Finite Group Databasebeta

← #14527342

Subgroups — #14527342

Order 2176.

264 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 69 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#11763354C211yesmaximal, characteristic
2#11762859C211yesmaximal, characteristic, complemented
3#11762673C211yesmaximal, characteristic, complemented
4#11762544C211yesmaximal, characteristic, complemented
5#11762543C211yesmaximal, characteristic, complemented
6#11762542C211yesmaximal, characteristic, complemented
7C17×#327C211yesmaximal, characteristic, complemented
8C2×#63254—21no—
9C17×#121C2211yescharacteristic
10#10587626C411yescharacteristic, complemented
11#10587625—21no—
12D4×Dic17C411yescharacteristic, complemented
13#10587616C2211yescharacteristic
14C2×#63225—21no—
15C2×C4×Dic17C2211yescharacteristic
16#10587579C411yescharacteristic, complemented
17#10587574C411yescharacteristic, complemented
18#10587571—21no—
19#10587567—21no—
20C17×C22⋊C8C2211yescharacteristic
21C17×#95C2211yescharacteristic
22#10587502—21no—
23#10587502C2211yescharacteristic
24C22⋊#3622C2211yescharacteristic
25#10587485—21no—
26C8×Dic17—21no—
27C2×D4×C17—21no—
28C2×D4×C17C2×C411yescharacteristic
29C22×C4×C17—21no—
30C22×C4×C17C2311yescharacteristic
31C2×C22⋊D17—21no—
32C2×C22⋊D17—21no—
33C2×C22⋊D17—21no—
34C22×Dic17—21no—
35C22×Dic17—21no—
36C22×Dic17D411yescharacteristic
37C22×Dic17D411yescharacteristic
38#63254—42no—
39#63254—42no—
40C2×D68C2×C411yescharacteristic
41C2×C4×D17—21no—
42C2×Dic34—21no—
43C2×C8×C17—21no—
44C4⋊C4×C17C2×C411yescharacteristic
45C22⋊C4×C17—21no—
46C22⋊Dic17—21no—
47#63227—21no—
48#63227—21no—
49C4⋊Dic17C2×C411yescharacteristic
50#63225—21no—
51#63225—42no—
52C4×Dic17C2×C411yescharacteristic
53C4×Dic17C2×C411yescharacteristic
54C4×Dic17D411yescharacteristic
55C4×Dic17D411yescharacteristic
56C2×#3622—21no—
57C23×C17—21no—
58C23×C17C2×D411yescharacteristic
59C22×D17—21no—
60D4×C17—41no—
61D4×C17—41no—
62D4×C17—41no—
63C2×C4×C17—21no—
64C2×C4×C17—41no—
65C2×C4×C17—41no—
66C2×C4×C17—41no—
67C2×C4×C17—41no—
68C2×C4×C17C22×C411yescharacteristic
69C2×C4×C17C2×D411yescharacteristic
70C22⋊D17—42no—
71C22⋊D17—41no—
72C22⋊D17—41no—
73C22⋊D17—42no—
74C2×Dic17—21no—
75C2×Dic17—41no—
76C2×Dic17—42no—
77C2×Dic17—41no—
78C2×Dic17—41no—
79C2×Dic17—41no—
80C2×Dic17C22⋊C411yescharacteristic
81C2×Dic17C22⋊C411yescharacteristic
82C2×Dic17C22⋊C411yescharacteristic
83C2×Dic17C22⋊C411yescharacteristic
84C2×Dic17D812yes—
85C2×Dic17SD1612yes—
86D68—41no—
87C4×D17—42no—
88Dic34—42no—
89C8×C17—41no—
90#3622—41no—
91#1517—171nomaximal, Sylow
92C22×C17—21no—
93C22×C17—21no—
94C22×C17—41no—
95C22×C17—41no—
96C22×C17—41no—
97C22×C17C2×C22⋊C411yescharacteristic
98C22×C17C2×D811yescharacteristic
99C22×C17C2×SD1611yescharacteristic
100C2×D17—41no—
101C2×D17—41no—
102C2×D17—41no—
103C4×C17—21no—
104C4×C17—21no—
105C4×C17—41no—
106C4×C17—81no—
107Dic17—21no—
108Dic17—21no—
109Dic17—41no—
110Dic17—81no—
111Dic17#10212yes—
112Dic17#10212yes—
113#532—171no—
114#431—171no—
115#382—171no—
116#327—171no—
117#327—171no—
118#327—171no—
119#327D1711yescharacteristic, complemented
120C2×C17—21no—
121C2×C17—21no—
122C2×C17—81no—
123C2×C17#41011yescharacteristic
124C2×C17C2×#10211yescharacteristic
125C2×C17#42111yescharacteristic
126D17—81no—
127C2×#41—341no—
128C4⋊Q8—171no—
129#127—171no—
130#124—341no—
131#121—171no—
132#121—341no—
133#121C2×D1711yescharacteristic
134C4×D4—171no—
135C4×D4—171no—
136C4×D4—341no—
137C2×C4⋊C4—341no—
138C2×C42—171no—
139C4⋊C8—341no—
140C22⋊C8—171no—
141C22⋊C8C2×D1711yescharacteristic
142C4×C8—341no—
143#95—171no—
144#95—341no—
145#95C2×D1711yescharacteristic
146C17#151711yescharacteristic, complemented, Sylow
147#41—682no—
148#41—682no—
149C2×Q8—341no—
150C2×D4—21no—
151C2×D4—171no—
152C2×D4—341no—
153C2×D4—341no—
154C2×D4—341no—
155C2×D4C4×D1711yescharacteristic
156C22×C4—21no—
157C22×C4—171no—
158C22×C4—171no—
159C22×C4—341no—
160C22×C4—341no—
161C22×C4—341no—
162C22×C4C22×D1711yescharacteristic
163C2×C8—21no—
164C2×C8—341no—
165C4⋊C4—171no—
166C4⋊C4—341no—
167C4⋊C4—682no—
168C4⋊C4C4×D1711yescharacteristic
169C22⋊C4—21no—
170C22⋊C4—341no—
171C22⋊C4—341no—
172C22⋊C4—341no—
173C42—171no—
174C42—171no—
175C42—171no—
176C42—171no—
177C23—21no—
178C23—341no—
179C23D4×D1711yescharacteristic
180Q8—682no—
181D4—41no—
182D4—41no—
183D4—41no—
184D4—681no—
185D4—682no—
186D4—682no—
187D4—681no—
188D4—681no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—171no—
195C2×C4—171no—
196C2×C4—172no—
197C2×C4—172no—
198C2×C4—171no—
199C2×C4—171no—
200C2×C4—341no—
201C2×C4—682no—
202C2×C4—682no—
203C2×C4—681no—
204C2×C4—681no—
205C2×C4—681no—
206C2×C4—681no—
207C2×C4C2×C4×D1711yescharacteristic
208C2×C4D4×D1711yescharacteristic
209C8—41no—
210C8—681no—
211C22—21no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—681no—
217C22—681no—
218C22—681no—
219C22C22⋊C4×D1711yescharacteristic
220C22D8×D1711yescharacteristic
221C22SD16×D1711yescharacteristic
222C4—21no—
223C4—21no—
224C4—41no—
225C4—81no—
226C4—172no—
227C4—172no—
228C4—341no—
229C4—341no—
230C4—681no—
231C4—1361no—
232C2—21no—
233C2—21no—
234C2—81no—
235C2—1361no—
236C2#1176278111yescharacteristic
237C2#102×D1711yescharacteristic
238C2#1176282411yescharacteristic
subgroupG/H
centerC22C22⋊C4×D17
commutatorC2×C4×C17C22×C4
FrattiniC22×C4C22×D17
FittingC17×#327C2
socleC22×C17C2×C22⋊C4
prime radical#11763354C2
cosocleC22×C4×C17C23
hypercenter#327D17
hypocenterC17#1517
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #11762859C2semidirect
3. #11762673C2semidirect
4. #11762544C2semidirect
5. #11762543C2semidirect
6. #11762542C2semidirect
7. C17×#327C2semidirect
10. #10587626C4semidirect
12. D4×Dic17C4semidirect
16. #10587579C4semidirect
17. #10587574C4semidirect
119. #327D17semidirect
146. C17#1517semidirect

Maximal subgroups

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

subgroupindex
#117633542
#117628592
#117626732
#117625442
#117625432
#117625422
C17×#3272
#151717