Finite Group Databasebeta

← #14527833

Subgroups — #14527833

Order 2176.

240 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 49 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#11763396C211yesmaximal, characteristic
2#11763145C211yesmaximal, characteristic, complemented
3#11763125C211yesmaximal, characteristic, complemented
4#11762746C211yesmaximal, characteristic, complemented
5#11762545C211yesmaximal, characteristic, complemented
6#11762532C211yesmaximal, characteristic, complemented
7C17×#327C211yesmaximal, characteristic, complemented
8C17×#121C2211yescharacteristic
9#10587625—21no—
10#10587623—21no—
11D4×Dic17—21no—
12C2×#63227—21no—
13C2×#63227C2211yescharacteristic
14C2×C4⋊Dic17—21no—
15C2×C4⋊Dic17C2211yescharacteristic
16#10587610C2211yescharacteristic
17#10587584—21no—
18#10587572—21no—
19#10587571—21no—
20C17×C22⋊C8C2211yescharacteristic
21C17×#95C2211yescharacteristic
22#10587503—21no—
23C22⋊#3622C2211yescharacteristic
24#10587489—21no—
25#10587487—21no—
26#10587479—21no—
27C23×D17D411yescharacteristic
28C2×D4×C17—21no—
29C2×D4×C17D411yescharacteristic
30C22×C4×C17—21no—
31C22×C4×C17C2311yescharacteristic
32C2×C22⋊D17—21no—
33C2×C22⋊D17—21no—
34C22×Dic17—21no—
35C22×Dic17—21no—
36C22×Dic17D411yescharacteristic
37C2×Dic34D411yescharacteristic
38C2×C8×C17—21no—
39C4⋊C4×C17D411yescharacteristic
40C22⋊C4×C17—21no—
41C22⋊Dic17—21no—
42C22⋊Dic17—21no—
43#63227—21no—
44#63227—21no—
45#63227—21no—
46#63227—21no—
47#63227—42no—
48C4⋊Dic17—21no—
49C4⋊Dic17—21no—
50C4⋊Dic17—21no—
51C4⋊Dic17—42no—
52C4⋊Dic17D411yescharacteristic
53#63225—21no—
54C4×Dic17—21no—
55C2×#3622—21no—
56C23×C17—21no—
57C23×C17C2×D411yescharacteristic
58C22×D17—21no—
59C22×D17—21no—
60C22×D17—21no—
61C22×D17—41no—
62C22×D17—41no—
63D4×C17—41no—
64D4×C17—41no—
65D4×C17—41no—
66C2×C4×C17—22no—
67C2×C4×C17—21no—
68C2×C4×C17—41no—
69C2×C4×C17—41no—
70C2×C4×C17—41no—
71C2×C4×C17C2×D411yescharacteristic
72C2×C4×C17C2×D411yescharacteristic
73C22⋊D17—81no—
74C22⋊D17—81no—
75C2×Dic17—21no—
76C2×Dic17—21no—
77C2×Dic17—21no—
78C2×Dic17—41no—
79C2×Dic17—41no—
80C2×Dic17—41no—
81C2×Dic17—41no—
82C2×Dic17—81no—
83Dic34—41no—
84C8×C17—41no—
85#3622—41no—
86#1618—171nomaximal, Sylow
87C22×C17—21no—
88C22×C17—21no—
89C22×C17—41no—
90C22×C17—41no—
91C22×C17—41no—
92C22×C17#12011yescharacteristic
93C22×C17#13511yescharacteristic
94C22×C17#13711yescharacteristic
95C2×D17—42no—
96C2×D17—41no—
97C2×D17—41no—
98C2×D17—41no—
99C2×D17—81no—
100C4×C17—21no—
101C4×C17—21no—
102C4×C17—42no—
103C4×C17—41no—
104Dic17—41no—
105Dic17—41no—
106Dic17—41no—
107Dic17—81no—
108#538—171no—
109#484—171no—
110#481—171no—
111#399—171no—
112#328—171no—
113#327D1711yescharacteristic, complemented
114#323—171no—
115C2×C17—21no—
116C2×C17—21no—
117C2×C17—81no—
118C2×C17#45211yescharacteristic
119C2×C17#45311yescharacteristic
120C2×C17#45811yescharacteristic
121D17—81no—
122#126—341no—
123#124—341no—
124#123—341no—
125C22⋊Q8—171no—
126#121—341no—
127#121C2×D1711yescharacteristic
128#120—341no—
129C4×D4—341no—
130C2×C4⋊C4—171no—
131C2×C4⋊C4—341no—
132C2×C22⋊C4—171no—
133C2×C22⋊C4—341no—
134#107—341no—
135#106—341no—
136#103—341no—
137#102—341no—
138C22⋊C8—171no—
139C22⋊C8C2×D1711yescharacteristic
140#95C2×D1711yescharacteristic
141C17#161811yescharacteristic, complemented, Sylow
142C24—171no—
143C2×Q8—171no—
144C2×D4—21no—
145C2×D4—341no—
146C2×D4—341no—
147C2×D4D6811yescharacteristic
148C22×C4—21no—
149C22×C4—171no—
150C22×C4—341no—
151C22×C4—341no—
152C22×C4C22×D1711yescharacteristic
153C2×C8—21no—
154C2×C8—341no—
155C4⋊C4—171no—
156C4⋊C4—341no—
157C4⋊C4—341no—
158C4⋊C4—341no—
159C4⋊C4—341no—
160C4⋊C4—682no—
161C4⋊C4D6811yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—341no—
164C22⋊C4—341no—
165C22⋊C4—341no—
166C22⋊C4—341no—
167C22⋊C4—341no—
168C22⋊C4—341no—
169C22⋊C4—682no—
170C42—341no—
171C23—21no—
172C23—341no—
173C23—341no—
174C23—341no—
175C23—681no—
176C23—681no—
177C23D4×D1711yescharacteristic
178Q8—681no—
179D4—41no—
180D4—41no—
181D4—41no—
182D4—1361no—
183D4—1361no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—341no—
190C2×C4—341no—
191C2×C4—341no—
192C2×C4—681no—
193C2×C4—681no—
194C2×C4—681no—
195C2×C4—681no—
196C2×C4—1361no—
197C2×C4C2×D6811yescharacteristic
198C2×C4D4×D1711yescharacteristic
199C8—41no—
200C8—681no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—682no—
207C22—681no—
208C22—681no—
209C22—681no—
210C22—1361no—
211C22#1058756811yescharacteristic
212C22#1058759811yescharacteristic
213C22#1058760111yescharacteristic
214C4—21no—
215C4—21no—
216C4—42no—
217C4—41no—
218C4—681no—
219C4—681no—
220C4—681no—
221C4—1361no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—1361no—
226C2#1176297211yescharacteristic
227C2#1176297811yescharacteristic
228C2#1176300011yescharacteristic
subgroupG/H
centerC22#10587568
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#327C2
socleC22×C17#120
prime radical#11763396C2
hypercenter#327D17
hypocenterC17#1618
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #11763145C2semidirect
3. #11763125C2semidirect
4. #11762746C2semidirect
5. #11762545C2semidirect
6. #11762532C2semidirect
7. C17×#327C2semidirect
113. #327D17semidirect
141. C17#1618semidirect

Maximal subgroups

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

subgroupindex
#117633962
#117631452
#117631252
#117627462
#117625452
#117625322
C17×#3272
#161817