Finite Group Databasebeta

← #30640420

Subgroups — #30640420

Order 16256.

262 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 55 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#22550444C211yesmaximal, characteristic
2#22549859C212yesmaximal, complemented
3#22549673C211yesmaximal, characteristic, complemented
4#22549672C212yesmaximal, complemented
5#355×C127C211yesmaximal, characteristic, complemented
6C2×#14523249C2211yescharacteristic
7C2×D4×D127C411yescharacteristic
8M16×D127—22no—
9#16861206C411yescharacteristic
10#16861205C411yescharacteristic
11#16861198C2211yescharacteristic
12#16861182—22no—
13#16861179—22no—
14#16861166C411yescharacteristic
15#16861128C2212yescomplemented
16#16861125—22no—
17#124×C127C2211yescharacteristic
18#101×C127C2212yescomplemented
19C23×D127—21no—
20#14523249—42no—
21C2×D508—22no—
22C2×D508C2×C411yescharacteristic
23C2×D508C2×C411yescharacteristic
24C2×D508D412yes—
25C2×C22⋊D127—22no—
26C2×C22⋊D127—21no—
27D4×D127—42no—
28C2×C4×D127C2×C411yescharacteristic
29C2×C4×D127D412yes—
30#14523232—22no—
31#14523231—42no—
32C8×D127—42no—
33Dic127⋊C4—21no—
34(C2×D127)⋊C4—21no—
35(C2×D127)⋊C4—22no—
36C4×Dic127C2×C411yescharacteristic
37C2×Q8×C127C2311yescharacteristic
38C2×D4×C127C2×C411yescharacteristic
39M16×C127—22no—
40C22⋊C4×C127—22no—
41C42×C127C2×C411yescharacteristic
42C22×D127—21no—
43C22×D127—42no—
44C22×D127—82no—
45C22×D127C22⋊C411yescharacteristic
46C22×D127C22⋊C412yes—
47D508—41no—
48D508—42no—
49D508—41no—
50D508—82no—
51C22⋊D127—42no—
52C22⋊D127—82no—
53C2×Dic127—21no—
54C2×Dic127C22⋊C411yescharacteristic
55C4×D127—22no—
56C4×D127—22no—
57C4×D127—22no—
58C127⋊C8—42no—
59C23×C127—21no—
60Q8×C127—41no—
61D4×C127—41no—
62C2×C4×C127—21no—
63C2×C4×C127C22×C411yescharacteristic
64C2×C4×C127C2×D412yes—
65C8×C127—42no—
66C22×C127—42no—
67C22×C127—41no—
68C22×C127C2×C22⋊C411yescharacteristic
69C2×D127—21no—
70C2×D127—22no—
71C2×D127—42no—
72C2×D127—41no—
73C2×D127—42no—
74C2×D127—41no—
75C2×D127—84no—
76C4×C127—22no—
77C4×C127—21no—
78C4×C127—41no—
79Dic127—41no—
80Dic127#9912yes—
81C2×C127—21no—
82C2×C127—81no—
83C2×C127C2×#9911yescharacteristic
84D127—42no—
85D127—41no—
86D127—81no—
87#2134—1271nomaximal, Sylow
88C127#213411yescharacteristic, complemented, Sylow
89#535—1271no—
90#413—1272no—
91#355—1271no—
92#355—1272no—
93#355D12711yescharacteristic, complemented
94C2×#41—1271no—
95C22×D4—1271no—
96C2×M16—2542no—
97#127—1271no—
98#127—1271no—
99#124—1271no—
100#124C2×D12711yescharacteristic
101#121—2542no—
102#120—2542no—
103#117—1271no—
104#101—1272no—
105#101C2×D12712yescomplemented
106#100—2542no—
107C24—2541no—
108#41—5082no—
109C2×Q8C22×D12711yescharacteristic
110C2×D4—1271no—
111C2×D4—1271no—
112C2×D4—1272no—
113C2×D4—5082no—
114C2×D4—2542no—
115C2×D4—2541no—
116C2×D4—2542no—
117C2×D4C4×D12711yescharacteristic
118C22×C4—1271no—
119C22×C4—1272no—
120M16—22no—
121M16—5082no—
122M16—2542no—
123C2×C8—5082no—
124C4⋊C4—2541no—
125C22⋊C4—22no—
126C22⋊C4—2541no—
127C22⋊C4—2542no—
128C42—1271no—
129C42C4×D12711yescharacteristic
130C23—21no—
131C23—1271no—
132C23—1272no—
133C23—10162no—
134C23—5082no—
135C23—2541no—
136Q8—41no—
137D4—41no—
138D4—10162no—
139D4—10162no—
140D4—5081no—
141D4—5082no—
142D4—5081no—
143D4—5082no—
144C2×C4—21no—
145C2×C4—1271no—
146C2×C4—2542no—
147C2×C4—2542no—
148C2×C4—2542no—
149C2×C4—2541no—
150C2×C4C2×C4×D12711yescharacteristic
151C2×C4D4×D12712yes—
152C8—42no—
153C8—5082no—
154C22—42no—
155C22—41no—
156C22—10164no—
157C22—5082no—
158C22—5082no—
159C22—5081no—
160C22—5081no—
161C22—2541no—
162C22—2542no—
163C22C22⋊C4×D12711yescharacteristic
164C4—22no—
165C4—21no—
166C4—41no—
167C4—1272no—
168C4—5081no—
169C2—21no—
170C2—81no—
171C2—10161no—
172C2—5081no—
173C2—5082no—
174C2#99×D12711yescharacteristic
subgroupG/H
centerC2#99×D127
commutatorC2×C4×C127C22×C4
FrattiniC2×Q8C22×D127
Fitting#355×C127C2
socleC2×C127C2×#99
prime radical#22550444C2
cosocleC2×Q8×C127C23
hypercenter#355D127
hypocenterC127#2134
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #22549859C2semidirect
3. #22549673C2semidirect
4. #22549672C2semidirect
5. #355×C127C2semidirect
15. #16861128C22semidirect
18. #101×C127C22semidirect
88. C127#2134semidirect
93. #355D127semidirect
105. #101C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504442
#225498592
#225496732
#225496722
#355×C1272
#2134127