Finite Group Databasebeta

← #30650171

Subgroups — #30650171

Order 16256.

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

normal subgroups only

This group has 346 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#22550481C211yesmaximal, characteristic
2#22550183C212yesmaximal, complemented
3#22550173C212yesmaximal, complemented
4#22550124C212yesmaximal, complemented
5#22550107C212yesmaximal, complemented
6#22550078C212yesmaximal, complemented
7#22550076C212yesmaximal, complemented
8#22549903C211yesmaximal, characteristic, complemented
9#541×C127C211yesmaximal, characteristic, complemented
10C2×#14523235C2212yes—
11C2×#14523235C2212yes—
12C2×#14523232C2211yescharacteristic
13C22×C127⋊C8C2211yescharacteristic
14#16861206C2211yescharacteristic
15#16861201C2211yescharacteristic
16#16861194—22no—
17#16861183C2212yes—
18#16861182—22no—
19C4×D508C2212yes—
20#16861142C2212yes—
21#16861140C2212yes—
22C4⋊(C127⋊C8)C2212yes—
23C4⋊(C127⋊C8)C2212yes—
24#16861133C2212yes—
25#16861133C2212yes—
26#16861129C2212yes—
27#16861129C2212yes—
28C4⋊Q8×C127C2211yescharacteristic
29#126×C127—22no—
30#125×C127—22no—
31#125×C127C2211yescharacteristic
32C22⋊Q8×C127C2212yes—
33C4×Q8×C127C2212yes—
34#117×C127C2211yescharacteristic
35C2×D508—22no—
36C2×D508C2312yes—
37C2×C22⋊D127—22no—
38C2×C4×D127—22no—
39#14523235—42no—
40#14523235—42no—
41#14523232—41no—
42C2×C127⋊C8—41no—
43C2×C127⋊C8C2312yes—
44C2×C127⋊C8C2312yes—
45C4⋊Dic127C2312yes—
46Dic127⋊C4—22no—
47(C2×D127)⋊C4—22no—
48(C2×D127)⋊C4—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
62C22×D127—22no—
63D508—22no—
64D508—22no—
65D508—82no—
66C22⋊D127—82no—
67C2×Dic127—22no—
68C4×D127—82no—
69C127⋊C8—22no—
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×D412yes—
79C2×C4×C127C2×D412yes—
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2411yescharacteristic
82C22×C127—21no—
83C22×C127—22no—
84C22×C127C22×D411yescharacteristic
85C2×D127—42no—
86C2×D127—42no—
87C2×D127—42no—
88C4×C127—21no—
89C4×C127—22no—
90C4×C127—22no—
91C4×C127—42no—
92C4×C127—42no—
93C4×C127H(5,𝔽2)11yescharacteristic
94C4×C1272−1+411yescharacteristic
95Dic127—42no—
96C2×C127—41no—
97C2×C127#53711yescharacteristic
98C2×C127#57712yes—
99D127—82no—
100#3260—1271nomaximal, Sylow
101C127#326011yescharacteristic, complemented, Sylow
102#541D12711yescharacteristic, complemented
103#540—1271no—
104#479—1272no—
105#478—1272no—
106#468—1272no—
107#465—1272no—
108#461—1272no—
109#460—1272no—
110#424—1271no—
111C2×SD16—1272no—
112C2×SD16—1272no—
113C2×M16—1271no—
114C22×C8—1271no—
115C4⋊Q8C2×D12711yescharacteristic
116#127—1271no—
117#126—22no—
118#125—22no—
119#125—1271no—
120#125C2×D12711yescharacteristic
121#123—2542no—
122C22⋊Q8C2×D12712yes—
123#121—1272no—
124#121—2542no—
125C4×Q8C2×D12712yes—
126C4×D4—1272no—
127#117C2×D12711yescharacteristic
128#107—1272no—
129#106—1272no—
130C4⋊C8—1272no—
131C4⋊C8—1272no—
132#103—1272no—
133#103—1272no—
134#102—1272no—
135#102—1272no—
136C2×Q8C22×D12712yes—
137C2×D4—1272no—
138C2×D4—2542no—
139C2×D4—2542no—
140C22×C4—2542no—
141C22×C4C22×D12711yescharacteristic
142SD16—5082no—
143SD16—5082no—
144M16—5081no—
145C2×C8—1272no—
146C2×C8—1272no—
147C2×C8—5081no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—1272no—
153C4⋊C4—2542no—
154C4⋊C4C22⋊D12712yes—
155C4⋊C4C22×D12712yes—
156C22⋊C4—22no—
157C22⋊C4—2542no—
158C22⋊C4—2542no—
159C22⋊C4C22⋊D12712yes—
160C42—22no—
161C42C22×D12711yescharacteristic
162C42C22×D12711yescharacteristic
163C23—2542no—
164C23C2×C22⋊D12711yescharacteristic
165Q8—22no—
166Q8—22no—
167D4—10162no—
168D4—10162no—
169D4—2542no—
170D4—2542no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—10162no—
176C2×C4—2542no—
177C2×C4C2×C22⋊D12712yes—
178C2×C4C2×C22⋊D12712yes—
179C2×C4C2×C22⋊D12711yescharacteristic
180C2×C4C23×D12711yescharacteristic
181C8—2542no—
182C8—2542no—
183C22—22no—
184C22—21no—
185C22—5082no—
186C22—5082no—
187C22—5082no—
188C22C22×C22⋊D12711yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—42no—
193C4—42no—
194C4—5082no—
195C4#1686125511yescharacteristic
196C4#1686125611yescharacteristic
197C2—41no—
198C2—10162no—
199C2#2255045411yescharacteristic
200C2#2255068412yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#541×C127C2
socleC22×C127C22×D4
prime radical#22550481C2
hypercenter#541D127
hypocenterC127#3260
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. #22550173C2semidirect
4. #22550124C2semidirect
5. #22550107C2semidirect
6. #22550078C2semidirect
7. #22550076C2semidirect
8. #22549903C2semidirect
9. #541×C127C2semidirect
101. C127#3260semidirect
102. #541D127semidirect

Maximal subgroups

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

subgroupindex
#225504812
#225501832
#225501732
#225501242
#225501072
#225500782
#225500762
#225499032
#541×C1272
#3260127