Finite Group Databasebeta

← C2×#22550149 #30648441

Subgroups — C2×#22550149 #30648441

Order 16256.

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

normal subgroups only

This group has 448 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×#16861239C211yesmaximal, characteristic
2C22×#14523238C211yesmaximal, characteristic, complemented
3C2×#16861216C211yesmaximal, characteristic
4#22550149C218yesmaximal, direct product factor
5C2×#16861144C211yesmaximal, characteristic, complemented
6C2×#16861128C211yesmaximal, characteristic, complemented
7C2×#16861127C211yesmaximal, characteristic, complemented
8C2×#137×C127C211yesmaximal, characteristic
9C22×Dic254C2211yescharacteristic
10#16861239—22no—
11#16861239C2214yes—
12C2×#14523238—21no—
13C2×#14523238—21no—
14C2×#14523238—22no—
15C2×#14523238—21no—
16C2×#14523238C2214yescomplemented
17C2×#14523236—21no—
18#16861216—22no—
19#16861216C2214yes—
20C2×#14523232—21no—
21C2×#14523232C2211yescharacteristic
22C2×#14523232C2211yescharacteristic
23C22×C127⋊C8—21no—
24C22×C127⋊C8C2211yescharacteristic
25#16861144C2214yescomplemented
26#16861128C2214yescomplemented
27#16861127C2214yescomplemented
28C2×#41×C127C2211yescharacteristic
29C22×Q8×C127C2211yescharacteristic
30#137×C127—22no—
31#137×C127C2214yes—
32C2×Q16×C127—21no—
33C2×SD16×C127—21no—
34C2×M16×C127C2211yescharacteristic
35C2×Dic254—21no—
36C2×Dic254—21no—
37C2×Dic254—21no—
38C2×Dic254—21no—
39C2×Dic254—21no—
40C2×Dic254C2312yes—
41C22×Dic127—21no—
42#14523238—24no—
43#14523238—24no—
44#14523238—24no—
45#14523236—24no—
46#14523232—24no—
47#14523232—21no—
48#14523232—21no—
49#14523232C2312yes—
50#14523232C2312yes—
51C2×C127⋊C8—24no—
52C2×C127⋊C8—21no—
53C2×C127⋊C8—21no—
54C2×C127⋊C8D411yescharacteristic
55C2×C127⋊C8D411yescharacteristic
56C2×C127⋊C8D412yes—
57C2×C127⋊C8C2312yes—
58#41×C127—21no—
59#41×C127—21no—
60#41×C127D412yes—
61#41×C127C2312yes—
62C2×Q8×C127—21no—
63C2×Q8×C127—21no—
64C2×Q8×C127—21no—
65C2×Q8×C127—21no—
66C2×Q8×C127—21no—
67C2×Q8×C127D411yescharacteristic
68C2×Q8×C127C2312yes—
69C2×D4×C127—21no—
70C2×D4×C127D411yescharacteristic
71C22×C4×C127—21no—
72C22×C4×C127—21no—
73C22×C4×C127C2311yescharacteristic
74Q16×C127—24no—
75SD16×C127—24no—
76M16×C127—21no—
77M16×C127C2312yes—
78C2×C8×C127—21no—
79Dic254—22no—
80Dic254—22no—
81Dic254—41no—
82Dic254—41no—
83C2×Dic127—22no—
84C2×Dic127—41no—
85C2×Dic127—41no—
86C127⋊C8—22no—
87C127⋊C8—22no—
88C127⋊C8C2×D412yes—
89C127⋊C8C2×D412yes—
90C23×C127—21no—
91C23×C127#4111yescharacteristic
92Q8×C127—21no—
93Q8×C127—22no—
94Q8×C127—22no—
95Q8×C127—41no—
96Q8×C127—41no—
97Q8×C127C2×D412yes—
98D4×C127—22no—
99D4×C127—22no—
100D4×C127—21no—
101D4×C127C2×D412yes—
102C2×C4×C127—21no—
103C2×C4×C127—22no—
104C2×C4×C127—22no—
105C2×C4×C127—22no—
106C2×C4×C127—41no—
107C2×C4×C127—41no—
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127#4111yescharacteristic
113C2×C4×C127C2411yescharacteristic
114C8×C127—22no—
115C22×C127—22no—
116C22×C127—22no—
117C22×C127—42no—
118C22×C127#12111yescharacteristic
119C22×C127#12111yescharacteristic
120C22×C127C2×#4111yescharacteristic
121C4×C127—22no—
122C4×C127—42no—
123C4×C127#12111yescharacteristic
124C4×C127#12111yescharacteristic
125C4×C127C22×D411yescharacteristic
126C4×C127C22×D411yescharacteristic
127Dic127—42no—
128C2×C127—21no—
129C2×C127—21no—
130C2×C127—42no—
131C2×C127#47312yescomplemented
132C2×C127C2×#12111yescharacteristic
133C2×#473—1271nomaximal, Sylow
134C127C2×#47311yescharacteristic, complemented, Sylow
135C2×#137—1271no—
136C2×#137D12711yescharacteristic
137C22×Q16—1271no—
138C2×#131—1271no—
139#473—1278no—
140C2×#108—1271no—
141C2×#101—1271no—
142C2×#101—1271no—
143C2×#41C2×D12711yescharacteristic
144C22×Q8—1271no—
145C22×Q8C2×D12711yescharacteristic
146#137—22no—
147#137—1274no—
148#137—2542no—
149#137C2×D12714yes—
150C2×Q16—21no—
151C2×Q16—1274no—
152C2×Q16—2541no—
153C2×Q16—2541no—
154C2×Q16—2542no—
155C2×Q16—2541no—
156C2×SD16—21no—
157C2×SD16—2541no—
158#131—1274no—
159#131—2542no—
160C2×M16—1271no—
161C2×M16—1271no—
162C2×M16—2541no—
163C2×M16C2×D12711yescharacteristic
164C22×C8—1271no—
165C22×C8—2541no—
166#108—1274no—
167#101—1274no—
168#101—1274no—
169#41—21no—
170#41—21no—
171#41C22⋊D12712yes—
172#41C22×D12712yes—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—1272no—
179C2×Q8—2541no—
180C2×Q8—2541no—
181C2×Q8—2541no—
182C2×Q8—2541no—
183C2×Q8—2541no—
184C2×Q8C22⋊D12711yescharacteristic
185C2×Q8C22×D12712yes—
186C2×D4—21no—
187C2×D4C22⋊D12711yescharacteristic
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—2541no—
191C22×C4C22×D12711yescharacteristic
192Q16—24no—
193Q16—2544no—
194Q16—2544no—
195Q16—2544no—
196SD16—24no—
197SD16—2544no—
198M16—21no—
199M16—1272no—
200M16—1272no—
201M16—2544no—
202M16—2541no—
203M16—2541no—
204M16C22×D12712yes—
205C2×C8—21no—
206C2×C8—1271no—
207C2×C8—1271no—
208C2×C8—1272no—
209C2×C8—1272no—
210C2×C8—2544no—
211C2×C8—2541no—
212C2×C8—2541no—
213C23—21no—
214C23#1452324711yescharacteristic
215Q8—21no—
216Q8—22no—
217Q8—22no—
218Q8—41no—
219Q8—41no—
220Q8—5081no—
221Q8—5081no—
222Q8—2542no—
223Q8—2542no—
224Q8C2×C22⋊D12712yes—
225D4—22no—
226D4—22no—
227D4—21no—
228D4C2×C22⋊D12712yes—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—5081no—
236C2×C4—5081no—
237C2×C4—2542no—
238C2×C4D4×D12711yescharacteristic
239C2×C4D4×D12711yescharacteristic
240C2×C4C2×C22⋊D12711yescharacteristic
241C2×C4C2×C22⋊D12711yescharacteristic
242C2×C4#1452324711yescharacteristic
243C2×C4C23×D12711yescharacteristic
244C8—22no—
245C8—1272no—
246C8—1272no—
247C8—2542no—
248C8—2542no—
249C22—22no—
250C22—22no—
251C22—42no—
252C22#1686118111yescharacteristic
253C22#1686118111yescharacteristic
254C22C2×#1452324711yescharacteristic
255C4—22no—
256C4—42no—
257C4—5082no—
258C4#1686118111yescharacteristic
259C4#1686118111yescharacteristic
260C4C2×D4×D12711yescharacteristic
261C4C22×C22⋊D12711yescharacteristic
262C2—21no—
263C2—21no—
264C2—42no—
265C2#2255014912yesdirect product factor
266C2C2×#1686118111yescharacteristic
subgroupG/H
centerC22#16861181
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#137×C127C2
socleC22×C127#121
prime radicalC2×#41×C127C22
hypercenterC2×#137D127
hypocenterC127C2×#473
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#14523238C2semidirect
4. #22550149C2direct (N × H)
5. C2×#16861144C2semidirect
6. C2×#16861128C2semidirect
7. C2×#16861127C2semidirect
16. C2×#14523238C22semidirect
25. #16861144C22semidirect
26. #16861128C22semidirect
27. #16861127C22semidirect
131. C2×C127#473semidirect
134. C127C2×#473semidirect
265. C2#22550149direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612392
C22×#145232382
C2×#168612162
#225501492
C2×#168611442
C2×#168611282
C2×#168611272
C2×#137×C1272
C2×#473127