Finite Group Databasebeta

← #30638963

Subgroups — #30638963

Order 16256.

286 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 93 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#22549919C211yesmaximal, characteristic, complemented
2C2×#16861136C211yesmaximal, characteristic
3#22549877C211yesmaximal, characteristic
4#22549741C211yesmaximal, characteristic, complemented
5#22549645C211yesmaximal, characteristic, complemented
6#22549635C211yesmaximal, characteristic, complemented
7#520×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C2×C4⋊Dic127C2211yescharacteristic
11C2×C4×Dic127C2211yescharacteristic
12#16861142—21no—
13#16861140—21no—
14#16861136C414yescomplemented
15#16861133—21no—
16#16861130—21no—
17C127⋊#95—22no—
18C2×#41×C127C2211yescharacteristic
19C4×Q8×C127—21no—
20C4×D4×C127—21no—
21C4×D4×C127—22no—
22#117×C127—21no—
23#117×C127C2211yescharacteristic
24C2×C4⋊C4×C127—21no—
25C2×C4⋊C4×C127C2211yescharacteristic
26C22×Dic127—21no—
27C22×Dic127—21no—
28#14523232—21no—
29#14523232—21no—
30#14523232—21no—
31#14523232C2×C412yescomplemented
32C2×C127⋊C8—21no—
33C2×C127⋊C8—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127—21no—
36C4×Dic127—21no—
37C4×Dic127C2×C412yescomplemented
38#41×C127—21no—
39#41×C127—21no—
40#41×C127C2×C412yes—
41#41×C127Q812yes—
42C2×Q8×C127D411yescharacteristic
43C2×D4×C127—21no—
44C2×D4×C127D411yescharacteristic
45C22×C4×C127—21no—
46C22×C4×C127—22no—
47C22×C4×C127D412yes—
48C22×C4×C127C2311yescharacteristic
49C4⋊C4×C127—22no—
50C4⋊C4×C127—21no—
51C4⋊C4×C127—21no—
52C4⋊C4×C127D411yescharacteristic
53C4⋊C4×C127D411yescharacteristic
54C22⋊C4×C127—22no—
55C22⋊C4×C127—21no—
56C42×C127—22no—
57C42×C127—21no—
58C2×Dic127—22no—
59C2×Dic127—41no—
60C2×Dic127—41no—
61C2×Dic127—42no—
62C2×Dic127—41no—
63C127⋊C8—22no—
64C127⋊C8—41no—
65C23×C127—21no—
66C23×C127C2×D411yescharacteristic
67Q8×C127—21no—
68Q8×C127C4⋊C412yes—
69D4×C127—22no—
70D4×C127—22no—
71D4×C127—21no—
72D4×C127C4⋊C412yes—
73C2×C4×C127—21no—
74C2×C4×C127—22no—
75C2×C4×C127—22no—
76C2×C4×C127—21no—
77C2×C4×C127—22no—
78C2×C4×C127—21no—
79C2×C4×C127—42no—
80C2×C4×C127—82no—
81C2×C4×C127C22×C411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×Q811yescharacteristic
85C2×C4×C127#4111yescharacteristic
86C2×C4×C127#4111yescharacteristic
87C22×C127—21no—
88C22×C127—21no—
89C22×C127—22no—
90C22×C127—41no—
91C22×C127—41no—
92C22×C127C4×D411yescharacteristic
93C22×C127#12011yescharacteristic
94C22×C127C22⋊Q811yescharacteristic
95C4×C127—22no—
96C4×C127—41no—
97C4×C127—42no—
98C4×C127—41no—
99C4×C127C2×C4⋊C411yescharacteristic
100C4×C127C4×D411yescharacteristic
101C4×C127C22⋊Q811yescharacteristic
102C4×C127#12311yescharacteristic
103Dic127—42no—
104Dic127—81no—
105C2×C127—21no—
106C2×C127—21no—
107C2×C127—42no—
108C2×C127#38511yescharacteristic
109C2×C127#45411yescharacteristic
110C2×C127#45511yescharacteristic
111#1860—1271nomaximal, Sylow
112C127#186011yescharacteristic, complemented, Sylow
113#520D12711yescharacteristic
114#428—1271no—
115C2×#104—1271no—
116#417—1271no—
117#382—1271no—
118#343—1271no—
119#339—1271no—
120C2×#41C2×D12711yescharacteristic
121C2×M16—1271no—
122C2×M16—1271no—
123C4×Q8—21no—
124C4×D4—21no—
125C4×D4—22no—
126#117—21no—
127#117C2×D12711yescharacteristic
128C2×C4⋊C4—21no—
129C2×C4⋊C4—1271no—
130C2×C4⋊C4C2×D12711yescharacteristic
131C2×C42—1271no—
132#107—2541no—
133#106—2541no—
134#104—1274no—
135#103—2541no—
136#102—2541no—
137#95—2542no—
138#41—21no—
139#41—21no—
140#41C4×D12712yes—
141#41Dic25412yes—
142C2×Q8C22⋊D12711yescharacteristic
143C2×D4—21no—
144C2×D4C22⋊D12711yescharacteristic
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—2541no—
148C22×C4—2541no—
149C22×C4C22⋊D12712yes—
150C22×C4C22×D12711yescharacteristic
151M16—1272no—
152M16—2541no—
153M16—2541no—
154M16—2541no—
155C2×C8—2541no—
156C2×C8—2541no—
157C4⋊C4—22no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—2541no—
161C4⋊C4—2541no—
162C4⋊C4C22⋊D12711yescharacteristic
163C4⋊C4C22⋊D12711yescharacteristic
164C22⋊C4—22no—
165C22⋊C4—21no—
166C42—22no—
167C42—21no—
168C42—1272no—
169C42—2541no—
170C23—21no—
171C23C2×C22⋊D12711yescharacteristic
172Q8—21no—
173Q8Dic127⋊C412yes—
174D4—22no—
175D4—22no—
176D4—21no—
177D4Dic127⋊C412yes—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—42no—
185C2×C4—82no—
186C2×C4—5082no—
187C2×C4—5081no—
188C2×C4—5081no—
189C2×C4—5081no—
190C2×C4—2542no—
191C2×C4C2×C4×D12711yescharacteristic
192C2×C4C2×C22⋊D12711yescharacteristic
193C2×C4C2×C22⋊D12711yescharacteristic
194C2×C4C2×Dic25411yescharacteristic
195C2×C4#1452324811yescharacteristic
196C2×C4#1452324811yescharacteristic
197C8—5081no—
198C8—2542no—
199C22—21no—
200C22—21no—
201C22—22no—
202C22—41no—
203C22—41no—
204C22C4×C22⋊D12711yescharacteristic
205C22#1686117811yescharacteristic
206C22#1686118811yescharacteristic
207C4—22no—
208C4—41no—
209C4—42no—
210C4—41no—
211C4—10161no—
212C4—5082no—
213C4C2×Dic127⋊C411yescharacteristic
214C4C4×C22⋊D12711yescharacteristic
215C4#1686118811yescharacteristic
216C4#1686119211yescharacteristic
217C2—21no—
218C2—21no—
219C2—42no—
220C2#2254974911yescharacteristic
221C2#2255004911yescharacteristic
222C2#2255005411yescharacteristic
subgroupG/H
centerC22#16861178
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#520×C127C2
socleC22×C127#120
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenter#520D127
hypocenterC127#1860
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #22549919C2semidirect
4. #22549741C2semidirect
5. #22549645C2semidirect
6. #22549635C2semidirect
14. #16861136C4semidirect
31. #14523232C2×C4semidirect
37. C4×Dic127C2×C4semidirect
112. C127#1860semidirect

Maximal subgroups

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

subgroupindex
#225499192
C2×#168611362
#225498772
#225497412
#225496452
#225496352
#520×C1272
#1860127