Finite Group Databasebeta

← #30640421

Subgroups — #30640421

Order 16256.

262 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 65 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#22550440C211yesmaximal, characteristic
2#22549860C211yesmaximal, characteristic, complemented
3#22549672C214yesmaximal, complemented
4#413×C127C211yesmaximal, characteristic, complemented
5C22×D508C411yescharacteristic
6C2×#14523232—21no—
7#16861205C412yes—
8#16861198C2212yes—
9#16861183—21no—
10#16861183—21no—
11#16861177—21no—
12#16861177—21no—
13#16861168C411yescharacteristic
14#16861128C2212yescomplemented
15#16861126—21no—
16C2×#41×C127C2211yescharacteristic
17C2×M16×C127—21no—
18#101×C127C2212yescomplemented
19#100×C127—21no—
20C23×D127—21no—
21C2×D508—21no—
22C2×D508—41no—
23C2×D508—41no—
24C2×D508C2×C412yes—
25C2×C22⋊D127—22no—
26C2×C22⋊D127—22no—
27#14523232—22no—
28#14523232—41no—
29C2×C127⋊C8—41no—
30C4⋊Dic127—21no—
31C22⋊Dic127—21no—
32(C2×D127)⋊C4—22no—
33(C2×D127)⋊C4—22no—
34C4×Dic127C2×C412yes—
35#41×C127—41no—
36#41×C127—41no—
37C2×Q8×C127C2311yescharacteristic
38C2×D4×C127C2×C411yescharacteristic
39C2×D4×C127D411yescharacteristic
40C2×D4×C127D411yescharacteristic
41C22×C4×C127C2×C411yescharacteristic
42C22×C4×C127D411yescharacteristic
43C22×C4×C127D411yescharacteristic
44M16×C127—22no—
45M16×C127—41no—
46C2×C8×C127—41no—
47C22×D127—22no—
48C22×D127—42no—
49C22×D127—82no—
50D508—42no—
51D508—41no—
52D508—41no—
53C22⋊D127—82no—
54C22⋊D127—82no—
55C2×Dic127—22no—
56C127⋊C8—42no—
57C23×C127C22⋊C411yescharacteristic
58C23×C127C22⋊C411yescharacteristic
59C23×C127C22⋊C411yescharacteristic
60Q8×C127—41no—
61D4×C127—41no—
62D4×C127—41no—
63D4×C127—41no—
64C2×C4×C127—21no—
65C2×C4×C127—21no—
66C2×C4×C127—21no—
67C2×C4×C127—21no—
68C2×C4×C127—21no—
69C2×C4×C127—21no—
70C2×C4×C127C22⋊C411yescharacteristic
71C2×C4×C127C22×C411yescharacteristic
72C2×C4×C127C2×D411yescharacteristic
73C2×C4×C127C2×D411yescharacteristic
74C8×C127—42no—
75C22×C127—21no—
76C22×C127—21no—
77C22×C127—21no—
78C22×C127—41no—
79C22×C127—41no—
80C22×C127—41no—
81C22×C127C2×C22⋊C411yescharacteristic
82C2×D127—42no—
83C2×D127—42no—
84C2×D127—42no—
85C2×D127—84no—
86C4×C127—21no—
87C4×C127—21no—
88C4×C127—21no—
89C4×C127#9911yescharacteristic
90C4×C127#9911yescharacteristic
91Dic127—42no—
92C2×C127—21no—
93C2×C127—41no—
94C2×C127—41no—
95C2×C127—41no—
96C2×C127C2×#9911yescharacteristic
97D127—82no—
98#2134—1271nomaximal, Sylow
99C127#213411yescharacteristic, complemented, Sylow
100#535—1271no—
101#413—1271no—
102#413D12711yescharacteristic, complemented
103#355—1274no—
104C2×#41C2×D12711yescharacteristic
105C22×D4—1271no—
106C2×M16—21no—
107C2×M16—2541no—
108#127—1272no—
109#124—1272no—
110#121—2541no—
111#121—2541no—
112#120—2541no—
113#120—2541no—
114#117—1271no—
115#101—1272no—
116#101C2×D12712yescomplemented
117#100—21no—
118#100—2541no—
119C24—2541no—
120#41—41no—
121#41—41no—
122C2×Q8C22×D12711yescharacteristic
123C2×D4—1272no—
124C2×D4—5081no—
125C2×D4—5081no—
126C2×D4—2542no—
127C2×D4—2542no—
128C2×D4—2541no—
129C2×D4C4×D12711yescharacteristic
130C2×D4C22⋊D12711yescharacteristic
131C2×D4D50811yescharacteristic
132C22×C4C4×D12711yescharacteristic
133C22×C4C22⋊D12711yescharacteristic
134C22×C4D50811yescharacteristic
135M16—22no—
136M16—41no—
137M16—5081no—
138M16—2542no—
139C2×C8—41no—
140C2×C8—5081no—
141C4⋊C4—2541no—
142C22⋊C4—2541no—
143C22⋊C4—2542no—
144C22⋊C4—2542no—
145C42—1272no—
146C23—10162no—
147C23—5082no—
148C23—2542no—
149C23(C2×D127)⋊C411yescharacteristic
150C23(C2×D127)⋊C411yescharacteristic
151C23(C2×D127)⋊C411yescharacteristic
152Q8—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—10162no—
157D4—10162no—
158D4—5082no—
159D4—5081no—
160D4—5081no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—2542no—
168C2×C4(C2×D127)⋊C411yescharacteristic
169C2×C4C2×C4×D12711yescharacteristic
170C2×C4C2×C22⋊D12711yescharacteristic
171C2×C4C2×D50811yescharacteristic
172C8—42no—
173C8—5082no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22—10164no—
181C22—5082no—
182C22—5082no—
183C22—5082no—
184C22C2×(C2×D127)⋊C411yescharacteristic
185C4—21no—
186C4—21no—
187C4—21no—
188C4—5082no—
189C4#1686112311yescharacteristic
190C4#1686112311yescharacteristic
191C2—21no—
192C2—41no—
193C2—41no—
194C2—41no—
195C2—10162no—
196C2C2×#1686112311yescharacteristic
subgroupG/H
centerC2C2×#16861123
commutatorC2×C4×C127C22×C4
FrattiniC2×Q8C22×D127
Fitting#413×C127C2
socleC2×C127C2×#99
prime radical#22550440C2
cosocleC2×Q8×C127C23
hypercenter#413D127
hypocenterC127#2134
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #22549860C2semidirect
3. #22549672C2semidirect
4. #413×C127C2semidirect
14. #16861128C22semidirect
18. #101×C127C22semidirect
99. C127#2134semidirect
102. #413D127semidirect
116. #101C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504402
#225498602
#225496722
#413×C1272
#2134127