Finite Group Databasebeta

← #30650121

Subgroups — #30650121

Order 16256.

342 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550433C211yesmaximal, characteristic
2#22550161C212yesmaximal, complemented
3#22550158C212yesmaximal, complemented
4#22550132C214yesmaximal
5#22550086C212yesmaximal, complemented
6#22550083C212yesmaximal, complemented
7#22549900C211yesmaximal, characteristic
8#533×C127C211yesmaximal, characteristic
9C2×#14523238C2214yes—
10C2×#14523232C2212yes—
11#16861209C2211yescharacteristic
12#16861209C2211yescharacteristic
13#16861201—21no—
14#16861188—21no—
15#16861188C2212yes—
16C4×Dic254—21no—
17C4×Dic254C2212yes—
18#16861168—21no—
19#16861140C2214yes—
20C4⋊(C127⋊C8)C2212yes—
21C4⋊(C127⋊C8)C2212yes—
22#16861133C2214yes—
23#16861132C2214yes—
24C4⋊Q8×C127C2211yescharacteristic
25C4⋊Q8×C127C2211yescharacteristic
26#125×C127—21no—
27C22⋊Q8×C127—21no—
28C22⋊Q8×C127C2212yes—
29C4×Q8×C127—21no—
30C4×Q8×C127C2212yes—
31#117×C127—21no—
32C2×C42×C127C2211yescharacteristic
33C2×Dic254—21no—
34C2×Dic254C2312yes—
35#14523238—44no—
36#14523232—42no—
37C2×C127⋊C8C2314yes—
38C4⋊Dic127—21no—
39C4⋊Dic127—21no—
40C4⋊Dic127—21no—
41C4⋊Dic127C2312yes—
42Dic127⋊C4—22no—
43Dic127⋊C4—22no—
44C22⋊Dic127—22no—
45C4×Dic127—22no—
46C2×Q8×C127—21no—
47C2×Q8×C127C2312yes—
48C22×C4×C127D411yescharacteristic
49C22×C4×C127D411yescharacteristic
50C22×C4×C127C2311yescharacteristic
51C4⋊C4×C127—21no—
52C4⋊C4×C127—21no—
53C4⋊C4×C127—21no—
54C4⋊C4×C127—22no—
55C4⋊C4×C127—22no—
56C4⋊C4×C127C2312yes—
57C22⋊C4×C127—22no—
58C42×C127—22no—
59C42×C127D411yescharacteristic
60C42×C127D411yescharacteristic
61C42×C127C2311yescharacteristic
62C42×C127C2311yescharacteristic
63Dic254—22no—
64Dic254—22no—
65Dic254—42no—
66C2×Dic127—22no—
67C2×Dic127—22no—
68C127⋊C8—24no—
69C23×C127C2×D411yescharacteristic
70Q8×C127—22no—
71Q8×C127—22no—
72Q8×C127—42no—
73C2×C4×C127—22no—
74C2×C4×C127—22no—
75C2×C4×C127—21no—
76C2×C4×C127—21no—
77C2×C4×C127—22no—
78C2×C4×C127—22no—
79C2×C4×C127C2×D411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2411yescharacteristic
85C22×C127—21no—
86C22×C127—21no—
87C22×C127—21no—
88C22×C127C22×D411yescharacteristic
89C4×C127—21no—
90C4×C127—21no—
91C4×C127—21no—
92C4×C127—42no—
93C4×C127—42no—
94C4×C127#13712yes—
95C4×C127#13712yes—
96C4×C127H(5,𝔽2)11yescharacteristic
97C4×C1272−1+411yescharacteristic
98Dic127—42no—
99Dic127—42no—
100C2×C127—41no—
101C2×C127#53711yescharacteristic
102C2×C127C2×#13711yescharacteristic
103C2×C127C2×#13711yescharacteristic
104#3255—1271nomaximal, Sylow
105C127#325511yescharacteristic, complemented, Sylow
106#533—1271no—
107#533D12711yescharacteristic
108#475—1272no—
109#475—1272no—
110#470—1274no—
111#462—1272no—
112#462—1272no—
113#423—1271no—
114C2×Q16—1274no—
115C2×M16—1272no—
116C4⋊Q8—1271no—
117C4⋊Q8—1271no—
118C4⋊Q8C2×D12711yescharacteristic
119C4⋊Q8C2×D12711yescharacteristic
120#125—21no—
121#125—2541no—
122C22⋊Q8—21no—
123C22⋊Q8—1272no—
124C22⋊Q8—2541no—
125C22⋊Q8C2×D12712yes—
126C4×Q8—21no—
127C4×Q8—1272no—
128C4×Q8—2541no—
129C4×Q8C2×D12712yes—
130#117—21no—
131#117—2541no—
132C2×C42C2×D12711yescharacteristic
133#106—1274no—
134C4⋊C8—1272no—
135C4⋊C8—1272no—
136#103—1274no—
137#103—1274no—
138C2×Q8—21no—
139C2×Q8—1272no—
140C2×Q8—2541no—
141C2×Q8C22×D12712yes—
142C22×C4C22⋊D12711yescharacteristic
143C22×C4C22⋊D12711yescharacteristic
144C22×C4C22×D12711yescharacteristic
145Q16—5084no—
146M16—5082no—
147C2×C8—1274no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—1272no—
154C4⋊C4—2542no—
155C4⋊C4—2542no—
156C4⋊C4—2541no—
157C4⋊C4—2541no—
158C4⋊C4—2541no—
159C4⋊C4C22×D12712yes—
160C22⋊C4—22no—
161C22⋊C4—2542no—
162C42—22no—
163C42—2542no—
164C42C22⋊D12711yescharacteristic
165C42C22⋊D12711yescharacteristic
166C42C22×D12711yescharacteristic
167C42C22×D12711yescharacteristic
168C23C2×C22⋊D12711yescharacteristic
169Q8—22no—
170Q8—22no—
171Q8—42no—
172Q8—5082no—
173Q8—2542no—
174Q8—2542no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—2542no—
182C2×C4—2542no—
183C2×C4C2×C22⋊D12711yescharacteristic
184C2×C4C2×C22⋊D12711yescharacteristic
185C2×C4C2×C22⋊D12711yescharacteristic
186C2×C4C2×C22⋊D12711yescharacteristic
187C2×C4C2×C22⋊D12711yescharacteristic
188C2×C4C23×D12711yescharacteristic
189C8—2544no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22C22×C22⋊D12711yescharacteristic
194C4—21no—
195C4—21no—
196C4—21no—
197C4—42no—
198C4—42no—
199C4—5082no—
200C4—5082no—
201C4#1686123912yes—
202C4#1686124012yes—
203C4#1686125511yescharacteristic
204C4#1686125611yescharacteristic
205C2—41no—
206C2#2255045411yescharacteristic
207C2C2×#1686123911yescharacteristic
208C2C2×#1686124011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#533×C127C2
socleC22×C127C22×D4
prime radicalC23×C127C2×D4
hypercenter#533D127
hypocenterC127#3255
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #22550161C2semidirect
3. #22550158C2semidirect
5. #22550086C2semidirect
6. #22550083C2semidirect
105. C127#3255semidirect

Maximal subgroups

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

subgroupindex
#225504332
#225501612
#225501582
#225501322
#225500862
#225500832
#225499002
#533×C1272
#3255127