Finite Group Databasebeta

← #14544975

Subgroups — #14544975

Order 2176.

324 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 72 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#11763751C211yesmaximal, characteristic
2C4×C22⋊#595C211yesmaximal, characteristic
3#11763687C212yesmaximal
4#11763687C212yesmaximal
5C2×#10587582C211yesmaximal, characteristic
6C22×C4×D17C2211yescharacteristic
7C22×Dic34C411yescharacteristic, complemented
8C2×C22⋊#595C2212yes—
9C2×C22⋊#595C2212yes—
10C2×C4⋊#595—21no—
11C2×C4×#595—21no—
12C2×C4⋊C4×C17C411yescharacteristic, complemented
13C2×#63227C2211yescharacteristic
14C2×#63227C2211yescharacteristic
15C2×#63225C411yescharacteristic, complemented
16C2×#63225C411yescharacteristic, complemented
17#10587582—21no—
18#10587582—21no—
19#10587582—21no—
20#10587582—21no—
21(C2×#595)⋊C4—21no—
22(C2×#595)⋊C4—21no—
23C23×D17C2311yescharacteristic
24C22×#595—22no—
25C22×#595—22no—
26C22×C4×C17C2×C411yescharacteristic
27C22×C4×C17C2×C411yescharacteristic
28C22×C4×C17C2×C411yescharacteristic
29C22×Dic17C2×C411yescharacteristic
30C22×Dic17C2×C411yescharacteristic
31C22×Dic17C2×C411yescharacteristic
32C2×C4×D17—42no—
33C2×C4×D17D411yescharacteristic
34C2×C4×D17D411yescharacteristic
35C2×C4×D17D411yescharacteristic
36C2×C4×D17D411yescharacteristic
37C2×Dic34—21no—
38C2×Dic34—21no—
39C2×Dic34—42no—
40C22⋊#595—22no—
41C22⋊#595—24no—
42C22⋊#595Q814yescomplemented
43C4⋊#595—42no—
44C4×#595—42no—
45C4⋊C4×C17—21no—
46C4⋊C4×C17—21no—
47#63227—21no—
48#63227—21no—
49#63227—21no—
50#63227—21no—
51#63225—21no—
52#63225—21no—
53#63225—21no—
54#63225—21no—
55C23×C17C22×C411yescharacteristic
56C22×D17—41no—
57C22×D17—41no—
58C22×D17C2×D411yescharacteristic
59C22×D17C2×D411yescharacteristic
60C22×D17C2×Q812yes—
61C22×D17#4112yes—
62C2×#595—24no—
63C2×#595—42no—
64C2×#595—42no—
65C2×#595—42no—
66C2×#595—42no—
67C2×#595—42no—
68C2×C4×C17—21no—
69C2×C4×C17—21no—
70C2×C4×C17—41no—
71C2×C4×C17—41no—
72C2×C4×C17—41no—
73C2×C4×C17C22⋊C411yescharacteristic
74C2×C4×C17C22⋊C411yescharacteristic
75C2×Dic17—22no—
76C2×Dic17—21no—
77C2×Dic17—21no—
78C2×Dic17—41no—
79C2×Dic17—41no—
80C2×Dic17C22⋊C411yescharacteristic
81C2×Dic17C22⋊C411yescharacteristic
82C4×D17—22no—
83C4×D17—22no—
84C4×D17—41no—
85C4×D17—41no—
86Dic34—42no—
87Dic34—42no—
88#1996—171nomaximal, Sylow
89C22×C17—21no—
90C22×C17—21no—
91C22×C17C2×C22⋊C411yescharacteristic
92C22×C17C4×Q812yes—
93C2×D17—21no—
94C2×D17—21no—
95C2×D17—21no—
96C2×D17—21no—
97C2×D17—42no—
98C2×D17—42no—
99C2×D17C22⋊Q812yes—
100C2×D17#12411yescharacteristic
101C2×D17C4⋊Q811yescharacteristic
102#595—44no—
103#595—82no—
104C4×C17—21no—
105C4×C17—21no—
106C4×C17—41no—
107C4×C17—41no—
108Dic17—21no—
109Dic17—41no—
110Dic17—41no—
111Dic17#9912yes—
112C2×C22⋊Q8—171no—
113#380—171no—
114C4×C22⋊C4—171no—
115#342—172no—
116#342—172no—
117C2×C17—22no—
118C2×C17#39111yescharacteristic
119C2×C17C2×#9911yescharacteristic
120C2×C17#41011yescharacteristic
121D17—22no—
122D17—21no—
123D17—21no—
124C22×Q8—171no—
125C23×C4—171no—
126C22⋊Q8—341no—
127C22⋊Q8—341no—
128C22⋊Q8—341no—
129C22⋊Q8—341no—
130C2×C4⋊C4—171no—
131C2×C4⋊C4—171no—
132C2×C4⋊C4—341no—
133C2×C4⋊C4#59511yescharacteristic, complemented
134C2×C22⋊C4—172no—
135C2×C22⋊C4—171no—
136C2×C22⋊C4—171no—
137C2×C22⋊C4—172no—
138C2×C42—341no—
139#95—341no—
140#95—341no—
141C17#199611yescharacteristic, complemented, Sylow
142C24—171no—
143C2×Q8—341no—
144C2×Q8—341no—
145C2×Q8—682no—
146C22×C4—171no—
147C22×C4—171no—
148C22×C4—171no—
149C22×C4—171no—
150C22×C4—171no—
151C22×C4—171no—
152C22×C4—171no—
153C22×C4—342no—
154C22×C4—342no—
155C22×C4—682no—
156C22×C4C2×#59511yescharacteristic
157C22×C4C2×#59511yescharacteristic
158C22×C4C2×#59511yescharacteristic
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—341no—
162C4⋊C4—341no—
163C4⋊C4—341no—
164C4⋊C4—341no—
165C4⋊C4—682no—
166C22⋊C4—174no—
167C22⋊C4—341no—
168C22⋊C4—341no—
169C22⋊C4—341no—
170C22⋊C4—341no—
171C22⋊C4—342no—
172C22⋊C4—344no—
173C42—682no—
174C23—171no—
175C23—171no—
176C23—172no—
177C23—172no—
178C23—681no—
179C23—681no—
180C23C22×#59511yescharacteristic
181Q8—682no—
182Q8—682no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—171no—
189C2×C4—171no—
190C2×C4—342no—
191C2×C4—342no—
192C2×C4—342no—
193C2×C4—344no—
194C2×C4—341no—
195C2×C4—341no—
196C2×C4—681no—
197C2×C4—681no—
198C2×C4—682no—
199C2×C4—682no—
200C2×C4—681no—
201C2×C4—681no—
202C2×C4—682no—
203C2×C4—682no—
204C2×C4—682no—
205C2×C4C22⋊#59511yescharacteristic
206C2×C4C22⋊#59511yescharacteristic
207C22—21no—
208C22—21no—
209C22—172no—
210C22—171no—
211C22—171no—
212C22—341no—
213C22—341no—
214C22—341no—
215C22—341no—
216C22—682no—
217C22—682no—
218C22Q8×#59512yes—
219C22C2×C22⋊#59511yescharacteristic
220C4—21no—
221C4—21no—
222C4—41no—
223C4—41no—
224C4—172no—
225C4—341no—
226C4—684no—
227C4—681no—
228C4—681no—
229C4—1362no—
230C2—22no—
231C2—342no—
232C2—341no—
233C2—341no—
234C2(C2×#595)⋊Q811yescharacteristic
235C2D17⋊#9911yescharacteristic
236C2#1176378811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC2×C4⋊C4×C17C4
socleC22×C17C2×C22⋊C4
prime radical, cosocleC23×D17C23
hypercenterC2×C4⋊C4#595
hypocenterC17#1996
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
7. C22×Dic34C4semidirect
12. C2×C4⋊C4×C17C4semidirect
15. C2×#63225C4semidirect
16. C2×#63225C4semidirect
42. C22⋊#595Q8semidirect
133. C2×C4⋊C4#595semidirect
141. C17#1996semidirect

Maximal subgroups

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

subgroupindex
#117637512
C4×C22⋊#5952
#117636872
C2×#105875822
#199617