Finite Group Databasebeta

← #329×D127 #30637309

Subgroups — #329×D127 #30637309

Order 16256.

284 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 67 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#124×D127C211yesmaximal, characteristic
2#97×D127C212yesmaximal, complemented
3#22549616C211yesmaximal, characteristic, complemented
4#22549615C212yesmaximal, complemented
5#329×C127C211yesmaximal, characteristic, complemented
6C2×Q8×D127C411yescharacteristic
7C2×D4×D127C411yescharacteristic
8C2×C8×D127—22no—
9#16861198C411yescharacteristic
10#16861197C2211yescharacteristic
11#16861196C411yescharacteristic
12#16861195—22no—
13C22⋊C4×D127—22no—
14C42×D127C2211yescharacteristic
15#16861120C2212yescomplemented
16#16861119—22no—
17#124×C127C2211yescharacteristic
18#97×C127C2212yescomplemented
19C23×D127—21no—
20C2×Dic254—21no—
21C2×Dic254C2×C411yescharacteristic
22Q8×D127—42no—
23C2×D508C2×C411yescharacteristic
24C2×C22⋊D127—21no—
25D4×D127—42no—
26C2×C4×D127—21no—
27C2×C4×D127C2×C411yescharacteristic
28C2×C4×D127D412yes—
29C2×C127⋊C8—22no—
30C8×D127—28no—
31C22⋊Dic127—22no—
32(C2×D127)⋊C4—22no—
33(C2×D127)⋊C4—22no—
34C4×Dic127C2×C411yescharacteristic
35C4×Dic127D412yes—
36C2×Q8×C127C2×C411yescharacteristic
37C2×D4×C127C2×C411yescharacteristic
38C2×C8×C127—22no—
39C22⋊C4×C127—22no—
40C42×C127C2311yescharacteristic
41C22×D127—21no—
42C22×D127—42no—
43C22×D127—44no—
44C22×D127C22⋊C411yescharacteristic
45Dic254—42no—
46Dic254—41no—
47D508—41no—
48C22⋊D127—42no—
49C2×Dic127—21no—
50C2×Dic127C22⋊C412yes—
51C2×Dic127C22⋊C411yescharacteristic
52C4×D127—22no—
53C4×D127—24no—
54C4×D127—42no—
55C127⋊C8—24no—
56C23×C127—21no—
57Q8×C127—41no—
58D4×C127—41no—
59C2×C4×C127—21no—
60C2×C4×C127C22×C411yescharacteristic
61C2×C4×C127C2×D412yes—
62C8×C127—24no—
63C22×C127—41no—
64C22×C127—42no—
65C22×C127C2×C22⋊C411yescharacteristic
66C2×D127—41no—
67C2×D127—42no—
68C2×D127—84no—
69C2×D127#10012yes—
70C2×D127#10414yes—
71C4×C127—21no—
72C4×C127—22no—
73C4×C127—41no—
74Dic127—21no—
75Dic127—22no—
76Dic127—41no—
77C2×C127—81no—
78C2×C127C2×#10011yescharacteristic
79C2×C127C2×#10412yes—
80D127—81no—
81D127#32914yesdirect product factor
82C2×#329—1271nomaximal, Sylow
83C127C2×#32911yescharacteristic, complemented, Sylow
84C2×#124—1271no—
85C2×#97—1272no—
86#329—1271no—
87#329—1272no—
88#329D12711yescharacteristic, direct product factor
89C22×Q8—1271no—
90C22×D4—1271no—
91C22×C8—2542no—
92#124—1271no—
93#124—1271no—
94#124—1271no—
95#124—2542no—
96#124C2×D12711yescharacteristic
97C2×C22⋊C4—2542no—
98C2×C42—1271no—
99#97—1272no—
100#97—2542no—
101#97C2×D12712yescomplemented
102C24—2541no—
103C2×Q8—1271no—
104C2×Q8—5082no—
105C2×Q8—2541no—
106C2×Q8C4×D12711yescharacteristic
107C2×D4—1271no—
108C2×D4—5082no—
109C2×D4—2541no—
110C2×D4C4×D12711yescharacteristic
111C22×C4—1271no—
112C22×C4—1272no—
113C22×C4—2541no—
114C2×C8—22no—
115C2×C8—2548no—
116C2×C8—2542no—
117C22⋊C4—22no—
118C22⋊C4—2542no—
119C22⋊C4—2542no—
120C22⋊C4—2542no—
121C42—1271no—
122C42—1272no—
123C42C22×D12711yescharacteristic
124C23—21no—
125C23—1271no—
126C23—5082no—
127C23—5084no—
128C23—2541no—
129Q8—41no—
130Q8—5081no—
131Q8—5082no—
132D4—41no—
133D4—5081no—
134D4—5082no—
135C2×C4—21no—
136C2×C4—1272no—
137C2×C4—1271no—
138C2×C4—5082no—
139C2×C4—2544no—
140C2×C4—2542no—
141C2×C4—2541no—
142C2×C4C2×C4×D12711yescharacteristic
143C2×C4D4×D12712yes—
144C8—24no—
145C8—2544no—
146C22—41no—
147C22—42no—
148C22—1272no—
149C22—1274no—
150C22—10164no—
151C22—5081no—
152C22—5082no—
153C22C22⋊C4×D12711yescharacteristic
154C4—21no—
155C4—22no—
156C4—41no—
157C4—5081no—
158C4—2541no—
159C4—2542no—
160C2—81no—
161C2—1274no—
162C2—10161no—
163C2#100×D12711yescharacteristic
164C2#104×D12712yes—
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#329×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC2×D4×D127C4
cosocleC42×C127C23
hypercenter#329D127
hypocenterC127C2×#329
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #97×D127C2semidirect
3. #22549616C2semidirect
4. #22549615C2semidirect
5. #329×C127C2semidirect
15. #16861120C22semidirect
18. #97×C127C22semidirect
81. D127#329direct (N × H)
83. C127C2×#329semidirect
88. #329D127direct (N × H)
101. #97C2×D127semidirect

Maximal subgroups

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

subgroupindex
#124×D1272
#97×D1272
#225496162
#225496152
#329×C1272
C2×#329127