Finite Group Databasebeta

← #30639572

Subgroups — #30639572

Order 16256.

292 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#16861207C211yesmaximal, characteristic
2#22549899C211yesmaximal, characteristic
3C2×#16861128C211yesmaximal, characteristic
4C2×#16861127C211yesmaximal, characteristic
5C2×#16861127C211yesmaximal, characteristic
6M16×Dic127C211yesmaximal, characteristic
7C2×#101×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C2×#14523232C2211yescharacteristic
11#16861207—21no—
12#16861207—21no—
13#16861207—22no—
14C2×Dic127⋊C4C414yes—
15C2×C4×Dic127C2211yescharacteristic
16Dic127⋊C8—22no—
17#16861128—22no—
18#16861127—22no—
19#16861127—22no—
20#16861119—21no—
21C8×Dic127—21no—
22C22×Q8×C127C2211yescharacteristic
23C2×M16×C127C2211yescharacteristic
24C2×M16×C127C2211yescharacteristic
25#101×C127—22no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254—21no—
32C22×Dic127C2×C412yes—
33C22×Dic127C2×C412yes—
34#14523232—21no—
35#14523232—21no—
36#14523232—21no—
37#14523232D412yes—
38C2×C127⋊C8—21no—
39C2×C127⋊C8—21no—
40Dic127⋊C4—24no—
41Dic127⋊C4—24no—
42C4×Dic127D411yescharacteristic
43C4×Dic127D411yescharacteristic
44C4×Dic127D412yes—
45C2×Q8×C127—21no—
46C2×Q8×C127—21no—
47C2×Q8×C127—21no—
48C2×Q8×C127—21no—
49C2×Q8×C127—21no—
50C2×Q8×C127—21no—
51C22×C4×C127C2×C412yes—
52C22×C4×C127C2311yescharacteristic
53M16×C127—21no—
54M16×C127—21no—
55M16×C127—21no—
56M16×C127D412yes—
57C2×C8×C127—21no—
58C2×C8×C127—21no—
59Dic254—41no—
60Dic254—41no—
61Dic254—41no—
62Dic254—41no—
63C2×Dic127—24no—
64C2×Dic127—22no—
65C2×Dic127—22no—
66C2×Dic127—22no—
67C2×Dic127C22⋊C412yes—
68C2×Dic127C22⋊C412yes—
69C127⋊C8—22no—
70C127⋊C8—41no—
71C23×C127C22×C411yescharacteristic
72Q8×C127—41no—
73Q8×C127—41no—
74Q8×C127—41no—
75Q8×C127—41no—
76C2×C4×C127—22no—
77C2×C4×C127—22no—
78C2×C4×C127—22no—
79C2×C4×C127C2×D411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127#4111yescharacteristic
84C2×C4×C127#4111yescharacteristic
85C8×C127—22no—
86C8×C127—41no—
87C22×C127—22no—
88C22×C127C2×C22⋊C411yescharacteristic
89C22×C127C4×D411yescharacteristic
90C22×C127C4×D411yescharacteristic
91C4×C127—42no—
92C4×C127#12111yescharacteristic
93C4×C127#12111yescharacteristic
94C4×C127#12411yescharacteristic
95C4×C127#12711yescharacteristic
96Dic127—22no—
97Dic127—42no—
98Dic127#10114yescomplemented
99C2×C127—21no—
100C2×C127—21no—
101C2×C127#39011yescharacteristic
102C2×C127C2×#10112yes—
103#1966—1271nomaximal, Sylow
104C127#196611yescharacteristic, complemented, Sylow
105C2×C4⋊Q8—1271no—
106#423—1271no—
107C2×#101—1271no—
108C2×#101—1271no—
109C2×#101—1271no—
110C2×#101D12711yescharacteristic
111C4×M16—1271no—
112C22×Q8—1271no—
113C22×Q8C2×D12711yescharacteristic
114C2×M16—1271no—
115C2×M16—1271no—
116C2×M16C2×D12711yescharacteristic
117C2×M16C2×D12711yescharacteristic
118C4⋊Q8—2541no—
119C4⋊Q8—2541no—
120C4⋊Q8—2542no—
121C2×C4⋊C4—1274no—
122C2×C42—1271no—
123C4⋊C8—2542no—
124#101—22no—
125#101—2542no—
126#101—2542no—
127#101—2542no—
128#97—2541no—
129C4×C8—2541no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—2541no—
137C2×Q8—2541no—
138C2×Q8—2541no—
139C2×Q8—2541no—
140C2×Q8—2541no—
141C2×Q8—2541no—
142C22×C4—1272no—
143C22×C4—1272no—
144C22×C4C4×D12712yes—
145C22×C4C22×D12711yescharacteristic
146M16—21no—
147M16—21no—
148M16—21no—
149M16—1272no—
150M16—2541no—
151M16—2541no—
152M16—2541no—
153M16C22⋊D12712yes—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—2541no—
157C2×C8—2541no—
158C4⋊C4—2544no—
159C4⋊C4—2544no—
160C42—1271no—
161C42—1271no—
162C42—1272no—
163C23C2×C4×D12711yescharacteristic
164Q8—41no—
165Q8—41no—
166Q8—41no—
167Q8—41no—
168Q8—5081no—
169Q8—5081no—
170Q8—5081no—
171Q8—5081no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—1272no—
176C2×C4—1272no—
177C2×C4—2544no—
178C2×C4—2542no—
179C2×C4—2542no—
180C2×C4—2542no—
181C2×C4D4×D12711yescharacteristic
182C2×C4D4×D12711yescharacteristic
183C2×C4D4×D12711yescharacteristic
184C2×C4C2×C22⋊D12711yescharacteristic
185C2×C4#1452324711yescharacteristic
186C2×C4#1452324811yescharacteristic
187C8—22no—
188C8—41no—
189C8—5081no—
190C8—2542no—
191C22—22no—
192C22C22⋊C4×D12711yescharacteristic
193C22(C22⋊D127)⋊C411yescharacteristic
194C22C4×C22⋊D12711yescharacteristic
195C4—42no—
196C4—1274no—
197C4—5082no—
198C4—2542no—
199C4#1686118111yescharacteristic
200C4#1686118211yescharacteristic
201C4#1686119511yescharacteristic
202C4#1686120511yescharacteristic
203C2—21no—
204C2—21no—
205C2#2254977611yescharacteristic
206C2#101×D12712yes—
subgroupG/H
centerC22C22⋊C4×D127
commutator, prime radicalC23×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#101×C127C2
socleC22×C127C2×C22⋊C4
cosocleC22×C4×C127C23
hypercenterC2×#101D127
hypocenterC127#1966
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
98. Dic127#101semidirect
104. C127#1966semidirect

Maximal subgroups

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

subgroupindex
C2×#168612072
#225498992
C2×#168611282
C2×#168611272
M16×Dic1272
C2×#101×C1272
#1966127