Finite Group Databasebeta

← #14544103

Subgroups — #14544103

Order 2176.

262 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 62 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#11763825C211yesmaximal, characteristic, complemented
2C4×#63247C211yesmaximal, characteristic, complemented
3#11763657C212yesmaximal, complemented
4D4⋊#3624C212yesmaximal, complemented
5#11763354C211yesmaximal, characteristic
6C2×#63254—21no—
7C2×#63247—21no—
8C17×#121C411yescharacteristic
9#10587626C411yescharacteristic
10#10587625—21no—
11D4×Dic17C2211yescharacteristic
12#10587616C411yescharacteristic
13C2×C4×Dic17C2211yescharacteristic
14#10587579C2211yescharacteristic
15#10587574C411yescharacteristic
16#10587571—21no—
17#10587567—21no—
18C22⋊#3624—21no—
19#10587542—21no—
20C4⋊#3624C2212yes—
21C4×#3624—21no—
22C4×#3624C2212yes—
23C2×D4×C17—21no—
24C2×D4×C17C2×C411yescharacteristic
25C22×C4×C17C2×C411yescharacteristic
26C2×C22⋊D17—21no—
27C2×C22⋊D17—21no—
28C2×C22⋊D17—21no—
29C22×Dic17—21no—
30C22×Dic17D411yescharacteristic
31C22×Dic17D411yescharacteristic
32#63254—44no—
33C2×D68C2×C411yescharacteristic
34C2×C4×D17—21no—
35C2×Dic34—21no—
36#63247—24no—
37C2×#3624—22no—
38C2×#3624—22no—
39C4⋊C4×C17C2×C411yescharacteristic
40C22⋊C4×C17—21no—
41C22⋊Dic17—21no—
42#63227—21no—
43#63227—21no—
44C4⋊Dic17C2×C411yescharacteristic
45#63225—21no—
46C4×Dic17C2×C411yescharacteristic
47C4×Dic17D411yescharacteristic
48C4×Dic17D411yescharacteristic
49C4×Dic17C2311yescharacteristic
50C23×C17—21no—
51C23×C17C22⋊C411yescharacteristic
52C22×D17—21no—
53D4×C17—22no—
54D4×C17—42no—
55C2×C4×C17—22no—
56C2×C4×C17—21no—
57C2×C4×C17C22⋊C411yescharacteristic
58C2×C4×C17C22×C411yescharacteristic
59C22⋊D17—42no—
60C22⋊D17—42no—
61C22⋊D17—42no—
62C2×Dic17—22no—
63C2×Dic17—22no—
64C2×Dic17—21no—
65C2×Dic17—42no—
66C2×Dic17C22⋊C411yescharacteristic
67C2×Dic17C22⋊C411yescharacteristic
68C2×Dic17C2×D411yescharacteristic
69C2×Dic17C2×D411yescharacteristic
70D68—22no—
71C4×D17—42no—
72Dic34—42no—
73#3624—24no—
74#3624—42no—
75#1484—171nomaximal, Sylow
76C22×C17—21no—
77C22×C17—21no—
78C22×C17—21no—
79C22×C17—41no—
80C22×C17—41no—
81C22×C17—41no—
82C22×C17C2×C22⋊C411yescharacteristic
83C2×D17—41no—
84C2×D17—41no—
85C2×D17—41no—
86C4×C17—21no—
87C4×C17—41no—
88C4×C17#13112yes—
89Dic17—21no—
90Dic17—21no—
91Dic17—41no—
92Dic17#10412yes—
93Dic17AGL(1,ℤ/8)12yes—
94#532—171no—
95#431—171no—
96C4×M16—171no—
97D4⋊C8—172no—
98D4⋊C8—172no—
99C2×C17—41no—
100C2×C17—81no—
101C2×C17#40811yescharacteristic
102C2×C17#41811yescharacteristic
103C2×C17C2×#10411yescharacteristic
104D17—81no—
105C2×#41—341no—
106C2×M16—341no—
107C4⋊Q8—171no—
108#127—171no—
109#124—341no—
110#121—171no—
111#121—341no—
112#121#59511yescharacteristic
113C4×D4—171no—
114C4×D4—171no—
115C4×D4—341no—
116C2×C42—171no—
117C4⋊C8—172no—
118C22⋊C8—341no—
119#97—341no—
120C4×C8—172no—
121C4×C8—341no—
122C17#148411yescharacteristic, complemented, Sylow
123#41—684no—
124C2×Q8—341no—
125C2×D4—21no—
126C2×D4—171no—
127C2×D4—341no—
128C2×D4—341no—
129C2×D4—341no—
130C2×D4C2×#59511yescharacteristic
131C22×C4—171no—
132C22×C4—171no—
133C22×C4—341no—
134C22×C4—341no—
135C22×C4C2×#59511yescharacteristic
136M16—344no—
137C2×C8—342no—
138C2×C8—342no—
139C4⋊C4—171no—
140C4⋊C4—341no—
141C4⋊C4C2×#59511yescharacteristic
142C22⋊C4—21no—
143C22⋊C4—341no—
144C22⋊C4—341no—
145C22⋊C4—341no—
146C42—171no—
147C42—171no—
148C42—171no—
149C42—171no—
150C23—21no—
151C23—341no—
152C23C22⋊#59511yescharacteristic
153Q8—682no—
154D4—22no—
155D4—42no—
156D4—342no—
157D4—682no—
158D4—682no—
159D4—682no—
160C2×C4—22no—
161C2×C4—21no—
162C2×C4—171no—
163C2×C4—171no—
164C2×C4—171no—
165C2×C4—171no—
166C2×C4—342no—
167C2×C4—342no—
168C2×C4—341no—
169C2×C4—682no—
170C2×C4—682no—
171C2×C4C22⋊#59511yescharacteristic
172C2×C4C22×#59511yescharacteristic
173C8—344no—
174C8—682no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—681no—
182C22—681no—
183C22—681no—
184C22C2×C22⋊#59511yescharacteristic
185C4—21no—
186C4—41no—
187C4—172no—
188C4—172no—
189C4—341no—
190C4—341no—
191C4—681no—
192C4#1058767812yes—
193C2—41no—
194C2—81no—
195C2—1361no—
196C2#1176378211yescharacteristic
197C2#1176380111yescharacteristic
198C2D17⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#121C4
socleC22×C17C2×C22⋊C4
prime radical#11763354C2
cosocleC4×Dic17C23
hypercenter#121#595
hypocenterC17#1484
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #11763825C2semidirect
2. C4×#63247C2semidirect
3. #11763657C2semidirect
4. D4⋊#3624C2semidirect
122. C17#1484semidirect

Maximal subgroups

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

subgroupindex
#117638252
C4×#632472
#117636572
D4⋊#36242
#117633542
#148417