Finite Group Databasebeta

← #14527243

Subgroups — #14527243

Order 2176.

230 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 67 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#11763346C211yesmaximal, characteristic
2#11762865C211yesmaximal, characteristic, complemented
3C4×C8×D17C211yesmaximal, characteristic, complemented
4Q8⋊#3622C211yesmaximal, characteristic, complemented
5#11762538C211yesmaximal, characteristic, complemented
6#11762536C211yesmaximal, characteristic, complemented
7C17×Q8⋊C8C211yesmaximal, characteristic, complemented
8C4×Q8×C17C2211yescharacteristic
9#10587633C411yescharacteristic
10#10587632C411yescharacteristic
11C2×C8×D17—21no—
12#10587584—21no—
13#10587581C411yescharacteristic
14#10587580—21no—
15#10587576C411yescharacteristic
16C4×D68—21no—
17C4×D68C2211yescharacteristic
18#10587558—21no—
19C42×D17C2211yescharacteristic
20C17×C4⋊C8C2211yescharacteristic
21C4×C8×C17C2211yescharacteristic
22#10587491—21no—
23#10587486—21no—
24C8×Dic17—21no—
25C4⋊#3622C2211yescharacteristic
26C4×#3622C2211yescharacteristic
27C2×Q8×C17C2×C411yescharacteristic
28C2×D68—21no—
29C2×D68C2×C411yescharacteristic
30C2×C4×D17—21no—
31C2×C4×D17C2×C411yescharacteristic
32C2×C4×D17D411yescharacteristic
33C2×C4×D17D411yescharacteristic
34C2×C8×C17—21no—
35C2×C8×C17—21no—
36C4⋊C4×C17—21no—
37C4⋊C4×C17C2×C411yescharacteristic
38C42×C17—21no—
39C42×C17C2311yescharacteristic
40#63227—21no—
41#63227—21no—
42#63227—21no—
43#63227—21no—
44#63227—21no—
45C4⋊Dic17—21no—
46C4⋊Dic17C2×C411yescharacteristic
47#63225—21no—
48#63225—21no—
49C4×Dic17C2×C411yescharacteristic
50C4×Dic17D411yescharacteristic
51C4×Dic17D411yescharacteristic
52C2×#3622—21no—
53C2×#3622—21no—
54C8×D17—24no—
55C22×D17—21no—
56C22×D17C22⋊C411yescharacteristic
57Q8×C17—21no—
58Q8×C17—21no—
59C2×C4×C17—21no—
60C2×C4×C17—21no—
61C2×C4×C17C22×C411yescharacteristic
62C2×C4×C17C2×D411yescharacteristic
63C2×C4×C17C2×D411yescharacteristic
64C2×Dic17—21no—
65C2×Dic17C22⋊C411yescharacteristic
66C2×Dic17C22⋊C411yescharacteristic
67C2×Dic17C22⋊C411yescharacteristic
68D68—21no—
69D68—21no—
70D68—42no—
71C4×D17—22no—
72C4×D17—21no—
73C4×D17—21no—
74C4×D17—21no—
75C4×D17—21no—
76C4×D17—41no—
77C4×D17—41no—
78C8×C17—22no—
79C8×C17—41no—
80#3622—22no—
81#3622—41no—
82#1494—171nomaximal, Sylow
83C22×C17C2×C22⋊C411yescharacteristic
84C2×D17—21no—
85C2×D17—21no—
86C2×D17—41no—
87C2×D17—41no—
88C2×D17—41no—
89C2×D17#10412yes—
90C4×C17—21no—
91C4×C17—41no—
92C4×C17—41no—
93C4×C17#13111yescharacteristic
94C4×C17#13111yescharacteristic
95C4×C17#13511yescharacteristic
96C4×C17#13511yescharacteristic
97Dic17—21no—
98Dic17—21no—
99Dic17—21no—
100Dic17—41no—
101#529—171no—
102#432—171no—
103C2×C4×C8—171no—
104Q8⋊C8—171no—
105Q8⋊C8D1711yescharacteristic, complemented
106D4⋊C8—171no—
107D4⋊C8—171no—
108C2×C17#40811yescharacteristic
109C2×C17#41611yescharacteristic
110C2×C17C2×#10411yescharacteristic
111D17—22no—
112D17—81no—
113C22×C8—341no—
114#126—341no—
115#125—171no—
116#124—171no—
117#123—341no—
118C22⋊Q8—171no—
119#121—171no—
120C4×Q8C2×D1711yescharacteristic
121C4×D4—171no—
122C4×D4—341no—
123#117—341no—
124C2×C42—171no—
125C4⋊C8—171no—
126C4⋊C8C2×D1711yescharacteristic
127C22⋊C8—341no—
128#97—341no—
129C4×C8—171no—
130C4×C8—341no—
131C4×C8C2×D1711yescharacteristic
132C17#149411yescharacteristic, complemented, Sylow
133C2×Q8C4×D1711yescharacteristic
134C2×D4—171no—
135C2×D4—341no—
136C22×C4—171no—
137C22×C4—171no—
138C22×C4—171no—
139C22×C4—341no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—344no—
143C2×C8—341no—
144C2×C8—341no—
145C4⋊C4—21no—
146C4⋊C4—171no—
147C4⋊C4—341no—
148C4⋊C4—341no—
149C4⋊C4—341no—
150C4⋊C4C4×D1711yescharacteristic
151C22⋊C4—341no—
152C22⋊C4—341no—
153C22⋊C4—341no—
154C22⋊C4—341no—
155C22⋊C4—341no—
156C42—21no—
157C42—171no—
158C42—171no—
159C42—171no—
160C42C22×D1711yescharacteristic
161C23—171no—
162C23—341no—
163Q8—21no—
164Q8—21no—
165D4—341no—
166D4—341no—
167D4—682no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—171no—
171C2×C4—171no—
172C2×C4—171no—
173C2×C4—342no—
174C2×C4—341no—
175C2×C4—341no—
176C2×C4—341no—
177C2×C4—341no—
178C2×C4—341no—
179C2×C4—681no—
180C2×C4—681no—
181C2×C4C2×C4×D1711yescharacteristic
182C2×C4D4×D1711yescharacteristic
183C2×C4D4×D1711yescharacteristic
184C8—22no—
185C8—41no—
186C8—342no—
187C8—681no—
188C22—172no—
189C22—341no—
190C22—341no—
191C22—681no—
192C22—681no—
193C22—681no—
194C22C22⋊C4×D1711yescharacteristic
195C4—21no—
196C4—41no—
197C4—41no—
198C4—341no—
199C4—341no—
200C4—341no—
201C4—681no—
202C4#1058758711yescharacteristic
203C4#1058759311yescharacteristic
204C4#1058760211yescharacteristic
205C4#1058760511yescharacteristic
206C2—342no—
207C2—1361no—
208C2#1176277411yescharacteristic
209C2#1176280411yescharacteristic
210C2#104×D1711yescharacteristic
subgroupG/H
centerC2×C4D4×D17
commutatorC2×C4×C17C22×C4
FrattiniC42C22×D17
FittingC17×Q8⋊C8C2
socleC22×C17C2×C22⋊C4
prime radical#10587581C4
cosocleC42×C17C23
hypercenterQ8⋊C8D17
hypocenterC17#1494
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #11762865C2semidirect
3. C4×C8×D17C2semidirect
4. Q8⋊#3622C2semidirect
5. #11762538C2semidirect
6. #11762536C2semidirect
7. C17×Q8⋊C8C2semidirect
105. Q8⋊C8D17semidirect
132. C17#1494semidirect

Maximal subgroups

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

subgroupindex
#117633462
#117628652
C4×C8×D172
Q8⋊#36222
#117625382
#117625362
C17×Q8⋊C82
#149417