Finite Group Databasebeta

← #30639214

Subgroups — #30639214

Order 16256.

346 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 79 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×#16861234C211yesmaximal, characteristic
2#22550350C211yesmaximal, characteristic, complemented
3#22549882C211yesmaximal, characteristic, complemented
4#22549850C211yesmaximal, characteristic, complemented
5#22549648C211yesmaximal, characteristic, complemented
6#22549632C211yesmaximal, characteristic, complemented
7#421×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9#16861234—22no—
10#16861234—22no—
11C2×#14523235C412yescomplemented
12C2×#14523233C412yescomplemented
13C2×#14523232C2211yescharacteristic
14C4×D508—22no—
15#16861171—22no—
16#16861168C2211yescharacteristic
17#16861168C2211yescharacteristic
18C22⋊C4×D127—22no—
19#16861131—22no—
20#16861124—22no—
21C2×#41×C127C2211yescharacteristic
22C2×M16×C127C2211yescharacteristic
23#117×C127C2211yescharacteristic
24#104×C127—22no—
25C23×D127—21no—
26C2×D508—21no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508C2×C412yes—
30C2×D508D412yes—
31C2×D508D412yes—
32C2×C4×D127—24no—
33#14523235—24no—
34#14523233—24no—
35#14523232—21no—
36#14523232—21no—
37C2×C127⋊C8C2×C412yescomplemented
38C4⋊Dic127—21no—
39C4⋊Dic127—21no—
40C22⋊Dic127—21no—
41C22⋊Dic127D412yes—
42(C2×D127)⋊C4—24no—
43C4×Dic127—21no—
44C4×Dic127—21no—
45#41×C127—21no—
46#41×C127—21no—
47#41×C127—21no—
48#41×C127—21no—
49C2×Q8×C127C2×C411yescharacteristic, complemented
50C2×D4×C127—21no—
51C2×D4×C127C2×C411yescharacteristic, complemented
52C22×C4×C127—21no—
53C22×C4×C127C2311yescharacteristic
54M16×C127—21no—
55M16×C127—21no—
56C2×C8×C127—21no—
57C4⋊C4×C127—21no—
58C22⋊C4×C127—21no—
59C42×C127D412yes—
60C22×D127—22no—
61C22×D127—22no—
62C22×D127—22no—
63C22×D127—82no—
64D508—22no—
65D508—22no—
66D508—22no—
67D508C22⋊C414yes—
68C2×Dic127—22no—
69C2×Dic127—41no—
70C4×D127—84no—
71C127⋊C8—22no—
72C23×C127—21no—
73C23×C127C2×D411yescharacteristic
74Q8×C127—21no—
75Q8×C127—21no—
76D4×C127—21no—
77D4×C127—21no—
78D4×C127—41no—
79D4×C127—41no—
80C2×C4×C127—21no—
81C2×C4×C127—22no—
82C2×C4×C127—41no—
83C2×C4×C127—41no—
84C2×C4×C127C22×C411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127#4111yescharacteristic
89C2×C4×C127#4111yescharacteristic
90C8×C127—41no—
91C22×C127—41no—
92C22×C127—41no—
93C22×C127—81no—
94C22×C127C4×D411yescharacteristic
95C22×C127#12011yescharacteristic
96C22×C127#12111yescharacteristic
97C2×D127—24no—
98C2×D127—44no—
99C2×D127—82no—
100C2×D127—82no—
101C4×C127—42no—
102C4×C127—41no—
103C4×C127C2×C22⋊C411yescharacteristic
104C4×C127C4×D411yescharacteristic
105C4×C127#12111yescharacteristic
106C4×C127#12311yescharacteristic
107Dic127—42no—
108Dic127—81no—
109C2×C127—21no—
110C2×C127—21no—
111C2×C127—21no—
112C2×C127—81no—
113C2×C127#38611yescharacteristic
114D127—44no—
115#1904—1271nomaximal, Sylow
116C127#190411yescharacteristic, complemented, Sylow
117C2×AGL(1,ℤ/8)—1271no—
118#518—1271no—
119#421D12711yescharacteristic, complemented
120#418—1271no—
121#410—1271no—
122#344—1271no—
123H(3,ℤ/4)—1271no—
124C2×#41C2×D12711yescharacteristic
125C22×D4—1271no—
126AGL(1,ℤ/8)—2542no—
127AGL(1,ℤ/8)—2542no—
128C2×SD16—1272no—
129C2×D8—1272no—
130C2×M16—1271no—
131C2×M16C2×D12711yescharacteristic
132C4×D4—2542no—
133C4×D4—2542no—
134#117—1271no—
135#117—1271no—
136#117C2×D12711yescharacteristic
137C2×C22⋊C4—2542no—
138#104—22no—
139#102—2542no—
140#99—2542no—
141C24—2541no—
142#41—21no—
143#41—21no—
144#41—21no—
145#41—21no—
146C2×Q8C4×D12711yescharacteristic, complemented
147C2×D4—21no—
148C2×D4—1272no—
149C2×D4—1272no—
150C2×D4—1272no—
151C2×D4—2541no—
152C2×D4—2541no—
153C2×D4—2541no—
154C2×D4C4×D12711yescharacteristic, complemented
155C22×C4—21no—
156C22×C4—2544no—
157C22×C4C22×D12711yescharacteristic
158SD16—2544no—
159D8—2544no—
160M16—21no—
161M16—21no—
162M16—2541no—
163M16—2541no—
164C2×C8—21no—
165C2×C8—1272no—
166C4⋊C4—21no—
167C4⋊C4—2541no—
168C4⋊C4—2541no—
169C22⋊C4—21no—
170C22⋊C4—1272no—
171C22⋊C4—2544no—
172C22⋊C4—2541no—
173C42—2541no—
174C42—2541no—
175C42C22⋊D12712yes—
176C23—21no—
177C23—10162no—
178C23—2542no—
179C23—2542no—
180C23—2542no—
181C23D4×D12711yescharacteristic
182Q8—21no—
183Q8—21no—
184D4—21no—
185D4—21no—
186D4—41no—
187D4—41no—
188D4—1274no—
189D4—2542no—
190D4—2542no—
191D4—2542no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—10164no—
197C2×C4—5081no—
198C2×C4—2542no—
199C2×C4C2×C4×D12711yescharacteristic
200C2×C4D4×D12711yescharacteristic
201C2×C4D4×D12711yescharacteristic
202C2×C4C2×C22⋊D12711yescharacteristic
203C2×C4#1452324711yescharacteristic
204C2×C4#1452324811yescharacteristic
205C8—41no—
206C8—2542no—
207C22—41no—
208C22—41no—
209C22—81no—
210C22—10162no—
211C22—10162no—
212C22—5084no—
213C22—2544no—
214C22(C22⋊D127)⋊C411yescharacteristic
215C22#1686117711yescharacteristic
216C22#1686118211yescharacteristic
217C4—42no—
218C4—41no—
219C4—10161no—
220C4—5082no—
221C4C22⋊C4×D12711yescharacteristic
222C4C4×C22⋊D12711yescharacteristic
223C4#1686118011yescharacteristic
224C4#1686119111yescharacteristic
225C2—21no—
226C2—21no—
227C2—21no—
228C2—81no—
229C2—5084no—
230C2#2254975211yescharacteristic
subgroupG/H
centerC2#22549752
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#421×C127C2
socleC2×C127#386
prime radicalC2×#16861234C2
cosocleC22×C4×C127C23
hypercenter#421D127
hypocenterC127#1904
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #22550350C2semidirect
3. #22549882C2semidirect
4. #22549850C2semidirect
5. #22549648C2semidirect
6. #22549632C2semidirect
7. #421×C127C2semidirect
11. C2×#14523235C4semidirect
12. C2×#14523233C4semidirect
37. C2×C127⋊C8C2×C4semidirect
49. C2×Q8×C127C2×C4semidirect
51. C2×D4×C127C2×C4semidirect
116. C127#1904semidirect
119. #421D127semidirect
146. C2×Q8C4×D127semidirect
154. C2×D4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
#225503502
#225498822
#225498502
#225496482
#225496322
#421×C1272
#1904127