Finite Group Databasebeta

← #30638688

Subgroups — #30638688

Order 16256.

552 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C2×#16861257C211yesmaximal, characteristic
2C2×#16861137C212yesmaximal, complemented
3#22549875C212yesmaximal, complemented
4C2×#16861128C211yesmaximal, characteristic, complemented
5#391×C127C211yesmaximal, characteristic, complemented
6#16861257—24no—
7#16861257—22no—
8#16861257C414yescomplemented
9C2×#14523249—22no—
10C2×#14523248—22no—
11C2×#14523248C2212yes—
12C2×Q8×D127—22no—
13C2×#14523232C2212yes—
14#16861137—24no—
15#16861132—22no—
16#16861129—22no—
17#16861128—22no—
18C22×Q8×C127C2211yescharacteristic
19C2×C4⋊C4×C127C2211yescharacteristic
20C2×C42×C127C2211yescharacteristic
21#95×C127—22no—
22#14523249—28no—
23#14523249—28no—
24#14523248—28no—
25#14523248—28no—
26#14523248—22no—
27#14523248—22no—
28#14523248C2×C414yes—
29#14523248D414yes—
30C2×Dic254—22no—
31C2×Dic254D412yes—
32Q8×D127—28no—
33Q8×D127—28no—
34C2×D508—22no—
35C2×D508D412yes—
36C2×C22⋊D127—22no—
37C2×C4×D127—24no—
38C2×C4×D127—22no—
39#14523232—22no—
40#14523232—22no—
41C2×C127⋊C8—22no—
42C2×Q8×C127—22no—
43C2×Q8×C127—22no—
44C2×Q8×C127C2×C412yes—
45C2×Q8×C127D412yes—
46C22×C4×C127—21no—
47C22×C4×C127—21no—
48C22×C4×C127D412yes—
49C22×C4×C127C2311yescharacteristic
50C4⋊C4×C127—22no—
51C42×C127—22no—
52C22×D127—22no—
53Dic254—28no—
54Dic254—22no—
55Dic254C22⋊C414yes—
56D508—28no—
57D508—22no—
58D508C22⋊C414yes—
59C22⋊D127—24no—
60C22⋊D127—24no—
61C2×Dic127—22no—
62C4×D127—216no—
63C4×D127—24no—
64C4×D127—24no—
65C127⋊C8—42no—
66C23×C127C2×D411yescharacteristic
67Q8×C127—24no—
68Q8×C127—24no—
69C2×C4×C127—22no—
70C2×C4×C127—22no—
71C2×C4×C127—22no—
72C2×C4×C127—41no—
73C2×C4×C127—41no—
74C2×C4×C127—42no—
75C2×C4×C127—41no—
76C2×C4×C127C22⋊C414yes—
77C2×C4×C127C22×C411yescharacteristic
78C2×C4×C127C2×D412yes—
79C2×C4×C127C2×D412yes—
80C2×C4×C127C2×D411yescharacteristic
81C22×C127—21no—
82C22×C127—21no—
83C22×C127C2×C22⋊C411yescharacteristic
84C22×C127#12011yescharacteristic
85C22×C127#12011yescharacteristic
86C2×D127—24no—
87C2×D127—42no—
88C2×D127—42no—
89C4×C127—24no—
90C4×C127—42no—
91C4×C127—81no—
92C4×C127C2×C22⋊C412yes—
93C4×C127#12012yes—
94Dic127—24no—
95C2×C127—21no—
96C2×C127—21no—
97C2×C127#37911yescharacteristic
98C2×C127#45411yescharacteristic
99C2×C127#45611yescharacteristic
100D127—44no—
101#1793—1271nomaximal, Sylow
102C127#179311yescharacteristic, complemented, Sylow
103C2×2−1+4—1271no—
104C2×#104—1272no—
105#417—1272no—
106C2×#101—1271no—
107#391D12711yescharacteristic, complemented
1082−1+4—1274no—
1092−1+4—2544no—
1102−1+4—2542no—
111C2×#41—1272no—
112C2×#41—2542no—
113C2×#41—2542no—
114C22×Q8—2542no—
115C22×Q8C2×D12711yescharacteristic
116C2×M16—1272no—
117C2×C4⋊C4C2×D12711yescharacteristic
118C2×C42C2×D12711yescharacteristic
119#104—2544no—
120#103—2542no—
121#102—2542no—
122#101—2542no—
123#95—22no—
124#41—1274no—
125#41—1274no—
126#41—2548no—
127#41—2548no—
128#41—2548no—
129#41—2548no—
130#41—2542no—
131#41—2542no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—1272no—
135C2×Q8—2542no—
136C2×Q8—2548no—
137C2×Q8—2548no—
138C2×Q8C4×D12712yes—
139C2×Q8D50812yes—
140C2×D4—1272no—
141C2×D4—2542no—
142C2×D4—2542no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—2544no—
146C22×C4—2542no—
147C22×C4C22⋊D12712yes—
148C22×C4C22×D12711yescharacteristic
149M16—2542no—
150M16—2542no—
151C2×C8—2542no—
152C4⋊C4—22no—
153C42—22no—
154C23—2542no—
155C23C2×C22⋊D12711yescharacteristic
156Q8—24no—
157Q8—24no—
158Q8—1274no—
159Q8—2548no—
160Q8—2542no—
161D4—1274no—
162D4—2548no—
163D4—2544no—
164D4—2544no—
165D4—2542no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—25416no—
174C2×C4—2542no—
175C2×C4—2544no—
176C2×C4—2544no—
177C2×C4(C2×D127)⋊C414yes—
178C2×C4C2×C4×D12711yescharacteristic
179C2×C4D4×D12712yes—
180C2×C4D4×D12712yes—
181C2×C4C2×D50811yescharacteristic
182C8—5082no—
183C22—21no—
184C22—21no—
185C22—5082no—
186C22—5082no—
187C22—2544no—
188C22C2×(C2×D127)⋊C411yescharacteristic
189C22#1686117711yescharacteristic
190C22#1686117711yescharacteristic
191C4—24no—
192C4—42no—
193C4—81no—
194C4—2544no—
195C4C22⋊C4×D12712yes—
196C4#1686117912yes—
197C2—21no—
198C2—21no—
199C2—5084no—
200C2#2254973111yescharacteristic
201C2#2255004811yescharacteristic
202C2#2255005811yescharacteristic
subgroupG/H
centerC22#16861177
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#391×C127C2
socleC22×C127#120
prime radicalC2×#16861257C2
cosocleC22×C4×C127C23
hypercenter#391D127
hypocenterC127#1793
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#16861137C2semidirect
3. #22549875C2semidirect
4. C2×#16861128C2semidirect
5. #391×C127C2semidirect
8. #16861257C4semidirect
102. C127#1793semidirect
107. #391D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612572
C2×#168611372
#225498752
C2×#168611282
#391×C1272
#1793127