Finite Group Databasebeta

← #30638655

Subgroups — #30638655

Order 16256.

584 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#14523247C211yesmaximal, characteristic
2C2×#16861127C212yesmaximal, complemented
3C2×#16861126C211yesmaximal, characteristic, complemented
4#22549839C212yesmaximal, complemented
5C2×#100×C127C211yesmaximal, characteristic, complemented
6C2×#14523247—24no—
7C2×#14523247—24no—
8C2×#14523247—22no—
9C2×#14523247—22no—
10C22×Dic254C2211yescharacteristic
11C22×C22⋊D127C412yes—
12C23×Dic127C412yes—
13C22×C4×D127C2211yescharacteristic
14C2×#14523232C2212yes—
15#16861127—24no—
16#16861126—22no—
17(C2×D127)⋊C8—24no—
18C22×D4×C127C2211yescharacteristic
19C2×M16×C127C2212yes—
20#100×C127—22no—
21C23×D127C2×C411yescharacteristic
22#14523247—216no—
23#14523247—216no—
24C2×Dic254—22no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254D414yes—
28C2×C22⋊D127—24no—
29C2×C22⋊D127—24no—
30C2×C22⋊D127—24no—
31C22×Dic127—24no—
32C22×Dic127—24no—
33C22×Dic127—24no—
34C22×Dic127C2×C411yescharacteristic
35C22×Dic127C2×C412yes—
36C2×C4×D127—22no—
37C2×C4×D127—21no—
38C2×C4×D127—21no—
39C2×C4×D127D414yes—
40#14523232—22no—
41#14523232—22no—
42C2×C127⋊C8—22no—
43C24×C127C2×C412yes—
44C2×D4×C127—22no—
45C2×D4×C127—22no—
46C2×D4×C127D412yes—
47C2×D4×C127D412yes—
48C22×C4×C127C2311yescharacteristic
49M16×C127—22no—
50M16×C127—22no—
51C2×C8×C127—22no—
52C22×D127—21no—
53C22×D127—21no—
54C22×D127—42no—
55C22×D127C22⋊C412yes—
56Dic254—24no—
57Dic254—24no—
58C22⋊D127—216no—
59C2×Dic127—216no—
60C2×Dic127—21no—
61C2×Dic127—21no—
62C2×Dic127—22no—
63C2×Dic127—22no—
64C2×Dic127C22⋊C412yes—
65C2×Dic127C22⋊C414yes—
66C4×D127—24no—
67C4×D127—24no—
68C127⋊C8—42no—
69C23×C127—22no—
70C23×C127—22no—
71C23×C127—44no—
72C23×C127C22⋊C414yes—
73C23×C127C22×C411yescharacteristic
74D4×C127—24no—
75D4×C127—24no—
76C2×C4×C127C2×D412yes—
77C2×C4×C127C2×D412yes—
78C2×C4×C127C2×D411yescharacteristic
79C2×C4×C127C2×D411yescharacteristic
80C8×C127—42no—
81C22×C127—22no—
82C22×C127—24no—
83C22×C127—44no—
84C22×C127—44no—
85C22×C127—44no—
86C22×C127C2×C22⋊C411yescharacteristic
87C22×C127C2×C22⋊C411yescharacteristic
88C22×C127C2×C22⋊C411yescharacteristic
89C2×D127—22no—
90C2×D127—24no—
91C2×D127—42no—
92C2×D127—42no—
93C4×C127#12012yes—
94C4×C127#12012yes—
95Dic127—22no—
96Dic127—24no—
97C2×C127—21no—
98C2×C127—21no—
99C2×C127—44no—
100C2×C127#37911yescharacteristic
101C2×C127#41312yes—
102D127—42no—
103#1785—1271nomaximal, Sylow
104C127#178511yescharacteristic, complemented, Sylow
105C22×#41—1271no—
106C2×#101—1272no—
107C2×#100—1271no—
108C2×#100D12711yescharacteristic, complemented
109#407—1272no—
110C2×#41—2544no—
111C2×#41—2544no—
112C2×#41—2542no—
113C2×#41—2542no—
114C22×Q8—1271no—
115C22×D4—1272no—
116C22×D4C2×D12711yescharacteristic
117C23×C4—1271no—
118C23×C4—1272no—
119C2×M16—1272no—
120C2×M16C2×D12712yes—
121#101—2544no—
122#100—22no—
123#100—2542no—
124C22⋊C8—2544no—
125C24—1271no—
126C24C4×D12712yes—
127#41—25416no—
128#41—25416no—
129C2×Q8—1274no—
130C2×Q8—2542no—
131C2×Q8—2541no—
132C2×Q8—2541no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—2544no—
136C2×D4—2544no—
137C2×D4—2544no—
138C2×D4C22⋊D12712yes—
139C2×D4D50812yes—
140C22×C4—1271no—
141C22×C4—1272no—
142C22×C4—1274no—
143C22×C4—2544no—
144C22×C4—2544no—
145C22×C4—2544no—
146C22×C4—2542no—
147C22×C4—2541no—
148C22×C4—2541no—
149C22×C4C22×D12711yescharacteristic
150M16—22no—
151M16—22no—
152M16—2542no—
153M16—2542no—
154C2×C8—22no—
155C2×C8—2542no—
156C23—22no—
157C23—22no—
158C23—44no—
159C23—1272no—
160C23—5082no—
161C23—2541no—
162C23—2541no—
163C23(C2×D127)⋊C414yes—
164C23C2×C4×D12711yescharacteristic
165Q8—2544no—
166Q8—2544no—
167D4—24no—
168D4—24no—
169D4—25416no—
170C2×C4—1274no—
171C2×C4—1272no—
172C2×C4—25416no—
173C2×C4—2541no—
174C2×C4—2542no—
175C2×C4—2541no—
176C2×C4—2542no—
177C2×C4—2544no—
178C2×C4—2544no—
179C2×C4D4×D12712yes—
180C2×C4D4×D12712yes—
181C2×C4C2×C22⋊D12711yescharacteristic
182C2×C4C2×D50811yescharacteristic
183C8—42no—
184C8—5082no—
185C22—22no—
186C22—24no—
187C22—44no—
188C22—44no—
189C22—44no—
190C22—5082no—
191C22—5082no—
192C22—2544no—
193C22—2542no—
194C22C22⋊C4×D12711yescharacteristic
195C22C22⋊C4×D12711yescharacteristic
196C22C2×(C2×D127)⋊C411yescharacteristic
197C4—2542no—
198C4—2544no—
199C4#1686117712yes—
200C4#1686117912yes—
201C2—21no—
202C2—21no—
203C2—44no—
204C2—5082no—
205C2#2254973111yescharacteristic
206C2#2254985712yes—
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#100×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×#14523247C2
cosocleC22×C4×C127C23
hypercenterC2×#100D127
hypocenterC127#1785
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. C2×#16861127C2semidirect
3. C2×#16861126C2semidirect
4. #22549839C2semidirect
5. C2×#100×C127C2semidirect
104. C127#1785semidirect
108. C2×#100D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232472
C2×#168611272
C2×#168611262
#225498392
C2×#100×C1272
#1785127