Finite Group Databasebeta

← #353×D127 #30640386

Subgroups — #353×D127 #30640386

Order 16256.

368 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 57 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#127×D127C211yesmaximal, characteristic
2#99×D127C212yesmaximal, complemented
3#22549669C211yesmaximal, characteristic, complemented
4#22549668C212yesmaximal, complemented
5#353×C127C211yesmaximal, characteristic, complemented
6C2×D4×D127—22no—
7C2×D4×D127C411yescharacteristic, complemented
8C2×D4×D127C2211yescharacteristic
9#16861206C411yescharacteristic, complemented
10#16861205—22no—
11#16861205C411yescharacteristic, complemented
12#16861205C2211yescharacteristic
13C22⋊C4×D127—22no—
14C42×D127C411yescharacteristic, complemented
15#16861124C2212yescomplemented
16#16861123—22no—
17#127×C127C2211yescharacteristic
18#99×C127C2212yescomplemented
19C23×D127—21no—
20C23×D127D412yes—
21C2×D508—22no—
22C2×D508C2×C411yescharacteristic
23C2×D508C2×C411yescharacteristic, complemented
24C2×C22⋊D127—22no—
25C2×C22⋊D127—21no—
26C2×C22⋊D127—22no—
27C2×C22⋊D127D412yes—
28D4×D127—41no—
29D4×D127—41no—
30D4×D127—41no—
31D4×D127—41no—
32D4×D127—48no—
33C2×C4×D127—21no—
34C2×C4×D127—42no—
35C2×C4×D127C2×C411yescharacteristic
36C22⋊Dic127—22no—
37(C2×D127)⋊C4—42no—
38C4×Dic127—21no—
39C4×Dic127C2×C411yescharacteristic, complemented
40C2×D4×C127—22no—
41C2×D4×C127C2×C411yescharacteristic, complemented
42C2×D4×C127C2311yescharacteristic
43C22⋊C4×C127—22no—
44C42×C127C2×C411yescharacteristic, complemented
45C22×D127—22no—
46C22×D127—22no—
47C22×D127—21no—
48C22×D127—44no—
49C22×D127—44no—
50C22×D127—41no—
51C22×D127—41no—
52C22×D127C22⋊C411yescharacteristic
53C22×D127C22⋊C412yes—
54D508—41no—
55D508—41no—
56D508—44no—
57C22⋊D127—42no—
58C22⋊D127—44no—
59C22⋊D127—44no—
60C22⋊D127—42no—
61C2×Dic127—21no—
62C2×Dic127—42no—
63C2×Dic127C22⋊C411yescharacteristic
64C4×D127—24no—
65C4×D127—21no—
66C4×D127—21no—
67C4×D127—84no—
68C23×C127—21no—
69C23×C127C2×D412yes—
70D4×C127—41no—
71D4×C127—41no—
72D4×C127—44no—
73C2×C4×C127—21no—
74C2×C4×C127—42no—
75C2×C4×C127C22×C411yescharacteristic
76C22×C127—22no—
77C22×C127—42no—
78C22×C127—42no—
79C22×C127—41no—
80C22×C127C2×C22⋊C411yescharacteristic
81C2×D127—22no—
82C2×D127—22no—
83C2×D127—44no—
84C2×D127—42no—
85C2×D127—42no—
86C2×D127—41no—
87C2×D127—82no—
88C2×D127—82no—
89C2×D127—82no—
90C2×D127#9911yescharacteristic
91C2×D127#9911yescharacteristic
92C4×C127—22no—
93C4×C127—21no—
94C4×C127—82no—
95Dic127—22no—
96Dic127—21no—
97Dic127—82no—
98C2×C127—21no—
99C2×C127—42no—
100C2×C127—81no—
101C2×C127C2×#9911yescharacteristic
102D127—21no—
103D127—42no—
104D127—81no—
105D127#35312yesdirect product factor
106C2×#353—1271nomaximal, Sylow
107C127C2×#35311yescharacteristic, complemented, Sylow
108C2×#127—1271no—
109C2×#99—1272no—
110#353—1271no—
111#353—1272no—
112#353D12711yescharacteristic, direct product factor
113C22×D4—1271no—
114C22×D4—1271no—
115C22×D4—2542no—
116#127—1271no—
117#127—1271no—
118#127—1271no—
119#127—2542no—
120#127C2×D12711yescharacteristic
121C2×C22⋊C4—2542no—
122C2×C42—1271no—
123#99—1272no—
124#99—2542no—
125#99C2×D12712yescomplemented
126C24—1272no—
127C24—2541no—
128C2×D4—22no—
129C2×D4—1272no—
130C2×D4—1271no—
131C2×D4—1271no—
132C2×D4—5081no—
133C2×D4—5081no—
134C2×D4—5088no—
135C2×D4—5081no—
136C2×D4—5081no—
137C2×D4—2542no—
138C2×D4—2541no—
139C2×D4—2542no—
140C2×D4—2542no—
141C2×D4C4×D12711yescharacteristic, complemented
142C2×D4C22×D12711yescharacteristic
143C22×C4—1271no—
144C22×C4—5082no—
145C22×C4—2541no—
146C22⋊C4—22no—
147C22⋊C4—5082no—
148C22⋊C4—2542no—
149C42—1271no—
150C42—2541no—
151C42C4×D12711yescharacteristic, complemented
152C23—21no—
153C23—1271no—
154C23—1272no—
155C23—5081no—
156C23—5081no—
157C23—5084no—
158C23—5084no—
159C23—2542no—
160C23—2542no—
161C23—2541no—
162C23D4×D12712yes—
163D4—41no—
164D4—41no—
165D4—44no—
166D4—5081no—
167D4—5084no—
168D4—5081no—
169D4—5082no—
170D4—5084no—
171D4—5084no—
172D4—5082no—
173C2×C4—21no—
174C2×C4—42no—
175C2×C4—1271no—
176C2×C4—10164no—
177C2×C4—5082no—
178C2×C4—2541no—
179C2×C4—2541no—
180C2×C4—2544no—
181C2×C4—2541no—
182C2×C4C2×C4×D12711yescharacteristic
183C22—22no—
184C22—42no—
185C22—42no—
186C22—41no—
187C22—1271no—
188C22—1271no—
189C22—10162no—
190C22—10162no—
191C22—10162no—
192C22—5082no—
193C22—5082no—
194C22—5084no—
195C22—5081no—
196C22—2542no—
197C22—2542no—
198C22C22⋊C4×D12711yescharacteristic
199C4—22no—
200C4—21no—
201C4—82no—
202C4—10162no—
203C4—2542no—
204C4—2541no—
205C2—21no—
206C2—42no—
207C2—81no—
208C2—1272no—
209C2—10161no—
210C2—5082no—
211C2—2541no—
212C2#99×D12711yescharacteristic
subgroupG/H
centerC2#99×D127
commutatorC2×C4×C127C22×C4
FrattiniC2×D4C22×D127
Fitting#353×C127C2
socleC2×C127C2×#99
prime radical#127×D127C2
cosocleC2×D4×C127C23
hypercenter#353D127
hypocenterC127C2×#353
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #99×D127C2semidirect
3. #22549669C2semidirect
4. #22549668C2semidirect
5. #353×C127C2semidirect
7. C2×D4×D127C4semidirect
9. #16861206C4semidirect
11. #16861205C4semidirect
14. C42×D127C4semidirect
15. #16861124C22semidirect
18. #99×C127C22semidirect
23. C2×D508C2×C4semidirect
39. C4×Dic127C2×C4semidirect
41. C2×D4×C127C2×C4semidirect
44. C42×C127C2×C4semidirect
105. D127#353direct (N × H)
107. C127C2×#353semidirect
112. #353D127direct (N × H)
125. #99C2×D127semidirect
141. C2×D4C4×D127semidirect
151. C42C4×D127semidirect

Maximal subgroups

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

subgroupindex
#127×D1272
#99×D1272
#225496692
#225496682
#353×C1272
C2×#353127