Finite Group Databasebeta

← #30650120

Subgroups — #30650120

Order 16256.

342 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 113 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#22550428C212yesmaximal
2#22550160C212yesmaximal, complemented
3#22550159C212yesmaximal, complemented
4#22550134C211yesmaximal, characteristic
5#22550132C212yesmaximal
6#22550085C212yesmaximal, complemented
7#22550084C212yesmaximal, complemented
8#22549899C211yesmaximal, characteristic
9#470×C127C211yesmaximal, characteristic
10C2×Dic508C2211yescharacteristic
11C2×#14523238C2212yes—
12C2×#14523232C2211yescharacteristic
13#16861208C2212yes—
14#16861207C2212yes—
15#16861199—22no—
16#16861188C2212yes—
17#16861186—22no—
18Q8×Dic127C2212yes—
19#16861174—22no—
20#16861174C2212yes—
21#16861165—22no—
22C2×C4×Dic127C2211yescharacteristic
23#16861141C2211yescharacteristic
24#16861140C2212yes—
25Dic127⋊C8C2212yes—
26Dic127⋊C8C2212yes—
27#16861134C2212yes—
28#16861133C2212yes—
29#16861132C2212yes—
30C2×Q16×C127C2211yescharacteristic
31C2×M16×C127C2211yescharacteristic
32C22⋊Q8×C127C2212yes—
33#106×C127C2211yescharacteristic
34#103×C127C2212yes—
35C2×Dic254—22no—
36C2×Dic254C2312yes—
37C22×Dic127D412yes—
38Dic508—41no—
39#14523238—42no—
40#14523232—41no—
41C2×C127⋊C8C2312yes—
42C4⋊Dic127—22no—
43C4⋊Dic127C2312yes—
44Dic127⋊C4—22no—
45Dic127⋊C4—22no—
46Dic127⋊C4—22no—
47Dic127⋊C4—22no—
48Dic127⋊C4—22no—
49C22⋊Dic127—22no—
50C4×Dic127—22no—
51C4×Dic127—22no—
52C4×Dic127D411yescharacteristic
53C4×Dic127D411yescharacteristic
54C4×Dic127C2312yes—
55C2×Q8×C127C2312yes—
56C22×C4×C127C2311yescharacteristic
57Q16×C127—41no—
58M16×C127—41no—
59C2×C8×C127C2311yescharacteristic
60C2×C8×C127C2311yescharacteristic
61C4⋊C4×C127—22no—
62C4⋊C4×C127C2312yes—
63C22⋊C4×C127—22no—
64Dic254—22no—
65Dic254—22no—
66Dic254—44no—
67C2×Dic127—24no—
68C2×Dic127—22no—
69C2×Dic127—22no—
70C2×Dic127C2×D412yes—
71C2×Dic127C2×D412yes—
72C127⋊C8—22no—
73C23×C127C2×D411yescharacteristic
74Q8×C127—22no—
75Q8×C127—22no—
76C2×C4×C127—21no—
77C2×C4×C127—21no—
78C2×C4×C127—22no—
79C2×C4×C127—22no—
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2411yescharacteristic
82C8×C127—21no—
83C8×C127—21no—
84C22×C127—21no—
85C22×C127—22no—
86C22×C127C22×D411yescharacteristic
87C4×C127—21no—
88C4×C127—42no—
89C4×C127—42no—
90C4×C127H(5,𝔽2)11yescharacteristic
91C4×C1272−1+411yescharacteristic
92Dic127—22no—
93Dic127—42no—
94Dic127—42no—
95Dic127#13714yes—
96C2×C127—41no—
97C2×C127#53711yescharacteristic
98C2×C127C2×#13712yes—
99#3255—1271nomaximal, Sylow
100C127#325511yescharacteristic, complemented, Sylow
101#533—1272no—
102#475—1272no—
103#475—1272no—
104#470—1271no—
105#470—1272no—
106#470D12711yescharacteristic
107#462—1272no—
108#462—1272no—
109#423—1271no—
110C2×Q16—1272no—
111C2×Q16—1271no—
112C2×Q16C2×D12711yescharacteristic
113C2×M16—1271no—
114C2×M16C2×D12711yescharacteristic
115C4⋊Q8—1272no—
116C4⋊Q8—1272no—
117#125—2542no—
118C22⋊Q8—1272no—
119C22⋊Q8—2542no—
120C22⋊Q8C2×D12712yes—
121C4×Q8—1272no—
122C4×Q8—1272no—
123C4×Q8—2542no—
124#117—2542no—
125C2×C42—1271no—
126#106—1271no—
127#106—1272no—
128#106C2×D12711yescharacteristic
129C4⋊C8—1272no—
130C4⋊C8—1272no—
131#103—1272no—
132#103—1272no—
133#103—1272no—
134#103C2×D12712yes—
135C2×Q8—1272no—
136C2×Q8—2542no—
137C2×Q8C22×D12712yes—
138C22×C4—1272no—
139C22×C4C22×D12711yescharacteristic
140Q16—41no—
141Q16—5081no—
142Q16—5082no—
143M16—41no—
144M16—5081no—
145C2×C8—1272no—
146C2×C8C22×D12711yescharacteristic
147C2×C8C22×D12711yescharacteristic
148C4⋊C4—22no—
149C4⋊C4—1272no—
150C4⋊C4—2542no—
151C4⋊C4—2542no—
152C4⋊C4—2542no—
153C4⋊C4—2542no—
154C4⋊C4—2542no—
155C4⋊C4—2542no—
156C4⋊C4C22×D12712yes—
157C22⋊C4—22no—
158C22⋊C4—2542no—
159C42—1271no—
160C42—1271no—
161C42—1272no—
162C42—2542no—
163C42—2542no—
164C23D4×D12711yescharacteristic
165Q8—22no—
166Q8—22no—
167Q8—5084no—
168Q8—2542no—
169Q8—2542no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—1272no—
175C2×C4—1272no—
176C2×C4—2544no—
177C2×C4—2542no—
178C2×C4—2542no—
179C2×C4D4×D12711yescharacteristic
180C2×C4C23×D12711yescharacteristic
181C8—21no—
182C8—21no—
183C8—2542no—
184C22—22no—
185C22—21no—
186C22C2×D4×D12711yescharacteristic
187C4—21no—
188C4—42no—
189C4—42no—
190C4—1274no—
191C4—5082no—
192C4—5082no—
193C4—2542no—
194C4#1686125411yescharacteristic
195C4#1686125611yescharacteristic
196C2—41no—
197C2#2255045311yescharacteristic
198C2#137×D12712yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#470×C127C2
socleC22×C127C22×D4
prime radicalC23×C127C2×D4
hypercenter#470D127
hypocenterC127#3255
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #22550160C2semidirect
3. #22550159C2semidirect
6. #22550085C2semidirect
7. #22550084C2semidirect
100. C127#3255semidirect

Maximal subgroups

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

subgroupindex
#225504282
#225501602
#225501592
#225501342
#225501322
#225500852
#225500842
#225498992
#470×C1272
#3255127