Finite Group Databasebeta

← #30638969

Subgroups — #30638969

Order 16256.

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

normal subgroups only

This group has 262 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#22549924C211yesmaximal, characteristic, complemented
3#22549892C211yesmaximal, characteristic
4C2×#16861137C211yesmaximal, characteristic
5#22549647C211yesmaximal, characteristic, complemented
6#22549636C211yesmaximal, characteristic, complemented
7#423×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861217—22no—
10#16861215—22no—
11C2×#14523232C2211yescharacteristic
12C2×#14523232C2211yescharacteristic
13M16×D127—22no—
14#16861168C2211yescharacteristic
15#16861144—22no—
16#16861137C412yescomplemented
17#16861137C412yescomplemented
18#16861135—22no—
19C2×M16×C127C2211yescharacteristic
20C2×M16×C127C2211yescharacteristic
21C2×C42×C127C2211yescharacteristic
22C4⋊C8×C127—22no—
23#14523248—21no—
24#14523248—21no—
25#14523248C2×C412yes—
26#14523248D412yes—
27C2×Dic254Q811yescharacteristic
28C2×D508Q811yescharacteristic
29C2×C22⋊D127—21no—
30C2×C4×D127—21no—
31#14523232—21no—
32#14523232—21no—
33#14523232C2×C412yes—
34#14523232D412yes—
35#14523231—24no—
36C2×C127⋊C8—21no—
37C2×C127⋊C8—21no—
38C8×D127—24no—
39C4⋊Dic127—21no—
40C22⋊Dic127—21no—
41C4×Dic127—21no—
42C22×C4×C127—21no—
43C22×C4×C127C2311yescharacteristic
44M16×C127—21no—
45M16×C127—21no—
46M16×C127—21no—
47M16×C127—21no—
48C2×C8×C127—21no—
49C2×C8×C127D412yes—
50C42×C127—21no—
51C42×C127C2×C411yescharacteristic
52C42×C127C2×C411yescharacteristic
53C22×D127—21no—
54Dic254—21no—
55Dic254C4⋊C412yes—
56D508—21no—
57D508C4⋊C412yes—
58C22⋊D127—22no—
59C22⋊D127—22no—
60C2×Dic127—21no—
61C2×Dic127—41no—
62C4×D127—22no—
63C4×D127—22no—
64C127⋊C8—22no—
65C127⋊C8—22no—
66C23×C127#4111yescharacteristic
67C2×C4×C127—21no—
68C2×C4×C127—21no—
69C2×C4×C127—42no—
70C2×C4×C127C22×C411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127C2×D411yescharacteristic
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127C2×Q811yescharacteristic
75C2×C4×C127#4111yescharacteristic
76C8×C127—22no—
77C8×C127—41no—
78C22×C127—41no—
79C22×C127C4×D411yescharacteristic
80C22×C127C22⋊Q811yescharacteristic
81C22×C127#12311yescharacteristic
82C2×D127—22no—
83C2×D127—81no—
84C4×C127—22no—
85C4×C127—41no—
86C4×C127C2×C4⋊C411yescharacteristic
87C4×C127C4×D411yescharacteristic
88C4×C127#12011yescharacteristic
89C4×C127C22⋊Q811yescharacteristic
90Dic127—22no—
91Dic127—81no—
92C2×C127—21no—
93C2×C127—21no—
94C2×C127—21no—
95C2×C127#38511yescharacteristic
96D127—42no—
97#1861—1271nomaximal, Sylow
98C127#186111yescharacteristic, complemented, Sylow
99#568—1271no—
100#430—1271no—
101#423D12711yescharacteristic, complemented
102#421—1271no—
103C2×#104—1271no—
104#344—1271no—
105#339—1271no—
106C2×#41—1271no—
107#131—2542no—
108#131—2542no—
109C2×M16—1271no—
110C2×M16—1271no—
111C2×M16—2542no—
112C2×M16C2×D12711yescharacteristic
113C2×M16C2×D12711yescharacteristic
114#117—1271no—
115C2×C42C2×D12711yescharacteristic
116#108—2542no—
117C4⋊C8—22no—
118#104—1272no—
119#104—1272no—
120#104—2542no—
121#41—1272no—
122#41—1272no—
123#41—2541no—
124#41—2541no—
125C2×Q8—1271no—
126C2×D4—1271no—
127C2×D4—2541no—
128C22×C4—21no—
129C22×C4—2541no—
130C22×C4C22×D12711yescharacteristic
131M16—21no—
132M16—21no—
133M16—21no—
134M16—21no—
135M16—1272no—
136M16—1272no—
137M16—2544no—
138M16—2541no—
139M16—2541no—
140C2×C8—21no—
141C2×C8—2544no—
142C2×C8—2541no—
143C2×C8—2541no—
144C2×C8C22⋊D12712yes—
145C4⋊C4—2541no—
146C22⋊C4—2541no—
147C42—21no—
148C42—2541no—
149C42C4×D12711yescharacteristic
150C42C4×D12711yescharacteristic
151C23—2541no—
152C23#1452324811yescharacteristic
153Q8—1272no—
154Q8—2541no—
155D4—1272no—
156D4—2542no—
157D4—2542no—
158D4—2541no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—5081no—
163C2×C4—2541no—
164C2×C4—2542no—
165C2×C4—2542no—
166C2×C4C2×C4×D12711yescharacteristic
167C2×C4D4×D12711yescharacteristic
168C2×C4D4×D12711yescharacteristic
169C2×C4C2×C22⋊D12711yescharacteristic
170C2×C4Q8×D12711yescharacteristic
171C2×C4#1452324911yescharacteristic
172C8—22no—
173C8—41no—
174C8—2542no—
175C8—2542no—
176C22—41no—
177C22—10161no—
178C22—2542no—
179C22C4×C22⋊D12711yescharacteristic
180C22#1686118711yescharacteristic
181C22#1686119411yescharacteristic
182C4—22no—
183C4—41no—
184C4—10161no—
185C4—2542no—
186C4C4⋊C4×D12711yescharacteristic
187C4#1686117111yescharacteristic
188C4#1686117711yescharacteristic
189C4#1686118511yescharacteristic
190C2—21no—
191C2—21no—
192C2—21no—
193C2—5082no—
194C2#2254974711yescharacteristic
subgroupG/H
centerC4#16861177
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#423×C127C2
socleC2×C127#385
prime radicalC2×#14523248C22
cosocleC22×C4×C127C23
hypercenter#423D127
hypocenterC127#1861
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #423

Complements

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

NH ≅ G/Ntype
2. #22549924C2semidirect
5. #22549647C2semidirect
6. #22549636C2semidirect
7. #423×C127C2semidirect
16. #16861137C4semidirect
17. #16861137C4semidirect
98. C127#1861semidirect
101. #423D127semidirect

Maximal subgroups

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

subgroupindex
#225506422
#225499242
#225498922
C2×#168611372
#225496472
#225496362
#423×C1272
#1861127