Finite Group Databasebeta

← #30637216

Subgroups — #30637216

Order 16256.

238 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 81 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#22549929C211yesmaximal, characteristic, complemented
2#22549722C211yesmaximal, characteristic, complemented
3#22549613C212yesmaximal, complemented
4#22549610C212yesmaximal
5#529×C127C211yesmaximal, characteristic
6C2×C4×Dic127—21no—
7C4⋊(C127⋊C8)—21no—
8C22⋊(C127⋊C8)C2212yes—
9C4×C127⋊C8—21no—
10C127⋊#95—21no—
11C127⋊#95C2212yes—
12#126×C127—21no—
13#125×C127C411yescharacteristic, complemented
14#124×C127C411yescharacteristic, complemented
15#123×C127—21no—
16C22⋊Q8×C127C2211yescharacteristic
17#121×C127C2211yescharacteristic
18C4×Q8×C127C411yescharacteristic, complemented
19C4×D4×C127—21no—
20C4×D4×C127C411yescharacteristic, complemented
21#117×C127—21no—
22C2×C42×C127C2211yescharacteristic
23C22×Dic127—22no—
24C2×C127⋊C8—22no—
25C4×Dic127—24no—
26C2×Q8×C127C2×C411yescharacteristic
27C2×D4×C127—21no—
28C2×D4×C127C2×C411yescharacteristic
29C22×C4×C127—21no—
30C22×C4×C127D411yescharacteristic
31C22×C4×C127D411yescharacteristic
32C22×C4×C127C2311yescharacteristic
33C4⋊C4×C127—21no—
34C4⋊C4×C127—21no—
35C4⋊C4×C127—21no—
36C4⋊C4×C127—21no—
37C4⋊C4×C127C2×C411yescharacteristic
38C4⋊C4×C127C2×C411yescharacteristic
39C22⋊C4×C127—21no—
40C22⋊C4×C127—21no—
41C22⋊C4×C127—21no—
42C22⋊C4×C127—21no—
43C22⋊C4×C127—21no—
44C42×C127—21no—
45C42×C127C2×C411yescharacteristic
46C42×C127C2×C411yescharacteristic
47C42×C127D411yescharacteristic
48C42×C127D411yescharacteristic
49C2×Dic127—24no—
50C2×Dic127—42no—
51C2×Dic127—42no—
52C127⋊C8—42no—
53C23×C127—21no—
54C23×C127C2×D411yescharacteristic
55Q8×C127—41no—
56D4×C127—41no—
57D4×C127—42no—
58C2×C4×C127—22no—
59C2×C4×C127—22no—
60C2×C4×C127—21no—
61C2×C4×C127—21no—
62C2×C4×C127—21no—
63C2×C4×C127—41no—
64C2×C4×C127—42no—
65C2×C4×C127C22⋊C411yescharacteristic
66C2×C4×C127C22⋊C411yescharacteristic
67C2×C4×C127C22⋊C411yescharacteristic
68C2×C4×C127C22⋊C411yescharacteristic
69C2×C4×C127C22×C411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C22×C127—21no—
72C22×C127—21no—
73C22×C127—41no—
74C22×C127—41no—
75C22×C127—41no—
76C22×C127C2×C22⋊C411yescharacteristic
77C22×C127#13512yes—
78C4×C127—21no—
79C4×C127—21no—
80C4×C127—21no—
81C4×C127—21no—
82C4×C127—21no—
83C4×C127—41no—
84C4×C127—41no—
85C4×C127—41no—
86C4×C127#10412yes—
87Dic127—44no—
88C2×C127—22no—
89C2×C127—81no—
90C2×C127#41011yescharacteristic
91C2×C127#41611yescharacteristic
92C2×C127C2×#10411yescharacteristic
93#1500—1271nomaximal, Sylow
94C127#150011yescharacteristic, complemented, Sylow
95#529D12711yescharacteristic
96#431—1271no—
97#376—1271no—
98#328—1272no—
99#327—1272no—
100#126—21no—
101#125Dic12711yescharacteristic, complemented
102#124Dic12711yescharacteristic, complemented
103#123—21no—
104C22⋊Q8C2×D12711yescharacteristic
105#121C2×D12711yescharacteristic
106C4×Q8Dic12711yescharacteristic, complemented
107C4×D4—21no—
108C4×D4Dic12711yescharacteristic, complemented
109#117—21no—
110C2×C42—2541no—
111C2×C42C2×D12711yescharacteristic
112C4⋊C8—2541no—
113C22⋊C8—1272no—
114C4×C8—2541no—
115#95—1272no—
116#95—2541no—
117C2×Q8C2×Dic12711yescharacteristic
118C2×D4—21no—
119C2×D4C2×Dic12711yescharacteristic
120C22×C4—21no—
121C22×C4—2542no—
122C22×C4C22⋊D12711yescharacteristic
123C22×C4C22⋊D12711yescharacteristic
124C22×C4C22×D12711yescharacteristic
125C2×C8—2542no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4C2×Dic12711yescharacteristic
131C4⋊C4C2×Dic12711yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—21no—
136C22⋊C4—21no—
137C42—21no—
138C42—2544no—
139C42C2×Dic12711yescharacteristic
140C42C2×Dic12711yescharacteristic
141C42C22⋊D12711yescharacteristic
142C42C22⋊D12711yescharacteristic
143C23—21no—
144C23C2×C22⋊D12711yescharacteristic
145Q8—41no—
146D4—41no—
147D4—42no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—5082no—
156C2×C4—5082no—
157C2×C4—2544no—
158C2×C4C22⋊Dic12711yescharacteristic
159C2×C4C22⋊Dic12711yescharacteristic
160C2×C4C22⋊Dic12711yescharacteristic
161C2×C4C22⋊Dic12711yescharacteristic
162C2×C4C22×Dic12711yescharacteristic
163C2×C4C2×C22⋊D12711yescharacteristic
164C8—5082no—
165C22—21no—
166C22—21no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22C2×C22⋊Dic12711yescharacteristic
171C22#1686122912yes—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—41no—
178C4—41no—
179C4—41no—
180C4—5084no—
181C4#1686113612yes—
182C2—22no—
183C2—81no—
184C2#2254985011yescharacteristic
185C2#2254987211yescharacteristic
186C2C2×#1686113611yescharacteristic
subgroupG/H
centerC2×C4C22⋊Dic127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#529×C127C2
socleC22×C127C2×C22⋊C4
prime radical#121×C127C22
cosocleC22×C4×C127C23
hypercenter#529D127
hypocenterC127#1500
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #22549929C2semidirect
2. #22549722C2semidirect
3. #22549613C2semidirect
13. #125×C127C4semidirect
14. #124×C127C4semidirect
18. C4×Q8×C127C4semidirect
20. C4×D4×C127C4semidirect
94. C127#1500semidirect
101. #125Dic127semidirect
102. #124Dic127semidirect
106. C4×Q8Dic127semidirect
108. C4×D4Dic127semidirect

Maximal subgroups

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

subgroupindex
#225499292
#225497222
#225496132
#225496102
#529×C1272
#1500127