Finite Group Databasebeta

← #14544949

Subgroups — #14544949

Order 2176.

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

normal subgroups only

This group has 336 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
1C2×#10587550C212yesmaximal, complemented
2C2×#10587548C212yesmaximal, complemented
3C4×#63243C211yesmaximal, characteristic, complemented
4C4×C4⋊#595C211yesmaximal, characteristic, complemented
5#11763358C211yesmaximal, characteristic
6C2×#63256C2211yescharacteristic
7C2×#63254C2211yescharacteristic
8C2×C4⋊#595—21no—
9C2×C4×#595C2212yes—
10C2×#63243C2212yes—
11C17×#124C411yescharacteristic, complemented
12#10587630C411yescharacteristic, complemented
13#10587626C411yescharacteristic, complemented
14#10587570—21no—
15#10587570—21no—
16#10587567—21no—
17#10587567—21no—
18#10587561C411yescharacteristic, complemented
19C42×D17C2211yescharacteristic
20#10587550—24no—
21#10587548—24no—
22(C2×#595)⋊C4—21no—
23#10587542—21no—
24C4×#3624—21no—
25C22×#595—22no—
26C2×Q8×C17C2×C411yescharacteristic
27C2×D4×C17C2×C411yescharacteristic
28C2×C22⋊D17—21no—
29C2×C22⋊D17—21no—
30C2×C22⋊D17—21no—
31C22×Dic17—21no—
32#63256—22no—
33#63256—22no—
34#63254—22no—
35#63254—22no—
36C2×D68—21no—
37C2×D68C2×C411yescharacteristic
38C2×C4×D17—21no—
39C2×C4×D17D411yescharacteristic
40C2×C4×D17D411yescharacteristic
41C2×C4×D17C2311yescharacteristic
42C2×Dic34C2×C411yescharacteristic
43C2×#3624—22no—
44C4⋊#595—24no—
45C4×#595—22no—
46C4×#595—22no—
47#63243—22no—
48#63243D414yescomplemented
49C22⋊C4×C17—21no—
50C22⋊C4×C17—21no—
51C42×C17C2×C411yescharacteristic
52#63227—21no—
53#63227—21no—
54#63225—21no—
55#63225—21no—
56C4×Dic17C2×C411yescharacteristic
57C4×Dic17D411yescharacteristic
58C4×Dic17D411yescharacteristic
59C23×C17—21no—
60C22×D17—21no—
61C22×D17C2×D411yescharacteristic
62C2×#595—24no—
63C2×#595—42no—
64C2×#595—42no—
65Q8×C17—22no—
66D4×C17—22no—
67C2×C4×C17—21no—
68C2×C4×C17C22⋊C411yescharacteristic
69C2×C4×C17C22⋊C411yescharacteristic
70C2×C4×C17C22×C411yescharacteristic
71C22⋊D17—42no—
72C22⋊D17—41no—
73C22⋊D17—41no—
74C22⋊D17—42no—
75C2×Dic17—21no—
76C2×Dic17—42no—
77C2×Dic17C22⋊C411yescharacteristic
78C2×Dic17C22⋊C411yescharacteristic
79C2×Dic17C2×D411yescharacteristic
80D68—22no—
81D68—41no—
82D68—41no—
83C4×D17—22no—
84C4×D17—41no—
85C4×D17—42no—
86C4×D17C2×D412yes—
87C4×D17#4112yes—
88Dic34—22no—
89#3624—24no—
90#1980—171nomaximal, Sylow
91C22×C17—41no—
92C22×C17—41no—
93C22×C17—41no—
94C22×C17C2×C22⋊C411yescharacteristic
95C2×D17—21no—
96C2×D17—21no—
97C2×D17—41no—
98C2×D17—41no—
99C2×D17—41no—
100C2×D17#12411yescharacteristic
101C2×D17#12711yescharacteristic
102#595—44no—
103C4×C17—21no—
104C4×C17—21no—
105C4×C17—41no—
106C4×C17C4×D412yes—
107Dic17—21no—
108Dic17—41no—
109Dic17#10412yescomplemented
110Dic17#12112yes—
111#532—171no—
112C2×#104—172no—
113C2×#104—172no—
114C4×M16—171no—
115C4×C4⋊C4—171no—
116C2×C17—81no—
117C2×C17#39011yescharacteristic
118C2×C17C2×#10411yescharacteristic
119C2×C17#42111yescharacteristic
120D17—21no—
121D17—21no—
122D17—81no—
123C2×#41—171no—
124C2×#41—171no—
125C2×M16—172no—
126C4⋊Q8—171no—
127#127—171no—
128#124—171no—
129#124#59511yescharacteristic, complemented
130#121—341no—
131#121—341no—
132C4×D4—341no—
133C4×D4—341no—
134C2×C4⋊C4—341no—
135C2×C42—171no—
136C2×C42—172no—
137#104—344no—
138#104—344no—
139#97—341no—
140C4×C8—341no—
141#95—341no—
142C17#198011yescharacteristic, complemented, Sylow
143#41—342no—
144#41—342no—
145#41—342no—
146#41—342no—
147C2×Q8—171no—
148C2×Q8C2×#59511yescharacteristic
149C2×D4—171no—
150C2×D4—341no—
151C2×D4—341no—
152C2×D4—341no—
153C2×D4—341no—
154C2×D4C2×#59511yescharacteristic
155C22×C4—171no—
156C22×C4—171no—
157C22×C4—171no—
158C22×C4—341no—
159C22×C4—341no—
160C22×C4—342no—
161M16—174no—
162M16—342no—
163C2×C8—342no—
164C4⋊C4—341no—
165C4⋊C4—341no—
166C4⋊C4—344no—
167C22⋊C4—21no—
168C22⋊C4—21no—
169C22⋊C4—341no—
170C22⋊C4—341no—
171C42—171no—
172C42—171no—
173C42—171no—
174C42—342no—
175C42—342no—
176C42C2×#59511yescharacteristic
177C23—21no—
178C23—171no—
179C23—341no—
180Q8—22no—
181Q8—342no—
182D4—22no—
183D4—342no—
184D4—681no—
185D4—681no—
186D4—682no—
187D4—682no—
188D4—681no—
189D4—681no—
190C2×C4—21no—
191C2×C4—171no—
192C2×C4—171no—
193C2×C4—172no—
194C2×C4—172no—
195C2×C4—171no—
196C2×C4—342no—
197C2×C4—344no—
198C2×C4—341no—
199C2×C4—681no—
200C2×C4—682no—
201C2×C4—682no—
202C2×C4—682no—
203C2×C4—682no—
204C2×C4C22⋊#59511yescharacteristic
205C2×C4C22⋊#59511yescharacteristic
206C2×C4C22×#59511yescharacteristic
207C8—344no—
208C22—41no—
209C22—41no—
210C22—41no—
211C22—171no—
212C22—171no—
213C22—341no—
214C22—341no—
215C22—681no—
216C22—681no—
217C22—681no—
218C22C2×C22⋊#59511yescharacteristic
219C4—21no—
220C4—21no—
221C4—41no—
222C4—172no—
223C4—172no—
224C4—341no—
225C4—684no—
226C4—681no—
227C4D4×#59512yes—
228C2—81no—
229C2—341no—
230C2—341no—
231C2—1361no—
232C2#1176376111yescharacteristic
233C2D17⋊#10411yescharacteristic
234C2#1176380611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#124C4
socleC22×C17C2×C22⋊C4
prime radical#11763358C2
cosocleC2×C4×D17C23
hypercenter#124#595
hypocenterC17#1980
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C2×#10587550C2semidirect
2. C2×#10587548C2semidirect
3. C4×#63243C2semidirect
4. C4×C4⋊#595C2semidirect
11. C17×#124C4semidirect
12. #10587630C4semidirect
13. #10587626C4semidirect
18. #10587561C4semidirect
48. #63243D4semidirect
109. Dic17#104semidirect
129. #124#595semidirect
142. C17#1980semidirect

Maximal subgroups

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

subgroupindex
C2×#105875502
C2×#105875482
C4×#632432
C4×C4⋊#5952
#117633582
#198017