Finite Group Databasebeta

← #30639899

Subgroups — #30639899

Order 16256.

244 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 67 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#22549894C211yesmaximal, characteristic
2C2×#16861137C211yesmaximal, characteristic
3#22549849C211yesmaximal, characteristic
4#22549646C211yesmaximal, characteristic, complemented
5#22549636C211yesmaximal, characteristic, complemented
6#22549632C211yesmaximal, characteristic, complemented
7#533×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C2×#14523232C2211yescharacteristic
11#16861168C2211yescharacteristic
12#16861137—21no—
13#16861137—21no—
14#16861137—22no—
15#16861123—22no—
16C4⋊Q8×C127—21no—
17#125×C127—21no—
18C22⋊Q8×C127—22no—
19C4×Q8×C127—22no—
20#117×C127C2211yescharacteristic
21#117×C127C2211yescharacteristic
22C2×C42×C127C2211yescharacteristic
23#14523248—21no—
24#14523248—21no—
25#14523248—21no—
26#14523248—21no—
27C2×Dic254Q811yescharacteristic, complemented
28C2×D508Q811yescharacteristic, complemented
29C2×C22⋊D127—21no—
30C2×C4×D127—21no—
31#14523232—21no—
32#14523232—21no—
33#14523232—21no—
34#14523232—21no—
35C2×C127⋊C8—21no—
36C2×C127⋊C8—21no—
37C4⋊Dic127—21no—
38C22⋊Dic127—21no—
39C4×Dic127—21no—
40C2×Q8×C127—22no—
41C22×C4×C127—21no—
42C22×C4×C127C2311yescharacteristic
43C4⋊C4×C127—24no—
44C4⋊C4×C127—22no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127—21no—
47C22⋊C4×C127—21no—
48C22⋊C4×C127D412yes—
49C42×C127—21no—
50C42×C127—21no—
51C42×C127D411yescharacteristic
52C42×C127D411yescharacteristic
53C42×C127D412yes—
54C22×D127—21no—
55Dic254—21no—
56Dic254—21no—
57D508—21no—
58D508—21no—
59C22⋊D127—41no—
60C22⋊D127—41no—
61C2×Dic127—21no—
62C2×Dic127—41no—
63C4×D127—41no—
64C4×D127—41no—
65C127⋊C8—41no—
66C127⋊C8—41no—
67C23×C127C2×D411yescharacteristic
68Q8×C127—44no—
69C2×C4×C127—21no—
70C2×C4×C127—21no—
71C2×C4×C127—22no—
72C2×C4×C127—22no—
73C2×C4×C127—42no—
74C2×C4×C127C2×D411yescharacteristic
75C2×C4×C127C2×D411yescharacteristic
76C2×C4×C127C2×Q811yescharacteristic
77C2×C4×C127#4111yescharacteristic
78C2×C4×C127#4111yescharacteristic
79C2×C4×C127#4111yescharacteristic
80C22×C127—41no—
81C22×C127#12011yescharacteristic
82C22×C127C22⋊Q811yescharacteristic
83C22×C127#12411yescharacteristic
84C2×D127—41no—
85C2×D127—81no—
86C4×C127—22no—
87C4×C127—41no—
88C4×C127—42no—
89C4×C127—42no—
90C4×C127C22⋊Q811yescharacteristic
91C4×C127C22⋊Q811yescharacteristic
92C4×C127#12411yescharacteristic
93C4×C127#12411yescharacteristic
94Dic127—41no—
95Dic127—81no—
96C2×C127—21no—
97C2×C127—21no—
98C2×C127—21no—
99C2×C127#39311yescharacteristic
100D127—81no—
101#2031—1271nomaximal, Sylow
102C127#203111yescharacteristic, complemented, Sylow
103#533D12711yescharacteristic, complemented
104#421—1271no—
105C2×#104—1271no—
106#410—1271no—
107#344—1271no—
108#339—1271no—
109H(3,ℤ/4)—1271no—
110C2×#41—1271no—
111C2×M16—1271no—
112C2×M16—1271no—
113C4⋊Q8—21no—
114#125—21no—
115C22⋊Q8—22no—
116C4×Q8—22no—
117#117—1271no—
118#117C2×D12711yescharacteristic
119#117C2×D12711yescharacteristic
120C2×C42C2×D12711yescharacteristic
121#104—2541no—
122#104—2541no—
123#104—2542no—
124#99—2542no—
125#41—2541no—
126#41—2541no—
127#41—2541no—
128#41—2541no—
129C2×Q8—22no—
130C2×Q8—1271no—
131C2×D4—1271no—
132C2×D4—2541no—
133C22×C4—21no—
134C22×C4—2541no—
135C22×C4C22×D12711yescharacteristic
136M16—2541no—
137M16—2541no—
138M16—2541no—
139M16—2541no—
140C2×C8—2541no—
141C2×C8—2541no—
142C4⋊C4—24no—
143C4⋊C4—22no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—2541no—
147C22⋊C4—21no—
148C22⋊C4—2541no—
149C22⋊C4C22⋊D12712yes—
150C42—21no—
151C42—21no—
152C42—2541no—
153C42C22⋊D12711yescharacteristic
154C42C22⋊D12711yescharacteristic
155C42C22⋊D12712yes—
156C23—2541no—
157C23C2×C22⋊D12711yescharacteristic
158Q8—44no—
159Q8—2541no—
160Q8—2541no—
161D4—5081no—
162D4—5081no—
163D4—2541no—
164D4—2541no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—42no—
170C2×C4—5081no—
171C2×C4—5081no—
172C2×C4—5081no—
173C2×C4—2541no—
174C2×C4C2×C22⋊D12711yescharacteristic
175C2×C4C2×C22⋊D12711yescharacteristic
176C2×C4Q8×D12711yescharacteristic
177C2×C4#1452324911yescharacteristic
178C2×C4#1452324911yescharacteristic
179C2×C4#1452324911yescharacteristic
180C8—5081no—
181C8—5081no—
182C22—41no—
183C22—10161no—
184C22—5081no—
185C22#1686117811yescharacteristic
186C22#1686118511yescharacteristic
187C22#1686119811yescharacteristic
188C4—22no—
189C4—41no—
190C4—42no—
191C4—42no—
192C4—10161no—
193C4—5081no—
194C4#1686118511yescharacteristic
195C4#1686118511yescharacteristic
196C4#1686119811yescharacteristic
197C4#1686119811yescharacteristic
198C2—21no—
199C2—21no—
200C2—21no—
201C2—10161no—
202C2#2254978711yescharacteristic
subgroupG/H
centerC4#16861198
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#533×C127C2
socleC2×C127#393
prime radicalC2×#14523248C22
hypercenter#533D127
hypocenterC127#2031
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #533

Complements

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

NH ≅ G/Ntype
4. #22549646C2semidirect
5. #22549636C2semidirect
6. #22549632C2semidirect
7. #533×C127C2semidirect
27. C2×Dic254Q8semidirect
28. C2×D508Q8semidirect
102. C127#2031semidirect
103. #533D127semidirect

Maximal subgroups

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

subgroupindex
#225498942
C2×#168611372
#225498492
#225496462
#225496362
#225496322
#533×C1272
#2031127