Finite Group Databasebeta

← C2×#22550085 #30648167

Subgroups — C2×#22550085 #30648167

Order 16256.

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

normal subgroups only

This group has 480 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
1C22×Dic508C211yesmaximal, characteristic
2C22×#14523238C211yesmaximal, characteristic
3C2×#16861207C211yesmaximal, characteristic
4C2×#16861174C211yesmaximal, characteristic
5#22550085C218yesmaximal, direct product factor
6C2×Dic127⋊C8C211yesmaximal, characteristic
7C2×#16861132C211yesmaximal, characteristic
8C2×#103×C127C211yesmaximal, characteristic
9C22×Dic254C2211yescharacteristic
10C22×Dic254C2211yescharacteristic
11C2×Dic508—24no—
12C2×Dic508C2214yes—
13C2×#14523238—24no—
14C2×#14523238C2214yes—
15C22×C127⋊C8C2211yescharacteristic
16#16861207—22no—
17#16861207C2214yes—
18#16861174—22no—
19#16861174C2214yes—
20C2×Dic127⋊C4—21no—
21C2×Dic127⋊C4—21no—
22C2×Dic127⋊C4—21no—
23C2×C4×Dic127—21no—
24C2×C4×Dic127C2211yescharacteristic
25Dic127⋊C8C2214yes—
26#16861132C2214yes—
27C22×Q8×C127C2211yescharacteristic
28C22×C8×C127C2211yescharacteristic
29C2×C4⋊C4×C127C2211yescharacteristic
30#103×C127C2214yes—
31C2×Dic254—21no—
32C2×Dic254—21no—
33C2×Dic254—22no—
34C2×Dic254—22no—
35C2×Dic254—22no—
36C2×Dic254D414yes—
37C2×Dic254C2312yes—
38C2×Dic254C2312yes—
39C22×Dic127—21no—
40C22×Dic127—21no—
41C22×Dic127D411yescharacteristic
42C22×Dic127D411yescharacteristic
43Dic508—28no—
44#14523238—28no—
45C2×C127⋊C8—22no—
46C2×C127⋊C8C2312yes—
47Dic127⋊C4—24no—
48Dic127⋊C4—24no—
49Dic127⋊C4—24no—
50C4×Dic127—24no—
51C4×Dic127D412yes—
52C4×Dic127C2312yes—
53C2×Q8×C127—21no—
54C2×Q8×C127—22no—
55C2×Q8×C127—22no—
56C2×Q8×C127C2312yes—
57C22×C4×C127—21no—
58C22×C4×C127—21no—
59C22×C4×C127C2311yescharacteristic
60C2×C8×C127—22no—
61C2×C8×C127C2312yes—
62C4⋊C4×C127—21no—
63C4⋊C4×C127C2312yes—
64Dic254—24no—
65Dic254—22no—
66Dic254—22no—
67Dic254—22no—
68Dic254—42no—
69Dic254C2×D414yes—
70C2×Dic127—24no—
71C2×Dic127—22no—
72C2×Dic127—22no—
73C2×Dic127—22no—
74C2×Dic127—42no—
75C2×Dic127Q1614yescomplemented
76C2×Dic127C2×D412yes—
77C2×Dic127C2×D412yes—
78C127⋊C8—22no—
79C23×C127C2×D411yescharacteristic
80Q8×C127—22no—
81Q8×C127—22no—
82Q8×C127—42no—
83C2×C4×C127—22no—
84C2×C4×C127—22no—
85C2×C4×C127—42no—
86C2×C4×C127—42no—
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2×D412yes—
89C2×C4×C127#4112yes—
90C2×C4×C127C2411yescharacteristic
91C8×C127—22no—
92C22×C127#12112yes—
93C22×C127C2×Q1612yes—
94C22×C127#13712yes—
95C22×C127C22×D411yescharacteristic
96C4×C127—42no—
97C4×C127—42no—
98C4×C127#12112yes—
99C4×C127C22×D411yescharacteristic
100C4×C127C2×#4111yescharacteristic
101Dic127—24no—
102Dic127—22no—
103Dic127—42no—
104Dic127C2×Q1614yescomplemented
105C2×C127#46214yescomplemented
106C2×C127C2×#12111yescharacteristic
107C2×C127C22×Q1611yescharacteristic
108C2×C127C2×#13711yescharacteristic
109C2×#462—1271nomaximal, Sylow
110C127C2×#46211yescharacteristic, complemented, Sylow
111C22×Q16—1271no—
112C22×Q16—1271no—
113C2×C4⋊Q8—1271no—
114C2×C4×Q8—1271no—
115#462—1278no—
116C2×C4⋊C8—1271no—
117C2×#103—1271no—
118C2×#103D12711yescharacteristic
119C22×Q8—1271no—
120C22×Q8—1271no—
121C22×Q8C2×D12711yescharacteristic
122C2×Q16—1274no—
123C2×Q16—1274no—
124C2×Q16—2544no—
125C2×Q16—2544no—
126C22×C8—1271no—
127C22×C8C2×D12711yescharacteristic
128C4⋊Q8—1274no—
129C4⋊Q8—2542no—
130C4×Q8—1274no—
131C4×Q8—2542no—
132C2×C4⋊C4—2541no—
133C2×C4⋊C4—2541no—
134C2×C4⋊C4—2541no—
135C2×C4⋊C4C2×D12711yescharacteristic
136C2×C42—1271no—
137C2×C42—2541no—
138C4⋊C8—1274no—
139#103—1274no—
140#103C2×D12714yes—
141C2×Q8—21no—
142C2×Q8—22no—
143C2×Q8—22no—
144C2×Q8—1274no—
145C2×Q8—1272no—
146C2×Q8—1272no—
147C2×Q8—2542no—
148C2×Q8—2542no—
149C2×Q8—2541no—
150C2×Q8—2541no—
151C2×Q8—2542no—
152C2×Q8C22×D12712yes—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—1271no—
156C22×C4—1271no—
157C22×C4—2541no—
158C22×C4—2541no—
159C22×C4C22×D12711yescharacteristic
160Q16—2548no—
161Q16—2548no—
162C2×C8—22no—
163C2×C8—1272no—
164C2×C8—2542no—
165C2×C8C22×D12712yes—
166C4⋊C4—21no—
167C4⋊C4—2544no—
168C4⋊C4—2544no—
169C4⋊C4—2544no—
170C4⋊C4C22×D12712yes—
171C42—1272no—
172C42—1272no—
173C42—2544no—
174C23D4×D12711yescharacteristic
175Q8—22no—
176Q8—22no—
177Q8—42no—
178Q8—1274no—
179Q8—5082no—
180Q8—2542no—
181Q8—2542no—
182Q8—2542no—
183Q8—2544no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—1274no—
189C2×C4—1272no—
190C2×C4—1272no—
191C2×C4—5082no—
192C2×C4—2542no—
193C2×C4—2542no—
194C2×C4—2542no—
195C2×C4—2544no—
196C2×C4D4×D12712yes—
197C2×C4D4×D12711yescharacteristic
198C2×C4#1452324812yes—
199C2×C4C23×D12711yescharacteristic
200C8—22no—
201C8—2542no—
202C22#1686118212yes—
203C22Q16×D12712yes—
204C22#1686123812yes—
205C22C2×D4×D12711yescharacteristic
206C4—42no—
207C4—42no—
208C4—1274no—
209C4—5082no—
210C4—2544no—
211C4—2542no—
212C4#1686118212yes—
213C4C2×D4×D12711yescharacteristic
214C4C2×#1452324811yescharacteristic
215C2#2255008514yesdirect product factor
216C2C2×#1686118211yescharacteristic
217C2C2×Q16×D12711yescharacteristic
218C2C2×#1686123811yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#103×C127C2
socle, prime radicalC23×C127C2×D4
hypercenterC2×#103D127
hypocenterC127C2×#462
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
5. #22550085C2direct (N × H)
75. C2×Dic127Q16semidirect
104. Dic127C2×Q16semidirect
105. C2×C127#462semidirect
110. C127C2×#462semidirect
215. C2#22550085direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Dic5082
C22×#145232382
C2×#168612072
C2×#168611742
#225500852
C2×Dic127⋊C82
C2×#168611322
C2×#103×C1272
C2×#462127