Finite Group Databasebeta

← C2×#22550217 #30648622

Subgroups — C2×#22550217 #30648622

Order 16256.

416 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861188C211yesmaximal, characteristic
2#22550217C218yesmaximal, direct product factor
3C2×#16861142C211yesmaximal, characteristic, complemented
4C2×#16861140C211yesmaximal, characteristic, complemented
5C2×#16861132C211yesmaximal, characteristic, complemented
6C2×#16861132C211yesmaximal, characteristic, complemented
7C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic
8C2×#117×C127C211yesmaximal, characteristic
9C22×Dic254C2211yescharacteristic
10C22×C127⋊C8C2211yescharacteristic
11C22×C127⋊C8C2211yescharacteristic
12#16861188—22no—
13#16861188C2214yes—
14C2×C4⋊Dic127C2211yescharacteristic
15C2×Dic127⋊C4—21no—
16C2×C22⋊Dic127—21no—
17#16861142C2214yescomplemented
18#16861140C2214yescomplemented
19#16861132C2214yescomplemented
20#16861132C2214yescomplemented
21C22⋊(C127⋊C8)C2214yes—
22C23×C4×C127C2211yescharacteristic
23#117×C127—22no—
24#117×C127C2214yes—
25C2×C4⋊C4×C127C2211yescharacteristic
26C2×C4⋊C4×C127C2211yescharacteristic
27C2×C22⋊C4×C127—21no—
28C2×C42×C127—21no—
29C2×Dic254—21no—
30C2×Dic254—22no—
31C2×Dic254—22no—
32C2×Dic254C2312yes—
33C22×Dic127—21no—
34C22×Dic127—21no—
35C2×C127⋊C8—22no—
36C2×C127⋊C8—22no—
37C2×C127⋊C8C2312yes—
38C2×C127⋊C8C2312yes—
39C4⋊Dic127—21no—
40C4⋊Dic127C2312yes—
41Dic127⋊C4—24no—
42C22⋊Dic127—24no—
43C24×C127D411yescharacteristic
44C22×C4×C127—22no—
45C22×C4×C127—22no—
46C22×C4×C127—21no—
47C22×C4×C127—21no—
48C22×C4×C127D412yes—
49C22×C4×C127D411yescharacteristic
50C22×C4×C127C2312yes—
51C22×C4×C127C2311yescharacteristic
52C4⋊C4×C127—21no—
53C4⋊C4×C127—21no—
54C4⋊C4×C127C2312yes—
55C4⋊C4×C127C2312yes—
56C22⋊C4×C127—24no—
57C42×C127—24no—
58Dic254—22no—
59Dic254—22no—
60Dic254—42no—
61C2×Dic127—22no—
62C2×Dic127—22no—
63C2×Dic127—42no—
64C2×Dic127—42no—
65C127⋊C8—22no—
66C127⋊C8—22no—
67C23×C127—22no—
68C23×C127—22no—
69C23×C127—22no—
70C23×C127C2×D412yes—
71C23×C127C2×D411yescharacteristic
72C2×C4×C127—22no—
73C2×C4×C127—24no—
74C2×C4×C127—22no—
75C2×C4×C127—22no—
76C2×C4×C127—22no—
77C2×C4×C127—22no—
78C2×C4×C127—42no—
79C2×C4×C127—42no—
80C2×C4×C127C2×D412yes—
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127#4112yes—
83C2×C4×C127#4112yes—
84C2×C4×C127C2411yescharacteristic
85C22×C127—22no—
86C22×C127—22no—
87C22×C127—22no—
88C22×C127—44no—
89C22×C127#12312yes—
90C22×C127#13512yes—
91C22×C127#13712yes—
92C22×C127C22×D411yescharacteristic
93C4×C127—22no—
94C4×C127—42no—
95C4×C127—42no—
96C4×C127#12312yes—
97C4×C127C2×#4111yescharacteristic
98C4×C127C2×#4111yescharacteristic
99Dic127—42no—
100Dic127—42no—
101C2×C127—42no—
102C2×C127#48514yescomplemented
103C2×C127C2×#12311yescharacteristic
104C2×C127C2×#13511yescharacteristic
105C2×C127C2×#13711yescharacteristic
106C2×#485—1271nomaximal, Sylow
107C127C2×#48511yescharacteristic, complemented, Sylow
108C2×C22⋊Q8—1271no—
109C2×#117D12711yescharacteristic
110#485—1278no—
111C2×#107—1271no—
112C2×#106—1271no—
113C2×#103—1271no—
114C2×#103—1271no—
115C2×C22⋊C8—1271no—
116C22×Q8—1271no—
117C23×C4C2×D12711yescharacteristic
118C22×C8—1271no—
119C22×C8—1271no—
120C22⋊Q8—1274no—
121C22⋊Q8—2542no—
122#117—22no—
123#117C2×D12714yes—
124C2×C4⋊C4—1271no—
125C2×C4⋊C4—2541no—
126C2×C4⋊C4C2×D12711yescharacteristic
127C2×C4⋊C4C2×D12711yescharacteristic
128C2×C22⋊C4—21no—
129C2×C22⋊C4—2541no—
130C2×C42—21no—
131#107—1274no—
132#106—1274no—
133#103—1274no—
134#103—1274no—
135C22⋊C8—1274no—
136C24C22⋊D12711yescharacteristic
137C2×Q8—1272no—
138C2×Q8—2542no—
139C2×Q8—2542no—
140C2×Q8—2541no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—2541no—
146C22×C4—2541no—
147C22×C4C22⋊D12712yes—
148C22×C4C22⋊D12711yescharacteristic
149C22×C4C22×D12712yes—
150C22×C4C22×D12711yescharacteristic
151C2×C8—1272no—
152C2×C8—1272no—
153C2×C8—2542no—
154C2×C8—2542no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—1272no—
158C4⋊C4—2544no—
159C4⋊C4—2541no—
160C4⋊C4C22×D12712yes—
161C4⋊C4C22×D12712yes—
162C22⋊C4—24no—
163C22⋊C4—2544no—
164C42—24no—
165C23—22no—
166C23—22no—
167C23—22no—
168C23C2×C22⋊D12712yes—
169C23C2×C22⋊D12711yescharacteristic
170Q8—5082no—
171Q8—2542no—
172Q8—2542no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—5082no—
182C2×C4—5082no—
183C2×C4—2542no—
184C2×C4—2542no—
185C2×C4C2×C22⋊D12712yes—
186C2×C4C2×C22⋊D12711yescharacteristic
187C2×C4#1452324812yes—
188C2×C4#1452324812yes—
189C2×C4C23×D12711yescharacteristic
190C8—2542no—
191C8—2542no—
192C22—22no—
193C22—22no—
194C22—22no—
195C22—44no—
196C22#1686119212yes—
197C22#1686122912yes—
198C22#1686123912yes—
199C22C22×C22⋊D12711yescharacteristic
200C4—22no—
201C4—42no—
202C4—42no—
203C4—5082no—
204C4—5082no—
205C4#1686119212yes—
206C4C2×#1452324811yescharacteristic
207C4C2×#1452324811yescharacteristic
208C2—42no—
209C2#2255021714yesdirect product factor
210C2C2×#1686119211yescharacteristic
211C2C2×#1686122911yescharacteristic
212C2C2×#1686123911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#117×C127C2
socleC23×C127C2×D4
prime radicalC24×C127D4
hypercenterC2×#117D127
hypocenterC127C2×#485
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #22550217C2direct (N × H)
3. C2×#16861142C2semidirect
4. C2×#16861140C2semidirect
5. C2×#16861132C2semidirect
6. C2×#16861132C2semidirect
17. #16861142C22semidirect
18. #16861140C22semidirect
19. #16861132C22semidirect
20. #16861132C22semidirect
102. C2×C127#485semidirect
107. C127C2×#485semidirect
209. C2#22550217direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168611882
#225502172
C2×#168611422
C2×#168611402
C2×#168611322
C2×C22⋊(C127⋊C8)2
C2×#117×C1272
C2×#485127