Finite Group Databasebeta

← #14544122

Subgroups — #14544122

Order 2176.

266 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 70 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#11763813C211yesmaximal, characteristic, complemented
2C4×#63243C211yesmaximal, characteristic, complemented
3Q8⋊#3624C212yesmaximal, complemented
4#11763657C212yesmaximal, complemented
5C4×#63256C211yesmaximal, characteristic
6C2×#63256C411yescharacteristic
7C2×#63243—21no—
8C2×#63243—21no—
9C4×Q8×C17C411yescharacteristic
10Q8×Dic17C2211yescharacteristic
11#10587579—21no—
12#10587579C2211yescharacteristic
13#10587578—21no—
14#10587578C411yescharacteristic
15C4×D68—21no—
16C4×D68C411yescharacteristic
17C42×D17—21no—
18C42×D17C2211yescharacteristic
19#10587542—21no—
20C4⋊#3624—21no—
21C4⋊#3624C2212yes—
22C4×#3624C2212yes—
23C2×Q8×C17C2×C411yescharacteristic
24#63256—22no—
25#63256—22no—
26C2×D68—21no—
27C2×D68C2×C411yescharacteristic
28C2×C4×D17—21no—
29C2×C4×D17—21no—
30C2×C4×D17—21no—
31C2×C4×D17C2×C411yescharacteristic
32C2×C4×D17D411yescharacteristic
33C2×C4×D17D411yescharacteristic
34C2×#3624—22no—
35C2×#3624—22no—
36#63243—24no—
37#63243—42no—
38C4⋊C4×C17—21no—
39C4⋊C4×C17C2×C411yescharacteristic
40C42×C17—21no—
41C42×C17C2×C411yescharacteristic
42#63227—21no—
43#63227—21no—
44#63227—21no—
45C4⋊Dic17—21no—
46C4⋊Dic17C2×C411yescharacteristic
47C4×Dic17—21no—
48C4×Dic17—21no—
49C4×Dic17D411yescharacteristic
50C4×Dic17D411yescharacteristic
51C4×Dic17C2311yescharacteristic
52C22×D17—21no—
53C22×D17C22⋊C411yescharacteristic
54Q8×C17—21no—
55Q8×C17M1612yescomplemented
56C2×C4×C17—21no—
57C2×C4×C17—21no—
58C2×C4×C17C22⋊C411yescharacteristic
59C2×C4×C17C22⋊C411yescharacteristic
60C2×C4×C17C22×C411yescharacteristic
61C2×Dic17—21no—
62C2×Dic17C22⋊C411yescharacteristic
63C2×Dic17C2×D411yescharacteristic
64C2×Dic17C2×D411yescharacteristic
65D68—24no—
66D68—21no—
67D68M1612yescomplemented
68C4×D17—24no—
69C4×D17—22no—
70C4×D17—21no—
71C4×D17—21no—
72C4×D17—21no—
73C4×D17—21no—
74C4×D17—42no—
75C4×D17—41no—
76C4×D17—41no—
77#3624—24no—
78#3624—42no—
79#1489—171nomaximal, Sylow
80C22×C17C2×C22⋊C411yescharacteristic
81C2×D17—21no—
82C2×D17—22no—
83C2×D17—21no—
84C2×D17—21no—
85C2×D17—41no—
86C2×D17—41no—
87C4×C17—22no—
88C4×C17—21no—
89C4×C17—41no—
90C4×C17#10412yes—
91C4×C17C2×M1611yescharacteristic
92C4×C17C2×M1611yescharacteristic
93Dic17—21no—
94Dic17—21no—
95Dic17—41no—
96Dic17AGL(1,ℤ/8)11yescharacteristic
97Dic17#13711yescharacteristic
98C4×#41—171no—
99#423—171no—
100C4×M16—171no—
101Q8⋊C8—172no—
102D4⋊C8—172no—
103C2×C17#40711yescharacteristic
104C2×C17#41711yescharacteristic
105C2×C17C2×#10411yescharacteristic
106D17—41no—
107D17—42no—
108C2×#41—171no—
109C2×M16—341no—
110C2×M16—341no—
111C4×Q8—171no—
112C4×Q8#59511yescharacteristic
113C4×D4—171no—
114C4×D4—171no—
115C4×D4—341no—
116C4×D4—341no—
117#117—171no—
118#117—341no—
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×Q8C2×#59511yescharacteristic
129C2×D4—171no—
130C2×D4—341no—
131C22×C4—171no—
132C22×C4—171no—
133C22×C4—171no—
134C22×C4—341no—
135C22×C4—341no—
136C22×C4—341no—
137M16—344no—
138M16—682no—
139C2×C8—342no—
140C2×C8—342no—
141C4⋊C4—21no—
142C4⋊C4—171no—
143C4⋊C4—341no—
144C4⋊C4C2×#59511yescharacteristic
145C22⋊C4—341no—
146C22⋊C4—341no—
147C22⋊C4—341no—
148C42—21no—
149C42—171no—
150C42—171no—
151C42—171no—
152C42—341no—
153C42—341no—
154C42C2×#59511yescharacteristic
155C23—171no—
156C23—341no—
157Q8—21no—
158Q8#6324312yescomplemented
159D4—172no—
160D4—344no—
161D4—341no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—171no—
165C2×C4—171no—
166C2×C4—171no—
167C2×C4—341no—
168C2×C4—341no—
169C2×C4—344no—
170C2×C4—341no—
171C2×C4—341no—
172C2×C4—342no—
173C2×C4—341no—
174C2×C4—681no—
175C2×C4—681no—
176C2×C4—682no—
177C2×C4C22⋊#59511yescharacteristic
178C2×C4C22⋊#59511yescharacteristic
179C2×C4C22×#59511yescharacteristic
180C8—344no—
181C8—682no—
182C22—342no—
183C22—341no—
184C22—341no—
185C22—341no—
186C22—681no—
187C22—681no—
188C22C2×C22⋊#59511yescharacteristic
189C4—22no—
190C4—21no—
191C4—41no—
192C4—171no—
193C4—171no—
194C4—341no—
195C4—341no—
196C4—681no—
197C4#1058755012yes—
198C4C2×#6324311yescharacteristic
199C4D17⋊M1611yescharacteristic
200C2—681no—
201C2—682no—
202C2#1176378011yescharacteristic
203C2#1176380011yescharacteristic
204C2C2×#1058755011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC4×Q8×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC2×#63256C4
cosocleC4×Dic17C23
hypercenterC4×Q8#595
hypocenterC17#1489
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. #11763813C2semidirect
2. C4×#63243C2semidirect
3. Q8⋊#3624C2semidirect
4. #11763657C2semidirect
55. Q8×C17M16semidirect
67. D68M16semidirect
125. C17#1489semidirect
158. Q8#63243semidirect

Maximal subgroups

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

subgroupindex
#117638132
C4×#632432
Q8⋊#36242
#117636572
C4×#632562
#148917