Finite Group Databasebeta

← C2×#22550059 #30648072

Subgroups — C2×#22550059 #30648072

Order 16256.

708 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16861239C212yesmaximal, complemented
2C2×#16861207C211yesmaximal, characteristic, complemented
3#22550059C218yesmaximal, direct product factor
4C2×#16861136C212yesmaximal, complemented
5C2×#16861128C211yesmaximal, characteristic, complemented
6C2×2−1+4×C127C211yesmaximal, characteristic
7C22×Dic254C2211yescharacteristic
8#16861239—24no—
9#16861239C2218yescomplemented
10C2×#14523238—22no—
11C2×#14523236—22no—
12C2×#14523232C2212yes—
13#16861207—22no—
14#16861207C2214yescomplemented
15C2×Dic127⋊C4—22no—
16C2×C4×Dic127C2211yescharacteristic
17#16861136C2218yescomplemented
18#16861128C2214yescomplemented
192−1+4×C127—24no—
202−1+4×C127—22no—
212−1+4×C127C2214yes—
22C2×#41×C127—22no—
23C2×#41×C127—22no—
24C2×#41×C127C2212yes—
25C22×Q8×C127—22no—
26C22×Q8×C127C2211yescharacteristic
27C2×Dic254—21no—
28C2×Dic254—22no—
29C2×Dic254—22no—
30C2×Dic254C2312yes—
31C22×Dic127—21no—
32C22×Dic127—21no—
33#14523238—28no—
34#14523236—28no—
35#14523232—22no—
36#14523232C2314yes—
37C2×C127⋊C8—22no—
38Dic127⋊C4—28no—
39C4×Dic127—21no—
40C4×Dic127C2312yes—
41#41×C127—28no—
42#41×C127—28no—
43#41×C127—28no—
44#41×C127—28no—
45#41×C127—22no—
46#41×C127—22no—
47#41×C127D414yes—
48#41×C127C2314yes—
49C2×Q8×C127—22no—
50C2×Q8×C127—28no—
51C2×Q8×C127—28no—
52C2×Q8×C127—22no—
53C2×Q8×C127—22no—
54C2×Q8×C127D412yes—
55C2×Q8×C127D412yes—
56C2×Q8×C127C2312yes—
57C2×D4×C127—22no—
58C2×D4×C127—22no—
59C2×D4×C127D412yes—
60C22×C4×C127—24no—
61C22×C4×C127—22no—
62C22×C4×C127D412yes—
63C22×C4×C127C2311yescharacteristic
64Dic254—24no—
65Dic254—42no—
66C2×Dic127—24no—
67C2×Dic127—22no—
68C2×Dic127—22no—
69C2×Dic127—41no—
70C2×Dic127—41no—
71C127⋊C8—24no—
72C23×C127—22no—
73C23×C127C2×D411yescharacteristic
74Q8×C127—28no—
75Q8×C127—22no—
76Q8×C127—24no—
77Q8×C127—24no—
78Q8×C127C2×D414yes—
79D4×C127—28no—
80D4×C127—24no—
81D4×C127—24no—
82D4×C127—22no—
83D4×C127C2×D414yes—
84C2×C4×C127—216no—
85C2×C4×C127—22no—
86C2×C4×C127—22no—
87C2×C4×C127—22no—
88C2×C4×C127—24no—
89C2×C4×C127—24no—
90C2×C4×C127C2×D414yes—
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D412yes—
93C2×C4×C127C2×D412yes—
94C2×C4×C127C2411yescharacteristic
95C22×C127—22no—
96C22×C127—24no—
97C22×C127—44no—
98C22×C127#12011yescharacteristic
99C22×C127#12011yescharacteristic
100C22×C127C22×D411yescharacteristic
101C4×C127—24no—
102C4×C127—24no—
103C4×C127#12012yes—
104C4×C127C22×D412yes—
105Dic127—24no—
106Dic127—42no—
107C2×C127—21no—
108C2×C127—21no—
109C2×C127—44no—
110C2×C127#45612yescomplemented
111C2×C127C2×#12011yescharacteristic
112C2×#456—1271nomaximal, Sylow
113C127C2×#45611yescharacteristic, complemented, Sylow
114C2×2−1+4D12711yescharacteristic
115C2×#137—1272no—
116C2×C4⋊Q8—1271no—
117#456—1278no—
118C2×#104—1272no—
119C2×#101—1271no—
1202−1+4—24no—
1212−1+4—22no—
1222−1+4C2×D12714yes—
123C2×#41—22no—
124C2×#41—22no—
125C2×#41C2×D12712yes—
126C22×Q8—22no—
127C22×Q8—1271no—
128C22×Q8C2×D12711yescharacteristic
129#137—1278no—
130#137—2544no—
131C2×Q16—2542no—
132C2×SD16—2542no—
133C2×M16—1272no—
134C4⋊Q8—1274no—
135C4⋊Q8—2542no—
136C2×C4⋊C4—2542no—
137C2×C42—1271no—
138#104—1278no—
139#101—1274no—
140#41—28no—
141#41—28no—
142#41—28no—
143#41—28no—
144#41—22no—
145#41—22no—
146#41C22⋊D12714yes—
147#41C22×D12714yes—
148C2×Q8—22no—
149C2×Q8—28no—
150C2×Q8—28no—
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8—1272no—
154C2×Q8—2542no—
155C2×Q8—2542no—
156C2×Q8—2541no—
157C2×Q8C22⋊D12712yes—
158C2×Q8C22⋊D12712yes—
159C2×Q8C22×D12712yes—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4C22⋊D12712yes—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—2541no—
166C22×C4—2541no—
167C22×C4C22⋊D12712yes—
168C22×C4C22×D12711yescharacteristic
169Q16—2548no—
170SD16—2548no—
171M16—1274no—
172M16—2542no—
173C2×C8—2542no—
174C4⋊C4—2548no—
175C42—1272no—
176C42—2541no—
177C23—22no—
178C23C2×C22⋊D12711yescharacteristic
179Q8—28no—
180Q8—24no—
181Q8—22no—
182Q8—24no—
183Q8—5082no—
184Q8—2544no—
185Q8C2×C22⋊D12714yes—
186D4—28no—
187D4—24no—
188D4—24no—
189D4—22no—
190D4C2×C22⋊D12714yes—
191C2×C4—216no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—5081no—
198C2×C4—5081no—
199C2×C4—2542no—
200C2×C4—2544no—
201C2×C4—2542no—
202C2×C4C2×C22⋊D12714yes—
203C2×C4C2×C22⋊D12712yes—
204C2×C4C2×C22⋊D12712yes—
205C2×C4C2×C22⋊D12711yescharacteristic
206C2×C4C23×D12711yescharacteristic
207C8—2544no—
208C22—22no—
209C22—24no—
210C22—44no—
211C22#1686117811yescharacteristic
212C22#1686117811yescharacteristic
213C22C22×C22⋊D12711yescharacteristic
214C4—24no—
215C4—24no—
216C4—5082no—
217C4—2544no—
218C4#1686117812yes—
219C4C22×C22⋊D12712yes—
220C2—21no—
221C2—21no—
222C2—44no—
223C2#2255005912yesdirect product factor
224C2C2×#1686117811yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radicalC2×2−1+4×C127C2
socleC22×C127#120
hypercenterC2×2−1+4D127
hypocenterC127C2×#456
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
2. C2×#16861207C2semidirect
3. #22550059C2direct (N × H)
4. C2×#16861136C2semidirect
5. C2×#16861128C2semidirect
9. #16861239C22semidirect
14. #16861207C22semidirect
17. #16861136C22semidirect
18. #16861128C22semidirect
110. C2×C127#456semidirect
113. C127C2×#456semidirect
223. C2#22550059direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612392
C2×#168612072
#225500592
C2×#168611362
C2×#168611282
C2×2−1+4×C1272
C2×#456127