Finite Group Databasebeta

← #30650177

Subgroups — #30650177

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#22550488C211yesmaximal, characteristic, complemented
2#22550481C211yesmaximal, characteristic
3#22550184C212yesmaximal, complemented
4#22550177C212yesmaximal, complemented
5#22550122C212yesmaximal, complemented
6#22550105C212yesmaximal, complemented
7#22550082C212yesmaximal, complemented
8#22550077C212yesmaximal, complemented
9#424×C127C211yesmaximal, characteristic, complemented
10C2×#14523237C2212yes—
11C2×#14523237C2212yes—
12#16861209C2211yescharacteristic
13#16861206C2211yescharacteristic
14#16861202—22no—
15#16861201C2211yescharacteristic
16#16861201C2211yescharacteristic
17#16861199—22no—
18#16861194—22no—
19#16861188C2212yes—
20#16861183C2212yes—
21#16861182—22no—
22C4×Dic254C2212yes—
23C4×D508C2212yes—
24#16861143C2212yes—
25#16861141C2212yes—
26#16861134C2212yes—
27#16861134C2212yes—
28#16861131C2212yes—
29#16861131C2212yes—
30C2×M16×C127C2211yescharacteristic
31C22×C8×C127C2211yescharacteristic
32#117×C127C2211yescharacteristic
33C4⋊C8×C127C2212yes—
34C4⋊C8×C127C2212yes—
35C2×Dic254C2312yes—
36C2×D508—22no—
37C2×D508C2312yes—
38C2×C22⋊D127—22no—
39C2×C4×D127—22no—
40#14523237—42no—
41#14523237—42no—
42C4⋊Dic127—22no—
43C4⋊Dic127C2312yes—
44C4⋊Dic127C2312yes—
45Dic127⋊C4—22no—
46Dic127⋊C4—22no—
47Dic127⋊C4—22no—
48Dic127⋊C4—22no—
49C22⋊Dic127—22no—
50(C2×D127)⋊C4—22no—
51(C2×D127)⋊C4—22no—
52C4×Dic127—22no—
53C22×C4×C127C2311yescharacteristic
54M16×C127—41no—
55C2×C8×C127—41no—
56C2×C8×C127C2312yes—
57C2×C8×C127C2312yes—
58C4⋊C4×C127D412yes—
59C22⋊C4×C127D412yes—
60C42×C127C2311yescharacteristic
61C42×C127C2311yescharacteristic
62C22×D127—22no—
63Dic254—22no—
64Dic254—22no—
65D508—22no—
66D508—22no—
67D508—82no—
68C22⋊D127—82no—
69C2×Dic127—22no—
70C2×Dic127—22no—
71C2×Dic127—22no—
72C4×D127—82no—
73C23×C127C2×D411yescharacteristic
74C2×C4×C127—21no—
75C2×C4×C127—21no—
76C2×C4×C127C2×D412yes—
77C2×C4×C127C2×D412yes—
78C2×C4×C127C2×D411yescharacteristic
79C2×C4×C127C2411yescharacteristic
80C8×C127—22no—
81C8×C127—22no—
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×C127H(5,𝔽2)11yescharacteristic
92C4×C1272−1+411yescharacteristic
93Dic127—42no—
94Dic127—42no—
95Dic127—42no—
96C2×C127—41no—
97C2×C127#53711yescharacteristic
98C2×C127#57712yes—
99D127—82no—
100#3260—1271nomaximal, Sylow
101C127#326011yescharacteristic, complemented, Sylow
102#541—1271no—
103#540—1271no—
104#479—1272no—
105#478—1272no—
106#468—1272no—
107#465—1272no—
108#461—1272no—
109#460—1272no—
110#424D12711yescharacteristic, complemented
111C2×SD16—1272no—
112C2×SD16—1272no—
113C2×M16C2×D12711yescharacteristic
114C22×C8C2×D12711yescharacteristic
115C4⋊Q8—1271no—
116#127—1271no—
117#126—2542no—
118#125—1271no—
119#125—1271no—
120#125—2542no—
121#123—2542no—
122C22⋊Q8—1272no—
123#121—1272no—
124#121—2542no—
125C4×Q8—1272no—
126C4×D4—1272no—
127#117C2×D12711yescharacteristic
128#107—1272no—
129#106—1272no—
130C4⋊C8C2×D12712yes—
131C4⋊C8C2×D12712yes—
132#103—1272no—
133#103—1272no—
134#102—1272no—
135#102—1272no—
136C2×Q8—1272no—
137C2×D4—1272no—
138C2×D4—2542no—
139C2×D4—2542no—
140C22×C4—2542no—
141C22×C4C22×D12711yescharacteristic
142SD16—5082no—
143SD16—5082no—
144M16—41no—
145C2×C8—41no—
146C2×C8C22×D12712yes—
147C2×C8C22×D12712yes—
148C4⋊C4—1272no—
149C4⋊C4—1272no—
150C4⋊C4—2542no—
151C4⋊C4—2542no—
152C4⋊C4—2542no—
153C4⋊C4—2542no—
154C4⋊C4—2542no—
155C4⋊C4D50812yes—
156C22⋊C4—2542no—
157C22⋊C4—2542no—
158C22⋊C4—2542no—
159C22⋊C4D50812yes—
160C42—2542no—
161C42C22×D12711yescharacteristic
162C42C22×D12711yescharacteristic
163C23—2542no—
164C23C2×D50811yescharacteristic
165Q8—2542no—
166Q8—2542no—
167D4—10162no—
168D4—10162no—
169D4—2542no—
170D4—2542no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—10162no—
174C2×C4—2542no—
175C2×C4—2542no—
176C2×C4—2542no—
177C2×C4C2×D50812yes—
178C2×C4C2×D50812yes—
179C2×C4C2×D50811yescharacteristic
180C2×C4C23×D12711yescharacteristic
181C8—22no—
182C8—22no—
183C22—22no—
184C22—21no—
185C22—5082no—
186C22—5082no—
187C22—5082no—
188C22C22×D50811yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—5082no—
193C4—5082no—
194C4—5082no—
195C4#1686125411yescharacteristic
196C4#1686125711yescharacteristic
197C2—41no—
198C2—10162no—
199C2#2255045711yescharacteristic
200C2#2255068812yes—
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#424×C127C2
socleC22×C127C22×D4
prime radical#22550481C2
hypercenter#424D127
hypocenterC127#3260
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #22550488C2semidirect
3. #22550184C2semidirect
4. #22550177C2semidirect
5. #22550122C2semidirect
6. #22550105C2semidirect
7. #22550082C2semidirect
8. #22550077C2semidirect
9. #424×C127C2semidirect
101. C127#3260semidirect
110. #424D127semidirect

Maximal subgroups

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

subgroupindex
#225504882
#225504812
#225501842
#225501772
#225501222
#225501052
#225500822
#225500772
#424×C1272
#3260127