Finite Group Databasebeta

← #30637330

Subgroups — #30637330

Order 16256.

212 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 61 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#125×D127C211yesmaximal, characteristic
2#22549931C211yesmaximal, characteristic
3#22549828C211yesmaximal, characteristic
4#22549618C211yesmaximal, characteristic, complemented
5#22549617C211yesmaximal, characteristic, complemented
6#22549617C211yesmaximal, characteristic, complemented
7#330×C127C211yesmaximal, characteristic, complemented
8C2×#14523231—21no—
9#16861201C2211yescharacteristic
10#16861200C411yescharacteristic
11#16861200C411yescharacteristic
12#16861199—21no—
13#16861199—21no—
14C4⋊C4×D127—21no—
15C4⋊C4×D127—21no—
16C4⋊C4×D127C411yescharacteristic
17C4⋊C4×D127C411yescharacteristic
18C42×D127C2211yescharacteristic
19Dic127⋊C8—21no—
20(C2×D127)⋊C8—21no—
21#16861120C2211yescharacteristic
22#16861120C2211yescharacteristic
23C8×Dic127—21no—
24#125×C127C2211yescharacteristic
25#97×C127C2211yescharacteristic
26#97×C127C2211yescharacteristic
27C2×C4×D127—21no—
28C2×C4×D127—21no—
29C2×C4×D127C2×C411yescharacteristic
30C2×C4×D127D411yescharacteristic
31C2×C4×D127D411yescharacteristic
32#14523231—24no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127C2×C411yescharacteristic
38C4⋊Dic127C2×C411yescharacteristic
39Dic127⋊C4—21no—
40Dic127⋊C4—21no—
41Dic127⋊C4—21no—
42Dic127⋊C4—21no—
43Dic127⋊C4—21no—
44Dic127⋊C4—21no—
45C4×Dic127C2×C411yescharacteristic
46C4×Dic127D411yescharacteristic
47C4×Dic127D411yescharacteristic
48C2×C8×C127—21no—
49C2×C8×C127—21no—
50C4⋊C4×C127—21no—
51C4⋊C4×C127—21no—
52C4⋊C4×C127C2×C411yescharacteristic
53C4⋊C4×C127C2×C411yescharacteristic
54C42×C127C2311yescharacteristic
55C22×D127C22⋊C411yescharacteristic
56C2×Dic127—21no—
57C2×Dic127—21no—
58C2×Dic127C22⋊C411yescharacteristic
59C2×Dic127C22⋊C411yescharacteristic
60C2×Dic127C22⋊C411yescharacteristic
61C4×D127—22no—
62C4×D127—22no—
63C4×D127—41no—
64C4×D127—42no—
65C4×D127—42no—
66C127⋊C8—22no—
67C127⋊C8—41no—
68C2×C4×C127—21no—
69C2×C4×C127—21no—
70C2×C4×C127C22×C411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127C2×D411yescharacteristic
73C8×C127—22no—
74C8×C127—41no—
75C22×C127C2×C22⋊C411yescharacteristic
76C2×D127—21no—
77C2×D127—21no—
78C2×D127#10412yes—
79C4×C127—21no—
80C4×C127—21no—
81C4×C127—21no—
82C4×C127—41no—
83C4×C127—41no—
84Dic127—21no—
85Dic127—21no—
86Dic127—41no—
87Dic127—41no—
88Dic127#10112yes—
89C2×C127C2×#10111yescharacteristic
90C2×C127C2×#10411yescharacteristic
91C2×C127#42111yescharacteristic
92D127—22no—
93#1528—1271nomaximal, Sylow
94C127#152811yescharacteristic, complemented, Sylow
95C2×#125—1271no—
96#432—1271no—
97C4×M16—1271no—
98#330—1271no—
99#330—1271no—
100#330—1271no—
101#330D12711yescharacteristic, complemented
102C2×M16—2541no—
103#125—1271no—
104#125—1271no—
105#125—1271no—
106#125—2541no—
107#125—2541no—
108#125C2×D12711yescharacteristic
109C2×C4⋊C4—1271no—
110C2×C4⋊C4—1271no—
111C2×C4⋊C4—2541no—
112C2×C4⋊C4—2541no—
113C2×C42—1271no—
114C4⋊C8—2541no—
115C22⋊C8—2541no—
116#97—1271no—
117#97—1271no—
118#97C2×D12711yescharacteristic
119#97C2×D12711yescharacteristic
120C4×C8—2541no—
121C22×C4—1271no—
122C22×C4—1271no—
123C22×C4—1271no—
124C22×C4—2541no—
125C22×C4—2541no—
126M16—2544no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—2541no—
130C2×C8—2541no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—1271no—
134C4⋊C4—1271no—
135C4⋊C4—2541no—
136C4⋊C4—2541no—
137C4⋊C4—2541no—
138C4⋊C4—2541no—
139C4⋊C4—2541no—
140C4⋊C4—2541no—
141C4⋊C4—2541no—
142C4⋊C4—2541no—
143C4⋊C4C4×D12711yescharacteristic
144C4⋊C4C4×D12711yescharacteristic
145C42—1271no—
146C42—1271no—
147C42—1271no—
148C42C22×D12711yescharacteristic
149C23—1271no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—1271no—
153C2×C4—1271no—
154C2×C4—1271no—
155C2×C4—5081no—
156C2×C4—5082no—
157C2×C4—5082no—
158C2×C4—2542no—
159C2×C4—2542no—
160C2×C4—2541no—
161C2×C4—2541no—
162C2×C4C2×C4×D12711yescharacteristic
163C2×C4D4×D12711yescharacteristic
164C2×C4D4×D12711yescharacteristic
165C8—22no—
166C8—41no—
167C8—5081no—
168C8—2542no—
169C22—1272no—
170C22—2541no—
171C22—2541no—
172C22C22⋊C4×D12711yescharacteristic
173C4—21no—
174C4—21no—
175C4—21no—
176C4—41no—
177C4—41no—
178C4—1272no—
179C4—5081no—
180C4—5081no—
181C4—2541no—
182C4—2541no—
183C2—2542no—
184C2#101×D12711yescharacteristic
185C2#104×D12711yescharacteristic
186C2#2254989111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#330×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×D127C22⋊C4
cosocleC42×C127C23
hypercenter#330D127
hypocenterC127#1528
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
4. #22549618C2semidirect
5. #22549617C2semidirect
6. #22549617C2semidirect
7. #330×C127C2semidirect
94. C127#1528semidirect
101. #330D127semidirect

Maximal subgroups

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

subgroupindex
#125×D1272
#225499312
#225498282
#225496182
#225496172
#330×C1272
#1528127