Finite Group Databasebeta

← #14527321

Subgroups — #14527321

Order 2176.

222 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 61 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#11763345C211yesmaximal, characteristic
2#11762863C211yesmaximal, characteristic, complemented
3#11762673C211yesmaximal, characteristic, complemented
4#11762546C211yesmaximal, characteristic, complemented
5#11762544C211yesmaximal, characteristic
6#11762542C211yesmaximal, characteristic
7C17×#328C211yesmaximal, characteristic, complemented
8C17×C22⋊Q8C2211yescharacteristic
9#10587633C411yescharacteristic, complemented
10Q8×Dic17C411yescharacteristic, complemented
11#10587616C2211yescharacteristic
12C2×#63225—21no—
13C2×C4×Dic17C2211yescharacteristic
14#10587584—21no—
15#10587579C411yescharacteristic, complemented
16#10587578—21no—
17#10587576C411yescharacteristic, complemented
18#10587572—21no—
19#10587567—21no—
20C17×C22⋊C8C2211yescharacteristic
21C17×#95C2211yescharacteristic
22#10587502—21no—
23#10587502C2211yescharacteristic
24C22⋊#3622C2211yescharacteristic
25#10587486—21no—
26#10587485—21no—
27C2×Q8×C17C2×C411yescharacteristic
28C22×C4×C17—21no—
29C22×C4×C17C2311yescharacteristic
30C2×C22⋊D17—21no—
31C22×Dic17—21no—
32C22×Dic17D411yescharacteristic
33C22×Dic17D411yescharacteristic
34C2×D68C2×C411yescharacteristic
35C2×C4×D17—21no—
36C2×C8×C17—21no—
37C4⋊C4×C17—21no—
38C4⋊C4×C17C2×C411yescharacteristic
39C22⋊C4×C17—21no—
40#63227—21no—
41#63227—21no—
42#63227—21no—
43#63227—21no—
44C4⋊Dic17—21no—
45C4⋊Dic17—21no—
46C4⋊Dic17C2×C411yescharacteristic
47#63225—21no—
48#63225—42no—
49C4×Dic17—21no—
50C4×Dic17C2×C411yescharacteristic
51C4×Dic17C2×C411yescharacteristic
52C4×Dic17D411yescharacteristic
53C4×Dic17D411yescharacteristic
54C2×#3622—21no—
55C23×C17C2×D411yescharacteristic
56C22×D17—21no—
57Q8×C17—41no—
58C2×C4×C17—21no—
59C2×C4×C17—21no—
60C2×C4×C17—41no—
61C2×C4×C17—41no—
62C2×C4×C17—41no—
63C2×C4×C17—41no—
64C2×C4×C17C22×C411yescharacteristic
65C2×C4×C17C2×D411yescharacteristic
66C22⋊D17—41no—
67C22⋊D17—41no—
68C2×Dic17—21no—
69C2×Dic17—21no—
70C2×Dic17—21no—
71C2×Dic17—41no—
72C2×Dic17—41no—
73C2×Dic17—41no—
74C2×Dic17—41no—
75C2×Dic17C22⋊C411yescharacteristic
76C2×Dic17C22⋊C411yescharacteristic
77C2×Dic17C22⋊C411yescharacteristic
78C2×Dic17C22⋊C411yescharacteristic
79D68—41no—
80C4×D17—42no—
81C8×C17—41no—
82#3622—41no—
83#1512—171nomaximal, Sylow
84C22×C17—21no—
85C22×C17—21no—
86C22×C17C2×C22⋊C411yescharacteristic
87C22×C17AGL(1,ℤ/8)11yescharacteristic
88C22×C17#13711yescharacteristic
89C2×D17—41no—
90C2×D17—41no—
91C2×D17—41no—
92C4×C17—21no—
93C4×C17—21no—
94C4×C17—41no—
95C4×C17—41no—
96C4×C17—81no—
97Dic17—21no—
98Dic17—21no—
99Dic17—21no—
100Dic17—21no—
101Dic17—41no—
102Dic17—81no—
103#529—171no—
104#432—171no—
105#382—171no—
106#328—171no—
107#328D1711yescharacteristic, complemented
108#327—171no—
109#327—171no—
110C2×C17—21no—
111C2×C17—21no—
112C2×C17#41011yescharacteristic
113C2×C17#41711yescharacteristic
114C2×C17#42111yescharacteristic
115D17—81no—
116#126—341no—
117#125—171no—
118#124—171no—
119#123—341no—
120C22⋊Q8C2×D1711yescharacteristic
121#121—171no—
122C4×Q8—171no—
123C4×D4—171no—
124C4×D4—341no—
125#117—341no—
126C2×C4⋊C4—341no—
127C2×C42—171no—
128C4⋊C8—341no—
129C22⋊C8—171no—
130C22⋊C8C2×D1711yescharacteristic
131#97—341no—
132#95—171no—
133#95—341no—
134#95C2×D1711yescharacteristic
135C17#151211yescharacteristic, complemented, Sylow
136C2×Q8C4×D1711yescharacteristic
137C2×D4—171no—
138C2×D4—341no—
139C22×C4—21no—
140C22×C4—171no—
141C22×C4—171no—
142C22×C4—341no—
143C22×C4—341no—
144C22×C4C22×D1711yescharacteristic
145C2×C8—21no—
146C2×C8—341no—
147C4⋊C4—21no—
148C4⋊C4—171no—
149C4⋊C4—341no—
150C4⋊C4—341no—
151C4⋊C4—341no—
152C4⋊C4—682no—
153C4⋊C4C4×D1711yescharacteristic
154C22⋊C4—21no—
155C22⋊C4—341no—
156C22⋊C4—341no—
157C22⋊C4—341no—
158C22⋊C4—341no—
159C42—171no—
160C42—171no—
161C42—171no—
162C42—171no—
163C42—341no—
164C23—341no—
165C23D4×D1711yescharacteristic
166Q8—41no—
167D4—681no—
168D4—681no—
169D4—681no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—171no—
177C2×C4—171no—
178C2×C4—171no—
179C2×C4—171no—
180C2×C4—341no—
181C2×C4—341no—
182C2×C4—341no—
183C2×C4—682no—
184C2×C4—681no—
185C2×C4—681no—
186C2×C4—681no—
187C2×C4—681no—
188C2×C4C2×C4×D1711yescharacteristic
189C2×C4D4×D1711yescharacteristic
190C8—41no—
191C8—681no—
192C22—21no—
193C22—21no—
194C22—681no—
195C22—681no—
196C22—681no—
197C22C22⋊C4×D1711yescharacteristic
198C22#1058760011yescharacteristic
199C22#1058760411yescharacteristic
200C4—21no—
201C4—21no—
202C4—41no—
203C4—41no—
204C4—81no—
205C4—341no—
206C4—341no—
207C4—341no—
208C4—341no—
209C4—681no—
210C4—1361no—
211C2—21no—
212C2—21no—
213C2—1361no—
214C2#1176278111yescharacteristic
215C2#1176280911yescharacteristic
216C2#1176282411yescharacteristic
subgroupG/H
centerC22C22⋊C4×D17
commutatorC2×C4×C17C22×C4
FrattiniC22×C4C22×D17
FittingC17×#328C2
socleC22×C17C2×C22⋊C4
prime radical#10587616C22
cosocleC22×C4×C17C23
hypercenter#328D17
hypocenterC17#1512
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #11762863C2semidirect
3. #11762673C2semidirect
4. #11762546C2semidirect
7. C17×#328C2semidirect
9. #10587633C4semidirect
10. Q8×Dic17C4semidirect
15. #10587579C4semidirect
17. #10587576C4semidirect
107. #328D17semidirect
135. C17#1512semidirect

Maximal subgroups

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

subgroupindex
#117633452
#117628632
#117626732
#117625462
#117625442
#117625422
C17×#3282
#151217