Finite Group Databasebeta

← #14527340

Subgroups — #14527340

Order 2176.

284 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C2×C22⋊#3622C211yesmaximal, characteristic, complemented
2#11762672C211yesmaximal, characteristic, complemented
3#11762546C212yesmaximal, complemented
4#11762543C212yesmaximal, complemented
5C17×#525C211yesmaximal, characteristic
6C2×#41×C17C411yescharacteristic, complemented
7C23×C4×C17C2211yescharacteristic
8C17×#123—21no—
9C17×C22⋊Q8C411yescharacteristic, complemented
10C17×C22⋊Q8C2211yescharacteristic
11C17×#121C411yescharacteristic, complemented
12C17×#121C2211yescharacteristic
13C17×#120—21no—
14C4×D4×C17—21no—
15C4×D4×C17—21no—
16C17×#117C411yescharacteristic, complemented
17C2×C22⋊Dic17—21no—
18C22×#3622—21no—
19#10587502—21no—
20#10587502C2212yes—
21C22⋊#3622—21no—
22C22⋊#3622C2212yes—
23C24×C17D411yescharacteristic
24#41×C17—42no—
25C2×Q8×C17C2×C411yescharacteristic
26C2×D4×C17—21no—
27C2×D4×C17—21no—
28C2×D4×C17—21no—
29C2×D4×C17C2×C411yescharacteristic
30C22×C4×C17—21no—
31C22×C4×C17—42no—
32C22×C4×C17C2×C411yescharacteristic
33C22×C4×C17C2×C411yescharacteristic
34C22×C4×C17D411yescharacteristic
35C22×C4×C17D411yescharacteristic
36C22×C4×C17D411yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C22×Dic17—22no—
39C4⋊C4×C17—21no—
40C4⋊C4×C17—21no—
41C4⋊C4×C17C2×C411yescharacteristic
42C4⋊C4×C17C2×C411yescharacteristic
43C22⋊C4×C17—21no—
44C22⋊C4×C17—21no—
45C22⋊C4×C17—21no—
46C22⋊C4×C17—21no—
47C22⋊C4×C17—21no—
48C42×C17—21no—
49C22⋊Dic17—42no—
50C2×#3622—22no—
51C2×#3622—42no—
52C23×C17—22no—
53C23×C17—22no—
54C23×C17—21no—
55C23×C17—41no—
56C23×C17C22⋊C411yescharacteristic
57C23×C17C22⋊C411yescharacteristic
58C23×C17C2×D411yescharacteristic
59Q8×C17—41no—
60D4×C17—41no—
61D4×C17—42no—
62D4×C17—42no—
63D4×C17—42no—
64C2×C4×C17—22no—
65C2×C4×C17—22no—
66C2×C4×C17—21no—
67C2×C4×C17—21no—
68C2×C4×C17—21no—
69C2×C4×C17—41no—
70C2×C4×C17—41no—
71C2×C4×C17—41no—
72C2×C4×C17—41no—
73C2×C4×C17—42no—
74C2×C4×C17C22⋊C411yescharacteristic
75C2×C4×C17C22⋊C411yescharacteristic
76C2×C4×C17C22×C411yescharacteristic
77C2×C4×C17C2×D411yescharacteristic
78C2×Dic17—42no—
79C2×Dic17—42no—
80C2×Dic17—42no—
81#3622—42no—
82#1516—171nomaximal, Sylow
83C22×C17—21no—
84C22×C17—21no—
85C22×C17—21no—
86C22×C17—21no—
87C22×C17—21no—
88C22×C17—21no—
89C22×C17—21no—
90C22×C17—44no—
91C22×C17—41no—
92C22×C17—41no—
93C22×C17—41no—
94C22×C17#9912yes—
95C22×C17C2×C22⋊C411yescharacteristic
96C22×C17#13512yes—
97C4×C17—21no—
98C4×C17—21no—
99C4×C17—21no—
100C4×C17—21no—
101C4×C17—41no—
102C4×C17—41no—
103C4×C17—41no—
104Dic17—82no—
105#525D1711yescharacteristic
106C2×C22⋊C8—171no—
107#381—171no—
108#328—172no—
109#327—172no—
110C2×C17—22no—
111C2×C17—22no—
112C2×C17—41no—
113C2×C17—81no—
114C2×C17C2×#9911yescharacteristic
115C2×C17#41611yescharacteristic
116C2×C17#42111yescharacteristic
117C2×#41Dic1711yescharacteristic, complemented
118C23×C4C2×D1711yescharacteristic
119C22×C8—341no—
120#123—21no—
121C22⋊Q8Dic1711yescharacteristic, complemented
122C22⋊Q8C2×D1711yescharacteristic
123#121Dic1711yescharacteristic, complemented
124#121C2×D1711yescharacteristic
125#120—21no—
126C4×D4—21no—
127C4×D4—21no—
128#117Dic1711yescharacteristic, complemented
129C2×C22⋊C4—341no—
130C22⋊C8—172no—
131C22⋊C8—341no—
132#95—172no—
133#95—341no—
134C17#151611yescharacteristic, complemented, Sylow
135C24C22⋊D1711yescharacteristic
136#41—42no—
137C2×Q8C2×Dic1711yescharacteristic
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4C2×Dic1711yescharacteristic
142C22×C4—21no—
143C22×C4—42no—
144C22×C4—342no—
145C22×C4C2×Dic1711yescharacteristic
146C22×C4C2×Dic1711yescharacteristic
147C22×C4C22⋊D1711yescharacteristic
148C22×C4C22⋊D1711yescharacteristic
149C22×C4C22⋊D1711yescharacteristic
150C22×C4C22×D1711yescharacteristic
151C2×C8—342no—
152C2×C8—682no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4C2×Dic1711yescharacteristic
156C4⋊C4C2×Dic1711yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—682no—
163C42—21no—
164C23—22no—
165C23—22no—
166C23—21no—
167C23—41no—
168C23C22⋊Dic1711yescharacteristic
169C23C22⋊Dic1711yescharacteristic
170C23C2×C22⋊D1711yescharacteristic
171Q8—41no—
172D4—41no—
173D4—42no—
174D4—42no—
175D4—42no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—682no—
187C2×C4—682no—
188C2×C4—682no—
189C2×C4C22⋊Dic1711yescharacteristic
190C2×C4C22⋊Dic1711yescharacteristic
191C2×C4C22×Dic1711yescharacteristic
192C2×C4C2×C22⋊D1711yescharacteristic
193C8—682no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22#1058750512yes—
206C22#1058763612yes—
207C22C2×C22⋊Dic1711yescharacteristic
208C4—21no—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—41no—
213C4—41no—
214C4—41no—
215C4—1362no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—81no—
220C2C2×#1058750511yescharacteristic
221C2#1176280511yescharacteristic
222C2#1176282611yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic17
commutatorC2×C4×C17C22×C4
FrattiniC22×C4C22×D17
Fitting, prime radicalC17×#525C2
socleC22×C17C2×C22⋊C4
cosocleC22×C4×C17C23
hypercenter#525D17
hypocenterC17#1516
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊#3622C2semidirect
2. #11762672C2semidirect
3. #11762546C2semidirect
4. #11762543C2semidirect
6. C2×#41×C17C4semidirect
9. C17×C22⋊Q8C4semidirect
11. C17×#121C4semidirect
16. C17×#117C4semidirect
117. C2×#41Dic17semidirect
121. C22⋊Q8Dic17semidirect
123. #121Dic17semidirect
128. #117Dic17semidirect
134. C17#1516semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊#36222
#117626722
#117625462
#117625432
C17×#5252
#151617