Finite Group Databasebeta

← #30638792

Subgroups — #30638792

Order 16256.

264 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 93 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×#16861216C211yesmaximal, characteristic
2C2×#16861144C211yesmaximal, characteristic, complemented
3C2×#16861142C211yesmaximal, characteristic, complemented
4#22549877C211yesmaximal, characteristic
5#22549645C211yesmaximal, characteristic, complemented
6#22549644C211yesmaximal, characteristic, complemented
7#417×C127C211yesmaximal, characteristic
8#16861216—22no—
9#16861216C414yescomplemented
10C2×#14523232—21no—
11C2×#14523232C2211yescharacteristic
12C2×#14523232C2211yescharacteristic
13C22×C127⋊C8—21no—
14C22×C127⋊C8C2211yescharacteristic
15C2×C4⋊Dic127C2211yescharacteristic
16#16861144—22no—
17#16861142—21no—
18#16861142—21no—
19#16861133—21no—
20#16861130—21no—
21C2×#41×C127C2211yescharacteristic
22C2×M16×C127C2211yescharacteristic
23C2×C4⋊C4×C127C2211yescharacteristic
24#103×C127—21no—
25#102×C127—21no—
26C22×Dic127—21no—
27#14523232—24no—
28#14523232—21no—
29#14523232—21no—
30#14523232—21no—
31#14523232C2×C412yes—
32C2×C127⋊C8—24no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C2×C127⋊C8C2×C412yes—
36C2×C127⋊C8D411yescharacteristic
37C2×C127⋊C8D411yescharacteristic
38C2×C127⋊C8D412yes—
39C4⋊Dic127—21no—
40C4⋊Dic127—21no—
41#41×C127—21no—
42#41×C127—21no—
43#41×C127C2×C412yes—
44#41×C127Q812yes—
45C2×Q8×C127D411yescharacteristic
46C2×D4×C127—21no—
47C2×D4×C127D411yescharacteristic
48C22×C4×C127—21no—
49C22×C4×C127—21no—
50C22×C4×C127C2311yescharacteristic
51M16×C127—21no—
52M16×C127—21no—
53C2×C8×C127—21no—
54C4⋊C4×C127—21no—
55C4⋊C4×C127—21no—
56C2×Dic127—41no—
57C2×Dic127—81no—
58C127⋊C8—22no—
59C127⋊C8—41no—
60C127⋊C8C22⋊C412yes—
61C127⋊C8C22⋊C412yes—
62C23×C127—21no—
63C23×C127#4111yescharacteristic
64Q8×C127—21no—
65Q8×C127C4⋊C412yes—
66D4×C127—22no—
67D4×C127—22no—
68D4×C127—21no—
69D4×C127C4⋊C412yes—
70C2×C4×C127—21no—
71C2×C4×C127—22no—
72C2×C4×C127—22no—
73C2×C4×C127—41no—
74C2×C4×C127—81no—
75C2×C4×C127C22×C411yescharacteristic
76C2×C4×C127C2×D411yescharacteristic
77C2×C4×C127C2×D411yescharacteristic
78C2×C4×C127C2×D411yescharacteristic
79C2×C4×C127C2×Q811yescharacteristic
80C2×C4×C127#4111yescharacteristic
81C8×C127—41no—
82C22×C127—21no—
83C22×C127—21no—
84C22×C127—22no—
85C22×C127—41no—
86C22×C127—41no—
87C22×C127#11711yescharacteristic
88C22×C127#12111yescharacteristic
89C22×C127C22⋊Q811yescharacteristic
90C4×C127—22no—
91C4×C127—81no—
92C4×C127C2×C22⋊C411yescharacteristic
93C4×C127C2×C4⋊C411yescharacteristic
94C4×C127#12111yescharacteristic
95C4×C127C22⋊Q811yescharacteristic
96Dic127—81no—
97C2×C127—21no—
98C2×C127—21no—
99C2×C127—42no—
100C2×C127#38011yescharacteristic
101C2×C127#47211yescharacteristic
102C2×C127#47311yescharacteristic
103#1816—1271nomaximal, Sylow
104C127#181611yescharacteristic, complemented, Sylow
105C2×#131—1271no—
106C2×#108—1271no—
107C2×#107—1271no—
108#417—1271no—
109#417D12711yescharacteristic
110#343—1271no—
111#343—1271no—
112C2×#41C2×D12711yescharacteristic
113#131—1274no—
114#131—2542no—
115C2×M16—1271no—
116C2×M16—1271no—
117C2×M16—2541no—
118C2×M16C2×D12711yescharacteristic
119C22×C8—1271no—
120C22×C8—2541no—
121C2×C4⋊C4—1271no—
122C2×C4⋊C4C2×D12711yescharacteristic
123#108—2542no—
124#107—2541no—
125#107—2541no—
126#103—21no—
127#103—2541no—
128#102—21no—
129#102—2541no—
130#41—21no—
131#41—21no—
132#41C4×D12712yes—
133#41Dic25412yes—
134C2×Q8C22⋊D12711yescharacteristic
135C2×D4—21no—
136C2×D4C22⋊D12711yescharacteristic
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—2541no—
140C22×C4C22×D12711yescharacteristic
141M16—21no—
142M16—21no—
143M16—1272no—
144M16—2544no—
145M16—2541no—
146M16—2541no—
147M16—2541no—
148C2×C8—21no—
149C2×C8—1272no—
150C2×C8—1271no—
151C2×C8—1271no—
152C2×C8—1272no—
153C2×C8—2544no—
154C2×C8—2541no—
155C2×C8—2541no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—2541no—
159C4⋊C4—2541no—
160C23—21no—
161C23#1452324711yescharacteristic
162Q8—21no—
163Q8Dic127⋊C412yes—
164D4—22no—
165D4—22no—
166D4—21no—
167D4Dic127⋊C412yes—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—41no—
172C2×C4—81no—
173C2×C4—10161no—
174C2×C4—5081no—
175C2×C4C2×C4×D12711yescharacteristic
176C2×C4D4×D12711yescharacteristic
177C2×C4D4×D12711yescharacteristic
178C2×C4C2×C22⋊D12711yescharacteristic
179C2×C4C2×Dic25411yescharacteristic
180C2×C4#1452324711yescharacteristic
181C8—41no—
182C8—1272no—
183C8—1272no—
184C8—5081no—
185C8—2542no—
186C22—21no—
187C22—21no—
188C22—22no—
189C22—41no—
190C22—41no—
191C22#1686116511yescharacteristic
192C22#1686118111yescharacteristic
193C22#1686118611yescharacteristic
194C4—22no—
195C4—81no—
196C4—10161no—
197C4C22⋊C4×D12711yescharacteristic
198C4C2×Dic127⋊C411yescharacteristic
199C4#1686118111yescharacteristic
200C4#1686118611yescharacteristic
201C2—21no—
202C2—21no—
203C2—42no—
204C2#2254973411yescharacteristic
205C2#2255014411yescharacteristic
206C2#2255014911yescharacteristic
subgroupG/H
centerC22#16861181
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#417×C127C2
socleC22×C127#121
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenter#417D127
hypocenterC127#1816
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#16861144C2semidirect
3. C2×#16861142C2semidirect
5. #22549645C2semidirect
6. #22549644C2semidirect
9. #16861216C4semidirect
104. C127#1816semidirect

Maximal subgroups

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

subgroupindex
C2×#168612162
C2×#168611442
C2×#168611422
#225498772
#225496452
#225496442
#417×C1272
#1816127