Finite Group Databasebeta

← #14529016

Subgroups — #14529016

Order 2176.

286 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 83 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#11763290C211yesmaximal, characteristic
2#11762849C211yesmaximal, characteristic, complemented
3C2×#10587476C211yesmaximal, characteristic
4#11762808C211yesmaximal, characteristic
5#11762578C211yesmaximal, characteristic, complemented
6#11762569C211yesmaximal, characteristic, complemented
7C17×#382C211yesmaximal, characteristic, complemented
8C2×#63252C2211yescharacteristic
9C2×C4⋊C4×C17C2211yescharacteristic
10C2×C4⋊C4×C17C2211yescharacteristic
11C2×C42×C17C2211yescharacteristic
12C4×C22⋊D17—22no—
13#10587612C2211yescharacteristic
14C2×#63223C2211yescharacteristic
15C2×#63223C2211yescharacteristic
16#10587579—21no—
17#10587578—21no—
18C4⋊C4×D17—21no—
19#10587573—21no—
20C17×#95—22no—
21#10587481—21no—
22#10587480—21no—
23#10587479—21no—
24#10587478—21no—
25#10587476C414yescomplemented
26C22×C4×C17—21no—
27C22×C4×C17—21no—
28C22×C4×C17D412yes—
29C22×C4×C17C2311yescharacteristic
30C2×C22⋊D17—21no—
31#63252—21no—
32#63252—21no—
33#63252C2×C412yes—
34#63252Q812yes—
35C2×D68D411yescharacteristic
36C2×C4×D17—21no—
37C2×C4×D17—22no—
38C2×Dic34D411yescharacteristic
39C4⋊C4×C17—21no—
40C4⋊C4×C17—21no—
41C4⋊C4×C17—21no—
42C4⋊C4×C17—21no—
43C42×C17—21no—
44C42×C17C2×C412yescomplemented
45C22⋊Dic17—21no—
46#63227—22no—
47C4⋊Dic17D411yescharacteristic
48C4⋊Dic17D411yescharacteristic
49#63225—22no—
50C4×Dic17—22no—
51C4×Dic17—21no—
52#63223—21no—
53#63223—21no—
54#63223—21no—
55#63223C2×C412yescomplemented
56C2×#3622—21no—
57C2×#3622—21no—
58C23×C17C2×D411yescharacteristic
59C22×D17—21no—
60C2×C4×C17—22no—
61C2×C4×C17—22no—
62C2×C4×C17—41no—
63C2×C4×C17—41no—
64C2×C4×C17—42no—
65C2×C4×C17—42no—
66C2×C4×C17—41no—
67C2×C4×C17C22×C411yescharacteristic
68C2×C4×C17C2×D411yescharacteristic
69C2×C4×C17C2×D411yescharacteristic
70C2×C4×C17C2×Q811yescharacteristic
71C2×C4×C17#4111yescharacteristic
72C2×C4×C17#4111yescharacteristic
73C22⋊D17—22no—
74C22⋊D17—22no—
75C2×Dic17—21no—
76C2×Dic17—21no—
77C2×Dic17—21no—
78D68—21no—
79D68C4⋊C412yes—
80C4×D17—22no—
81C4×D17—22no—
82C4×D17—82no—
83Dic34—21no—
84Dic34C4⋊C412yes—
85#3622—22no—
86#3622—41no—
87#1860—171nomaximal, Sylow
88C22×C17—21no—
89C22×C17—21no—
90C22×C17C4×D411yescharacteristic
91C22×C17#12011yescharacteristic
92C22×C17C22⋊Q811yescharacteristic
93C2×D17—22no—
94C2×D17—41no—
95C2×D17—41no—
96C4×C17—42no—
97C4×C17—42no—
98C4×C17—81no—
99C4×C17C2×C4⋊C411yescharacteristic
100C4×C17C4×D411yescharacteristic
101C4×C17C22⋊Q811yescharacteristic
102C4×C17#12311yescharacteristic
103Dic17—22no—
104Dic17—41no—
105Dic17—41no—
106#520—171no—
107#428—171no—
108C2×#104—171no—
109#417—171no—
110#382D1711yescharacteristic, complemented
111#343—171no—
112#339—171no—
113C2×C17—21no—
114C2×C17—21no—
115C2×C17#38511yescharacteristic
116C2×C17#45411yescharacteristic
117C2×C17#45511yescharacteristic
118D17—42no—
119C2×#41—171no—
120C2×M16—171no—
121C2×M16—171no—
122C4×Q8—341no—
123C4×D4—341no—
124C4×D4—342no—
125#117—171no—
126#117—341no—
127C2×C4⋊C4—341no—
128C2×C4⋊C4C2×D1711yescharacteristic
129C2×C4⋊C4C2×D1711yescharacteristic
130C2×C42C2×D1711yescharacteristic
131#107—341no—
132#106—341no—
133#104—174no—
134#103—341no—
135#102—341no—
136#95—22no—
137C17#186011yescharacteristic, complemented, Sylow
138#41—172no—
139#41—172no—
140#41—341no—
141#41—341no—
142C2×Q8—171no—
143C2×D4—171no—
144C2×D4—341no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—342no—
148C22×C4—341no—
149C22×C4C22⋊D1712yes—
150C22×C4C22×D1711yescharacteristic
151M16—172no—
152M16—341no—
153M16—341no—
154M16—341no—
155C2×C8—341no—
156C2×C8—341no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—171no—
162C4⋊C4—171no—
163C4⋊C4—342no—
164C22⋊C4—342no—
165C22⋊C4—341no—
166C42—21no—
167C42—342no—
168C42—341no—
169C42C4×D1712yescomplemented
170C23—341no—
171C23C2×C22⋊D1711yescharacteristic
172Q8—172no—
173Q8—341no—
174D4—172no—
175D4—342no—
176D4—342no—
177D4—341no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—341no—
186C2×C4—342no—
187C2×C4—342no—
188C2×C4—341no—
189C2×C4—341no—
190C2×C4—1362no—
191C2×C4C2×C4×D1711yescharacteristic
192C2×C4#6325211yescharacteristic
193C2×C4D4×D1711yescharacteristic
194C2×C4D4×D1711yescharacteristic
195C2×C4Q8×D1711yescharacteristic
196C2×C4#6325611yescharacteristic
197C8—342no—
198C8—681no—
199C22—21no—
200C22—21no—
201C22—342no—
202C22—681no—
203C22—681no—
204C22C4×C22⋊D1711yescharacteristic
205C22#1058762311yescharacteristic
206C22#1058763211yescharacteristic
207C4—42no—
208C4—42no—
209C4—81no—
210C4—342no—
211C4—681no—
212C4—681no—
213C4C4⋊C4×D1711yescharacteristic
214C4#1058757911yescharacteristic
215C4#1058758011yescharacteristic
216C4#1058758211yescharacteristic
217C2—21no—
218C2—21no—
219C2—682no—
220C2#1176268011yescharacteristic
221C2#1176298111yescharacteristic
222C2#1176298611yescharacteristic
subgroupG/H
centerC22#10587623
commutatorC2×C4×C17C22×C4
FrattiniC22×C4C22×D17
FittingC17×#382C2
socleC22×C17#120
prime radicalC2×#63252C22
cosocleC22×C4×C17C23
hypercenter#382D17
hypocenterC17#1860
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #382

Complements

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

NH ≅ G/Ntype
2. #11762849C2semidirect
5. #11762578C2semidirect
6. #11762569C2semidirect
7. C17×#382C2semidirect
25. #10587476C4semidirect
44. C42×C17C2×C4semidirect
55. #63223C2×C4semidirect
110. #382D17semidirect
137. C17#1860semidirect
169. C42C4×D17semidirect

Maximal subgroups

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

subgroupindex
#117632902
#117628492
C2×#105874762
#117628082
#117625782
#117625692
C17×#3822
#186017