Finite Group Databasebeta

← #14544182

Subgroups — #14544182

Order 2176.

294 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 60 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#11763781C211yesmaximal, characteristic
2C4×C22⋊#595C211yesmaximal, characteristic
3#11763663C212yesmaximal, complemented
4#11763662C212yesmaximal, complemented
5C22⋊Q8×D17C211yesmaximal, characteristic
6C2×Q8×D17C411yescharacteristic, complemented
7C22×C4×D17C2211yescharacteristic
8C2×C22⋊#595—21no—
9C2×#63247—21no—
10C2×C4×#595—21no—
11C17×C22⋊Q8C411yescharacteristic, complemented
12#10587632C2211yescharacteristic
13#10587610C411yescharacteristic, complemented
14#10587583C2211yescharacteristic
15#10587582—21no—
16C4⋊C4×D17—21no—
17C4⋊C4×D17C411yescharacteristic, complemented
18C22⋊C4×D17—21no—
19#10587564—21no—
20(C2×#595)⋊C4C2212yes—
21#10587543—21no—
22#10587543C2212yes—
23C23×D17D411yescharacteristic
24C22×#595—22no—
25C2×Q8×C17C2×C411yescharacteristic
26C22×C4×C17C2×C411yescharacteristic
27C22×Dic17D411yescharacteristic
28Q8×D17—42no—
29C2×C4×D17—21no—
30C2×C4×D17—21no—
31C2×C4×D17—42no—
32C2×C4×D17C2×C411yescharacteristic
33C2×C4×D17D411yescharacteristic
34C2×C4×D17D411yescharacteristic
35C2×C4×D17C2311yescharacteristic
36C2×Dic34—21no—
37C2×Dic34C2×C411yescharacteristic
38C22⋊#595—24no—
39#63247—42no—
40C2×#3624—22no—
41C4×#595—42no—
42C4⋊C4×C17—21no—
43C4⋊C4×C17C2×C411yescharacteristic
44C22⋊C4×C17—21no—
45C22⋊Dic17—21no—
46#63227—21no—
47#63227—21no—
48C4⋊Dic17—21no—
49C4⋊Dic17C2×C411yescharacteristic
50#63225—21no—
51#63225—21no—
52#63225—21no—
53C23×C17C22⋊C411yescharacteristic
54C22×D17—22no—
55C22×D17—22no—
56C22×D17—41no—
57C22×D17C22⋊C411yescharacteristic
58C22×D17C2×D411yescharacteristic
59C2×#595—24no—
60C2×#595—42no—
61C2×#595—42no—
62Q8×C17—41no—
63C2×C4×C17—21no—
64C2×C4×C17—21no—
65C2×C4×C17—41no—
66C2×C4×C17C22⋊C411yescharacteristic
67C2×C4×C17C22×C411yescharacteristic
68C2×Dic17—22no—
69C2×Dic17—21no—
70C2×Dic17—21no—
71C2×Dic17C22⋊C411yescharacteristic
72C2×Dic17C2×D411yescharacteristic
73C4×D17—22no—
74C4×D17—22no—
75C4×D17—41no—
76C4×D17—41no—
77C4×D17—42no—
78C4×D17—42no—
79Dic34—42no—
80Dic34—41no—
81#3624—42no—
82#1508—171nomaximal, Sylow
83C22×C17—21no—
84C22×C17—21no—
85C22×C17—21no—
86C22×C17C2×C22⋊C411yescharacteristic
87C2×D17—21no—
88C2×D17—21no—
89C2×D17—21no—
90C2×D17—21no—
91C2×D17—44no—
92C2×D17#10412yes—
93C2×D17#13712yes—
94#595—44no—
95C4×C17—21no—
96C4×C17—21no—
97C4×C17—41no—
98C4×C17—41no—
99Dic17—21no—
100Dic17—41no—
101Dic17—41no—
102Dic17#9912yes—
103C2×C22⋊Q8—171no—
104#407—171no—
105C4×C22⋊C4—171no—
106#328—172no—
107#328—172no—
108C2×C17—41no—
109C2×C17C2×#9911yescharacteristic
110C2×C17#41911yescharacteristic
111C2×C17C2×#10411yescharacteristic
112D17—22no—
113D17—22no—
114C22×Q8—171no—
115C23×C4—171no—
116C2×M16—341no—
117C22⋊Q8—171no—
118C22⋊Q8—171no—
119C22⋊Q8—171no—
120C22⋊Q8—341no—
121C22⋊Q8—341no—
122C22⋊Q8#59511yescharacteristic, complemented
123C2×C4⋊C4—171no—
124C2×C4⋊C4—341no—
125C2×C22⋊C4—341no—
126C2×C22⋊C4—341no—
127C2×C42—341no—
128C22⋊C8—172no—
129C22⋊C8—341no—
130#95—172no—
131C17#150811yescharacteristic, complemented, Sylow
132C24—171no—
133C2×Q8—171no—
134C2×Q8—341no—
135C2×Q8—682no—
136C2×Q8C2×#59511yescharacteristic
137C22×C4—171no—
138C22×C4—171no—
139C22×C4—171no—
140C22×C4—171no—
141C22×C4—171no—
142C22×C4—341no—
143C22×C4—341no—
144C22×C4—342no—
145C22×C4—682no—
146C22×C4C2×#59511yescharacteristic
147M16—682no—
148C2×C8—342no—
149C4⋊C4—21no—
150C4⋊C4—171no—
151C4⋊C4—341no—
152C4⋊C4—341no—
153C4⋊C4—341no—
154C4⋊C4—341no—
155C4⋊C4C2×#59511yescharacteristic
156C22⋊C4—21no—
157C22⋊C4—341no—
158C22⋊C4—341no—
159C22⋊C4—344no—
160C22⋊C4—341no—
161C42—682no—
162C23—171no—
163C23—171no—
164C23—342no—
165C23—342no—
166C23—681no—
167C23C22⋊#59511yescharacteristic
168Q8—41no—
169Q8—681no—
170Q8—682no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—41no—
174C2×C4—171no—
175C2×C4—171no—
176C2×C4—342no—
177C2×C4—342no—
178C2×C4—342no—
179C2×C4—344no—
180C2×C4—341no—
181C2×C4—341no—
182C2×C4—681no—
183C2×C4—681no—
184C2×C4—682no—
185C2×C4—682no—
186C2×C4—682no—
187C2×C4—682no—
188C2×C4C22⋊#59511yescharacteristic
189C2×C4C22×#59511yescharacteristic
190C8—682no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—172no—
195C22—172no—
196C22—341no—
197C22—341no—
198C22—341no—
199C22—341no—
200C22—684no—
201C22C2×C22⋊#59511yescharacteristic
202C4—21no—
203C4—21no—
204C4—41no—
205C4—41no—
206C4—172no—
207C4—341no—
208C4—684no—
209C4—681no—
210C4—681no—
211C2—41no—
212C2—342no—
213C2—342no—
214C2D17⋊#9911yescharacteristic
215C2#1176380211yescharacteristic
216C2D17⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×C22⋊Q8C4
socleC22×C17C2×C22⋊C4
prime radicalC23×D17D4
cosocleC2×C4×D17C23
hypercenterC22⋊Q8#595
hypocenterC17#1508
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
3. #11763663C2semidirect
4. #11763662C2semidirect
6. C2×Q8×D17C4semidirect
11. C17×C22⋊Q8C4semidirect
13. #10587610C4semidirect
17. C4⋊C4×D17C4semidirect
122. C22⋊Q8#595semidirect
131. C17#1508semidirect

Maximal subgroups

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

subgroupindex
#117637812
C4×C22⋊#5952
#117636632
#117636622
C22⋊Q8×D172
#150817