Finite Group Databasebeta

← #30650180

Subgroups — #30650180

Order 16256.

322 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 117 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#22550488C211yesmaximal, characteristic
2#22550183C212yesmaximal, complemented
3#22550180C212yesmaximal, complemented
4#22550132C212yesmaximal
5#22550115C212yesmaximal
6#22550086C212yesmaximal, complemented
7#22550083C212yesmaximal, complemented
8#22549903C211yesmaximal, characteristic
9#541×C127C211yesmaximal, characteristic
10C2×#14523238C2212yes—
11C2×#14523238C2212yes—
12C2×#14523232C2211yescharacteristic
13C22×C127⋊C8C2211yescharacteristic
14#16861209C2211yescharacteristic
15#16861202—22no—
16#16861201C2211yescharacteristic
17#16861199—22no—
18#16861188C2212yes—
19C4×Dic254C2212yes—
20#16861142C2212yes—
21#16861140C2212yes—
22C4⋊(C127⋊C8)C2212yes—
23C4⋊(C127⋊C8)C2212yes—
24#16861133C2212yes—
25#16861133C2212yes—
26#16861132C2212yes—
27#16861132C2212yes—
28C4⋊Q8×C127C2211yescharacteristic
29#126×C127—22no—
30#125×C127—22no—
31#125×C127C2211yescharacteristic
32C22⋊Q8×C127C2212yes—
33C4×Q8×C127C2212yes—
34#117×C127C2211yescharacteristic
35C2×Dic254C2312yes—
36#14523238—42no—
37#14523238—42no—
38#14523232—41no—
39C2×C127⋊C8—41no—
40C2×C127⋊C8C2312yes—
41C2×C127⋊C8C2312yes—
42C4⋊Dic127—22no—
43C4⋊Dic127C2312yes—
44Dic127⋊C4—22no—
45Dic127⋊C4—22no—
46Dic127⋊C4—22no—
47C22⋊Dic127—22no—
48C4×Dic127—22no—
49C2×Q8×C127C2312yes—
50C22×C4×C127C2311yescharacteristic
51C4⋊C4×C127—22no—
52C4⋊C4×C127—22no—
53C4⋊C4×C127—22no—
54C4⋊C4×C127—22no—
55C4⋊C4×C127D412yes—
56C4⋊C4×C127C2312yes—
57C22⋊C4×C127—22no—
58C22⋊C4×C127D412yes—
59C42×C127—22no—
60C42×C127C2311yescharacteristic
61C42×C127C2311yescharacteristic
62Dic254—22no—
63Dic254—22no—
64C2×Dic127—22no—
65C2×Dic127—22no—
66C127⋊C8—22no—
67C127⋊C8—22no—
68C23×C127C2×D411yescharacteristic
69Q8×C127—22no—
70Q8×C127—22no—
71C2×C4×C127—21no—
72C2×C4×C127—21no—
73C2×C4×C127—22no—
74C2×C4×C127—22no—
75C2×C4×C127C2×D412yes—
76C2×C4×C127C2×D412yes—
77C2×C4×C127C2×D411yescharacteristic
78C2×C4×C127C2411yescharacteristic
79C22×C127—21no—
80C22×C127—22no—
81C22×C127C22×D411yescharacteristic
82C4×C127—21no—
83C4×C127—22no—
84C4×C127—22no—
85C4×C127—42no—
86C4×C127—42no—
87C4×C127H(5,𝔽2)11yescharacteristic
88C4×C1272−1+411yescharacteristic
89Dic127—42no—
90Dic127—42no—
91C2×C127—41no—
92C2×C127#53711yescharacteristic
93C2×C127#57812yes—
94#3261—1271nomaximal, Sylow
95C127#326111yescharacteristic, complemented, Sylow
96#541—1271no—
97#541D12711yescharacteristic
98#479—1272no—
99#479—1272no—
100#470—1272no—
101#467—1272no—
102#462—1272no—
103#462—1272no—
104#424—1271no—
105C2×Q16—1272no—
106C2×Q16—1272no—
107C2×M16—1271no—
108C22×C8—1271no—
109C4⋊Q8—1271no—
110C4⋊Q8C2×D12711yescharacteristic
111#126—22no—
112#126—2542no—
113#125—22no—
114#125—1271no—
115#125—2542no—
116#125C2×D12711yescharacteristic
117C22⋊Q8—1272no—
118C22⋊Q8C2×D12712yes—
119C4×Q8—1272no—
120C4×Q8C2×D12712yes—
121#117C2×D12711yescharacteristic
122#107—1272no—
123#106—1272no—
124C4⋊C8—1272no—
125C4⋊C8—1272no—
126#103—1272no—
127#103—1272no—
128#103—1272no—
129#103—1272no—
130C2×Q8—1272no—
131C2×Q8C22×D12712yes—
132C22×C4C22×D12711yescharacteristic
133Q16—5082no—
134Q16—5082no—
135M16—5081no—
136C2×C8—1272no—
137C2×C8—1272no—
138C2×C8—5081no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—1272no—
144C4⋊C4—2542no—
145C4⋊C4—2542no—
146C4⋊C4—2542no—
147C4⋊C4—2542no—
148C4⋊C4C22⋊D12712yes—
149C4⋊C4C22×D12712yes—
150C22⋊C4—22no—
151C22⋊C4—2542no—
152C22⋊C4C22⋊D12712yes—
153C42—22no—
154C42—2542no—
155C42C22×D12711yescharacteristic
156C42C22×D12711yescharacteristic
157C23C2×C22⋊D12711yescharacteristic
158Q8—22no—
159Q8—22no—
160Q8—2542no—
161Q8—2542no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—2542no—
167C2×C4—2542no—
168C2×C4C2×C22⋊D12712yes—
169C2×C4C2×C22⋊D12712yes—
170C2×C4C2×C22⋊D12711yescharacteristic
171C2×C4C23×D12711yescharacteristic
172C8—2542no—
173C8—2542no—
174C22—22no—
175C22—21no—
176C22C22×C22⋊D12711yescharacteristic
177C4—22no—
178C4—22no—
179C4—21no—
180C4—42no—
181C4—42no—
182C4—5082no—
183C4—5082no—
184C4#1686125511yescharacteristic
185C4#1686125611yescharacteristic
186C2—41no—
187C2#2255045411yescharacteristic
188C2#2255069112yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#541×C127C2
socleC22×C127C22×D4
prime radicalC23×C127C2×D4
hypercenter#541D127
hypocenterC127#3261
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #22550183C2semidirect
3. #22550180C2semidirect
6. #22550086C2semidirect
7. #22550083C2semidirect
95. C127#3261semidirect

Maximal subgroups

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

subgroupindex
#225504882
#225501832
#225501802
#225501322
#225501152
#225500862
#225500832
#225499032
#541×C1272
#3261127