Finite Group Databasebeta

← #14544252

Subgroups — #14544252

Order 2176.

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

normal subgroups only

This group has 216 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#11763829C211yesmaximal, characteristic, complemented
2C4×#63247C211yesmaximal, characteristic, complemented
3#11763666C212yesmaximal, complemented
4#11763665C212yesmaximal, complemented
5#11763345C211yesmaximal, characteristic
6C2×#63247—21no—
7C17×C22⋊Q8C411yescharacteristic
8#10587633C2211yescharacteristic
9Q8×Dic17C411yescharacteristic
10#10587616C411yescharacteristic
11C2×C4×Dic17C2211yescharacteristic
12#10587584—21no—
13#10587579C411yescharacteristic
14#10587578—21no—
15#10587576C2211yescharacteristic
16#10587572—21no—
17#10587567—21no—
18C22⋊#3624—21no—
19#10587544—21no—
20#10587542C2212yes—
21#10587542C2212yes—
22C4×#3624—21no—
23C2×Q8×C17C2×C411yescharacteristic
24C22×C4×C17C2×C411yescharacteristic
25C2×C22⋊D17—21no—
26C22×Dic17D411yescharacteristic
27C22×Dic17D411yescharacteristic
28C2×D68C2×C411yescharacteristic
29C2×C4×D17—21no—
30#63247—24no—
31C2×#3624—22no—
32C2×#3624—22no—
33C4⋊C4×C17—21no—
34C4⋊C4×C17C2×C411yescharacteristic
35C22⋊C4×C17—21no—
36#63227—21no—
37#63227—21no—
38#63227—21no—
39#63227—21no—
40C4⋊Dic17—21no—
41C4⋊Dic17—21no—
42C4⋊Dic17C2×C411yescharacteristic
43#63225—21no—
44C4×Dic17—21no—
45C4×Dic17C2×C411yescharacteristic
46C4×Dic17D411yescharacteristic
47C4×Dic17D411yescharacteristic
48C4×Dic17C2311yescharacteristic
49C23×C17C22⋊C411yescharacteristic
50C22×D17—21no—
51Q8×C17—41no—
52C2×C4×C17—21no—
53C2×C4×C17—21no—
54C2×C4×C17—41no—
55C2×C4×C17C22⋊C411yescharacteristic
56C2×C4×C17C22×C411yescharacteristic
57C22⋊D17—42no—
58C2×Dic17—22no—
59C2×Dic17—21no—
60C2×Dic17—41no—
61C2×Dic17C22⋊C411yescharacteristic
62C2×Dic17C22⋊C411yescharacteristic
63C2×Dic17C2×D411yescharacteristic
64C2×Dic17C2×D411yescharacteristic
65D68—41no—
66C4×D17—42no—
67#3624—24no—
68#3624—42no—
69#1529—171nomaximal, Sylow
70C22×C17—21no—
71C22×C17—21no—
72C22×C17—21no—
73C22×C17C2×C22⋊C411yescharacteristic
74C2×D17—41no—
75C2×D17—41no—
76C2×D17—41no—
77C4×C17—21no—
78C4×C17—21no—
79C4×C17—41no—
80C4×C17—41no—
81Dic17—21no—
82Dic17—21no—
83Dic17—21no—
84Dic17—41no—
85Dic17#10412yes—
86#529—171no—
87#432—171no—
88C4×M16—171no—
89#330—172no—
90#329—172no—
91C2×C17—41no—
92C2×C17#41311yescharacteristic
93C2×C17C2×#10411yescharacteristic
94C2×C17#42111yescharacteristic
95D17—81no—
96C2×M16—341no—
97#126—341no—
98#125—171no—
99#124—171no—
100#123—341no—
101C22⋊Q8#59511yescharacteristic
102#121—171no—
103C4×Q8—171no—
104C4×D4—171no—
105C4×D4—341no—
106#117—341no—
107C2×C42—171no—
108C4⋊C8—341no—
109C22⋊C8—341no—
110#97—172no—
111#97—172no—
112C4×C8—341no—
113C17#152911yescharacteristic, complemented, Sylow
114C2×Q8C2×#59511yescharacteristic
115C2×D4—171no—
116C2×D4—341no—
117C22×C4—171no—
118C22×C4—171no—
119C22×C4—341no—
120C22×C4C2×#59511yescharacteristic
121M16—344no—
122C2×C8—342no—
123C2×C8—342no—
124C4⋊C4—21no—
125C4⋊C4—171no—
126C4⋊C4—341no—
127C4⋊C4—341no—
128C4⋊C4—341no—
129C4⋊C4C2×#59511yescharacteristic
130C22⋊C4—21no—
131C22⋊C4—341no—
132C22⋊C4—341no—
133C22⋊C4—341no—
134C22⋊C4—341no—
135C42—171no—
136C42—171no—
137C42—171no—
138C42—171no—
139C42—341no—
140C23—341no—
141C23C22⋊#59511yescharacteristic
142Q8—41no—
143D4—681no—
144D4—682no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—171no—
149C2×C4—171no—
150C2×C4—171no—
151C2×C4—171no—
152C2×C4—342no—
153C2×C4—341no—
154C2×C4—682no—
155C2×C4—681no—
156C2×C4C22⋊#59511yescharacteristic
157C2×C4C22×#59511yescharacteristic
158C8—344no—
159C8—682no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—681no—
164C22—681no—
165C22—681no—
166C22C2×C22⋊#59511yescharacteristic
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—172no—
172C4—341no—
173C4—341no—
174C4—341no—
175C4—681no—
176C2—41no—
177C2—1361no—
178C2#1176379611yescharacteristic
179C2D17⋊#10411yescharacteristic
180C2#1176380811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×C22⋊Q8C4
socleC22×C17C2×C22⋊C4
prime radical#10587616C4
cosocleC4×Dic17C23
hypercenterC22⋊Q8#595
hypocenterC17#1529
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #11763829C2semidirect
2. C4×#63247C2semidirect
3. #11763666C2semidirect
4. #11763665C2semidirect
113. C17#1529semidirect

Maximal subgroups

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

subgroupindex
#117638292
C4×#632472
#117636662
#117636652
#117633452
#152917