Finite Group Databasebeta

← C2×#22549601 #30637039

Subgroups — C2×#22549601 #30637039

Order 16256.

328 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 109 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×Dic127⋊C4C211yesmaximal, characteristic
2C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic
3#22549601C214yesmaximal, direct product factor
4C2×C22⋊C8×C127C211yesmaximal, characteristic
5C23×Dic127C411yescharacteristic
6C23×Dic127C411yescharacteristic
7C22×C127⋊C8—21no—
8C2×Dic127⋊C4—21no—
9C2×Dic127⋊C4—21no—
10C2×Dic127⋊C4—21no—
11C2×Dic127⋊C4—21no—
12C2×Dic127⋊C4C412yes—
13C2×Dic127⋊C4C2212yes—
14C22⋊(C127⋊C8)—21no—
15C22⋊(C127⋊C8)C2212yes—
16C23×C4×C127C2211yescharacteristic
17C22×C8×C127—21no—
18C22⋊C8×C127—21no—
19C22⋊C8×C127C2212yes—
20C22×Dic127—21no—
21C22×Dic127—21no—
22C22×Dic127—21no—
23C22×Dic127—21no—
24C22×Dic127—22no—
25C22×Dic127—42no—
26C22×Dic127C814yescomplemented
27C22×Dic127C2×C412yes—
28C22×Dic127C2×C412yes—
29C2×C127⋊C8—22no—
30C2×C127⋊C8—42no—
31Dic127⋊C4—22no—
32Dic127⋊C4—22no—
33Dic127⋊C4—22no—
34Dic127⋊C4—22no—
35C24×C127C2×C411yescharacteristic
36C22×C4×C127—42no—
37C22×C4×C127C2×C411yescharacteristic
38C22×C4×C127D411yescharacteristic
39C22×C4×C127D411yescharacteristic
40C22×C4×C127D411yescharacteristic
41C22×C4×C127D411yescharacteristic
42C22×C4×C127C2311yescharacteristic
43C2×C8×C127—22no—
44C2×C8×C127—42no—
45C2×Dic127—24no—
46C2×Dic127—24no—
47C2×Dic127—22no—
48C2×Dic127—22no—
49C2×Dic127—22no—
50C2×Dic127—42no—
51C2×Dic127—42no—
52C2×Dic127—42no—
53C2×Dic127C2×C814yescomplemented
54C127⋊C8—42no—
55C23×C127—22no—
56C23×C127—22no—
57C23×C127C22⋊C411yescharacteristic
58C23×C127C22⋊C411yescharacteristic
59C23×C127C2×C812yes—
60C23×C127M1612yes—
61C23×C127C22×C411yescharacteristic
62C2×C4×C127—22no—
63C2×C4×C127—22no—
64C2×C4×C127—41no—
65C2×C4×C127—41no—
66C2×C4×C127—41no—
67C2×C4×C127—41no—
68C2×C4×C127C22⋊C411yescharacteristic
69C2×C4×C127C22⋊C411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C8×C127—42no—
73C22×C127—24no—
74C22×C127—24no—
75C22×C127—21no—
76C22×C127—21no—
77C22×C127—21no—
78C22×C127—21no—
79C22×C127C22⋊C812yes—
80C22×C127C22⋊C812yes—
81C22×C127#9912yes—
82C22×C127#10112yes—
83C22×C127C2×C22⋊C411yescharacteristic
84C22×C127C22×C811yescharacteristic
85C22×C127C2×M1611yescharacteristic
86C4×C127—21no—
87C4×C127—21no—
88C4×C127—21no—
89C4×C127—21no—
90Dic127—24no—
91Dic127—42no—
92C2×C127—22no—
93C2×C127—21no—
94C2×C127—21no—
95C2×C127#32414yescomplemented
96C2×C127C2×C22⋊C811yescharacteristic
97C2×C127C2×#9911yescharacteristic
98C2×C127C2×#10111yescharacteristic
99C2×#324—1271nomaximal, Sylow
100C127C2×#32411yescharacteristic, complemented, Sylow
101C22×C4⋊C4—1271no—
102C2×C22⋊C8—1271no—
103C2×C22⋊C8D12711yescharacteristic
104#324—1274no—
105C23×C4—1271no—
106C23×C4—1271no—
107C23×C4C2×D12711yescharacteristic
108C22×C8—21no—
109C22×C8—2541no—
110C2×C4⋊C4—1272no—
111C2×C4⋊C4—1272no—
112C2×C4⋊C4—2541no—
113C2×C4⋊C4—2541no—
114C2×C4⋊C4—2541no—
115C2×C4⋊C4—2541no—
116C22⋊C8—21no—
117C22⋊C8—1272no—
118C22⋊C8—2541no—
119C22⋊C8C2×D12712yes—
120C24C4×D12711yescharacteristic
121C22×C4—42no—
122C22×C4—1274no—
123C22×C4—1272no—
124C22×C4—1272no—
125C22×C4—5082no—
126C22×C4—2541no—
127C22×C4—2541no—
128C22×C4—2541no—
129C22×C4—2541no—
130C22×C4—2542no—
131C22×C4C4×D12711yescharacteristic
132C22×C4C22⋊D12711yescharacteristic
133C22×C4C22⋊D12711yescharacteristic
134C22×C4D50811yescharacteristic
135C22×C4D50811yescharacteristic
136C22×C4C22×D12711yescharacteristic
137C2×C8—22no—
138C2×C8—42no—
139C2×C8—5082no—
140C2×C8—2542no—
141C4⋊C4—2542no—
142C4⋊C4—2542no—
143C4⋊C4—2542no—
144C4⋊C4—2542no—
145C23—22no—
146C23—22no—
147C23(C2×D127)⋊C411yescharacteristic
148C23(C2×D127)⋊C411yescharacteristic
149C23C8×D12712yes—
150C23#1452323112yes—
151C23C2×C4×D12711yescharacteristic
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—1274no—
159C2×C4—5082no—
160C2×C4—5082no—
161C2×C4—5082no—
162C2×C4—2542no—
163C2×C4—2542no—
164C2×C4—2544no—
165C2×C4—2544no—
166C2×C4—2542no—
167C2×C4(C2×D127)⋊C411yescharacteristic
168C2×C4(C2×D127)⋊C411yescharacteristic
169C2×C4C2×C22⋊D12711yescharacteristic
170C2×C4C2×D50811yescharacteristic
171C8—42no—
172C8—5082no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—24no—
178C22—24no—
179C22(C2×D127)⋊C812yes—
180C22(C2×D127)⋊C812yes—
181C22#1686112312yes—
182C22#1686112712yes—
183C22C2×(C2×D127)⋊C411yescharacteristic
184C22C2×C8×D12711yescharacteristic
185C22C2×#1452323111yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—21no—
190C4—5082no—
191C4—2544no—
192C2—21no—
193C2—21no—
194C2—22no—
195C2#2254960114yesdirect product factor
196C2C2×(C2×D127)⋊C811yescharacteristic
197C2C2×#1686112311yescharacteristic
198C2C2×#1686112711yescharacteristic
subgroupG/H
centerC23(C2×D127)⋊C4
commutatorC22×C127C22×C8
FrattiniC22×C4C22×D127
FittingC2×C22⋊C8×C127C2
socleC23×C127C22⋊C4
prime radicalC24×C127C2×C4
cosocleC22×C4×C127C23
hypercenterC2×C22⋊C8D127
hypocenterC127C2×#324
LCSG ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
3. #22549601C2direct (N × H)
26. C22×Dic127C8semidirect
53. C2×Dic127C2×C8semidirect
95. C2×C127#324semidirect
100. C127C2×#324semidirect
195. C2#22549601direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Dic127⋊C42
C2×C22⋊(C127⋊C8)2
#225496012
C2×C22⋊C8×C1272
C2×#324127