Finite Group Databasebeta

← #14527273

Subgroups — #14527273

Order 2176.

304 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 77 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#11763320C211yesmaximal, characteristic
2C2×C22⋊#3622C211yesmaximal, characteristic, complemented
3#11762545C212yesmaximal, complemented
4#11762542C212yesmaximal, complemented
5C4×C22⋊C4×C17C211yesmaximal, characteristic, complemented
6C23×C4×C17C2211yescharacteristic
7C2×#63252C411yescharacteristic, complemented
8C2×C22⋊C4×C17—21no—
9C2×C42×C17—21no—
10#10587639—21no—
11#10587616C411yescharacteristic, complemented
12#10587616C2211yescharacteristic
13#10587615—21no—
14C4×C22⋊D17—21no—
15C4×C22⋊D17—21no—
16#10587612C411yescharacteristic, complemented
17#10587610C411yescharacteristic, complemented
18#10587610C2211yescharacteristic
19C22×#3622—21no—
20C17×#95C2212yes—
21C22⋊#3622—21no—
22C22⋊#3622C2212yes—
23C24×C17D411yescharacteristic
24C22×C4×C17—22no—
25C22×C4×C17—42no—
26C22×C4×C17C2×C411yescharacteristic
27C22×C4×C17C2×C411yescharacteristic
28C22×C4×C17D411yescharacteristic
29C22×C4×C17D411yescharacteristic
30C22×C4×C17D411yescharacteristic
31C22×C4×C17C2311yescharacteristic
32C2×C22⋊D17—21no—
33C2×C22⋊D17—21no—
34C2×C22⋊D17—21no—
35#63252—42no—
36C2×D68C2×C411yescharacteristic
37C2×C4×D17—21no—
38C2×Dic34C2×C411yescharacteristic
39C22⋊C4×C17—24no—
40C42×C17—42no—
41C22⋊Dic17—21no—
42C22⋊Dic17—21no—
43C22⋊Dic17—21no—
44#63227—21no—
45#63227—21no—
46C4⋊Dic17C2×C411yescharacteristic
47C4⋊Dic17C2×C411yescharacteristic
48#63225—21no—
49#63225—21no—
50C4×Dic17—21no—
51C2×#3622—22no—
52C2×#3622—42no—
53C23×C17—22no—
54C23×C17—22no—
55C23×C17—41no—
56C23×C17C22⋊C411yescharacteristic
57C23×C17C22⋊C411yescharacteristic
58C23×C17C2×D411yescharacteristic
59C22×D17—21no—
60C2×C4×C17—22no—
61C2×C4×C17—22no—
62C2×C4×C17—22no—
63C2×C4×C17—24no—
64C2×C4×C17—41no—
65C2×C4×C17—41no—
66C2×C4×C17—41no—
67C2×C4×C17—42no—
68C2×C4×C17—42no—
69C2×C4×C17C22⋊C411yescharacteristic
70C2×C4×C17C22⋊C411yescharacteristic
71C2×C4×C17C22×C411yescharacteristic
72C2×C4×C17C2×D411yescharacteristic
73C22⋊D17—42no—
74C22⋊D17—42no—
75C22⋊D17—42no—
76C2×Dic17—21no—
77C2×Dic17—21no—
78C2×Dic17—21no—
79D68—41no—
80C4×D17—42no—
81Dic34—41no—
82#3622—42no—
83#1501—171nomaximal, Sylow
84C22×C17—21no—
85C22×C17—21no—
86C22×C17—21no—
87C22×C17—21no—
88C22×C17—21no—
89C22×C17—21no—
90C22×C17—21no—
91C22×C17—44no—
92C22×C17#10412yes—
93C22×C17C2×C22⋊C411yescharacteristic
94C22×C17#13512yes—
95C2×D17—41no—
96C2×D17—41no—
97C2×D17—41no—
98C4×C17—21no—
99C4×C17—21no—
100C4×C17—21no—
101C4×C17—44no—
102C4×C17#9912yes—
103Dic17—41no—
104Dic17—41no—
105Dic17—41no—
106#525—171no—
107C2×C22⋊C8—171no—
108C4×C22⋊C4D1711yescharacteristic, complemented
109#328—172no—
110#327—172no—
111C2×C17—22no—
112C2×C17—22no—
113C2×C17—41no—
114C2×C17C2×#9911yescharacteristic
115C2×C17#41611yescharacteristic
116C2×C17C2×#10411yescharacteristic
117D17—81no—
118C2×#41—171no—
119C23×C4C2×D1711yescharacteristic
120C22×C8—341no—
121#123—341no—
122C22⋊Q8—171no—
123C22⋊Q8—171no—
124#121—171no—
125#121—171no—
126#120—341no—
127C4×D4—341no—
128C4×D4—341no—
129#117—171no—
130C2×C22⋊C4—21no—
131C2×C42—21no—
132C22⋊C8—172no—
133C22⋊C8—341no—
134#95C2×D1712yes—
135C17#150111yescharacteristic, complemented, Sylow
136C24C22⋊D1711yescharacteristic
137#41—682no—
138C2×Q8—171no—
139C2×D4—171no—
140C2×D4—341no—
141C2×D4—341no—
142C2×D4—341no—
143C22×C4—22no—
144C22×C4—42no—
145C22×C4—341no—
146C22×C4C4×D1711yescharacteristic
147C22×C4C4×D1711yescharacteristic
148C22×C4D6811yescharacteristic
149C22×C4D6811yescharacteristic
150C22×C4C22⋊D1711yescharacteristic
151C22×C4C22×D1711yescharacteristic
152C2×C8—342no—
153C2×C8—682no—
154C4⋊C4—171no—
155C4⋊C4—171no—
156C4⋊C4—341no—
157C4⋊C4—341no—
158C22⋊C4—24no—
159C22⋊C4—341no—
160C22⋊C4—341no—
161C22⋊C4—341no—
162C22⋊C4—341no—
163C22⋊C4—341no—
164C42—42no—
165C42—341no—
166C23—22no—
167C23—22no—
168C23—41no—
169C23—341no—
170C23#6322711yescharacteristic
171C23#6322711yescharacteristic
172C23C2×C22⋊D1711yescharacteristic
173Q8—681no—
174D4—681no—
175D4—682no—
176D4—682no—
177D4—682no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—341no—
188C2×C4—341no—
189C2×C4—341no—
190C2×C4—682no—
191C2×C4#6322711yescharacteristic
192C2×C4#6322711yescharacteristic
193C2×C4C2×C4×D1711yescharacteristic
194C2×C4C2×D6811yescharacteristic
195C8—682no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—44no—
204C22—681no—
205C22—681no—
206C22—681no—
207C22#1058747612yes—
208C22C2×#6322711yescharacteristic
209C22#1058763612yes—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—44no—
214C4—681no—
215C4—681no—
216C4—681no—
217C4#1058747712yes—
218C2—22no—
219C2—22no—
220C2—41no—
221C2—1361no—
222C2C2×#1058747711yescharacteristic
223C2#1176280311yescharacteristic
224C2C2×#1058747611yescharacteristic
subgroupG/H
centerC2×C4#63227
commutatorC2×C4×C17C22×C4
FrattiniC22×C4C22×D17
FittingC4×C22⋊C4×C17C2
socleC22×C17C2×C22⋊C4
prime radical#11763320C2
cosocleC22×C4×C17C23
hypercenterC4×C22⋊C4D17
hypocenterC17#1501
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊#3622C2semidirect
3. #11762545C2semidirect
4. #11762542C2semidirect
5. C4×C22⋊C4×C17C2semidirect
7. C2×#63252C4semidirect
11. #10587616C4semidirect
16. #10587612C4semidirect
17. #10587610C4semidirect
108. C4×C22⋊C4D17semidirect
135. C17#1501semidirect

Maximal subgroups

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

subgroupindex
#117633202
C2×C22⋊#36222
#117625452
#117625422
C4×C22⋊C4×C172
#150117