Finite Group Databasebeta

← C2×#22550225 #30649039

Subgroups — C2×#22550225 #30649039

Order 16256.

424 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 157 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×#16861208C211yesmaximal, characteristic
2C2×#16861207C211yesmaximal, characteristic
3#22550225C218yesmaximal, direct product factor
4C2×#16861134C211yesmaximal, characteristic
5C2×#16861133C211yesmaximal, characteristic
6C2×#16861132C211yesmaximal, characteristic
7C2×C8×Dic127C211yesmaximal, characteristic
8C2×#103×C127C211yesmaximal, characteristic
9C22×Dic254—21no—
10C22×Dic254C2211yescharacteristic
11C22×C127⋊C8C2211yescharacteristic
12#16861208—22no—
13#16861208C2214yes—
14#16861207—22no—
15#16861207C2214yes—
16C2×C4⋊Dic127C2211yescharacteristic
17C2×Dic127⋊C4—21no—
18C2×Dic127⋊C4—21no—
19C2×Dic127⋊C4—21no—
20C2×C4×Dic127C2211yescharacteristic
21#16861134C2214yes—
22#16861133C2214yes—
23#16861132C2214yes—
24C8×Dic127C2214yes—
25C22×Q8×C127C2211yescharacteristic
26C22×C8×C127C2211yescharacteristic
27C2×C4⋊C4×C127C2211yescharacteristic
28#103×C127C2214yes—
29C2×Dic254—24no—
30C2×Dic254—21no—
31C2×Dic254—22no—
32C2×Dic254—22no—
33C2×Dic254—44no—
34C2×Dic254C2312yes—
35C22×Dic127—21no—
36C22×Dic127—21no—
37C22×Dic127D411yescharacteristic
38C22×Dic127D411yescharacteristic
39C2×C127⋊C8—22no—
40C2×C127⋊C8C2312yes—
41C4⋊Dic127—21no—
42C4⋊Dic127C2312yes—
43Dic127⋊C4—24no—
44Dic127⋊C4—24no—
45Dic127⋊C4—24no—
46C4×Dic127D412yes—
47C4×Dic127C2312yes—
48C2×Q8×C127—21no—
49C2×Q8×C127—22no—
50C2×Q8×C127—22no—
51C2×Q8×C127C2312yes—
52C22×C4×C127—21no—
53C22×C4×C127—21no—
54C22×C4×C127C2311yescharacteristic
55C2×C8×C127—22no—
56C2×C8×C127C2312yes—
57C4⋊C4×C127—21no—
58C4⋊C4×C127C2312yes—
59Dic254—22no—
60Dic254—22no—
61Dic254—42no—
62Dic254—48no—
63C2×Dic127—22no—
64C2×Dic127—22no—
65C2×Dic127—42no—
66C2×Dic127—42no—
67C2×Dic127SD1614yes—
68C2×Dic127Q1614yes—
69C2×Dic127C2×D412yes—
70C2×Dic127C2×D412yes—
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#12412yes—
86C22×C127C2×SD1612yes—
87C22×C127C2×Q1612yes—
88C22×C127C22×D411yescharacteristic
89C4×C127—42no—
90C4×C127—42no—
91C4×C127#12412yes—
92C4×C127C2×#4111yescharacteristic
93C4×C127C2×#4111yescharacteristic
94Dic127—42no—
95Dic127—42no—
96Dic127C2×SD1614yes—
97Dic127C2×Q1614yes—
98C2×C127#48714yescomplemented
99C2×C127C2×#12411yescharacteristic
100C2×C127C22×SD1611yescharacteristic
101C2×C127C22×Q1611yescharacteristic
102C2×#487—1271nomaximal, Sylow
103C127C2×#48711yescharacteristic, complemented, Sylow
104C2×C4⋊Q8—1271no—
105C2×C4⋊Q8—1271no—
106#487—1278no—
107C2×#103—1271no—
108C2×#103—1271no—
109C2×#103—1271no—
110C2×#103D12711yescharacteristic
111C2×C4×C8—1271no—
112C22×Q8—1271no—
113C22×Q8—2541no—
114C22×Q8C2×D12711yescharacteristic
115C22×C8—1271no—
116C22×C8C2×D12711yescharacteristic
117C4⋊Q8—1274no—
118C4⋊Q8—1274no—
119C4⋊Q8—2542no—
120C4⋊Q8—2542no—
121C2×C4⋊C4—1271no—
122C2×C4⋊C4—2541no—
123C2×C4⋊C4—2541no—
124C2×C4⋊C4—2541no—
125C2×C4⋊C4C2×D12711yescharacteristic
126C2×C42—1271no—
127#103—1274no—
128#103—1274no—
129#103—1274no—
130#103C2×D12714yes—
131C4×C8—1274no—
132C2×Q8—21no—
133C2×Q8—22no—
134C2×Q8—22no—
135C2×Q8—1272no—
136C2×Q8—5084no—
137C2×Q8—2544no—
138C2×Q8—2542no—
139C2×Q8—2542no—
140C2×Q8—2541no—
141C2×Q8C22×D12712yes—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—1271no—
145C22×C4—1271no—
146C22×C4—2541no—
147C22×C4—2541no—
148C22×C4C22×D12711yescharacteristic
149C2×C8—22no—
150C2×C8—1272no—
151C2×C8—2542no—
152C2×C8C22×D12712yes—
153C4⋊C4—21no—
154C4⋊C4—1272no—
155C4⋊C4—2544no—
156C4⋊C4—2544no—
157C4⋊C4—2544no—
158C4⋊C4—2541no—
159C4⋊C4C22×D12712yes—
160C42—1272no—
161C42—1272no—
162C23D4×D12711yescharacteristic
163Q8—22no—
164Q8—22no—
165Q8—42no—
166Q8—5088no—
167Q8—5082no—
168Q8—2542no—
169Q8—2542no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—1274no—
175C2×C4—1274no—
176C2×C4—1272no—
177C2×C4—1272no—
178C2×C4—5082no—
179C2×C4—5082no—
180C2×C4—2542no—
181C2×C4—2542no—
182C2×C4D4×D12711yescharacteristic
183C2×C4#1452324712yes—
184C2×C4#1452324812yes—
185C2×C4C23×D12711yescharacteristic
186C8—22no—
187C8—2542no—
188C22#1686119512yes—
189C22SD16×D12712yes—
190C22Q16×D12712yes—
191C22C2×D4×D12711yescharacteristic
192C4—42no—
193C4—42no—
194C4—1274no—
195C4—1274no—
196C4—5082no—
197C4—5082no—
198C4#1686119512yes—
199C4C2×#1452324711yescharacteristic
200C4C2×#1452324811yescharacteristic
201C2#2255022514yesdirect product factor
202C2C2×#1686119511yescharacteristic
203C2C2×SD16×D12711yescharacteristic
204C2C2×Q16×D12711yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#103×C127C2
socle, prime radicalC23×C127C2×D4
hypercenterC2×#103D127
hypocenterC127C2×#487
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. #22550225C2direct (N × H)
98. C2×C127#487semidirect
103. C127C2×#487semidirect
201. C2#22550225direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612082
C2×#168612072
#225502252
C2×#168611342
C2×#168611332
C2×#168611322
C2×C8×Dic1272
C2×#103×C1272
C2×#487127