Finite Group Databasebeta

← #14544225

Subgroups — #14544225

Order 2176.

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

normal subgroups only

This group has 218 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#11763828C211yesmaximal, characteristic, complemented
2#11763754C211yesmaximal, characteristic, complemented
3#11763663C212yesmaximal, complemented
4#11763660C212yesmaximal
5#11763338C211yesmaximal, characteristic
6C17×#124C411yescharacteristic, complemented
7#10587632C2211yescharacteristic
8Q8×Dic17C411yescharacteristic, complemented
9#10587624C2211yescharacteristic
10D4×Dic17C411yescharacteristic, complemented
11#10587569—21no—
12#10587567—21no—
13#10587565—21no—
14#10587563—21no—
15C42×D17C2211yescharacteristic
16#10587556C411yescharacteristic, complemented
17(C2×#595)⋊C4—21no—
18(C2×#595)⋊C4—21no—
19(C2×#595)⋊C4C2212yes—
20#10587543C2212yes—
21C4⋊#3624—21no—
22C4×#3624—21no—
23C22×#595—22no—
24C2×Q8×C17C2×C411yescharacteristic
25C2×D4×C17C2×C411yescharacteristic
26C2×C22⋊D17—21no—
27C22×Dic17—21no—
28C2×C4×D17D411yescharacteristic
29C2×C4×D17D411yescharacteristic
30C2×C4×D17C2311yescharacteristic
31C2×#3624—22no—
32C22⋊C4×C17—21no—
33C22⋊C4×C17—21no—
34C42×C17C2×C411yescharacteristic
35C22⋊Dic17—21no—
36C22⋊Dic17—21no—
37#63227—21no—
38C4⋊Dic17—21no—
39C4⋊Dic17C2×C411yescharacteristic
40C4⋊Dic17C2×C411yescharacteristic
41#63225—21no—
42#63225—21no—
43#63225—21no—
44C4×Dic17—21no—
45C4×Dic17C2×C411yescharacteristic
46C4×Dic17D411yescharacteristic
47C4×Dic17D411yescharacteristic
48C23×C17—21no—
49C22×D17C2×D411yescharacteristic
50C2×#595—42no—
51C2×#595—42no—
52C2×#595—42no—
53Q8×C17—41no—
54D4×C17—41no—
55C2×C4×C17—21no—
56C2×C4×C17C22⋊C411yescharacteristic
57C2×C4×C17C22⋊C411yescharacteristic
58C2×C4×C17C22×C411yescharacteristic
59C22⋊D17—42no—
60C2×Dic17—21no—
61C2×Dic17—21no—
62C2×Dic17—42no—
63C2×Dic17C22⋊C411yescharacteristic
64C2×Dic17C22⋊C411yescharacteristic
65C2×Dic17C2×D411yescharacteristic
66C4×D17—22no—
67C4×D17—41no—
68C4×D17—41no—
69#3624—42no—
70#1521—171nomaximal, Sylow
71C22×C17—41no—
72C22×C17—41no—
73C22×C17—41no—
74C22×C17C2×C22⋊C411yescharacteristic
75C2×D17—21no—
76C2×D17—21no—
77C2×D17#13512yes—
78#595—82no—
79C4×C17—21no—
80C4×C17—21no—
81C4×C17—21no—
82C4×C17—41no—
83Dic17—21no—
84Dic17—21no—
85Dic17—21no—
86Dic17—41no—
87Dic17—41no—
88#529—171no—
89#431—171no—
90#383—171no—
91#328—172no—
92#327—172no—
93C2×C17—81no—
94C2×C17#41011yescharacteristic
95C2×C17#41611yescharacteristic
96C2×C17#42111yescharacteristic
97D17—22no—
98#126—341no—
99#125—171no—
100#124#59511yescharacteristic, complemented
101#123—341no—
102C22⋊Q8—171no—
103#121—171no—
104C4×Q8—171no—
105C4×D4—171no—
106C4×D4—341no—
107#117—341no—
108C2×C42—171no—
109C4⋊C8—341no—
110C22⋊C8—172no—
111C4×C8—341no—
112#95—172no—
113#95—341no—
114#95—341no—
115C17#152111yescharacteristic, complemented, Sylow
116C2×Q8C2×#59511yescharacteristic
117C2×D4—341no—
118C2×D4C2×#59511yescharacteristic
119C22×C4—171no—
120C22×C4—171no—
121C22×C4—171no—
122C22×C4—341no—
123C22×C4—342no—
124C2×C8—342no—
125C4⋊C4—171no—
126C4⋊C4—171no—
127C4⋊C4—341no—
128C4⋊C4—341no—
129C4⋊C4—341no—
130C4⋊C4—341no—
131C22⋊C4—21no—
132C22⋊C4—21no—
133C22⋊C4—341no—
134C22⋊C4—341no—
135C22⋊C4—341no—
136C42—171no—
137C42—171no—
138C42—171no—
139C42—341no—
140C42C2×#59511yescharacteristic
141C23—21no—
142C23—171no—
143Q8—41no—
144D4—41no—
145D4—682no—
146C2×C4—21no—
147C2×C4—171no—
148C2×C4—171no—
149C2×C4—171no—
150C2×C4—342no—
151C2×C4—341no—
152C2×C4—341no—
153C2×C4—681no—
154C2×C4—681no—
155C2×C4—682no—
156C2×C4—682no—
157C2×C4—682no—
158C2×C4—682no—
159C2×C4C22⋊#59511yescharacteristic
160C2×C4C22⋊#59511yescharacteristic
161C2×C4C22×#59511yescharacteristic
162C8—682no—
163C22—41no—
164C22—41no—
165C22—41no—
166C22—172no—
167C22—341no—
168C22—341no—
169C22C2×C22⋊#59511yescharacteristic
170C4—21no—
171C4—21no—
172C4—21no—
173C4—41no—
174C4—341no—
175C4—341no—
176C4—341no—
177C4—681no—
178C4—681no—
179C4—1362no—
180C2—81no—
181C2—342no—
182C2#1176378911yescharacteristic
183C2#1176379911yescharacteristic
184C2#1176380611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#124C4
socleC22×C17C2×C22⋊C4
prime radical#10587624C22
cosocleC2×C4×D17C23
hypercenter#124#595
hypocenterC17#1521
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #11763828C2semidirect
2. #11763754C2semidirect
3. #11763663C2semidirect
6. C17×#124C4semidirect
8. Q8×Dic17C4semidirect
10. D4×Dic17C4semidirect
16. #10587556C4semidirect
100. #124#595semidirect
115. C17#1521semidirect

Maximal subgroups

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

subgroupindex
#117638282
#117637542
#117636632
#117636602
#117633382
#152117