Finite Group Databasebeta

← #14545038

Subgroups — #14545038

Order 2176.

264 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 52 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#11763796C211yesmaximal, characteristic, complemented
2#11763789C211yesmaximal, characteristic, complemented
3#11763711C212yesmaximal, complemented
4#11763710C212yesmaximal, complemented
5#11763317C211yesmaximal, characteristic
6C2×#63254C2211yescharacteristic
7C22×C4×D17C411yescharacteristic, complemented
8C2×#63247—21no—
9#10587677—21no—
10C17×#123C411yescharacteristic, complemented
11#10587623C411yescharacteristic, complemented
12#10587615C2211yescharacteristic
13#10587582—21no—
14#10587579—21no—
15#10587570—21no—
16#10587568C2211yescharacteristic
17#10587567—21no—
18#10587563C411yescharacteristic, complemented
19#10587553C2212yes—
20#10587551—21no—
21#10587551C2212yes—
22#10587545—21no—
23C23×D17C2×C411yescharacteristic
24C2×D4×C17C2×C411yescharacteristic
25C22×C4×C17C2×C411yescharacteristic
26C2×C22⋊D17—21no—
27C2×C22⋊D17D411yescharacteristic
28C2×C22⋊D17C2311yescharacteristic
29C22×Dic17C2×C411yescharacteristic
30C22×Dic17D411yescharacteristic
31#63254—41no—
32#63254—41no—
33C2×D68—21no—
34C2×C4×D17—21no—
35C2×C4×D17—21no—
36C2×C4×D17—41no—
37C2×C4×D17—41no—
38C2×C4×D17D411yescharacteristic
39C2×Dic34D411yescharacteristic
40C22⋊#595—22no—
41#63247—22no—
42#63247—41no—
43C2×#3624—41no—
44C4⋊#595—41no—
45C4×#595—41no—
46C4⋊C4×C17—21no—
47C22⋊C4×C17—21no—
48C22⋊C4×C17C2×C411yescharacteristic
49C22⋊Dic17C2×C411yescharacteristic
50#63227—21no—
51#63227—21no—
52#63227—21no—
53#63225—21no—
54#63225—21no—
55C4×Dic17—21no—
56C23×C17C22⋊C411yescharacteristic
57C23×C17C22×C411yescharacteristic
58C22×D17—21no—
59C22×D17—42no—
60C22×D17—41no—
61C22×D17C2×D411yescharacteristic
62C2×#595—42no—
63D4×C17—41no—
64C2×C4×C17—21no—
65C2×C4×C17—21no—
66C2×C4×C17—41no—
67C2×C4×C17C22⋊C411yescharacteristic
68C22⋊D17—41no—
69C22⋊D17—41no—
70C22⋊D17—42no—
71C2×Dic17—21no—
72C2×Dic17—21no—
73C2×Dic17—21no—
74C2×Dic17—21no—
75C2×Dic17—21no—
76C2×Dic17C22⋊C411yescharacteristic
77C2×Dic17C22⋊C411yescharacteristic
78C2×Dic17C2×D411yescharacteristic
79D68—42no—
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—21no—
90C22×C17—41no—
91C22×C17—41no—
92C22×C17C2×C22⋊C411yescharacteristic
93C2×D17—22no—
94C2×D17—21no—
95C2×D17—41no—
96C2×D17—42no—
97C2×D17—41no—
98C2×D17—41no—
99C2×D17—42no—
100#595—82no—
101C4×C17—21no—
102C4×C17—41no—
103C4×C17—41no—
104Dic17—21no—
105Dic17—21no—
106Dic17—41no—
107Dic17#9911yescharacteristic
108Dic17#9911yescharacteristic
109#525—171no—
110#413—171no—
111#410—171no—
112#352—172no—
113AΣL(1,𝔽16)—172no—
114C2×C17—21no—
115C2×C17—41no—
116C2×C17—41no—
117C2×C17C2×#9911yescharacteristic
118D17—41no—
119D17—42no—
120C2×#41—171no—
121C23×C4—171no—
122C2×M16—341no—
123#123—171no—
124#123#59511yescharacteristic, complemented
125C22⋊Q8—341no—
126#121—341no—
127#120—171no—
128#120—171no—
129C4×D4—341no—
130C4×D4—341no—
131#117—171no—
132#117—341no—
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—341no—
145C2×D4—341no—
146C2×D4C2×#59511yescharacteristic
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—21no—
163C22⋊C4—171no—
164C22⋊C4—341no—
165C22⋊C4—341no—
166C22⋊C4—341no—
167C22⋊C4—342no—
168C22⋊C4C2×#59511yescharacteristic
169C42—341no—
170C42—681no—
171C23—171no—
172C23—341no—
173C23—681no—
174C23—682no—
175C23C22⋊#59511yescharacteristic
176C23C22×#59511yescharacteristic
177Q8—681no—
178D4—41no—
179D4—682no—
180D4—681no—
181D4—681no—
182D4—682no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—171no—
187C2×C4—171no—
188C2×C4—171no—
189C2×C4—341no—
190C2×C4—341no—
191C2×C4—341no—
192C2×C4—341no—
193C2×C4—341no—
194C2×C4—341no—
195C2×C4—341no—
196C2×C4—681no—
197C2×C4—681no—
198C2×C4—682no—
199C2×C4—682no—
200C2×C4C22⋊#59511yescharacteristic
201C8—682no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—342no—
207C22—341no—
208C22—681no—
209C22—681no—
210C22—682no—
211C22—682no—
212C22—681no—
213C22C2×C22⋊#59511yescharacteristic
214C4—21no—
215C4—41no—
216C4—41no—
217C4—171no—
218C4—171no—
219C4—341no—
220C4—341no—
221C4—681no—
222C4—1362no—
223C2—21no—
224C2—41no—
225C2—41no—
226C2—681no—
227C2—682no—
228C2D17⋊#9911yescharacteristic
subgroupG/H
centerC2D17⋊#99
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC17×#123C4
socleC2×C17C2×#99
prime radical#11763317C2
cosocleC2×C22⋊D17C23
hypercenter#123#595
hypocenterC17#2120
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
1. #11763796C2semidirect
2. #11763789C2semidirect
3. #11763711C2semidirect
4. #11763710C2semidirect
7. C22×C4×D17C4semidirect
10. C17×#123C4semidirect
11. #10587623C4semidirect
18. #10587563C4semidirect
124. #123#595semidirect
137. C17#2120semidirect

Maximal subgroups

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

subgroupindex
#117637962
#117637892
#117637112
#117637102
#117633172
#212017