Finite Group Databasebeta

← #30651337

Subgroups — #30651337

Order 16256.

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

normal subgroups only

This group has 360 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#22550576C212yesmaximal, complemented
2#22550278C211yesmaximal, characteristic, complemented
3#22550198C212yesmaximal, complemented
4#22550188C212yesmaximal, complemented
5#22550176C212yesmaximal, complemented
6#22550126C212yesmaximal, complemented
7#22549955C211yesmaximal, characteristic, complemented
8#22549917C212yesmaximal, complemented
9C4×D8×C127C211yesmaximal, characteristic, complemented
10C2×#14523231C2212yes—
11#16861202—22no—
12#16861201C2212yes—
13#16861192—22no—
14#16861182—22no—
15#16861180C2212yes—
16#16861180C2212yes—
17C4×D508C2211yescharacteristic
18C4×C22⋊D127—22no—
19#16861167C2212yes—
20C4⋊C4×D127C2212yes—
21#16861143C2211yescharacteristic
22#16861143C2211yescharacteristic
23#16861142C2212yes—
24#16861141C2212yes—
25#16861140C2212yes—
26C4⋊(C127⋊C8)C2211yescharacteristic
27#16861130C2212yes—
28#16861130C2212yes—
29(C2×D127)⋊C8C2212yes—
30C2×D8×C127C2211yescharacteristic
31C4×D4×C127C2212yes—
32#107×C127C2211yescharacteristic
33#102×C127C2212yes—
34C4×C8×C127C2211yescharacteristic
35C2×D508D411yescharacteristic
36C2×C22⋊D127—22no—
37C2×C22⋊D127—22no—
38C2×C4×D127—22no—
39C2×C4×D127C2312yes—
40#14523231—24no—
41C2×C127⋊C8C2312yes—
42C4⋊Dic127D411yescharacteristic
43C4⋊Dic127C2312yes—
44C4⋊Dic127C2312yes—
45Dic127⋊C4—22no—
46Dic127⋊C4—22no—
47Dic127⋊C4—22no—
48C22⋊Dic127—22no—
49C22⋊Dic127—22no—
50(C2×D127)⋊C4—22no—
51(C2×D127)⋊C4D412yes—
52C4×Dic127—22no—
53C2×D4×C127C2312yes—
54C22×C4×C127—22no—
55D8×C127—22no—
56C2×C8×C127C2311yescharacteristic
57C2×C8×C127C2311yescharacteristic
58C4⋊C4×C127C2312yes—
59C22⋊C4×C127—22no—
60C42×C127C2311yescharacteristic
61C22×D127C2×D412yes—
62D508—41no—
63C22⋊D127—44no—
64C22⋊D127—82no—
65C2×Dic127—22no—
66C2×Dic127—22no—
67C2×Dic127C2×D412yes—
68C4×D127—22no—
69C4×D127—22no—
70C4×D127—82no—
71C127⋊C8—22no—
72C23×C127—22no—
73D4×C127—22no—
74D4×C127—22no—
75C2×C4×C127—22no—
76C2×C4×C127—82no—
77C2×C4×C127C2×D411yescharacteristic
78C2×C4×C127C2×D411yescharacteristic
79C2×C4×C127C2411yescharacteristic
80C8×C127—21no—
81C8×C127#4112yes—
82C22×C127—42no—
83C22×C127—42no—
84C22×C127—42no—
85C22×C127C22×D411yescharacteristic
86C2×D127—22no—
87C2×D127—22no—
88C2×D127—22no—
89C4×C127—21no—
90C4×C127—21no—
91C4×C127—42no—
92C4×C127C2×#4111yescharacteristic
93C4×C1272−1+411yescharacteristic
94Dic127—22no—
95Dic127—42no—
96Dic127—42no—
97C2×C127—82no—
98C2×C127#54711yescharacteristic
99C2×C127#57511yescharacteristic
100C2×C127#57611yescharacteristic
101D127—42no—
102#3355—1271nomaximal, Sylow
103C127#335511yescharacteristic, complemented, Sylow
104#553—1272no—
105#499—1271no—
106#482—1272no—
107#480—1272no—
108#478—1272no—
109#469—1272no—
110C4×D8D12711yescharacteristic, complemented
111#436—1271no—
112#428—1272no—
113C2×D8C2×D12711yescharacteristic
114C2×M16—1272no—
115#126—2542no—
116#125—1272no—
117#123—2542no—
118#121—1272no—
119#121—1272no—
120#121—2542no—
121C4×D4—1271no—
122C4×D4—2542no—
123C4×D4C2×D12712yes—
124#117—1272no—
125C2×C4⋊C4—1272no—
126#107—1271no—
127#107—1271no—
128#107—1272no—
129#107C2×D12711yescharacteristic
130#106—1272no—
131#106—1272no—
132C4⋊C8—1271no—
133#102—1272no—
134#102—1272no—
135#102C2×D12712yes—
136C22⋊C8—1272no—
137C4×C8C2×D12711yescharacteristic
138C2×D4—1271no—
139C2×D4—2542no—
140C2×D4—2542no—
141C2×D4C22×D12712yes—
142C22×C4—22no—
143C22×C4—1272no—
144C22×C4—2542no—
145D8—22no—
146M16—2544no—
147C2×C8—1272no—
148C2×C8C22×D12711yescharacteristic
149C2×C8C22×D12711yescharacteristic
150C4⋊C4—1271no—
151C4⋊C4—1272no—
152C4⋊C4—1272no—
153C4⋊C4—2542no—
154C4⋊C4—2542no—
155C4⋊C4—2542no—
156C4⋊C4C22×D12712yes—
157C22⋊C4—22no—
158C22⋊C4—1272no—
159C22⋊C4—2542no—
160C22⋊C4—2542no—
161C22⋊C4—2542no—
162C42—2542no—
163C42C22×D12711yescharacteristic
164C23—22no—
165C23—1272no—
166D4—22no—
167D4—22no—
168D4—10162no—
169D4—5081no—
170D4—5084no—
171C2×C4—22no—
172C2×C4—82no—
173C2×C4—1272no—
174C2×C4—10162no—
175C2×C4—2542no—
176C2×C4—2542no—
177C2×C4—2542no—
178C2×C4—2542no—
179C2×C4D4×D12711yescharacteristic
180C2×C4D4×D12711yescharacteristic
181C2×C4C23×D12711yescharacteristic
182C8—21no—
183C8—2542no—
184C8#1452324812yes—
185C22—42no—
186C22—42no—
187C22—42no—
188C22—2542no—
189C22—2542no—
190C22—2542no—
191C22C2×D4×D12711yescharacteristic
192C4—21no—
193C4—21no—
194C4—42no—
195C4—5082no—
196C4—5082no—
197C4—2542no—
198C4C2×#1452324811yescharacteristic
199C4#1686125611yescharacteristic
200C2—82no—
201C2—5082no—
202C2#2255053111yescharacteristic
203C2#2255067311yescharacteristic
204C2#2255067711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC4×D8×C127C2
socleC22×C127C22×D4
hypercenterC4×D8D127
hypocenterC127#3355
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #22550576C2semidirect
2. #22550278C2semidirect
3. #22550198C2semidirect
4. #22550188C2semidirect
5. #22550176C2semidirect
6. #22550126C2semidirect
7. #22549955C2semidirect
8. #22549917C2semidirect
9. C4×D8×C127C2semidirect
103. C127#3355semidirect
110. C4×D8D127semidirect

Maximal subgroups

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

subgroupindex
#225505762
#225502782
#225501982
#225501882
#225501762
#225501262
#225499552
#225499172
C4×D8×C1272
#3355127