Finite Group Databasebeta

← #30638656

Subgroups — #30638656

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×#14523248C211yesmaximal, characteristic
2C2×#16861127C212yesmaximal, complemented
3C2×#16861125C212yesmaximal, complemented
4#22549840C211yesmaximal, characteristic, complemented
5#407×C127C211yesmaximal, characteristic, complemented
6C2×#14523248—24no—
7C2×#14523248—24no—
8C2×#14523248—22no—
9C2×#14523248—22no—
10C22×Dic254C2211yescharacteristic
11C22×D508C2211yescharacteristic
12C22×C22⋊D127C412yes—
13C22×C4×D127C412yes—
14C2×#14523232C2212yes—
15#16861127—24no—
16#16861125—24no—
17C22⋊(C127⋊C8)—22no—
18C23×C4×C127C2211yescharacteristic
19C2×M16×C127C2212yes—
20C22⋊C8×C127—22no—
21C23×D127C2×C412yes—
22#14523248—216no—
23#14523248—216no—
24C2×Dic254—22no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254D414yes—
28C2×D508—22no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508D414yes—
32C2×C22⋊D127—24no—
33C2×C22⋊D127—24no—
34C2×C22⋊D127—24no—
35C22×Dic127C2×C412yes—
36C2×C4×D127—24no—
37C2×C4×D127—24no—
38C2×C4×D127—24no—
39#14523232—22no—
40#14523232—22no—
41C2×C127⋊C8—22no—
42C24×C127C2×C411yescharacteristic
43C22×C4×C127—22no—
44C22×C4×C127—22no—
45C22×C4×C127C2×C411yescharacteristic
46C22×C4×C127D412yes—
47C22×C4×C127D412yes—
48C22×C4×C127C2311yescharacteristic
49M16×C127—22no—
50M16×C127—22no—
51C2×C8×C127—22no—
52C22×D127—22no—
53C22×D127—22no—
54C22×D127—44no—
55C22×D127C22⋊C414yes—
56Dic254—24no—
57Dic254—24no—
58D508—24no—
59D508—24no—
60C22⋊D127—216no—
61C2×Dic127—22no—
62C2×Dic127—22no—
63C2×Dic127C22⋊C414yes—
64C4×D127—216no—
65C127⋊C8—42no—
66C23×C127—21no—
67C23×C127—21no—
68C23×C127—42no—
69C23×C127C22⋊C412yes—
70C23×C127C22×C411yescharacteristic
71C2×C4×C127—21no—
72C2×C4×C127—21no—
73C2×C4×C127—24no—
74C2×C4×C127—24no—
75C2×C4×C127C22⋊C412yes—
76C2×C4×C127C2×D412yes—
77C2×C4×C127C2×D412yes—
78C2×C4×C127C2×D411yescharacteristic
79C2×C4×C127C2×D411yescharacteristic
80C8×C127—42no—
81C22×C127—22no—
82C22×C127—22no—
83C22×C127—24no—
84C22×C127—42no—
85C22×C127—42no—
86C22×C127C2×C22⋊C411yescharacteristic
87C22×C127C2×C22⋊C411yescharacteristic
88C22×C127C2×C22⋊C411yescharacteristic
89C2×D127—24no—
90C2×D127—44no—
91C2×D127—44no—
92C2×D127—44no—
93C4×C127—22no—
94C4×C127#12012yes—
95C4×C127#12012yes—
96Dic127—24no—
97C2×C127—21no—
98C2×C127—21no—
99C2×C127—42no—
100C2×C127#37911yescharacteristic
101C2×C127#41312yes—
102D127—44no—
103#1785—1271nomaximal, Sylow
104C127#178511yescharacteristic, complemented, Sylow
105C22×#41—1271no—
106C2×#101—1272no—
107C2×#100—1272no—
108#407—1271no—
109#407D12711yescharacteristic, complemented
110C2×#41—2544no—
111C2×#41—2544no—
112C2×#41—2542no—
113C2×#41—2542no—
114C22×Q8—1271no—
115C22×D4—1272no—
116C22×D4—1271no—
117C23×C4—1272no—
118C23×C4C2×D12711yescharacteristic
119C2×M16—1272no—
120C2×M16C2×D12712yes—
121#101—2544no—
122#100—2544no—
123C22⋊C8—22no—
124C22⋊C8—2542no—
125C24—1272no—
126C24C4×D12711yescharacteristic
127#41—25416no—
128#41—25416no—
129C2×Q8—1274no—
130C2×Q8—2542no—
131C2×Q8—2541no—
132C2×Q8—2541no—
133C2×D4—1274no—
134C2×D4—2544no—
135C2×D4—2544no—
136C2×D4—2544no—
137C2×D4—2542no—
138C2×D4—2541no—
139C2×D4—2541no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—1272no—
143C22×C4—2544no—
144C22×C4—2544no—
145C22×C4—2544no—
146C22×C4C4×D12711yescharacteristic
147C22×C4C22⋊D12712yes—
148C22×C4D50812yes—
149C22×C4C22×D12711yescharacteristic
150M16—22no—
151M16—22no—
152M16—2542no—
153M16—2542no—
154C2×C8—22no—
155C2×C8—2542no—
156C23—21no—
157C23—21no—
158C23—42no—
159C23—1274no—
160C23—5084no—
161C23—2542no—
162C23—2542no—
163C23(C2×D127)⋊C412yes—
164C23C2×C4×D12711yescharacteristic
165Q8—2544no—
166Q8—2544no—
167D4—25416no—
168D4—2544no—
169D4—2544no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—24no—
173C2×C4—24no—
174C2×C4—1274no—
175C2×C4—25416no—
176C2×C4—2542no—
177C2×C4—2542no—
178C2×C4(C2×D127)⋊C412yes—
179C2×C4D4×D12712yes—
180C2×C4D4×D12712yes—
181C2×C4C2×C22⋊D12711yescharacteristic
182C2×C4C2×D50811yescharacteristic
183C8—42no—
184C8—5082no—
185C22—22no—
186C22—24no—
187C22—22no—
188C22—42no—
189C22—42no—
190C22—5084no—
191C22—5084no—
192C22—5084no—
193C22—2544no—
194C22C22⋊C4×D12711yescharacteristic
195C22C22⋊C4×D12711yescharacteristic
196C22C2×(C2×D127)⋊C411yescharacteristic
197C4—22no—
198C4—2544no—
199C4#1686117712yes—
200C4#1686117912yes—
201C2—21no—
202C2—21no—
203C2—42no—
204C2—5084no—
205C2#2254973111yescharacteristic
206C2#2254985812yes—
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#407×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×#14523248C2
cosocleC22×C4×C127C23
hypercenter#407D127
hypocenterC127#1785
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#16861127C2semidirect
3. C2×#16861125C2semidirect
4. #22549840C2semidirect
5. #407×C127C2semidirect
104. C127#1785semidirect
109. #407D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232482
C2×#168611272
C2×#168611252
#225498402
#407×C1272
#1785127