Finite Group Databasebeta

← #30650181

Subgroups — #30650181

Order 16256.

322 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 109 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#22550486C212yesmaximal
2#22550186C212yesmaximal, complemented
3#22550181C212yesmaximal, complemented
4#22550134C211yesmaximal, characteristic
5#22550115C212yesmaximal
6#22550085C212yesmaximal, complemented
7#22550084C212yesmaximal, complemented
8#22549904C211yesmaximal, characteristic
9#470×C127C211yesmaximal, characteristic
10C2×Dic508C2211yescharacteristic
11C2×#14523238C2212yes—
12C22×C127⋊C8C2211yescharacteristic
13#16861207C2212yes—
14#16861202—22no—
15#16861202—22no—
16#16861200—22no—
17#16861200C2212yes—
18#16861199—22no—
19#16861188C2212yes—
20Q8×Dic127C2212yes—
21#16861174C2212yes—
22#16861168C2211yescharacteristic
23#16861142C2212yes—
24#16861141C2211yescharacteristic
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×Dic254C2312yes—
36Dic508—41no—
37#14523238—42no—
38C2×C127⋊C8—41no—
39C2×C127⋊C8C2312yes—
40C4⋊Dic127—22no—
41C4⋊Dic127—22no—
42C4⋊Dic127D412yes—
43C4⋊Dic127C2312yes—
44Dic127⋊C4—22no—
45Dic127⋊C4—22no—
46Dic127⋊C4—22no—
47Dic127⋊C4—22no—
48Dic127⋊C4—22no—
49C22⋊Dic127—22no—
50C22⋊Dic127D411yescharacteristic
51C22⋊Dic127D411yescharacteristic
52C4×Dic127—22no—
53C4×Dic127—22no—
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—
66C2×Dic127—22no—
67C2×Dic127—22no—
68C2×Dic127C2×D412yes—
69C2×Dic127C2×D412yes—
70C127⋊C8—22no—
71C23×C127C2×D411yescharacteristic
72Q8×C127—22no—
73Q8×C127—22no—
74C2×C4×C127—21no—
75C2×C4×C127—21no—
76C2×C4×C127—22no—
77C2×C4×C127—22no—
78C2×C4×C127C2×D411yescharacteristic
79C2×C4×C127C2411yescharacteristic
80C8×C127—21no—
81C8×C127—21no—
82C22×C127—21no—
83C22×C127—22no—
84C22×C127C22×D411yescharacteristic
85C4×C127—21no—
86C4×C127—42no—
87C4×C127—42no—
88C4×C127H(5,𝔽2)11yescharacteristic
89C4×C1272−1+411yescharacteristic
90Dic127—22no—
91Dic127—22no—
92Dic127—42no—
93Dic127—42no—
94C2×C127—41no—
95C2×C127#53711yescharacteristic
96C2×C127#57812yes—
97#3261—1271nomaximal, Sylow
98C127#326111yescharacteristic, complemented, Sylow
99#541—1272no—
100#479—1272no—
101#479—1272no—
102#470—1271no—
103#470D12711yescharacteristic
104#467—1272no—
105#462—1272no—
106#462—1272no—
107#424—1271no—
108C2×Q16—1272no—
109C2×Q16—1271no—
110C2×Q16C2×D12711yescharacteristic
111C2×M16C2×D12711yescharacteristic
112C22×C8—1271no—
113C4⋊Q8—1272no—
114#126—2542no—
115#126—2542no—
116#125—1272no—
117#125—2542no—
118#125—2542no—
119C22⋊Q8—1272no—
120C22⋊Q8C2×D12712yes—
121C4×Q8—1272no—
122C4×Q8—1272no—
123#117—1271no—
124#107—1272no—
125#106—1271no—
126#106C2×D12711yescharacteristic
127C4⋊C8—1272no—
128C4⋊C8—1272no—
129#103—1272no—
130#103—1272no—
131#103—1272no—
132#103C2×D12712yes—
133C2×Q8—1272no—
134C2×Q8C22×D12712yes—
135C22×C4C22×D12711yescharacteristic
136Q16—41no—
137Q16—5081no—
138Q16—5082no—
139M16—41no—
140C2×C8—1272no—
141C2×C8—5081no—
142C2×C8C22×D12711yescharacteristic
143C2×C8C22×D12711yescharacteristic
144C4⋊C4—22no—
145C4⋊C4—1272no—
146C4⋊C4—1272no—
147C4⋊C4—2542no—
148C4⋊C4—2542no—
149C4⋊C4—2542no—
150C4⋊C4—2542no—
151C4⋊C4—2542no—
152C4⋊C4—2542no—
153C4⋊C4—2542no—
154C4⋊C4C22×D12712yes—
155C22⋊C4—22no—
156C22⋊C4—1271no—
157C22⋊C4—1271no—
158C22⋊C4—2542no—
159C42—1272no—
160C42—2542no—
161C42—2542no—
162C23D4×D12711yescharacteristic
163Q8—22no—
164Q8—22no—
165Q8—2542no—
166Q8—2542no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—1272no—
172C2×C4—1272no—
173C2×C4—2542no—
174C2×C4—2542no—
175C2×C4D4×D12711yescharacteristic
176C2×C4C23×D12711yescharacteristic
177C8—21no—
178C8—21no—
179C8—2542no—
180C22—22no—
181C22—21no—
182C22C2×D4×D12711yescharacteristic
183C4—21no—
184C4—42no—
185C4—42no—
186C4—5082no—
187C4—5082no—
188C4—2542no—
189C4—2542no—
190C4#1686125411yescharacteristic
191C4#1686125611yescharacteristic
192C2—41no—
193C2#2255045311yescharacteristic
194C2#2255069212yes—
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#3261
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. #22550186C2semidirect
3. #22550181C2semidirect
6. #22550085C2semidirect
7. #22550084C2semidirect
98. C127#3261semidirect

Maximal subgroups

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

subgroupindex
#225504862
#225501862
#225501812
#225501342
#225501152
#225500852
#225500842
#225499042
#470×C1272
#3261127