Finite Group Databasebeta

← #14530447

Subgroups — #14530447

Order 2176.

258 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 65 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#11763355C211yesmaximal, characteristic
2#11762783C211yesmaximal, characteristic, complemented
3#11762603C212yesmaximal, complemented
4#11762601C212yesmaximal, complemented
5C17×#410C211yesmaximal, characteristic, complemented
6C2×#41×C17C2211yescharacteristic
7C2×#63252C411yescharacteristic, complemented
8C17×#117—21no—
9#10587633C411yescharacteristic, complemented
10#10587630C411yescharacteristic, complemented
11#10587626C2211yescharacteristic
12#10587625—21no—
13#10587625—21no—
14#10587622C2211yescharacteristic
15C4×C22⋊D17—21no—
16C4×C22⋊D17—21no—
17#10587612—21no—
18C2×C4×Dic17C411yescharacteristic, complemented
19C17×#99—21no—
20C17×#99C2212yes—
21#10587505—21no—
22#10587505C2212yes—
23#41×C17—41no—
24#41×C17—41no—
25C2×Q8×C17C2×C411yescharacteristic
26C2×D4×C17D411yescharacteristic
27C2×D4×C17D411yescharacteristic
28C2×D4×C17C2311yescharacteristic
29C22×C4×C17C2×C411yescharacteristic
30C22×C4×C17D411yescharacteristic
31C22×C4×C17D411yescharacteristic
32C2×C22⋊D17—21no—
33C2×C22⋊D17—21no—
34C2×C22⋊D17—21no—
35C22×Dic17—21no—
36#63252—41no—
37#63252—41no—
38C2×D68C2×C411yescharacteristic
39C2×C4×D17—21no—
40C2×Dic34C2×C411yescharacteristic
41C4⋊C4×C17—41no—
42C22⋊C4×C17—22no—
43C42×C17—41no—
44C22⋊Dic17—21no—
45C22⋊Dic17—21no—
46C22⋊Dic17—22no—
47#63227—21no—
48#63227—21no—
49C4⋊Dic17—41no—
50#63225—21no—
51#63225—21no—
52C4×Dic17—21no—
53C4×Dic17—41no—
54C4×Dic17C2×C411yescharacteristic
55C4×Dic17C2×C411yescharacteristic
56C23×C17C22⋊C411yescharacteristic
57C23×C17C2×D411yescharacteristic
58C23×C17C2×D411yescharacteristic
59C22×D17—21no—
60Q8×C17—41no—
61D4×C17—41no—
62D4×C17—41no—
63D4×C17—41no—
64C2×C4×C17—21no—
65C2×C4×C17—21no—
66C2×C4×C17—21no—
67C2×C4×C17—21no—
68C2×C4×C17—21no—
69C2×C4×C17—21no—
70C2×C4×C17—42no—
71C2×C4×C17C22⋊C411yescharacteristic
72C2×C4×C17C22⋊C411yescharacteristic
73C2×C4×C17C22⋊C411yescharacteristic
74C2×C4×C17C22×C411yescharacteristic
75C22⋊D17—42no—
76C22⋊D17—42no—
77C22⋊D17—42no—
78C2×Dic17—21no—
79C2×Dic17—21no—
80C2×Dic17—21no—
81C2×Dic17—42no—
82C2×Dic17—42no—
83D68—41no—
84C4×D17—41no—
85C4×D17—41no—
86Dic34—41no—
87#2126—171nomaximal, Sylow
88C22×C17—21no—
89C22×C17—21no—
90C22×C17—21no—
91C22×C17—41no—
92C22×C17—41no—
93C22×C17—41no—
94C22×C17C2×C22⋊C411yescharacteristic
95C2×D17—41no—
96C2×D17—41no—
97C2×D17—41no—
98C4×C17—21no—
99C4×C17—21no—
100C4×C17—21no—
101C4×C17—82no—
102C4×C17#9911yescharacteristic
103C4×C17#9911yescharacteristic
104Dic17—22no—
105Dic17—41no—
106Dic17—41no—
107Dic17—82no—
108#532—171no—
109#410—171no—
110#410D1711yescharacteristic, complemented
111#354—172no—
112#353—172no—
113C2×C17—21no—
114C2×C17—41no—
115C2×C17—41no—
116C2×C17—41no—
117C2×C17C2×#9911yescharacteristic
118D17—81no—
119C2×#41—171no—
120C2×#41C2×D1711yescharacteristic
121C4⋊Q8—171no—
122#127—171no—
123#124—171no—
124#124—171no—
125#121—341no—
126#121—341no—
127C4×D4—341no—
128C4×D4—341no—
129#117—21no—
130#117—341no—
131C2×C42—171no—
132#99—21no—
133#99—172no—
134#99—341no—
135#99C2×D1712yes—
136C17#212611yescharacteristic, complemented, Sylow
137#41—41no—
138#41—41no—
139#41—681no—
140#41—681no—
141C2×Q8—171no—
142C2×Q8C4×D1711yescharacteristic
143C2×D4—171no—
144C2×D4—341no—
145C2×D4—341no—
146C2×D4—341no—
147C2×D4D6811yescharacteristic
148C2×D4C22⋊D1711yescharacteristic
149C2×D4C22×D1711yescharacteristic
150C22×C4—341no—
151C22×C4—341no—
152C22×C4C4×D1711yescharacteristic
153C22×C4D6811yescharacteristic
154C22×C4C22⋊D1711yescharacteristic
155C4⋊C4—41no—
156C4⋊C4—341no—
157C4⋊C4—341no—
158C4⋊C4—681no—
159C22⋊C4—22no—
160C22⋊C4—341no—
161C22⋊C4—341no—
162C22⋊C4—341no—
163C22⋊C4—341no—
164C22⋊C4—342no—
165C42—41no—
166C42—171no—
167C42—171no—
168C42—341no—
169C42—681no—
170C23—341no—
171C23#6322711yescharacteristic
172C23C2×D6811yescharacteristic
173C23C2×C22⋊D1711yescharacteristic
174Q8—41no—
175Q8—681no—
176D4—41no—
177D4—41no—
178D4—41no—
179D4—681no—
180D4—682no—
181D4—682no—
182D4—682no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—341no—
191C2×C4—341no—
192C2×C4—341no—
193C2×C4—682no—
194C2×C4—681no—
195C2×C4—681no—
196C2×C4—682no—
197C2×C4#6322711yescharacteristic
198C2×C4#6322711yescharacteristic
199C2×C4#6322711yescharacteristic
200C2×C4C2×C4×D1711yescharacteristic
201C22—21no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—681no—
208C22—681no—
209C22—681no—
210C22C2×#6322711yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—82no—
215C4—342no—
216C4—681no—
217C4—681no—
218C4—1362no—
219C4#1058747711yescharacteristic
220C4#1058747711yescharacteristic
221C2—21no—
222C2—41no—
223C2—41no—
224C2—41no—
225C2—1361no—
226C2C2×#1058747711yescharacteristic
subgroupG/H
centerC4#10587477
commutatorC2×C4×C17C22×C4
FrattiniC2×D4C22×D17
FittingC17×#410C2
socleC2×C17C2×#99
prime radical#11763355C2
cosocleC2×D4×C17C23
hypercenter#410D17
hypocenterC17#2126
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #11762783C2semidirect
3. #11762603C2semidirect
4. #11762601C2semidirect
5. C17×#410C2semidirect
7. C2×#63252C4semidirect
9. #10587633C4semidirect
10. #10587630C4semidirect
18. C2×C4×Dic17C4semidirect
110. #410D17semidirect
136. C17#2126semidirect

Maximal subgroups

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

subgroupindex
#117633552
#117627832
#117626032
#117626012
C17×#4102
#212617