Finite Group Databasebeta

← #30641869

Subgroups — #30641869

Order 16256.

688 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#16861178C211yesmaximal, characteristic
2#22549776C213yesmaximal, complemented
3#22549770C213yesmaximal, complemented
4#22549764C213yesmaximal, complemented
5#22549752C213yesmaximal, complemented
6#22549732C211yesmaximal, characteristic, complemented
7#379×C127C211yesmaximal, characteristic, complemented
8C22×C22⋊D127C2213yes—
9C22×C4×D127C2211yescharacteristic, complemented
10#16861178C418yescomplemented
11C2×Dic127⋊C4C2213yescomplemented
12C2×C22⋊Dic127C2213yes—
13C2×C22⋊Dic127C2213yes—
14C2×C22⋊Dic127C2213yes—
15C2×(C2×D127)⋊C4C2213yescomplemented
16C2×C4×Dic127C2213yescomplemented
17C127⋊#95C2213yescomplemented
18C127⋊#95C2213yescomplemented
19C25×C127C2211yescharacteristic
20C2×C22⋊C4×C127C2213yes—
21C2×C22⋊C4×C127C2213yes—
22C23×D127C2311yescharacteristic
23C2×C22⋊D127—43no—
24C2×C22⋊D127—43no—
25C2×C22⋊D127C2×C4112yes—
26C22×Dic127C2311yescharacteristic
27C22×Dic127C2313yes—
28C22×Dic127C2313yes—
29C2×C4×D127—43no—
30Dic127⋊C4—43no—
31C22⋊Dic127—43no—
32C22⋊Dic127—43no—
33C22⋊Dic127C2×C4112yescomplemented
34(C2×D127)⋊C4—43no—
35C4×Dic127—43no—
36C24×C127—43no—
37C24×C127—43no—
38C24×C127C2×C414yescomplemented
39C24×C127C2313yes—
40C22×C4×C127C2311yescharacteristic
41C22×C4×C127C2313yes—
42C22⋊C4×C127—43no—
43C22⋊C4×C127—43no—
44C22×D127—23no—
45C22×D127—23no—
46C22×D127C22×C412yes—
47C22⋊D127—412no—
48C2×Dic127—23no—
49C2×Dic127—23no—
50C2×Dic127—23no—
51C2×Dic127—23no—
52C2×Dic127—23no—
53C2×Dic127—23no—
54C2×Dic127C22×C416yes—
55C4×D127—44no—
56C23×C127—23no—
57C23×C127—23no—
58C23×C127—23no—
59C23×C127—23no—
60C23×C127—23no—
61C23×C127—23no—
62C23×C127—412no—
63C23×C127—412no—
64C23×C127—44no—
65C23×C127C22×C416yes—
66C23×C127C2411yescharacteristic
67C2×C4×C127—23no—
68C2×C4×C127—23no—
69C2×C4×C127—23no—
70C2×C4×C127—23no—
71C22×C127—26no—
72C22×C127—26no—
73C22×C127—26no—
74C22×C127—416no—
75C22×C127—43no—
76C22×C127—43no—
77C22×C127—43no—
78C22×C127—43no—
79C22×C127C23×C411yescharacteristic
80C22×C127H(5,𝔽2)13yes—
81C22×C127H(5,𝔽2)13yes—
82C2×D127—26no—
83C2×D127—43no—
84C2×D127—41no—
85C4×C127—43no—
86C4×C127—41no—
87Dic127—26no—
88Dic127—43no—
89Dic127—41no—
90C2×C127—46no—
91C2×C127#51813yes—
92C2×C127#56013yes—
93C2×C127H(3,𝔽4)11yescharacteristic
94D127—42no—
95#2375—1271nomaximal, Sylow
96C127#237511yescharacteristic, complemented, Sylow
97C2×#120—1271no—
98#390—1273no—
99#388—1273no—
100#388—1273no—
101#386—1273no—
102#379—1271no—
103#379D12711yescharacteristic, complemented
104C25C2×D12711yescharacteristic
105C22×D4—1273no—
106C23×C4—1271no—
107#120—1278no—
108C2×C4⋊C4—1273no—
109C2×C22⋊C4—1273no—
110C2×C22⋊C4—1273no—
111C2×C22⋊C4—1273no—
112C2×C22⋊C4—1273no—
113C2×C22⋊C4C2×D12713yes—
114C2×C22⋊C4C2×D12713yes—
115C2×C42—1273no—
116#95—1273no—
117#95—1273no—
118C24—43no—
119C24—43no—
120C24—1271no—
121C24C4×D12714yescomplemented
122C24C22×D12713yes—
123C2×D4—12712no—
124C2×D4—5083no—
125C2×D4—5083no—
126C22×C4—1273no—
127C22×C4—1271no—
128C22×C4—1273no—
129C22×C4—5083no—
130C22×C4C22×D12711yescharacteristic
131C22×C4C22×D12713yes—
132C4⋊C4—5083no—
133C22⋊C4—43no—
134C22⋊C4—43no—
135C22⋊C4—12712no—
136C22⋊C4—5083no—
137C22⋊C4—5083no—
138C22⋊C4—5083no—
139C42—5083no—
140C23—23no—
141C23—23no—
142C23—23no—
143C23—23no—
144C23—23no—
145C23—23no—
146C23—412no—
147C23—412no—
148C23—44no—
149C23—1272no—
150C23—2543no—
151C23—2543no—
152C23C2×C4×D12716yes—
153C23C23×D12711yescharacteristic
154D4—50812no—
155C2×C4—23no—
156C2×C4—23no—
157C2×C4—23no—
158C2×C4—23no—
159C2×C4—1276no—
160C2×C4—5084no—
161C2×C4—2543no—
162C2×C4—2543no—
163C2×C4—2543no—
164C2×C4—2543no—
165C2×C4—2543no—
166C2×C4—2543no—
167C22—26no—
168C22—26no—
169C22—26no—
170C22—416no—
171C22—43no—
172C22—43no—
173C22—43no—
174C22—43no—
175C22—5083no—
176C22—5081no—
177C22—2546no—
178C22C22×C4×D12711yescharacteristic
179C22#1686125413yes—
180C22#1686125413yes—
181C4—43no—
182C4—41no—
183C4—5083no—
184C4—5081no—
185C4—2546no—
186C2—46no—
187C2—5082no—
188C2#2255034913yes—
189C2#2255061513yes—
190C2#2255061911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#379×C127C2
socle, cosocleC23×C127C24
prime radicalC2×#16861178C2
hypercenter#379D127
hypocenterC127#2375
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. #22549776C2semidirect
3. #22549770C2semidirect
4. #22549764C2semidirect
5. #22549752C2semidirect
6. #22549732C2semidirect
7. #379×C127C2semidirect
9. C22×C4×D127C22semidirect
10. #16861178C4semidirect
11. C2×Dic127⋊C4C22semidirect
15. C2×(C2×D127)⋊C4C22semidirect
16. C2×C4×Dic127C22semidirect
17. C127⋊#95C22semidirect
18. C127⋊#95C22semidirect
33. C22⋊Dic127C2×C4semidirect
38. C24×C127C2×C4semidirect
96. C127#2375semidirect
103. #379D127semidirect
121. C24C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611782
#225497762
#225497702
#225497642
#225497522
#225497322
#379×C1272
#2375127