Finite Group Databasebeta

← #14528752

Subgroups — #14528752

Order 2176.

588 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 97 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×#10587696C211yesmaximal, characteristic
2#11762827C212yesmaximal, complemented
3#11762813C212yesmaximal, complemented
4#11762793C211yesmaximal, characteristic, complemented
5C17×#410C211yesmaximal, characteristic, complemented
6C2×#41×C17C2211yescharacteristic
7#10587696—24no—
8#10587696—24no—
9C2×#63256—21no—
10C2×D4×D17—22no—
11C2×D4×D17—21no—
12C2×#63252C412yescomplemented
13C2×#63252C2211yescharacteristic
14C22×D68C412yescomplemented
15C22×D68C2211yescharacteristic
16C17×#117C2212yescomplemented
17C2×#63223C2212yescomplemented
18C17×#99—22no—
19#10587507—21no—
20#10587504—21no—
21#10587480—24no—
22#10587476—24no—
23C23×D17—22no—
24C23×D17—21no—
25#41×C17—22no—
26#41×C17—22no—
27C2×Q8×C17D411yescharacteristic
28C2×D4×C17D411yescharacteristic
29C2×D4×C17D412yes—
30C22×C4×C17C2×C412yes—
31C22×C4×C17C2311yescharacteristic
32C2×C22⋊D17—22no—
33C2×C22⋊D17—21no—
34#63256—28no—
35D4×D17—216no—
36D4×D17—28no—
37#63252—22no—
38#63252—24no—
39#63252—24no—
40#63252D414yes—
41C2×D68—22no—
42C2×D68—21no—
43C2×D68—22no—
44C2×D68—24no—
45C2×D68—24no—
46C2×D68C2×C411yescharacteristic
47C2×D68C2×C412yes—
48C2×D68D414yes—
49C2×C4×D17—22no—
50C2×C4×D17—21no—
51C2×Dic34C2×C411yescharacteristic
52C4⋊C4×C17—22no—
53C22⋊C4×C17—22no—
54C42×C17—22no—
55#63223—22no—
56#63223—22no—
57C2×#3622—22no—
58C23×C17C22⋊C412yes—
59C23×C17C2×D411yescharacteristic
60C22×D17—22no—
61C22×D17—21no—
62C22×D17—28no—
63C22×D17—24no—
64C22×D17—82no—
65C22×D17—84no—
66Q8×C17—22no—
67D4×C17—24no—
68D4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—42no—
72C2×C4×C17C22⋊C412yes—
73C2×C4×C17C22×C411yescharacteristic
74C2×C4×C17C2×D411yescharacteristic
75C2×C4×C17C2×D412yes—
76C2×C4×C17C2×D412yes—
77C22⋊D17—28no—
78C22⋊D17—24no—
79C2×Dic17—21no—
80D68—28no—
81D68—21no—
82D68—22no—
83D68—24no—
84D68C22⋊C412yes—
85D68C22⋊C414yes—
86C4×D17—28no—
87C4×D17—24no—
88Dic34—21no—
89Dic34C22⋊C412yes—
90#3622—42no—
91#1795—171nomaximal, Sylow
92C22×C17—22no—
93C22×C17—41no—
94C22×C17—42no—
95C22×C17C2×C22⋊C411yescharacteristic
96C22×C17#12012yes—
97C2×D17—24no—
98C2×D17—22no—
99C2×D17—44no—
100C2×D17—42no—
101C2×D17—84no—
102C2×D17—88no—
103C4×C17—22no—
104C4×C17—82no—
105C4×C17C2×C22⋊C411yescharacteristic
106C4×C17C2×C22⋊C411yescharacteristic
107C4×C17#12012yes—
108Dic17—22no—
109C2×H(5,𝔽2)—171no—
110#421—172no—
111#418—172no—
112#413—171no—
113#410D1711yescharacteristic, complemented
114C2×C17—22no—
115C2×C17—21no—
116C2×C17—42no—
117C2×C17#37911yescharacteristic
118D17—44no—
119D17—42no—
120H(5,𝔽2)—344no—
121H(5,𝔽2)—344no—
122C2×#41—172no—
123C2×#41—171no—
124C2×#41—341no—
125C2×#41C2×D1711yescharacteristic
126C22×D4—172no—
127C22×D4—171no—
128C22×D4—341no—
129C22×D4—342no—
130C2×M16—172no—
131#117C2×D1712yescomplemented
132#104—344no—
133#102—344no—
134#101—341no—
135#100—341no—
136#99—22no—
137C17#179511yescharacteristic, complemented, Sylow
138C24—341no—
139C24—342no—
140#41—22no—
141#41—22no—
142#41—174no—
143#41—344no—
144#41—344no—
145#41—348no—
146#41—342no—
147C2×Q8—171no—
148C2×Q8D6811yescharacteristic
149C2×D4—172no—
150C2×D4—171no—
151C2×D4—174no—
152C2×D4—344no—
153C2×D4—344no—
154C2×D4—3416no—
155C2×D4—348no—
156C2×D4—341no—
157C2×D4—342no—
158C2×D4—341no—
159C2×D4—342no—
160C2×D4—342no—
161C2×D4D6811yescharacteristic
162C2×D4C22⋊D1712yes—
163C22×C4—341no—
164C22×C4—342no—
165C22×C4C4×D1712yes—
166C22×C4C22×D1711yescharacteristic
167M16—342no—
168M16—342no—
169C2×C8—342no—
170C4⋊C4—22no—
171C22⋊C4—22no—
172C42—22no—
173C23—348no—
174C23—341no—
175C23—342no—
176C23—344no—
177C23—1362no—
178C23—1364no—
179C23#6322712yes—
180C23C2×C22⋊D1711yescharacteristic
181Q8—22no—
182Q8—172no—
183Q8—341no—
184D4—24no—
185D4—22no—
186D4—174no—
187D4—172no—
188D4—348no—
189D4—348no—
190D4—344no—
191D4—341no—
192D4—342no—
193D4—344no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—348no—
198C2×C4—341no—
199C2×C4—344no—
200C2×C4#6322712yes—
201C2×C4C2×C4×D1711yescharacteristic
202C2×C4C2×D6811yescharacteristic
203C2×C4D4×D1712yes—
204C2×C4D4×D1712yes—
205C8—682no—
206C22—22no—
207C22—41no—
208C22—42no—
209C22—344no—
210C22—342no—
211C22—684no—
212C22—682no—
213C22—1364no—
214C22—1368no—
215C22C2×#6322711yescharacteristic
216C22#1058762312yes—
217C4—22no—
218C4—82no—
219C4—342no—
220C4C22⋊C4×D1711yescharacteristic
221C4C22⋊C4×D1711yescharacteristic
222C4#1058756812yes—
223C2—22no—
224C2—21no—
225C2—42no—
226C2—684no—
227C2—682no—
228C2#1176266411yescharacteristic
subgroupG/H
centerC2#11762664
commutatorC2×C4×C17C22×C4
FrattiniC22×C4C22×D17
FittingC17×#410C2
socleC2×C17#379
prime radicalC2×#10587696C2
cosocleC22×C4×C17C23
hypercenter#410D17
hypocenterC17#1795
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #11762827C2semidirect
3. #11762813C2semidirect
4. #11762793C2semidirect
5. C17×#410C2semidirect
12. C2×#63252C4semidirect
14. C22×D68C4semidirect
16. C17×#117C22semidirect
17. C2×#63223C22semidirect
113. #410D17semidirect
131. #117C2×D17semidirect
137. C17#1795semidirect

Maximal subgroups

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

subgroupindex
C2×#105876962
#117628272
#117628132
#117627932
C17×#4102
#179517