Finite Group Databasebeta

← #30638657

Subgroups — #30638657

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×#14523249C211yesmaximal, characteristic
2C2×#16861128C211yesmaximal, characteristic, complemented
3C2×#16861125C212yesmaximal, complemented
4#22549839C212yesmaximal, complemented
5C2×#101×C127C211yesmaximal, characteristic, complemented
6C2×#14523249—24no—
7C2×#14523249—24no—
8C2×#14523249—22no—
9C2×#14523249—22no—
10C22×D508C412yes—
11C22×D508C2211yescharacteristic
12C22×C4×D127C412yes—
13C22×C4×D127C2211yescharacteristic
14C2×#14523232C2212yes—
15#16861128—22no—
16#16861125—24no—
17(C2×D127)⋊C8—24no—
18C22×Q8×C127C2211yescharacteristic
19C2×M16×C127C2212yes—
20#101×C127—22no—
21C23×D127C2×C412yes—
22C23×D127C2×C411yescharacteristic
23#14523249—216no—
24#14523249—216no—
25C2×D508—24no—
26C2×D508—24no—
27C2×D508—24no—
28C2×D508—22no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508D414yes—
32C22×Dic127C2×C411yescharacteristic
33C2×C4×D127—24no—
34C2×C4×D127—24no—
35C2×C4×D127—24no—
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—
43C2×Q8×C127—22no—
44C2×Q8×C127—22no—
45C2×Q8×C127D412yes—
46C2×Q8×C127D412yes—
47C22×C4×C127C2×C412yes—
48C22×C4×C127C2311yescharacteristic
49M16×C127—22no—
50M16×C127—22no—
51C2×C8×C127—22no—
52C22×D127—21no—
53C22×D127—22no—
54C22×D127—22no—
55C22×D127—21no—
56C22×D127—42no—
57C22×D127—44no—
58C22×D127C22⋊C412yes—
59C22×D127C22⋊C414yes—
60D508—216no—
61D508—24no—
62D508—24no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127C22⋊C412yes—
66C4×D127—216no—
67C4×D127—24no—
68C4×D127—24no—
69C127⋊C8—42no—
70C23×C127C22×C411yescharacteristic
71Q8×C127—24no—
72Q8×C127—24no—
73C2×C4×C127—22no—
74C2×C4×C127—22no—
75C2×C4×C127C22⋊C414yes—
76C2×C4×C127C2×D412yes—
77C2×C4×C127C2×D412yes—
78C2×C4×C127C2×D411yescharacteristic
79C2×C4×C127C2×D411yescharacteristic
80C8×C127—42no—
81C22×C127—22no—
82C22×C127C2×C22⋊C411yescharacteristic
83C22×C127C2×C22⋊C411yescharacteristic
84C22×C127C2×C22⋊C411yescharacteristic
85C2×D127—22no—
86C2×D127—24no—
87C2×D127—24no—
88C2×D127—42no—
89C2×D127—42no—
90C2×D127—44no—
91C2×D127—44no—
92C2×D127—44no—
93C4×C127—24no—
94C4×C127#12012yes—
95C4×C127#12012yes—
96Dic127—22no—
97C2×C127—21no—
98C2×C127—21no—
99C2×C127#37911yescharacteristic
100C2×C127#41312yes—
101D127—42no—
102D127—44no—
103#1785—1271nomaximal, Sylow
104C127#178511yescharacteristic, complemented, Sylow
105C22×#41—1271no—
106C2×#101—1271no—
107C2×#101D12711yescharacteristic, complemented
108C2×#100—1272no—
109#407—1272no—
110C2×#41—2544no—
111C2×#41—2544no—
112C2×#41—2542no—
113C2×#41—2542no—
114C22×Q8C2×D12711yescharacteristic
115C22×D4—1272no—
116C22×D4—1271no—
117C23×C4—1271no—
118C23×C4—1272no—
119C2×M16—1272no—
120C2×M16C2×D12712yes—
121#101—22no—
122#101—2542no—
123#100—2544no—
124C22⋊C8—2544no—
125C24—1272no—
126C24—1271no—
127#41—25416no—
128#41—25416no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8C22⋊D12712yes—
132C2×Q8D50812yes—
133C2×D4—1274no—
134C2×D4—2544no—
135C2×D4—2544no—
136C2×D4—2544no—
137C2×D4—2542no—
138C2×D4—2541no—
139C2×D4—2541no—
140C22×C4—1271no—
141C22×C4—1274no—
142C22×C4—2544no—
143C22×C4—2544no—
144C22×C4—2544no—
145C22×C4—2542no—
146C22×C4—2541no—
147C22×C4—2541no—
148C22×C4C4×D12712yes—
149C22×C4C22×D12711yescharacteristic
150M16—22no—
151M16—22no—
152M16—2542no—
153M16—2542no—
154C2×C8—22no—
155C2×C8—2542no—
156C23—1274no—
157C23—1272no—
158C23—5084no—
159C23—5082no—
160C23—2541no—
161C23—2542no—
162C23—2542no—
163C23—2541no—
164C23C2×C4×D12711yescharacteristic
165Q8—24no—
166Q8—24no—
167D4—25416no—
168D4—2544no—
169D4—2544no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—1272no—
173C2×C4—25416no—
174C2×C4—2541no—
175C2×C4—2541no—
176C2×C4—2544no—
177C2×C4—2544no—
178C2×C4(C2×D127)⋊C414yes—
179C2×C4D4×D12712yes—
180C2×C4D4×D12712yes—
181C2×C4C2×C22⋊D12711yescharacteristic
182C2×C4C2×D50811yescharacteristic
183C8—42no—
184C8—5082no—
185C22—22no—
186C22—5084no—
187C22—5082no—
188C22—5084no—
189C22—5082no—
190C22—5084no—
191C22—2544no—
192C22—2544no—
193C22—2542no—
194C22C22⋊C4×D12711yescharacteristic
195C22C22⋊C4×D12711yescharacteristic
196C22C2×(C2×D127)⋊C411yescharacteristic
197C4—24no—
198C4—2542no—
199C4#1686117712yes—
200C4#1686117912yes—
201C2—21no—
202C2—21no—
203C2—5084no—
204C2—5082no—
205C2#2254973111yescharacteristic
206C2#2254985912yes—
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#101×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×#14523249C2
cosocleC22×C4×C127C23
hypercenterC2×#101D127
hypocenterC127#1785
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. C2×#16861128C2semidirect
3. C2×#16861125C2semidirect
4. #22549839C2semidirect
5. C2×#101×C127C2semidirect
104. C127#1785semidirect
107. C2×#101D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232492
C2×#168611282
C2×#168611252
#225498392
C2×#101×C1272
#1785127