Finite Group Databasebeta

← #30650191

Subgroups — #30650191

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#22550497C212yesmaximal, complemented
2#22550172C212yesmaximal, complemented
3#22550157C212yesmaximal, complemented
4#22550122C212yesmaximal, complemented
5#22550117C211yesmaximal, characteristic, complemented
6#22550112C211yesmaximal, characteristic
7#22550099C212yesmaximal, complemented
8#22550092C212yesmaximal, complemented
9#424×C127C211yesmaximal, characteristic, complemented
10C2×Dic508C2211yescharacteristic
11C2×#14523237C2212yes—
12C2×D1016C2211yescharacteristic
13#16861209C2212yes—
14#16861204—22no—
15#16861197C2212yes—
16#16861195—22no—
17#16861191—22no—
18#16861188C2212yes—
19#16861187—22no—
20#16861183C2212yes—
21C4×Dic254C2212yes—
22C4×D508C2212yes—
23#16861143C2211yescharacteristic
24#16861143C2211yescharacteristic
25#16861143C2212yes—
26#16861134C2212yes—
27#16861134C2212yes—
28#16861131C2212yes—
29#16861131C2212yes—
30C2×M16×C127C2211yescharacteristic
31C22×C8×C127C2211yescharacteristic
32#117×C127C2211yescharacteristic
33C4⋊C8×C127C2212yes—
34C4⋊C8×C127C2212yes—
35C2×Dic254—22no—
36C2×Dic254C2312yes—
37C2×D508C2312yes—
38C2×C22⋊D127—22no—
39C2×C4×D127—22no—
40Dic508—41no—
41#14523237—42no—
42D1016—41no—
43C4⋊Dic127—22no—
44C4⋊Dic127C2312yes—
45C4⋊Dic127C2312yes—
46Dic127⋊C4—22no—
47Dic127⋊C4—22no—
48C22⋊Dic127—22no—
49(C2×D127)⋊C4—22no—
50(C2×D127)⋊C4—22no—
51(C2×D127)⋊C4—22no—
52(C2×D127)⋊C4—22no—
53C4×Dic127—22no—
54C22×C4×C127C2311yescharacteristic
55M16×C127—41no—
56C2×C8×C127—41no—
57C2×C8×C127C2311yescharacteristic
58C2×C8×C127C2312yes—
59C2×C8×C127C2311yescharacteristic
60C4⋊C4×C127D411yescharacteristic
61C4⋊C4×C127D411yescharacteristic
62C22⋊C4×C127D412yes—
63C42×C127C2312yes—
64C22×D127—22no—
65Dic254—22no—
66Dic254—22no—
67Dic254—82no—
68D508—22no—
69D508—22no—
70C22⋊D127—82no—
71C2×Dic127—22no—
72C2×Dic127—22no—
73C2×Dic127—22no—
74C4×D127—82no—
75C23×C127C2×D411yescharacteristic
76C2×C4×C127—21no—
77C2×C4×C127—21no—
78C2×C4×C127C2×D412yes—
79C2×C4×C127C2×D412yes—
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2411yescharacteristic
82C8×C127—21no—
83C8×C127—21no—
84C8×C127—22no—
85C22×C127—21no—
86C22×C127—21no—
87C22×C127—21no—
88C22×C127C22×D411yescharacteristic
89C2×D127—42no—
90C2×D127—42no—
91C2×D127—42no—
92C4×C127—21no—
93C4×C127—22no—
94C4×C127—22no—
95C4×C127H(5,𝔽2)11yescharacteristic
96C4×C1272−1+411yescharacteristic
97Dic127—42no—
98Dic127—42no—
99Dic127—42no—
100C2×C127—41no—
101C2×C127#53711yescharacteristic
102C2×C127#57611yescharacteristic
103C2×C127#57811yescharacteristic
104D127—82no—
105#3262—1271nomaximal, Sylow
106C127#326211yescharacteristic, complemented, Sylow
107#542—1272no—
108#477—1272no—
109#474—1272no—
110#468—1272no—
111#467—1271no—
112#466—1271no—
113#464—1272no—
114#463—1272no—
115#424D12711yescharacteristic, complemented
116C2×Q16—1271no—
117C2×SD16—1272no—
118C2×D8—1271no—
119C2×M16C2×D12711yescharacteristic
120C22×C8C2×D12711yescharacteristic
121C4⋊Q8—1272no—
122#126—2542no—
123#124—1272no—
124#124—2542no—
125#123—2542no—
126C22⋊Q8—1272no—
127C22⋊Q8—2542no—
128#121—1272no—
129C4×Q8—1272no—
130C4×D4—1272no—
131#117C2×D12711yescharacteristic
132#107—1271no—
133#107—1271no—
134#107—1272no—
135C4⋊C8C2×D12712yes—
136C4⋊C8C2×D12712yes—
137#103—1272no—
138#103—1272no—
139#102—1272no—
140#102—1272no—
141C2×Q8—1272no—
142C2×Q8—2542no—
143C2×D4—1272no—
144C2×D4—2542no—
145C22×C4—2542no—
146C22×C4C22×D12711yescharacteristic
147Q16—5081no—
148SD16—5082no—
149D8—5081no—
150M16—41no—
151C2×C8—41no—
152C2×C8C22×D12711yescharacteristic
153C2×C8C22×D12712yes—
154C2×C8C22×D12711yescharacteristic
155C4⋊C4—1272no—
156C4⋊C4—1272no—
157C4⋊C4—2542no—
158C4⋊C4—2542no—
159C4⋊C4—2542no—
160C4⋊C4D50811yescharacteristic
161C4⋊C4D50811yescharacteristic
162C22⋊C4—2542no—
163C22⋊C4—2542no—
164C22⋊C4—2542no—
165C22⋊C4—2542no—
166C22⋊C4—2542no—
167C22⋊C4D50812yes—
168C42—2542no—
169C42C22×D12712yes—
170C23—2542no—
171C23C2×D50811yescharacteristic
172Q8—10162no—
173Q8—2542no—
174Q8—2542no—
175D4—10162no—
176D4—2542no—
177D4—2542no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—10162no—
181C2×C4—2542no—
182C2×C4—2542no—
183C2×C4—2542no—
184C2×C4C2×D50812yes—
185C2×C4C2×D50812yes—
186C2×C4C2×D50811yescharacteristic
187C2×C4C23×D12711yescharacteristic
188C8—21no—
189C8—21no—
190C8—22no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—5082no—
195C22—5082no—
196C22—5082no—
197C22C22×D50811yescharacteristic
198C4—22no—
199C4—22no—
200C4—21no—
201C4—5082no—
202C4—5082no—
203C4—5082no—
204C4#1686125411yescharacteristic
205C4#1686125711yescharacteristic
206C2—41no—
207C2—10162no—
208C2#2255045711yescharacteristic
209C2#2255068111yescharacteristic
210C2#2255069311yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#424×C127C2
socleC22×C127C22×D4
prime radical#22550112C2
hypercenter#424D127
hypocenterC127#3262
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. #22550497C2semidirect
2. #22550172C2semidirect
3. #22550157C2semidirect
4. #22550122C2semidirect
5. #22550117C2semidirect
7. #22550099C2semidirect
8. #22550092C2semidirect
9. #424×C127C2semidirect
106. C127#3262semidirect
115. #424D127semidirect

Maximal subgroups

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

subgroupindex
#225504972
#225501722
#225501572
#225501222
#225501172
#225501122
#225500992
#225500922
#424×C1272
#3262127