Finite Group Databasebeta

← #30638740

Subgroups — #30638740

Order 16256.

356 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 125 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×C4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
2C2×#16861130C212yesmaximal
3#22549742C211yesmaximal, characteristic, complemented
4#22549638C212yesmaximal, complemented
5C2×C4×D4×C127C211yesmaximal, characteristic
6C22×C127⋊C8C2212yes—
7C2×C4⋊Dic127C2212yes—
8C4⋊(C127⋊C8)—22no—
9#16861130—24no—
10C127⋊#95—21no—
11C127⋊#95—21no—
12C22×D4×C127C2211yescharacteristic
13C23×C4×C127—21no—
14C4×D4×C127—22no—
15C4×D4×C127C414yescomplemented
16C2×C4⋊C4×C127C2211yescharacteristic
17C2×C22⋊C4×C127—21no—
18C2×C42×C127C2211yescharacteristic
19C22×Dic127—22no—
20C2×C127⋊C8—22no—
21C2×C127⋊C8—22no—
22C2×C127⋊C8—22no—
23C4⋊Dic127—22no—
24C4⋊Dic127—22no—
25C24×C127—21no—
26C2×D4×C127—21no—
27C2×D4×C127—21no—
28C2×D4×C127—21no—
29C2×D4×C127C2×C412yes—
30C2×D4×C127D412yes—
31C2×D4×C127Q812yes—
32C22×C4×C127—21no—
33C22×C4×C127—24no—
34C22×C4×C127—44no—
35C22×C4×C127D411yescharacteristic
36C22×C4×C127D411yescharacteristic
37C22×C4×C127C2311yescharacteristic
38C4⋊C4×C127—21no—
39C4⋊C4×C127C2×C412yes—
40C22⋊C4×C127—24no—
41C42×C127C2×C412yes—
42C42×C127D412yes—
43C2×Dic127—42no—
44C2×Dic127—42no—
45C2×Dic127—42no—
46C127⋊C8—42no—
47C23×C127—22no—
48C23×C127—22no—
49C23×C127—22no—
50C23×C127—42no—
51C23×C127—42no—
52C23×C127C2×D411yescharacteristic
53D4×C127—22no—
54D4×C127—22no—
55D4×C127—22no—
56D4×C127C4⋊C414yes—
57C2×C4×C127—22no—
58C2×C4×C127—22no—
59C2×C4×C127—22no—
60C2×C4×C127—48no—
61C2×C4×C127—42no—
62C2×C4×C127C22⋊C412yes—
63C2×C4×C127C22⋊C412yes—
64C2×C4×C127C22×C411yescharacteristic
65C2×C4×C127C2×D411yescharacteristic
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127C2×Q811yescharacteristic
68C2×C4×C127#4111yescharacteristic
69C2×C4×C127#4111yescharacteristic
70C22×C127—24no—
71C22×C127—42no—
72C22×C127—42no—
73C22×C127—44no—
74C22×C127—44no—
75C22×C127C2×C22⋊C411yescharacteristic
76C22×C127#12111yescharacteristic
77C22×C127C22⋊Q811yescharacteristic
78C22×C127#13512yes—
79C22×C127AGL(1,ℤ/8)12yes—
80C4×C127—22no—
81C4×C127—22no—
82C4×C127—42no—
83C4×C127C2×C4⋊C411yescharacteristic
84C4×C127#11711yescharacteristic
85C4×C127#12111yescharacteristic
86C4×C127C22⋊Q811yescharacteristic
87Dic127—82no—
88C2×C127—44no—
89C2×C127#38011yescharacteristic
90C2×C127#41611yescharacteristic
91C2×C127#41811yescharacteristic
92C2×C127#46312yes—
93C2×C127#47812yes—
94#1804—1271nomaximal, Sylow
95C127#180411yescharacteristic, complemented, Sylow
96C2×C4×D4D12711yescharacteristic
97C2×C4⋊C8—1271no—
98C2×#102—1272no—
99#382—1271no—
100#340—1272no—
101C22×D4C2×D12711yescharacteristic
102C23×C4—21no—
103C22×C8—1272no—
104C4×D4—22no—
105C4×D4Dic12714yescomplemented
106C2×C4⋊C4—1272no—
107C2×C4⋊C4C2×D12711yescharacteristic
108C2×C22⋊C4—21no—
109C2×C42C2×D12711yescharacteristic
110C4⋊C8—2542no—
111#102—2544no—
112#95—2541no—
113#95—2541no—
114C24—21no—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4C2×Dic12712yes—
119C2×D4D50812yes—
120C2×D4Dic25412yes—
121C22×C4—21no—
122C22×C4—24no—
123C22×C4—44no—
124C22×C4—2542no—
125C22×C4C22⋊D12711yescharacteristic
126C22×C4C22⋊D12711yescharacteristic
127C22×C4C22×D12711yescharacteristic
128C2×C8—2542no—
129C2×C8—2542no—
130C2×C8—2542no—
131C4⋊C4—21no—
132C4⋊C4—2542no—
133C4⋊C4—2542no—
134C4⋊C4C2×Dic12712yes—
135C22⋊C4—24no—
136C42C2×Dic12712yes—
137C42C22⋊D12712yes—
138C23—22no—
139C23—22no—
140C23—22no—
141C23—42no—
142C23—42no—
143C23C2×C22⋊D12711yescharacteristic
144D4—22no—
145D4—22no—
146D4—22no—
147D4C4⋊Dic12714yes—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—48no—
152C2×C4—42no—
153C2×C4—5082no—
154C2×C4—5082no—
155C2×C4—5082no—
156C2×C4C22⋊Dic12712yes—
157C2×C4C22⋊Dic12712yes—
158C2×C4C22×Dic12711yescharacteristic
159C2×C4C2×C22⋊D12711yescharacteristic
160C2×C4C2×D50811yescharacteristic
161C2×C4C2×Dic25411yescharacteristic
162C2×C4#1452324811yescharacteristic
163C2×C4#1452324811yescharacteristic
164C8—5082no—
165C22—24no—
166C22—44no—
167C22—42no—
168C22—42no—
169C22—44no—
170C22C2×C22⋊Dic12711yescharacteristic
171C22#1686118311yescharacteristic
172C22#1686118811yescharacteristic
173C22#1686122912yes—
174C22#1686123512yes—
175C4—22no—
176C4—22no—
177C4—42no—
178C4—10162no—
179C4C2×C4⋊Dic12711yescharacteristic
180C4#1686116811yescharacteristic
181C4#1686118311yescharacteristic
182C4#1686118811yescharacteristic
183C2—44no—
184C2#2254973611yescharacteristic
185C2#2254987211yescharacteristic
186C2#2254988111yescharacteristic
187C2#2255009112yes—
188C2#2255017612yes—
subgroupG/H
centerC23C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×C4×D4×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
cosocleC22×C4×C127C23
hypercenterC2×C4×D4D127
hypocenterC127#1804
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊(C127⋊C8)C2semidirect
3. #22549742C2semidirect
4. #22549638C2semidirect
15. C4×D4×C127C4semidirect
95. C127#1804semidirect
105. C4×D4Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C127⋊C8)2
C2×#168611302
#225497422
#225496382
C2×C4×D4×C1272
#1804127