no—
145C22×C4—342no—
146C22×C4—682no—
147C22×C4C2×#59511yescharacteristic
148C2×C8—342no—
149C2×C8—682no—
150C4⋊C4—21no—
151C4⋊C4—682no—
152C4⋊C4C2×#59512yes—
153C22⋊C4—342no—
154C22⋊C4—342no—
155C22⋊C4—682no—
156C23—171no—
157C23—171no—
158C23—172no—
159C23—172no—
160C23—342no—
161C23—681no—
162C23—681no—
163C23—682no—
164C23—682no—
165C23—682no—
166C23C22⋊#59511yescharacteristic
167D4—682no—
168D4—682no—
169D4—684no—
170D4—682no—
171D4—682no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—42no—
175C2×C4—171no—
176C2×C4—171no—
177C2×C4—342no—
178C2×C4—342no—
179C2×C4—681no—
180C2×C4—681no—
181C2×C4—681no—
182C2×C4—682no—
183C2×C4—682no—
184C2×C4—682no—
185C2×C4C22⋊#59511yescharacteristic
186C2×C4C22×#59511yescharacteristic
187C8—682no—
188C22—21no—
189C22—21no—
190C22—171no—
191C22—172no—
192C22—171no—
193C22—341no—
194C22—341no—
195C22—341no—
196C22—341no—
197C22—682no—
198C22—682no—
199C22—682no—
200C22—682no—
201C22—682no—
202C22—1364no—
203C22#1058754712yes—
204C22C2×C22⋊#59511yescharacteristic
205C4—21no—
206C4—42no—
207C4—341no—
208C4—341no—
209C4—1362no—
210C4#1058753912yes—
211C2—22no—
212C2—341no—
213C2—342no—
214C2—341no—
215C2—1362no—
216C2C2×#1058753911yescharacteristic
217C2C2×#1058754711yescharacteristic
218C2#1176380811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC2×C4⋊C4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC2×#10587581C2
cosocleC2×C4×D17C23
hypercenterC2×C4⋊C4#595
hypocenterC17#1519
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11763777C2semidirect
2. #11763751C2semidirect
3. #11763661C2semidirect
6. C22×D68C4semidirect
11. C2×C4⋊C4×C17C4semidirect
15. #10587581C4semidirect
18. (C2×#595)⋊C4C22semidirect
19. #10587543C22semidirect
123. C2×C4⋊C4#595semidirect
129. C17#1519semidirect

Maximal subgroups

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

subgroupindex
#117637772
#117637512
#117636612
C2×#105875812
#151917