Finite Group Databasebeta

← #14544308

Subgroups — #14544308

Order 2176.

242 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 58 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#11763829C211yesmaximal, characteristic, complemented
2#11763826C211yesmaximal, characteristic, complemented
3#11763668C212yesmaximal, complemented
4#11763667C212yesmaximal, complemented
5#11763354C211yesmaximal, characteristic
6C2×#63254—21no—
7C17×#121C411yescharacteristic
8#10587626C2211yescharacteristic
9#10587625—21no—
10D4×Dic17C411yescharacteristic
11#10587616C411yescharacteristic
12C2×C4×Dic17C2211yescharacteristic
13#10587579C411yescharacteristic
14#10587574C2211yescharacteristic
15#10587571—21no—
16#10587567—21no—
17C22⋊#3624—21no—
18C22⋊#3624—21no—
19#10587544C2212yes—
20#10587544C2212yes—
21#10587542—21no—
22C4×#3624—21no—
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—
36C2×#3624—22no—
37C2×#3624—22no—
38C4⋊C4×C17C2×C411yescharacteristic
39C22⋊C4×C17—21no—
40C22⋊Dic17—21no—
41#63227—21no—
42#63227—21no—
43C4⋊Dic17C2×C411yescharacteristic
44#63225—21no—
45C4×Dic17C2×C411yescharacteristic
46C4×Dic17D411yescharacteristic
47C4×Dic17D411yescharacteristic
48C4×Dic17C2311yescharacteristic
49C23×C17—21no—
50C23×C17C22⋊C411yescharacteristic
51C22×D17—21no—
52D4×C17—41no—
53D4×C17—42no—
54C2×C4×C17—21no—
55C2×C4×C17—41no—
56C2×C4×C17C22⋊C411yescharacteristic
57C2×C4×C17C22×C411yescharacteristic
58C22⋊D17—42no—
59C22⋊D17—42no—
60C22⋊D17—42no—
61C2×Dic17—22no—
62C2×Dic17—22no—
63C2×Dic17—21no—
64C2×Dic17—42no—
65C2×Dic17C22⋊C411yescharacteristic
66C2×Dic17C22⋊C411yescharacteristic
67C2×Dic17C2×D411yescharacteristic
68C2×Dic17C2×D411yescharacteristic
69D68—41no—
70C4×D17—42no—
71Dic34—42no—
72#3624—42no—
73#3624—42no—
74#1553—171nomaximal, Sylow
75C22×C17—21no—
76C22×C17—21no—
77C22×C17—21no—
78C22×C17—41no—
79C22×C17—41no—
80C22×C17—41no—
81C22×C17C2×C22⋊C411yescharacteristic
82C2×D17—41no—
83C2×D17—41no—
84C2×D17—41no—
85C4×C17—21no—
86C4×C17—21no—
87C4×C17—41no—
88Dic17—21no—
89Dic17—21no—
90Dic17—41no—
91Dic17#13512yes—
92Dic17AGL(1,ℤ/8)12yes—
93#532—171no—
94#432—171no—
95#431—171no—
96#332—172no—
97#331—172no—
98C2×C17—41no—
99C2×C17—81no—
100C2×C17#41311yescharacteristic
101C2×C17#41611yescharacteristic
102C2×C17#41811yescharacteristic
103D17—81no—
104C2×#41—341no—
105C4⋊Q8—171no—
106#127—171no—
107#124—341no—
108#121—171no—
109#121—341no—
110#121#59511yescharacteristic
111C4×D4—171no—
112C4×D4—171no—
113C4×D4—341no—
114C2×C42—171no—
115C4⋊C8—172no—
116C4⋊C8—172no—
117C22⋊C8—341no—
118C22⋊C8—341no—
119#97—341no—
120C4×C8—341no—
121C17#155311yescharacteristic, complemented, Sylow
122#41—684no—
123C2×Q8—341no—
124C2×D4—21no—
125C2×D4—171no—
126C2×D4—341no—
127C2×D4—341no—
128C2×D4—341no—
129C2×D4C2×#59511yescharacteristic
130C22×C4—171no—
131C22×C4—171no—
132C22×C4—341no—
133C22×C4—341no—
134C22×C4C2×#59511yescharacteristic
135C2×C8—342no—
136C2×C8—342no—
137C4⋊C4—171no—
138C4⋊C4—341no—
139C4⋊C4C2×#59511yescharacteristic
140C22⋊C4—21no—
141C22⋊C4—341no—
142C22⋊C4—341no—
143C22⋊C4—341no—
144C42—171no—
145C42—171no—
146C42—171no—
147C42—171no—
148C23—21no—
149C23—341no—
150C23C22⋊#59511yescharacteristic
151Q8—682no—
152D4—41no—
153D4—42no—
154D4—681no—
155D4—682no—
156D4—682no—
157D4—682no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—171no—
161C2×C4—171no—
162C2×C4—171no—
163C2×C4—171no—
164C2×C4—342no—
165C2×C4—342no—
166C2×C4—341no—
167C2×C4—682no—
168C2×C4—682no—
169C2×C4C22⋊#59511yescharacteristic
170C2×C4C22×#59511yescharacteristic
171C8—682no—
172C8—682no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—681no—
180C22—681no—
181C22—681no—
182C22C2×C22⋊#59511yescharacteristic
183C4—21no—
184C4—21no—
185C4—41no—
186C4—172no—
187C4—172no—
188C4—341no—
189C4—341no—
190C4—681no—
191C2—41no—
192C2—81no—
193C2—1361no—
194C2#1176379611yescharacteristic
195C2#1176379911yescharacteristic
196C2#1176380111yescharacteristic
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#1553
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #11763829C2semidirect
2. #11763826C2semidirect
3. #11763668C2semidirect
4. #11763667C2semidirect
121. C17#1553semidirect

Maximal subgroups

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

subgroupindex
#117638292
#117638262
#117636682
#117636672
#117633542
#155317