Finite Group Databasebeta

← #30638951

Subgroups — #30638951

Order 16256.

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

normal subgroups only

This group has 252 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#22549925C211yesmaximal, characteristic, complemented
3#22549924C211yesmaximal, characteristic, complemented
4#22549858C211yesmaximal, characteristic
5#22549842C211yesmaximal, characteristic
6#22549641C211yesmaximal, characteristic, complemented
7#341×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861217—22no—
10#16861215—22no—
11C2×#14523232C2211yescharacteristic
12C2×#14523232C2211yescharacteristic
13C2×#14523232C2211yescharacteristic
14M16×D127—22no—
15#16861145—22no—
16#16861144—22no—
17#16861127C412yes—
18#16861125C412yes—
19(C2×D127)⋊C8—22no—
20C2×M16×C127C2211yescharacteristic
21C2×M16×C127C2211yescharacteristic
22C22×C8×C127C2211yescharacteristic
23#14523248—21no—
24#14523248—21no—
25#14523248—21no—
26#14523248—21no—
27C2×Dic254C2×C411yescharacteristic
28C2×D508C2×C411yescharacteristic
29C2×C22⋊D127Q812yes—
30C2×C4×D127D412yes—
31#14523232—21no—
32#14523232—21no—
33#14523232—21no—
34#14523232—21no—
35#14523232C2×C412yes—
36#14523232D412yes—
37#14523231—24no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8—21no—
40C2×C127⋊C8—21no—
41C8×D127—24no—
42C22×C4×C127C2311yescharacteristic
43M16×C127—21no—
44M16×C127—21no—
45M16×C127C2×C412yes—
46M16×C127D412yes—
47C2×C8×C127—21no—
48C2×C8×C127—21no—
49C2×C8×C127—21no—
50C2×C8×C127—21no—
51C2×C8×C127—21no—
52C22×D127C4⋊C412yes—
53Dic254—21no—
54Dic254—21no—
55D508—21no—
56D508—21no—
57C22⋊D127—22no—
58C22⋊D127—22no—
59C2×Dic127C4⋊C412yes—
60C4×D127—22no—
61C4×D127—22no—
62C127⋊C8—22no—
63C127⋊C8—22no—
64C127⋊C8—41no—
65C23×C127C2×Q811yescharacteristic
66C2×C4×C127C22×C411yescharacteristic
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127#4111yescharacteristic
71C2×C4×C127#4111yescharacteristic
72C8×C127—22no—
73C8×C127—22no—
74C8×C127—41no—
75C22×C127—41no—
76C22×C127C2×C4⋊C411yescharacteristic
77C22×C127C22⋊Q811yescharacteristic
78C22×C127C22⋊Q811yescharacteristic
79C2×D127—22no—
80C2×D127—42no—
81C4×C127C4×D411yescharacteristic
82C4×C127C4×D411yescharacteristic
83C4×C127#12011yescharacteristic
84C4×C127#12311yescharacteristic
85Dic127—22no—
86C2×C127—21no—
87C2×C127—21no—
88C2×C127—21no—
89C2×C127#38511yescharacteristic
90D127—42no—
91#1858—1271nomaximal, Sylow
92C127#185811yescharacteristic, complemented, Sylow
93#568—1271no—
94#430—1271no—
95#430—1271no—
96#413—1271no—
97#408—1271no—
98#341—1271no—
99#341D12711yescharacteristic, complemented
100C2×#41—1271no—
101#131—2542no—
102#131—2542no—
103C2×M16—1271no—
104C2×M16—1271no—
105C2×M16—1271no—
106C2×M16—2542no—
107C2×M16C2×D12711yescharacteristic
108C2×M16C2×D12711yescharacteristic
109C22×C8C2×D12711yescharacteristic
110#108—2542no—
111#108—2542no—
112#101—1272no—
113#100—1272no—
114C22⋊C8—2542no—
115#41—2541no—
116#41—2541no—
117#41—2541no—
118#41—2541no—
119C2×Q8—1271no—
120C2×D4—1272no—
121C2×D4—1271no—
122C22×C4—1272no—
123C22×C4C22×D12711yescharacteristic
124M16—21no—
125M16—21no—
126M16—1272no—
127M16—1272no—
128M16—2544no—
129M16—2541no—
130M16—2541no—
131M16—2541no—
132M16—2541no—
133M16C4×D12712yes—
134M16D50812yes—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—2544no—
141C2×C8—2541no—
142C2×C8—2541no—
143C2×C8—2541no—
144C23—1272no—
145C23Q8×D12711yescharacteristic
146Q8—2541no—
147Q8—2541no—
148D4—2542no—
149D4—2542no—
150D4—2541no—
151D4—2541no—
152C2×C4—1272no—
153C2×C4—2542no—
154C2×C4—2542no—
155C2×C4C2×C4×D12711yescharacteristic
156C2×C4D4×D12711yescharacteristic
157C2×C4D4×D12711yescharacteristic
158C2×C4C2×D50811yescharacteristic
159C2×C4#1452324711yescharacteristic
160C2×C4#1452324811yescharacteristic
161C8—22no—
162C8—22no—
163C8—41no—
164C8—5081no—
165C8—2542no—
166C8—2542no—
167C22—41no—
168C22—5082no—
169C22—2542no—
170C22C4⋊C4×D12711yescharacteristic
171C22#1686118711yescharacteristic
172C22#1686118911yescharacteristic
173C4—2542no—
174C4(C22⋊D127)⋊C411yescharacteristic
175C4C4×D50811yescharacteristic
176C4#1686117911yescharacteristic
177C4#1686119111yescharacteristic
178C2—21no—
179C2—21no—
180C2—21no—
181C2—5082no—
182C2#2254975011yescharacteristic
subgroupG/H
centerC4#16861179
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#341×C127C2
socleC2×C127#385
prime radicalC2×#14523248C22
cosocleC22×C4×C127C23
hypercenter#341D127
hypocenterC127#1858
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #341

Complements

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

NH ≅ G/Ntype
2. #22549925C2semidirect
3. #22549924C2semidirect
6. #22549641C2semidirect
7. #341×C127C2semidirect
92. C127#1858semidirect
99. #341D127semidirect

Maximal subgroups

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

subgroupindex
#225506422
#225499252
#225499242
#225498582
#225498422
#225496412
#341×C1272
#1858127