Finite Group Databasebeta

← #14544121

Subgroups — #14544121

Order 2176.

266 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 66 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#11763812C211yesmaximal, characteristic, complemented
2C4×#63247C211yesmaximal, characteristic, complemented
3#11763659C212yesmaximal, complemented
4#11763657C212yesmaximal, complemented
5#11763277C211yesmaximal, characteristic
6C2×#63252C411yescharacteristic
7C2×#63247—21no—
8C2×#63247—21no—
9C17×#117C411yescharacteristic
10C2×C4×Dic17C2211yescharacteristic
11#10587579C411yescharacteristic
12#10587579C2211yescharacteristic
13#10587573C411yescharacteristic
14#10587573C2211yescharacteristic
15#10587567—21no—
16#10587567—21no—
17#10587558—21no—
18C42×D17—21no—
19#10587544—21no—
20#10587542—21no—
21C4⋊#3624C2212yes—
22C4×#3624C2212yes—
23C22×C4×C17C2×C411yescharacteristic
24C2×C22⋊D17—21no—
25C22×Dic17D411yescharacteristic
26C22×Dic17D411yescharacteristic
27#63252—22no—
28#63252—22no—
29C2×D68C2×C411yescharacteristic
30C2×C4×D17—21no—
31C2×C4×D17—21no—
32C2×C4×D17—21no—
33C2×Dic34C2×C411yescharacteristic
34#63247—24no—
35#63247—42no—
36C2×#3624—22no—
37C2×#3624—22no—
38C4⋊C4×C17C2×C411yescharacteristic
39C4⋊C4×C17C2×C411yescharacteristic
40C22⋊C4×C17—21no—
41C42×C17—21no—
42#63227—21no—
43#63227—21no—
44#63225—21no—
45#63225—21no—
46C4×Dic17—21no—
47C4×Dic17—21no—
48C4×Dic17C2×C411yescharacteristic
49C4×Dic17D411yescharacteristic
50C4×Dic17D411yescharacteristic
51C4×Dic17C2311yescharacteristic
52C23×C17C22⋊C411yescharacteristic
53C22×D17—21no—
54C2×C4×C17—21no—
55C2×C4×C17—21no—
56C2×C4×C17—21no—
57C2×C4×C17—21no—
58C2×C4×C17C22⋊C411yescharacteristic
59C2×C4×C17C22×C411yescharacteristic
60C22⋊D17—24no—
61C2×Dic17—21no—
62C2×Dic17—22no—
63C2×Dic17—21no—
64C2×Dic17—21no—
65C2×Dic17C22⋊C411yescharacteristic
66C2×Dic17C22⋊C411yescharacteristic
67C2×Dic17C2×D411yescharacteristic
68C2×Dic17C2×D411yescharacteristic
69D68—21no—
70D68M1612yescomplemented
71C4×D17—24no—
72C4×D17—42no—
73C4×D17—41no—
74C4×D17—41no—
75Dic34—21no—
76Dic34M1612yescomplemented
77#3624—24no—
78#3624—42no—
79#1489—171nomaximal, Sylow
80C22×C17—21no—
81C22×C17—21no—
82C22×C17—21no—
83C22×C17C2×C22⋊C411yescharacteristic
84C2×D17—22no—
85C2×D17—41no—
86C2×D17—41no—
87C4×C17—21no—
88C4×C17—41no—
89C4×C17—41no—
90C4×C17C2×M1611yescharacteristic
91C4×C17C2×M1611yescharacteristic
92Dic17—22no—
93Dic17—21no—
94Dic17—21no—
95Dic17#10412yes—
96Dic17AGL(1,ℤ/8)11yescharacteristic
97Dic17#13711yescharacteristic
98C4×#41—171no—
99#423—171no—
100C4×M16—171no—
101Q8⋊C8—172no—
102D4⋊C8—172no—
103C2×C17—41no—
104C2×C17#40711yescharacteristic
105C2×C17#41711yescharacteristic
106C2×C17C2×#10411yescharacteristic
107D17—42no—
108C2×#41—171no—
109C2×M16—341no—
110C2×M16—341no—
111C4×Q8—171no—
112C4×Q8—171no—
113C4×D4—171no—
114C4×D4—171no—
115C4×D4—341no—
116C4×D4—341no—
117#117—341no—
118#117#59511yescharacteristic
119C2×C42—171no—
120C2×C42—341no—
121C4⋊C8—172no—
122C4⋊C8—341no—
123#97—341no—
124C4×C8—172no—
125C17#148911yescharacteristic, complemented, Sylow
126#41—342no—
127#41—342no—
128C2×Q8—171no—
129C2×D4—171no—
130C2×D4—341no—
131C22×C4—171no—
132C22×C4—171no—
133C22×C4—341no—
134C22×C4—341no—
135C22×C4—341no—
136C22×C4C2×#59511yescharacteristic
137M16—344no—
138M16—682no—
139C2×C8—342no—
140C2×C8—342no—
141C4⋊C4—341no—
142C4⋊C4—341no—
143C4⋊C4C2×#59511yescharacteristic
144C4⋊C4C2×#59511yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—341no—
147C22⋊C4—341no—
148C42—21no—
149C42—171no—
150C42—171no—
151C42—171no—
152C42—171no—
153C42—341no—
154C42—341no—
155C23—341no—
156C23C22⋊#59511yescharacteristic
157Q8—172no—
158Q8—341no—
159D4—172no—
160D4—344no—
161D4—341no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—171no—
167C2×C4—171no—
168C2×C4—171no—
169C2×C4—171no—
170C2×C4—344no—
171C2×C4—342no—
172C2×C4—341no—
173C2×C4—341no—
174C2×C4—341no—
175C2×C4—681no—
176C2×C4—681no—
177C2×C4—682no—
178C2×C4C22⋊#59511yescharacteristic
179C2×C4C22×#59511yescharacteristic
180C8—344no—
181C8—682no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—342no—
186C22—681no—
187C22—681no—
188C22C2×C22⋊#59511yescharacteristic
189C4—21no—
190C4—41no—
191C4—41no—
192C4—171no—
193C4—171no—
194C4—172no—
195C4—342no—
196C4—341no—
197C4—341no—
198C4D17⋊M1611yescharacteristic
199C4D17⋊M1611yescharacteristic
200C2—41no—
201C2—682no—
202C2#1176378111yescharacteristic
203C2#1176380011yescharacteristic
204C2D17⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#117C4
socleC22×C17C2×C22⋊C4
prime radicalC2×#63252C4
cosocleC4×Dic17C23
hypercenter#117#595
hypocenterC17#1489
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #11763812C2semidirect
2. C4×#63247C2semidirect
3. #11763659C2semidirect
4. #11763657C2semidirect
70. D68M16semidirect
76. Dic34M16semidirect
125. C17#1489semidirect

Maximal subgroups

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

subgroupindex
#117638122
C4×#632472
#117636592
#117636572
#117632772
#148917