Finite Group Databasebeta

← #14544196

Subgroups — #14544196

Order 2176.

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

normal subgroups only

This group has 222 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#11763831C211yesmaximal, characteristic, complemented
2#11763753C211yesmaximal, characteristic, complemented
3#11763663C212yesmaximal, complemented
4#11763660C212yesmaximal
5#11763338C211yesmaximal, characteristic
6C2×C4⋊#595—21no—
7C17×#124C411yescharacteristic, complemented
8#10587632C2211yescharacteristic
9Q8×Dic17C411yescharacteristic, complemented
10#10587624C2211yescharacteristic
11D4×Dic17C411yescharacteristic, complemented
12#10587569—21no—
13#10587567—21no—
14#10587565—21no—
15#10587563—21no—
16C42×D17C2211yescharacteristic
17#10587556C411yescharacteristic, complemented
18(C2×#595)⋊C4—21no—
19(C2×#595)⋊C4C2212yes—
20#10587543C2212yes—
21#10587542—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—
32C4⋊#595—42no—
33C22⋊C4×C17—21no—
34C22⋊C4×C17—21no—
35C42×C17C2×C411yescharacteristic
36C22⋊Dic17—21no—
37C22⋊Dic17—21no—
38#63227—21no—
39C4⋊Dic17—21no—
40C4⋊Dic17C2×C411yescharacteristic
41C4⋊Dic17C2×C411yescharacteristic
42#63225—21no—
43#63225—21no—
44#63225—21no—
45C4×Dic17—21no—
46C4×Dic17C2×C411yescharacteristic
47C4×Dic17D411yescharacteristic
48C4×Dic17D411yescharacteristic
49C23×C17—21no—
50C22×D17C2×D411yescharacteristic
51C2×#595—42no—
52C2×#595—42no—
53C2×#595—42no—
54Q8×C17—41no—
55D4×C17—41no—
56C2×C4×C17—21no—
57C2×C4×C17C22⋊C411yescharacteristic
58C2×C4×C17C22⋊C411yescharacteristic
59C2×C4×C17C22×C411yescharacteristic
60C22⋊D17—41no—
61C22⋊D17—41no—
62C2×Dic17—21no—
63C2×Dic17—21no—
64C2×Dic17—42no—
65C2×Dic17C22⋊C411yescharacteristic
66C2×Dic17C22⋊C411yescharacteristic
67C2×Dic17C2×D411yescharacteristic
68C4×D17—21no—
69C4×D17—21no—
70C4×D17—41no—
71C4×D17—41no—
72#3624—42no—
73#1512—171nomaximal, Sylow
74C22×C17—41no—
75C22×C17—41no—
76C22×C17—41no—
77C22×C17C2×C22⋊C411yescharacteristic
78C2×D17—21no—
79C2×D17—21no—
80C2×D17AGL(1,ℤ/8)11yescharacteristic
81C2×D17#13711yescharacteristic
82#595—82no—
83C4×C17—21no—
84C4×C17—21no—
85C4×C17—21no—
86C4×C17—41no—
87Dic17—21no—
88Dic17—21no—
89Dic17—21no—
90Dic17—41no—
91Dic17—41no—
92#529—171no—
93#432—171no—
94#382—171no—
95#328—172no—
96#327—172no—
97C2×C17—81no—
98C2×C17#41011yescharacteristic
99C2×C17#41711yescharacteristic
100C2×C17#42111yescharacteristic
101D17—21no—
102D17—21no—
103#126—341no—
104#125—171no—
105#124#59511yescharacteristic, complemented
106#123—341no—
107C22⋊Q8—171no—
108#121—171no—
109C4×Q8—171no—
110C4×D4—171no—
111C4×D4—341no—
112#117—341no—
113C2×C4⋊C4—341no—
114C2×C42—171no—
115C4⋊C8—341no—
116C22⋊C8—172no—
117#97—341no—
118#95—172no—
119#95—341no—
120C17#151211yescharacteristic, complemented, Sylow
121C2×Q8C2×#59511yescharacteristic
122C2×D4—341no—
123C2×D4C2×#59511yescharacteristic
124C22×C4—171no—
125C22×C4—171no—
126C22×C4—171no—
127C22×C4—341no—
128C22×C4—342no—
129C2×C8—342no—
130C4⋊C4—171no—
131C4⋊C4—171no—
132C4⋊C4—341no—
133C4⋊C4—341no—
134C4⋊C4—341no—
135C4⋊C4—341no—
136C4⋊C4—682no—
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—341no—
140C22⋊C4—341no—
141C22⋊C4—341no—
142C42—171no—
143C42—171no—
144C42—171no—
145C42—341no—
146C42C2×#59511yescharacteristic
147C23—21no—
148C23—171no—
149Q8—41no—
150D4—41no—
151D4—681no—
152D4—681no—
153C2×C4—21no—
154C2×C4—171no—
155C2×C4—171no—
156C2×C4—171no—
157C2×C4—341no—
158C2×C4—341no—
159C2×C4—341no—
160C2×C4—341no—
161C2×C4—681no—
162C2×C4—681no—
163C2×C4—682no—
164C2×C4—682no—
165C2×C4—682no—
166C2×C4—682no—
167C2×C4C22⋊#59511yescharacteristic
168C2×C4C22⋊#59511yescharacteristic
169C2×C4C22×#59511yescharacteristic
170C8—682no—
171C22—41no—
172C22—41no—
173C22—41no—
174C22—171no—
175C22—171no—
176C22—341no—
177C22—341no—
178C22C2×C22⋊#59511yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—41no—
183C4—341no—
184C4—341no—
185C4—341no—
186C4—681no—
187C4—681no—
188C4—1362no—
189C2—81no—
190C2—341no—
191C2—341no—
192C2#1176378911yescharacteristic
193C2#1176380011yescharacteristic
194C2#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#1512
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. #11763831C2semidirect
2. #11763753C2semidirect
3. #11763663C2semidirect
7. C17×#124C4semidirect
9. Q8×Dic17C4semidirect
11. D4×Dic17C4semidirect
17. #10587556C4semidirect
105. #124#595semidirect
120. C17#1512semidirect

Maximal subgroups

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

subgroupindex
#117638312
#117637532
#117636632
#117636602
#117633382
#151217