Finite Group Databasebeta

← #14544203

Subgroups — #14544203

Order 2176.

252 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 56 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#11763831C211yesmaximal, characteristic, complemented
2#11763754C211yesmaximal, characteristic, complemented
3#11763661C212yesmaximal, complemented
4#11763660C212yesmaximal, complemented
5#11763356C211yesmaximal, characteristic
6C2×#63252—21no—
7C4×D4×C17C411yescharacteristic, complemented
8#10587626C411yescharacteristic, complemented
9#10587624C2211yescharacteristic
10C4×C22⋊D17—21no—
11#10587581C2211yescharacteristic
12#10587574C411yescharacteristic, complemented
13#10587571—21no—
14#10587570—21no—
15C4×D68C411yescharacteristic, complemented
16C42×D17C2211yescharacteristic
17(C2×#595)⋊C4—21no—
18(C2×#595)⋊C4—21no—
19(C2×#595)⋊C4C2212yes—
20#10587543C2212yes—
21#10587542—21no—
22C4⋊#3624—21no—
23C22×#595—22no—
24C2×D4×C17C2×C411yescharacteristic
25C22×C4×C17—21no—
26C2×C22⋊D17—21no—
27C2×C22⋊D17—21no—
28C2×C22⋊D17—21no—
29#63252—44no—
30C2×D68—21no—
31C2×D68C2×C411yescharacteristic
32C2×C4×D17—21no—
33C2×C4×D17D411yescharacteristic
34C2×C4×D17D411yescharacteristic
35C2×C4×D17C2311yescharacteristic
36C2×Dic34—21no—
37C2×#3624—22no—
38C4⋊C4×C17C2×C411yescharacteristic
39C22⋊C4×C17—21no—
40C42×C17C2×C411yescharacteristic
41C22⋊Dic17—21no—
42#63227—21no—
43#63227—21no—
44C4⋊Dic17C2×C411yescharacteristic
45#63225—21no—
46C4×Dic17C2×C411yescharacteristic
47C4×Dic17D411yescharacteristic
48C4×Dic17D411yescharacteristic
49C23×C17—21no—
50C22×D17—21no—
51C22×D17C2×D411yescharacteristic
52C2×#595—42no—
53C2×#595—42no—
54C2×#595—42no—
55D4×C17—41no—
56C2×C4×C17—21no—
57C2×C4×C17—42no—
58C2×C4×C17C22⋊C411yescharacteristic
59C2×C4×C17C22⋊C411yescharacteristic
60C2×C4×C17C22×C411yescharacteristic
61C22⋊D17—42no—
62C22⋊D17—42no—
63C22⋊D17—42no—
64C2×Dic17—21no—
65C2×Dic17C22⋊C411yescharacteristic
66C2×Dic17C22⋊C411yescharacteristic
67C2×Dic17C2×D411yescharacteristic
68D68—41no—
69D68—42no—
70C4×D17—22no—
71C4×D17—41no—
72C4×D17—41no—
73C4×D17—42no—
74Dic34—42no—
75#3624—42no—
76#1514—171nomaximal, Sylow
77C22×C17—41no—
78C22×C17—41no—
79C22×C17—41no—
80C22×C17C2×C22⋊C411yescharacteristic
81C2×D17—21no—
82C2×D17—21no—
83C2×D17—41no—
84C2×D17—41no—
85C2×D17—41no—
86C2×D17AGL(1,ℤ/8)12yes—
87#595—82no—
88C4×C17—21no—
89C4×C17—21no—
90C4×C17—21no—
91C4×C17—41no—
92Dic17—21no—
93Dic17—21no—
94Dic17—21no—
95Dic17—41no—
96#532—171no—
97#432—171no—
98#383—171no—
99#327—172no—
100#327—172no—
101C2×C17—81no—
102C2×C17#41011yescharacteristic
103C2×C17#41811yescharacteristic
104C2×C17#42111yescharacteristic
105D17—22no—
106D17—81no—
107C2×#41—341no—
108C4⋊Q8—171no—
109#127—171no—
110#124—341no—
111#121—171no—
112#121—171no—
113#121—341no—
114C4×D4—171no—
115C4×D4—341no—
116C4×D4#59511yescharacteristic, complemented
117C2×C42—171no—
118C4⋊C8—341no—
119C22⋊C8—172no—
120#97—341no—
121#95—172no—
122#95—341no—
123#95—341no—
124C17#151411yescharacteristic, complemented, Sylow
125#41—684no—
126C2×Q8—341no—
127C2×D4—171no—
128C2×D4—341no—
129C2×D4—341no—
130C2×D4—341no—
131C2×D4—341no—
132C2×D4C2×#59511yescharacteristic
133C22×C4—21no—
134C22×C4—171no—
135C22×C4—171no—
136C22×C4—171no—
137C22×C4—341no—
138C22×C4—342no—
139C2×C8—342no—
140C4⋊C4—171no—
141C4⋊C4—341no—
142C4⋊C4C2×#59511yescharacteristic
143C22⋊C4—21no—
144C22⋊C4—341no—
145C22⋊C4—341no—
146C22⋊C4—341no—
147C42—171no—
148C42—171no—
149C42—171no—
150C42C2×#59511yescharacteristic
151C23—21no—
152C23—171no—
153C23—341no—
154Q8—682no—
155D4—41no—
156D4—681no—
157D4—682no—
158D4—682no—
159D4—682no—
160D4—682no—
161C2×C4—21no—
162C2×C4—42no—
163C2×C4—171no—
164C2×C4—171no—
165C2×C4—171no—
166C2×C4—342no—
167C2×C4—341no—
168C2×C4—681no—
169C2×C4—681no—
170C2×C4—682no—
171C2×C4—682no—
172C2×C4—682no—
173C2×C4—682no—
174C2×C4C22⋊#59511yescharacteristic
175C2×C4C22⋊#59511yescharacteristic
176C2×C4C22×#59511yescharacteristic
177C8—682no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—172no—
182C22—341no—
183C22—341no—
184C22—681no—
185C22—681no—
186C22—681no—
187C22C2×C22⋊#59511yescharacteristic
188C4—21no—
189C4—21no—
190C4—21no—
191C4—41no—
192C4—341no—
193C4—341no—
194C4—341no—
195C4—681no—
196C4—1362no—
197C2—81no—
198C2—342no—
199C2—1361no—
200C2#1176378811yescharacteristic
201C2#1176380111yescharacteristic
202C2#1176380611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC4×D4×C17C4
socleC22×C17C2×C22⋊C4
prime radical#11763356C2
cosocleC2×C4×D17C23
hypercenterC4×D4#595
hypocenterC17#1514
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #11763831C2semidirect
2. #11763754C2semidirect
3. #11763661C2semidirect
4. #11763660C2semidirect
7. C4×D4×C17C4semidirect
8. #10587626C4semidirect
12. #10587574C4semidirect
15. C4×D68C4semidirect
116. C4×D4#595semidirect
124. C17#1514semidirect

Maximal subgroups

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

subgroupindex
#117638312
#117637542
#117636612
#117636602
#117633562
#151417