Finite Group Databasebeta

← #30639249

Subgroups — #30639249

Order 16256.

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

normal subgroups only

This group has 312 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
1#22550642C211yesmaximal, characteristic
2#22550450C211yesmaximal, characteristic
3#22549885C211yesmaximal, characteristic, complemented
4#22549849C211yesmaximal, characteristic
5#22549633C211yesmaximal, characteristic, complemented
6#22549632C211yesmaximal, characteristic, complemented
7#419×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861217—22no—
10#16861215—22no—
11C2×#14523232C2211yescharacteristic
12M16×D127—22no—
13#16861207C412yescomplemented
14#16861198C412yescomplemented
15#16861192—22no—
16#16861185—22no—
17#16861168C2211yescharacteristic
18#16861168C2211yescharacteristic
19#16861133—22no—
20#16861123—22no—
21C22×Q8×C127C2211yescharacteristic
22C2×M16×C127C2211yescharacteristic
23#117×C127C2211yescharacteristic
24#103×C127—22no—
25#14523248—21no—
26#14523248—21no—
27#14523248—21no—
28#14523248—21no—
29C2×Dic254C2×C411yescharacteristic
30C2×D508C2×C411yescharacteristic
31C2×C22⋊D127D412yes—
32C2×C4×D127D412yes—
33#14523232—21no—
34#14523232—21no—
35#14523231—24no—
36C2×C127⋊C8D412yes—
37C8×D127—24no—
38C4⋊Dic127—21no—
39C4⋊Dic127—21no—
40Dic127⋊C4—24no—
41C22⋊Dic127—21no—
42C22⋊Dic127—21no—
43(C2×D127)⋊C4—24no—
44C4×Dic127—21no—
45C4×Dic127C2×C412yes—
46C2×Q8×C127—21no—
47C2×Q8×C127—21no—
48C2×Q8×C127—21no—
49C2×Q8×C127—21no—
50C2×Q8×C127—21no—
51C2×Q8×C127C2×C412yes—
52C22×C4×C127—21no—
53C22×C4×C127C2311yescharacteristic
54M16×C127—21no—
55M16×C127—21no—
56C2×C8×C127D412yes—
57C4⋊C4×C127—21no—
58C22⋊C4×C127—21no—
59C42×C127—21no—
60C22×D127C22⋊C412yes—
61Dic254—21no—
62Dic254—21no—
63D508—21no—
64D508—21no—
65C22⋊D127—22no—
66C22⋊D127—22no—
67C2×Dic127—22no—
68C2×Dic127—41no—
69C2×Dic127C22⋊C412yes—
70C4×D127—22no—
71C4×D127—22no—
72C127⋊C8—22no—
73C23×C127C2×D411yescharacteristic
74Q8×C127—22no—
75Q8×C127—22no—
76Q8×C127—41no—
77Q8×C127—41no—
78C2×C4×C127—22no—
79C2×C4×C127—41no—
80C2×C4×C127—41no—
81C2×C4×C127—41no—
82C2×C4×C127C22×C411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127#4111yescharacteristic
87C2×C4×C127#4111yescharacteristic
88C8×C127—22no—
89C22×C127—41no—
90C22×C127C2×C22⋊C411yescharacteristic
91C22×C127#12111yescharacteristic
92C22×C127#12111yescharacteristic
93C2×D127—22no—
94C2×D127—42no—
95C4×C127—42no—
96C4×C127—81no—
97C4×C127C4×D411yescharacteristic
98C4×C127C4×D411yescharacteristic
99C4×C127#12011yescharacteristic
100C4×C127#12311yescharacteristic
101Dic127—22no—
102Dic127—42no—
103Dic127—81no—
104C2×C127—21no—
105C2×C127—21no—
106C2×C127—21no—
107C2×C127#38611yescharacteristic
108D127—42no—
109#1909—1271nomaximal, Sylow
110C127#190911yescharacteristic, complemented, Sylow
111#568—1271no—
112#536—1271no—
113#419—1271no—
114#419D12711yescharacteristic, complemented
115#410—1271no—
116H(3,ℤ/4)—1271no—
117H(3,ℤ/4)—1271no—
118C2×#41—1271no—
119C22×Q8C2×D12711yescharacteristic
120#131—2542no—
121#131—2542no—
122C2×M16—1271no—
123C2×M16—2542no—
124C2×M16C2×D12711yescharacteristic
125C4⋊Q8—1272no—
126#124—1272no—
127#123—2542no—
128C22⋊Q8—2542no—
129#117—1271no—
130#117—1271no—
131#117C2×D12711yescharacteristic
132#103—22no—
133#103—2542no—
134#99—2542no—
135#41—2541no—
136#41—2541no—
137#41—2541no—
138#41—2541no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—1271no—
145C2×Q8C4×D12712yes—
146C2×D4—1272no—
147C2×D4—1271no—
148C22×C4—21no—
149C22×C4—1272no—
150C22×C4C22×D12711yescharacteristic
151M16—21no—
152M16—21no—
153M16—2544no—
154M16—2541no—
155M16—2541no—
156C2×C8—1272no—
157C2×C8—2544no—
158C2×C8C22⋊D12712yes—
159C4⋊C4—21no—
160C4⋊C4—2544no—
161C4⋊C4—2541no—
162C4⋊C4—2541no—
163C22⋊C4—21no—
164C22⋊C4—2544no—
165C22⋊C4—2541no—
166C22⋊C4—2541no—
167C42—21no—
168C42—1272no—
169C42—2541no—
170C23—1272no—
171C23D4×D12711yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—2541no—
177Q8—2541no—
178D4—2542no—
179D4—2542no—
180D4—2541no—
181D4—2541no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—1272no—
187C2×C4—5081no—
188C2×C4—2542no—
189C2×C4—2542no—
190C2×C4—2542no—
191C2×C4C2×C4×D12711yescharacteristic
192C2×C4D4×D12711yescharacteristic
193C2×C4D4×D12711yescharacteristic
194C2×C4C2×C22⋊D12711yescharacteristic
195C2×C4#1452324711yescharacteristic
196C2×C4#1452324811yescharacteristic
197C8—22no—
198C8—2542no—
199C22—41no—
200C22—5082no—
201C22—2542no—
202C22C22⋊C4×D12711yescharacteristic
203C22#1686118011yescharacteristic
204C22#1686118211yescharacteristic
205C4—42no—
206C4—81no—
207C4—10161no—
208C4—5082no—
209C4—2542no—
210C4(C22⋊D127)⋊C411yescharacteristic
211C4C4×C22⋊D12711yescharacteristic
212C4#1686117711yescharacteristic
213C4#1686119111yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—5082no—
218C2#2254975211yescharacteristic
subgroupG/H
centerC2#22549752
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#419×C127C2
socleC2×C127#386
prime radicalC2×#14523248C22
cosocleC22×C4×C127C23
hypercenter#419D127
hypocenterC127#1909
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #22549885C2semidirect
5. #22549633C2semidirect
6. #22549632C2semidirect
7. #419×C127C2semidirect
13. #16861207C4semidirect
14. #16861198C4semidirect
110. C127#1909semidirect
114. #419D127semidirect

Maximal subgroups

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

subgroupindex
#225506422
#225504502
#225498852
#225498492
#225496332
#225496322
#419×C1272
#1909127