Finite Group Databasebeta

← #14544223

Subgroups — #14544223

Order 2176.

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

normal subgroups only

This group has 288 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#11763777C211yesmaximal, characteristic
2#11763751C211yesmaximal, characteristic
3#11763662C214yesmaximal, complemented
4C2×#10587583C211yesmaximal, characteristic
5C22×C4×D17C2211yescharacteristic
6C22×Dic34C411yescharacteristic, complemented
7C2×C22⋊#595—21no—
8C2×C4⋊#595—21no—
9C2×#63242—21no—
10C2×C4⋊C4×C17C411yescharacteristic, complemented
11C2×#63227—21no—
12C2×C4⋊Dic17—21no—
13#10587583—21no—
14#10587583—21no—
15#10587583C412yescomplemented
16#10587583C2212yes—
17C22⋊#3624—21no—
18(C2×#595)⋊C4C2212yes—
19#10587543C2212yes—
20C23×D17D411yescharacteristic
21C22×#595—22no—
22C22×C4×C17—21no—
23C22×C4×C17C2×C411yescharacteristic
24C22×Dic17—21no—
25C22×Dic17D411yescharacteristic
26C2×C4×D17—42no—
27C2×C4×D17C2×C411yescharacteristic
28C2×C4×D17D411yescharacteristic
29C2×C4×D17D411yescharacteristic
30C2×C4×D17C2311yescharacteristic
31C2×Dic34—21no—
32C2×Dic34—42no—
33C2×Dic34C2×C412yes—
34C22⋊#595—42no—
35C2×#3624—22no—
36C4⋊#595—42no—
37#63242—42no—
38C4⋊C4×C17—21no—
39C4⋊C4×C17C2×C412yes—
40#63227—22no—
41#63227—22no—
42C4⋊Dic17—22no—
43C4⋊Dic17—22no—
44C23×C17C22⋊C411yescharacteristic
45C22×D17—41no—
46C22×D17—41no—
47C22×D17C22⋊C411yescharacteristic
48C22×D17SD1612yes—
49C22×D17Q1612yes—
50C22×D17C2×D411yescharacteristic
51C2×#595—42no—
52C2×#595—42no—
53C2×#595—42no—
54C2×C4×C17—22no—
55C2×C4×C17—22no—
56C2×C4×C17—42no—
57C2×C4×C17C22⋊C411yescharacteristic
58C2×C4×C17C22×C411yescharacteristic
59C2×Dic17—22no—
60C2×Dic17—41no—
61C2×Dic17—42no—
62C2×Dic17C22⋊C411yescharacteristic
63C2×Dic17C2×D411yescharacteristic
64C4×D17—22no—
65C4×D17—22no—
66C4×D17—41no—
67C4×D17—41no—
68Dic34—42no—
69Dic34—42no—
70#3624—42no—
71#1520—171nomaximal, Sylow
72C22×C17—21no—
73C22×C17—21no—
74C22×C17#10312yes—
75C22×C17C2×C22⋊C411yescharacteristic
76C2×D17—21no—
77C2×D17—21no—
78C2×D17—21no—
79C2×D17—21no—
80C2×D17—42no—
81C2×D17—42no—
82C2×D17#10312yes—
83C2×D17C2×SD1611yescharacteristic
84C2×D17C2×Q1611yescharacteristic
85#595—82no—
86C4×C17—21no—
87C4×C17—42no—
88C4×C17#9912yes—
89Dic17—21no—
90Dic17—21no—
91Dic17—42no—
92C2×C22⋊Q8—171no—
93C2×C22⋊C8—171no—
94#380—171no—
95#328—174no—
96C2×C17—22no—
97C2×C17C2×#9911yescharacteristic
98C2×C17C2×#10311yescharacteristic
99C2×C17#42111yescharacteristic
100D17—21no—
101D17—22no—
102D17—21no—
103C22×Q8—171no—
104C23×C4—171no—
105C22×C8—341no—
106C22⋊Q8—172no—
107C22⋊Q8—172no—
108C22⋊Q8—341no—
109C22⋊Q8—341no—
110C2×C4⋊C4—341no—
111C2×C4⋊C4—341no—
112C2×C4⋊C4#59511yescharacteristic, complemented
113C2×C22⋊C4—341no—
114C2×C22⋊C4—341no—
115C22⋊C8—172no—
116C22⋊C8—341no—
117#95—172no—
118C17#152011yescharacteristic, complemented, Sylow
119C24—171no—
120C2×Q8—172no—
121C2×Q8—341no—
122C2×Q8—682no—
123C22×C4—21no—
124C22×C4—171no—
125C22×C4—171no—
126C22×C4—171no—
127C22×C4—171no—
128C22×C4—171no—
129C22×C4—341no—
130C22×C4—342no—
131C22×C4—682no—
132C22×C4C2×#59511yescharacteristic
133C2×C8—342no—
134C2×C8—682no—
135C4⋊C4—21no—
136C4⋊C4—342no—
137C4⋊C4—342no—
138C4⋊C4—682no—
139C4⋊C4C2×#59512yes—
140C22⋊C4—342no—
141C22⋊C4—342no—
142C22⋊C4—682no—
143C23—171no—
144C23—171no—
145C23—172no—
146C23—172no—
147C23—681no—
148C23—681no—
149C23C22⋊#59511yescharacteristic
150Q8—682no—
151Q8—682no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—42no—
155C2×C4—171no—
156C2×C4—171no—
157C2×C4—342no—
158C2×C4—342no—
159C2×C4—342no—
160C2×C4—681no—
161C2×C4—681no—
162C2×C4—681no—
163C2×C4—682no—
164C2×C4—682no—
165C2×C4—682no—
166C2×C4—682no—
167C2×C4C22⋊#59511yescharacteristic
168C2×C4C22×#59511yescharacteristic
169C8—682no—
170C22—21no—
171C22—21no—
172C22—171no—
173C22—172no—
174C22—171no—
175C22—341no—
176C22—341no—
177C22—341no—
178C22—341no—
179C22—682no—
180C22—682no—
181C22#1058754912yes—
182C22C2×C22⋊#59511yescharacteristic
183C4—21no—
184C4—42no—
185C4—341no—
186C4—341no—
187C4—682no—
188C4—1362no—
189C4#1058753912yes—
190C2—22no—
191C2—341no—
192C2—342no—
193C2—341no—
194C2C2×#1058753911yescharacteristic
195C2C2×#1058754911yescharacteristic
196C2#1176380711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC2×C4⋊C4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC23×D17D4
cosocleC2×C4×D17C23
hypercenterC2×C4⋊C4#595
hypocenterC17#1520
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #11763662C2semidirect
6. C22×Dic34C4semidirect
10. C2×C4⋊C4×C17C4semidirect
15. #10587583C4semidirect
112. C2×C4⋊C4#595semidirect
118. C17#1520semidirect

Maximal subgroups

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

subgroupindex
#117637772
#117637512
#117636622
C2×#105875832
#152017