Finite Group Databasebeta

← #30639578

Subgroups — #30639578

Order 16256.

312 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16861198C211yesmaximal, characteristic
2#22549899C211yesmaximal, characteristic, complemented
3C2×#16861128C211yesmaximal, characteristic, complemented
4C2×#16861125C211yesmaximal, characteristic
5C2×#16861125C211yesmaximal, characteristic
6M16×Dic127C211yesmaximal, characteristic, complemented
7C2×#101×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C2×#14523232C2211yescharacteristic
11#16861198—21no—
12#16861198—21no—
13#16861198—22no—
14C2×(C2×D127)⋊C4C414yes—
15C2×C4×Dic127C2211yescharacteristic
16Dic127⋊C8—22no—
17#16861128—22no—
18#16861125—22no—
19#16861125—22no—
20#16861119—21no—
21C8×Dic127—21no—
22C22×Q8×C127C2211yescharacteristic
23C2×M16×C127C2211yescharacteristic
24C2×M16×C127C2211yescharacteristic
25#101×C127—22no—
26C23×D127C2×C412yes—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508—21no—
32C2×D508—21no—
33C22×Dic127C2×C412yes—
34#14523232—21no—
35#14523232—21no—
36#14523232—21no—
37#14523232D412yes—
38C2×C127⋊C8—21no—
39C2×C127⋊C8—21no—
40(C2×D127)⋊C4—24no—
41(C2×D127)⋊C4—24no—
42C4×Dic127D411yescharacteristic
43C4×Dic127D411yescharacteristic
44C4×Dic127D412yes—
45C2×Q8×C127—21no—
46C2×Q8×C127—21no—
47C2×Q8×C127—21no—
48C2×Q8×C127—21no—
49C2×Q8×C127—21no—
50C2×Q8×C127—21no—
51C22×C4×C127C2×C412yes—
52C22×C4×C127C2311yescharacteristic
53M16×C127—21no—
54M16×C127—21no—
55M16×C127—21no—
56M16×C127D412yes—
57C2×C8×C127—21no—
58C2×C8×C127—21no—
59C22×D127—22no—
60C22×D127—22no—
61C22×D127—22no—
62C22×D127—44no—
63D508—41no—
64D508—41no—
65D508—41no—
66D508—41no—
67C2×Dic127—42no—
68C2×Dic127C22⋊C412yes—
69C2×Dic127C22⋊C412yes—
70C127⋊C8—22no—
71C127⋊C8—41no—
72C23×C127C22×C411yescharacteristic
73Q8×C127—41no—
74Q8×C127—41no—
75Q8×C127—41no—
76Q8×C127—41no—
77C2×C4×C127—22no—
78C2×C4×C127—22no—
79C2×C4×C127—22no—
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127#4111yescharacteristic
85C2×C4×C127#4111yescharacteristic
86C8×C127—22no—
87C8×C127—41no—
88C22×C127—22no—
89C22×C127C2×C22⋊C411yescharacteristic
90C22×C127C4×D411yescharacteristic
91C22×C127C4×D411yescharacteristic
92C2×D127—42no—
93C2×D127—44no—
94C2×D127—82no—
95C2×D127—82no—
96C4×C127—42no—
97C4×C127#12111yescharacteristic
98C4×C127#12111yescharacteristic
99C4×C127#12411yescharacteristic
100C4×C127#12711yescharacteristic
101Dic127—22no—
102Dic127—22no—
103C2×C127—21no—
104C2×C127—21no—
105C2×C127#39011yescharacteristic
106C2×C127#41312yes—
107D127—82no—
108#1967—1271nomaximal, Sylow
109C127#196711yescharacteristic, complemented, Sylow
110C2×#124—1271no—
111#423—1271no—
112C2×#101—1271no—
113C2×#101D12711yescharacteristic, complemented
114C2×#100—1271no—
115C2×#100—1271no—
116C4×M16—1271no—
117C22×Q8C2×D12711yescharacteristic
118C22×D4—1271no—
119C2×M16—1271no—
120C2×M16—1271no—
121C2×M16C2×D12711yescharacteristic
122C2×M16C2×D12711yescharacteristic
123#124—2541no—
124#124—2541no—
125#124—2542no—
126C2×C22⋊C4—1274no—
127C2×C42—1271no—
128C4⋊C8—2542no—
129#101—22no—
130#101—2542no—
131#100—2542no—
132#100—2542no—
133#97—2541no—
134C4×C8—2541no—
135C24—1272no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×D4—2541no—
143C2×D4—2541no—
144C2×D4—2541no—
145C2×D4—2541no—
146C2×D4—2541no—
147C2×D4—2541no—
148C22×C4—1272no—
149C22×C4C4×D12712yes—
150C22×C4C22×D12711yescharacteristic
151M16—21no—
152M16—21no—
153M16—21no—
154M16—1272no—
155M16—2541no—
156M16—2541no—
157M16—2541no—
158M16C22⋊D12712yes—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—2541no—
162C2×C8—2541no—
163C22⋊C4—2544no—
164C22⋊C4—2544no—
165C42—1271no—
166C42—1271no—
167C42—1272no—
168C23—5084no—
169C23—2542no—
170C23—2542no—
171C23—2542no—
172C23C2×C4×D12711yescharacteristic
173Q8—41no—
174Q8—41no—
175Q8—41no—
176Q8—41no—
177D4—5081no—
178D4—5081no—
179D4—5081no—
180D4—5081no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—1272no—
185C2×C4—1272no—
186C2×C4—5082no—
187C2×C4D4×D12711yescharacteristic
188C2×C4D4×D12711yescharacteristic
189C2×C4D4×D12711yescharacteristic
190C2×C4C2×C22⋊D12711yescharacteristic
191C2×C4#1452324711yescharacteristic
192C2×C4#1452324811yescharacteristic
193C8—22no—
194C8—41no—
195C8—5081no—
196C8—2542no—
197C22—22no—
198C22—10162no—
199C22—10162no—
200C22—5082no—
201C22—5084no—
202C22C22⋊C4×D12711yescharacteristic
203C22(C22⋊D127)⋊C411yescharacteristic
204C22C4×C22⋊D12711yescharacteristic
205C4—42no—
206C4—2542no—
207C4—2542no—
208C4#1686118111yescharacteristic
209C4#1686118211yescharacteristic
210C4#1686119511yescharacteristic
211C4#1686120511yescharacteristic
212C2—21no—
213C2—21no—
214C2—10162no—
215C2#2254977611yescharacteristic
216C2#2254985912yes—
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#101×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×D508C22
cosocleC22×C4×C127C23
hypercenterC2×#101D127
hypocenterC127#1967
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. #22549899C2semidirect
3. C2×#16861128C2semidirect
6. M16×Dic127C2semidirect
7. C2×#101×C127C2semidirect
109. C127#1967semidirect
113. C2×#101D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611982
#225498992
C2×#168611282
C2×#168611252
M16×Dic1272
C2×#101×C1272
#1967127