Finite Group Databasebeta

← C2×#22550216 #30648621

Subgroups — C2×#22550216 #30648621

Order 16256.

416 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16861185C211yesmaximal, characteristic
2C2×#16861167C211yesmaximal, characteristic
3#22550216C218yesmaximal, direct product factor
4C2×#16861143C211yesmaximal, characteristic, complemented
5C2×#16861140C211yesmaximal, characteristic, complemented
6C2×#16861133C211yesmaximal, characteristic, complemented
7C2×(C2×D127)⋊C8C211yesmaximal, characteristic
8C2×#103×C127C211yesmaximal, characteristic, complemented
9C22×C4×D127C2211yescharacteristic
10C22×C127⋊C8C2211yescharacteristic
11#16861185—22no—
12#16861185C2214yes—
13#16861167—22no—
14#16861167C2214yes—
15C2×C4⋊Dic127C2211yescharacteristic
16C2×C4⋊Dic127C2211yescharacteristic
17C2×Dic127⋊C4—21no—
18C2×(C2×D127)⋊C4—21no—
19C2×(C2×D127)⋊C4—21no—
20C2×C4×Dic127—21no—
21#16861143C2214yescomplemented
22#16861140C2214yescomplemented
23#16861133C2214yescomplemented
24(C2×D127)⋊C8C2214yes—
25C22×Q8×C127C2211yescharacteristic
26C22×C8×C127C2211yescharacteristic
27C2×C4⋊C4×C127C2211yescharacteristic
28#103×C127C2214yescomplemented
29C23×D127D411yescharacteristic
30C22×Dic127—21no—
31C22×Dic127—21no—
32C22×Dic127D411yescharacteristic
33C2×C4×D127—22no—
34C2×C4×D127—22no—
35C2×C4×D127D412yes—
36C2×C4×D127C2312yes—
37C2×C127⋊C8—22no—
38C2×C127⋊C8C2312yes—
39C4⋊Dic127—21no—
40C4⋊Dic127—21no—
41C4⋊Dic127C2312yes—
42C4⋊Dic127C2312yes—
43Dic127⋊C4—24no—
44(C2×D127)⋊C4—24no—
45(C2×D127)⋊C4—24no—
46C4×Dic127—24no—
47C2×Q8×C127—21no—
48C2×Q8×C127—22no—
49C2×Q8×C127—22no—
50C2×Q8×C127C2312yes—
51C22×C4×C127—21no—
52C22×C4×C127—21no—
53C22×C4×C127C2311yescharacteristic
54C2×C8×C127—22no—
55C2×C8×C127C2312yes—
56C4⋊C4×C127—21no—
57C4⋊C4×C127C2312yes—
58C22×D127—22no—
59C22×D127—22no—
60C22×D127—22no—
61C22×D127C2×D412yes—
62C2×Dic127—22no—
63C2×Dic127—22no—
64C2×Dic127—22no—
65C2×Dic127—42no—
66C2×Dic127—42no—
67C2×Dic127C2×D412yes—
68C4×D127—24no—
69C4×D127—22no—
70C4×D127—22no—
71C127⋊C8—22no—
72C23×C127C2×D411yescharacteristic
73Q8×C127—22no—
74Q8×C127—22no—
75Q8×C127—42no—
76C2×C4×C127—22no—
77C2×C4×C127—22no—
78C2×C4×C127—42no—
79C2×C4×C127—42no—
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127#4112yes—
82C2×C4×C127#4112yes—
83C2×C4×C127C2411yescharacteristic
84C8×C127—22no—
85C22×C127#12312yes—
86C22×C127#13512yes—
87C22×C127#13712yes—
88C22×C127C22×D411yescharacteristic
89C2×D127—22no—
90C2×D127—22no—
91C2×D127—22no—
92C2×D127—44no—
93C4×C127—42no—
94C4×C127—42no—
95C4×C127#12312yes—
96C4×C127C2×#4111yescharacteristic
97C4×C127C2×#4111yescharacteristic
98Dic127—22no—
99Dic127—42no—
100Dic127—42no—
101C2×C127#48514yescomplemented
102C2×C127C2×#12311yescharacteristic
103C2×C127C2×#13511yescharacteristic
104C2×C127C2×#13711yescharacteristic
105D127—42no—
106C2×#485—1271nomaximal, Sylow
107C127C2×#48511yescharacteristic, complemented, Sylow
108C2×C22⋊Q8—1271no—
109C2×#117—1271no—
110#485—1278no—
111C2×#107—1271no—
112C2×#106—1271no—
113C2×#103—1271no—
114C2×#103D12711yescharacteristic, complemented
115C2×C22⋊C8—1271no—
116C22×Q8C2×D12711yescharacteristic
117C23×C4—1271no—
118C22×C8—1271no—
119C22×C8C2×D12711yescharacteristic
120C22⋊Q8—1274no—
121C22⋊Q8—2542no—
122#117—1274no—
123#117—2542no—
124C2×C4⋊C4—1271no—
125C2×C4⋊C4—1271no—
126C2×C4⋊C4—2541no—
127C2×C4⋊C4C2×D12711yescharacteristic
128C2×C22⋊C4—2541no—
129C2×C22⋊C4—2541no—
130C2×C42—2541no—
131#107—1274no—
132#106—1274no—
133#103—1274no—
134#103C2×D12714yescomplemented
135C22⋊C8—1274no—
136C24—1271no—
137C2×Q8—21no—
138C2×Q8—22no—
139C2×Q8—22no—
140C2×Q8C22×D12712yes—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—1271no—
144C22×C4—1272no—
145C22×C4—1272no—
146C22×C4—2541no—
147C22×C4—2541no—
148C22×C4—2542no—
149C22×C4—2542no—
150C22×C4C22×D12711yescharacteristic
151C2×C8—22no—
152C2×C8—1272no—
153C2×C8—2542no—
154C2×C8C22×D12712yes—
155C4⋊C4—21no—
156C4⋊C4—1272no—
157C4⋊C4—1272no—
158C4⋊C4—2544no—
159C4⋊C4—2541no—
160C4⋊C4—2541no—
161C4⋊C4C22×D12712yes—
162C22⋊C4—2544no—
163C22⋊C4—2544no—
164C42—2544no—
165C23—1272no—
166C23—2542no—
167C23—2542no—
168C23—2542no—
169C23D4×D12711yescharacteristic
170Q8—22no—
171Q8—22no—
172Q8—42no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—1272no—
178C2×C4—5082no—
179C2×C4—5082no—
180C2×C4—2542no—
181C2×C4—2542no—
182C2×C4—2542no—
183C2×C4—2544no—
184C2×C4—2542no—
185C2×C4—2542no—
186C2×C4D4×D12711yescharacteristic
187C2×C4#1452324712yes—
188C2×C4#1452324812yes—
189C2×C4C23×D12711yescharacteristic
190C8—22no—
191C8—2542no—
192C22—5084no—
193C22—2542no—
194C22—2542no—
195C22—2542no—
196C22#1686119112yes—
197C22#1686123212yes—
198C22#1686123812yes—
199C22C2×D4×D12711yescharacteristic
200C4—42no—
201C4—42no—
202C4—5082no—
203C4—5082no—
204C4—2542no—
205C4#1686119112yes—
206C4C2×#1452324711yescharacteristic
207C4C2×#1452324811yescharacteristic
208C2—5082no—
209C2#2255021614yesdirect product factor
210C2C2×#1686119111yescharacteristic
211C2C2×#1686123211yescharacteristic
212C2C2×#1686123811yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#103×C127C2
socleC23×C127C2×D4
prime radicalC23×D127D4
hypercenterC2×#103D127
hypocenterC127C2×#485
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. #22550216C2direct (N × H)
4. C2×#16861143C2semidirect
5. C2×#16861140C2semidirect
6. C2×#16861133C2semidirect
8. C2×#103×C127C2semidirect
21. #16861143C22semidirect
22. #16861140C22semidirect
23. #16861133C22semidirect
28. #103×C127C22semidirect
101. C2×C127#485semidirect
107. C127C2×#485semidirect
114. C2×#103D127semidirect
134. #103C2×D127semidirect
209. C2#22550216direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168611852
C2×#168611672
#225502162
C2×#168611432
C2×#168611402
C2×#168611332
C2×(C2×D127)⋊C82
C2×#103×C1272
C2×#485127