Finite Group Databasebeta

← #14545037

Subgroups — #14545037

Order 2176.

264 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 54 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#11763795C211yesmaximal, characteristic, complemented
2#11763788C211yesmaximal, characteristic, complemented
3#11763712C212yesmaximal, complemented
4#11763710C212yesmaximal, complemented
5#11763321C211yesmaximal, characteristic
6C2×#63252C2211yescharacteristic
7C22×C4×D17C411yescharacteristic, complemented
8#10587677—21no—
9D17⋊M16—21no—
10C17×#117C411yescharacteristic, complemented
11#10587583—21no—
12#10587581—21no—
13#10587572C411yescharacteristic, complemented
14#10587572C2211yescharacteristic
15#10587568C411yescharacteristic, complemented
16#10587568C2211yescharacteristic
17C4×D68—21no—
18C4×D68—21no—
19#10587553C2212yes—
20#10587551C2212yes—
21#10587545—21no—
22#10587539—21no—
23C23×D17C2×C411yescharacteristic
24C22×C4×C17C2×C411yescharacteristic
25C2×C22⋊D17C2×C411yescharacteristic
26C2×C22⋊D17C2311yescharacteristic
27C22×Dic17C2×C411yescharacteristic
28#63252—41no—
29#63252—41no—
30C2×D68—21no—
31C2×D68—21no—
32C2×D68D411yescharacteristic
33C2×C4×D17—21no—
34C2×C4×D17—21no—
35C2×C4×D17—41no—
36C2×C4×D17—41no—
37C2×C4×D17D411yescharacteristic
38C2×C4×D17D411yescharacteristic
39C2×Dic34D411yescharacteristic
40C22⋊#595—22no—
41#63247—22no—
42C4⋊#595—41no—
43C4×#595—41no—
44#63243—41no—
45#63242—41no—
46C4⋊C4×C17—21no—
47C22⋊C4×C17C2×C411yescharacteristic
48C22⋊C4×C17C2×C411yescharacteristic
49C42×C17—21no—
50#63227—21no—
51#63227—21no—
52#63227—21no—
53#63227—21no—
54C4⋊Dic17—21no—
55C4⋊Dic17—21no—
56C23×C17C22×C411yescharacteristic
57C22×D17—21no—
58C22×D17—42no—
59C22×D17—41no—
60C22×D17C22⋊C411yescharacteristic
61C22×D17C2×D411yescharacteristic
62C2×#595—42no—
63C2×C4×C17—21no—
64C2×C4×C17—21no—
65C2×C4×C17—21no—
66C2×C4×C17—21no—
67C2×C4×C17C22⋊C411yescharacteristic
68C2×C4×C17C22⋊C411yescharacteristic
69C22⋊D17—41no—
70C22⋊D17—41no—
71C2×Dic17—21no—
72C2×Dic17—41no—
73C2×Dic17C22⋊C411yescharacteristic
74C2×Dic17C2×D411yescharacteristic
75D68—41no—
76D68—42no—
77D68—42no—
78C4×D17—21no—
79C4×D17—21no—
80C4×D17—21no—
81C4×D17—21no—
82C4×D17—42no—
83C4×D17—41no—
84C4×D17—41no—
85Dic34—41no—
86#3624—42no—
87#2120—171nomaximal, Sylow
88C22×C17—21no—
89C22×C17—41no—
90C22×C17C2×C22⋊C411yescharacteristic
91C2×D17—22no—
92C2×D17—21no—
93C2×D17—21no—
94C2×D17—41no—
95C2×D17—42no—
96C2×D17—41no—
97C2×D17—41no—
98C2×D17—41no—
99C2×D17—42no—
100#595—82no—
101C4×C17—21no—
102C4×C17—41no—
103C4×C17—41no—
104C4×C17#9911yescharacteristic
105C4×C17#9911yescharacteristic
106Dic17—21no—
107Dic17—21no—
108Dic17—41no—
109#525—171no—
110#413—171no—
111#410—171no—
112#352—172no—
113AΣL(1,𝔽16)—172no—
114C2×C17—21no—
115C2×C17—41no—
116C2×C17C2×#9911yescharacteristic
117D17—41no—
118D17—41no—
119D17—42no—
120C2×#41—171no—
121C23×C4—171no—
122C2×M16—341no—
123#123—171no—
124#123—171no—
125C22⋊Q8—341no—
126#121—341no—
127#120—171no—
128#120—171no—
129C4×D4—341no—
130C4×D4—341no—
131#117—341no—
132#117#59511yescharacteristic, complemented
133#101—341no—
134#100—172no—
135#99—172no—
136#99—341no—
137C17#212011yescharacteristic, complemented, Sylow
138C24—171no—
139#41—681no—
140#41—681no—
141C2×Q8—171no—
142C2×D4—171no—
143C2×D4—171no—
144C2×D4—171no—
145C2×D4—341no—
146C2×D4—341no—
147C22×C4—171no—
148C22×C4—171no—
149C22×C4—171no—
150C22×C4—341no—
151C22×C4—341no—
152C22×C4—681no—
153C22×C4—681no—
154C22×C4C2×#59511yescharacteristic
155M16—342no—
156M16—681no—
157C2×C8—681no—
158C4⋊C4—21no—
159C4⋊C4—341no—
160C4⋊C4—341no—
161C4⋊C4—681no—
162C22⋊C4—341no—
163C22⋊C4—341no—
164C22⋊C4—341no—
165C22⋊C4—341no—
166C22⋊C4—342no—
167C22⋊C4C2×#59511yescharacteristic
168C22⋊C4C2×#59511yescharacteristic
169C42—21no—
170C42—681no—
171C23—171no—
172C23—171no—
173C23—341no—
174C23—681no—
175C23—682no—
176C23C22×#59511yescharacteristic
177Q8—681no—
178D4—681no—
179D4—682no—
180D4—682no—
181D4—681no—
182D4—681no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—171no—
188C2×C4—171no—
189C2×C4—341no—
190C2×C4—341no—
191C2×C4—341no—
192C2×C4—341no—
193C2×C4—341no—
194C2×C4—682no—
195C2×C4—681no—
196C2×C4—681no—
197C2×C4—681no—
198C2×C4—682no—
199C2×C4C22⋊#59511yescharacteristic
200C2×C4C22⋊#59511yescharacteristic
201C8—682no—
202C22—21no—
203C22—41no—
204C22—342no—
205C22—341no—
206C22—341no—
207C22—681no—
208C22—681no—
209C22—681no—
210C22—682no—
211C22—682no—
212C22—681no—
213C22C2×C22⋊#59511yescharacteristic
214C4—21no—
215C4—41no—
216C4—41no—
217C4—341no—
218C4—341no—
219C4—681no—
220C4—1362no—
221C4#1058753911yescharacteristic
222C4#1058753911yescharacteristic
223C2—21no—
224C2—41no—
225C2—681no—
226C2—682no—
227C2—681no—
228C2C2×#1058753911yescharacteristic
subgroupG/H
centerC4#10587539
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC17×#117C4
socleC2×C17C2×#99
prime radical#11763321C2
cosocleC2×C22⋊D17C23
hypercenter#117#595
hypocenterC17#2120
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #11763795C2semidirect
2. #11763788C2semidirect
3. #11763712C2semidirect
4. #11763710C2semidirect
7. C22×C4×D17C4semidirect
10. C17×#117C4semidirect
13. #10587572C4semidirect
15. #10587568C4semidirect
132. #117#595semidirect
137. C17#2120semidirect

Maximal subgroups

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

subgroupindex
#117637952
#117637882
#117637122
#117637102
#117633212
#212017