Finite Group Databasebeta

← #30639937

Subgroups — #30639937

Order 16256.

326 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 71 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, complemented
3#22549877C211yesmaximal, characteristic, complemented
4C2×#16861126C211yesmaximal, characteristic, complemented
5#22549779C211yesmaximal, characteristic, complemented
6#22549635C211yesmaximal, characteristic, complemented
7#535×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C2×C4⋊Dic127C2211yescharacteristic
11C2×C22⋊Dic127—22no—
12C2×C4×Dic127C2211yescharacteristic
13#16861142—21no—
14#16861140—21no—
15#16861136—22no—
16#16861133—21no—
17#16861130—21no—
18#16861126—22no—
19C2×#41×C127C2211yescharacteristic
20C22×D4×C127C2211yescharacteristic
21#127×C127—21no—
22#124×C127—21no—
23#121×C127—21no—
24#121×C127—21no—
25#120×C127—22no—
26#117×C127C2211yescharacteristic
27C22×Dic127—21no—
28C22×Dic127—21no—
29#14523232—21no—
30#14523232—21no—
31#14523232—21no—
32#14523232D412yescomplemented
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C22⋊Dic127—44no—
38C4×Dic127—21no—
39C4×Dic127—21no—
40C24×C127D412yes—
41#41×C127—21no—
42#41×C127—21no—
43#41×C127—21no—
44#41×C127—21no—
45C2×Q8×C127D411yescharacteristic
46C2×D4×C127—21no—
47C2×D4×C127—22no—
48C2×D4×C127—22no—
49C2×D4×C127—21no—
50C2×D4×C127—21no—
51C2×D4×C127—21no—
52C2×D4×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127D411yescharacteristic
56C22×C4×C127—21no—
57C22×C4×C127C2311yescharacteristic
58C4⋊C4×C127D411yescharacteristic
59C4⋊C4×C127D411yescharacteristic
60C22⋊C4×C127—22no—
61C22⋊C4×C127—22no—
62C22⋊C4×C127—21no—
63C42×C127—21no—
64C2×Dic127—22no—
65C2×Dic127—41no—
66C2×Dic127—41no—
67C2×Dic127—42no—
68C2×Dic127—41no—
69C127⋊C8—22no—
70C127⋊C8—41no—
71C23×C127—22no—
72C23×C127—21no—
73C23×C127—22no—
74C23×C127—22no—
75C23×C127—42no—
76C23×C127—42no—
77C23×C127C2×D411yescharacteristic
78Q8×C127—21no—
79Q8×C127—21no—
80D4×C127—21no—
81D4×C127—21no—
82D4×C127—41no—
83D4×C127—41no—
84D4×C127—41no—
85D4×C127—41no—
86D4×C127—41no—
87D4×C127—41no—
88D4×C127—82no—
89D4×C127—82no—
90C2×C4×C127—21no—
91C2×C4×C127—21no—
92C2×C4×C127—21no—
93C2×C4×C127—41no—
94C2×C4×C127—41no—
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127#4111yescharacteristic
99C2×C4×C127#4111yescharacteristic
100C2×C4×C127#4111yescharacteristic
101C22×C127—21no—
102C22×C127—21no—
103C22×C127—44no—
104C22×C127—41no—
105C22×C127—42no—
106C22×C127—41no—
107C22×C127—42no—
108C22×C127—41no—
109C22×C127—42no—
110C22×C127—82no—
111C22×C127#12011yescharacteristic
112C22×C127#12111yescharacteristic
113C22×C127#12311yescharacteristic
114C4×C127—41no—
115C4×C127—41no—
116C4×C127—41no—
117C4×C127#12111yescharacteristic
118C4×C127#12111yescharacteristic
119C4×C127#12311yescharacteristic
120C4×C127#12411yescharacteristic
121Dic127—42no—
122Dic127—81no—
123C2×C127—21no—
124C2×C127—21no—
125C2×C127—81no—
126C2×C127—82no—
127C2×C127#39411yescharacteristic
128C2×C127#45311yescharacteristic
129C2×C127#45511yescharacteristic
130#2037—1271nomaximal, Sylow
131C127#203711yescharacteristic, complemented, Sylow
132#535D12711yescharacteristic
133#428—1271no—
134C2×#104—1271no—
135#417—1271no—
136C2×#100—1271no—
137#390—1271no—
138#339—1271no—
139C2×#41C2×D12711yescharacteristic
140C22×D4C2×D12711yescharacteristic
141C2×M16—1271no—
142C2×M16—1271no—
143#127—21no—
144#124—21no—
145#121—21no—
146#121—21no—
147#120—22no—
148#117C2×D12711yescharacteristic
149C2×C4⋊C4—1271no—
150C2×C22⋊C4—2542no—
151C2×C42—1271no—
152#107—2541no—
153#106—2541no—
154#104—2542no—
155#103—2541no—
156#102—2541no—
157#100—2542no—
158C24C22⋊D12712yes—
159#41—21no—
160#41—21no—
161#41—21no—
162#41—21no—
163C2×Q8C22⋊D12711yescharacteristic
164C2×D4—21no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4C22⋊D12711yescharacteristic
174C22×C4—21no—
175C22×C4—2541no—
176C22×C4—2541no—
177C22×C4C22×D12711yescharacteristic
178M16—1272no—
179M16—2541no—
180M16—2541no—
181M16—2541no—
182C2×C8—2541no—
183C2×C8—2541no—
184C4⋊C4—2541no—
185C4⋊C4—2541no—
186C4⋊C4C22⋊D12711yescharacteristic
187C4⋊C4C22⋊D12711yescharacteristic
188C22⋊C4—22no—
189C22⋊C4—22no—
190C22⋊C4—21no—
191C22⋊C4—5084no—
192C42—21no—
193C42—2541no—
194C42—2541no—
195C23—22no—
196C23—22no—
197C23—21no—
198C23—22no—
199C23—42no—
200C23—42no—
201C23C2×C22⋊D12711yescharacteristic
202Q8—21no—
203Q8—21no—
204D4—21no—
205D4—21no—
206D4—41no—
207D4—41no—
208D4—41no—
209D4—41no—
210D4—41no—
211D4—41no—
212D4—82no—
213D4—82no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—5082no—
220C2×C4—5081no—
221C2×C4—5081no—
222C2×C4—5081no—
223C2×C4—2542no—
224C2×C4D4×D12711yescharacteristic
225C2×C4C2×C22⋊D12711yescharacteristic
226C2×C4C2×C22⋊D12711yescharacteristic
227C2×C4#1452324711yescharacteristic
228C2×C4#1452324711yescharacteristic
229C2×C4#1452324711yescharacteristic
230C8—5081no—
231C8—2542no—
232C22—21no—
233C22—21no—
234C22—44no—
235C22—41no—
236C22—42no—
237C22—41no—
238C22—42no—
239C22—41no—
240C22—42no—
241C22—82no—
242C22#1686117811yescharacteristic
243C22#1686118111yescharacteristic
244C22#1686119011yescharacteristic
245C4—41no—
246C4—41no—
247C4—41no—
248C4—10161no—
249C4—5082no—
250C4#1686118011yescharacteristic
251C4#1686118111yescharacteristic
252C4#1686119011yescharacteristic
253C4#1686119611yescharacteristic
254C2—21no—
255C2—21no—
256C2—82no—
257C2—81no—
258C2#2254979011yescharacteristic
259C2#2255004411yescharacteristic
260C2#2255005411yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting, prime radical#535×C127C2
socleC22×C127#120
hypercenter#535D127
hypocenterC127#2037
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #22549919C2semidirect
2. C2×#16861136C2semidirect
3. #22549877C2semidirect
4. C2×#16861126C2semidirect
5. #22549779C2semidirect
6. #22549635C2semidirect
32. #14523232D4semidirect
131. C127#2037semidirect

Maximal subgroups

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

subgroupindex
#225499192
C2×#168611362
#225498772
C2×#168611262
#225497792
#225496352
#535×C1272
#2037127