Finite Group Databasebeta

← #30639576

Subgroups — #30639576

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×#16861196C211yesmaximal, characteristic
2#22549899C211yesmaximal, characteristic, complemented
3C2×#16861127C211yesmaximal, characteristic, complemented
4C2×#16861127C211yesmaximal, characteristic, complemented
5C2×#16861126C211yesmaximal, characteristic
6M16×Dic127C211yesmaximal, characteristic, complemented
7C2×#100×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C2×#14523232C2211yescharacteristic
11#16861196—21no—
12#16861196—21no—
13#16861196—22no—
14C2×C22⋊Dic127C414yes—
15C2×C4×Dic127C2211yescharacteristic
16Dic127⋊C8—22no—
17#16861127—22no—
18#16861127—22no—
19#16861126—22no—
20#16861119—21no—
21C8×Dic127—21no—
22C22×D4×C127C2211yescharacteristic
23C2×M16×C127C2211yescharacteristic
24C2×M16×C127C2211yescharacteristic
25#100×C127—22no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254—21no—
32C22×Dic127C2×C412yes—
33C22×Dic127C2×C412yes—
34#14523232—21no—
35#14523232—21no—
36#14523232—21no—
37#14523232D412yes—
38C2×C127⋊C8—21no—
39C2×C127⋊C8—21no—
40C22⋊Dic127—24no—
41C22⋊Dic127—24no—
42C4×Dic127D411yescharacteristic
43C4×Dic127D411yescharacteristic
44C4×Dic127D412yes—
45C24×C127C2×C412yes—
46C2×D4×C127—21no—
47C2×D4×C127—21no—
48C2×D4×C127—21no—
49C2×D4×C127—21no—
50C2×D4×C127—21no—
51C2×D4×C127—21no—
52C22×C4×C127C2311yescharacteristic
53M16×C127—21no—
54M16×C127—21no—
55M16×C127—21no—
56M16×C127D412yes—
57C2×C8×C127—21no—
58C2×C8×C127—21no—
59Dic254—41no—
60Dic254—41no—
61Dic254—41no—
62Dic254—41no—
63C2×Dic127—22no—
64C2×Dic127—22no—
65C2×Dic127—22no—
66C2×Dic127—42no—
67C2×Dic127C22⋊C412yes—
68C2×Dic127C22⋊C412yes—
69C127⋊C8—22no—
70C127⋊C8—41no—
71C23×C127—22no—
72C23×C127—22no—
73C23×C127—22no—
74C23×C127—44no—
75C23×C127C22×C411yescharacteristic
76D4×C127—41no—
77D4×C127—41no—
78D4×C127—41no—
79D4×C127—41no—
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×C127—42no—
90C22×C127—44no—
91C22×C127—82no—
92C22×C127—82no—
93C22×C127C2×C22⋊C411yescharacteristic
94C22×C127C4×D411yescharacteristic
95C22×C127C4×D411yescharacteristic
96C4×C127#12111yescharacteristic
97C4×C127#12111yescharacteristic
98C4×C127#12411yescharacteristic
99C4×C127#12711yescharacteristic
100Dic127—22no—
101Dic127—22no—
102Dic127—42no—
103C2×C127—21no—
104C2×C127—21no—
105C2×C127—82no—
106C2×C127#39011yescharacteristic
107C2×C127#41312yes—
108#1967—1271nomaximal, Sylow
109C127#196711yescharacteristic, complemented, Sylow
110C2×#124—1271no—
111#423—1271no—
112C2×#101—1271no—
113C2×#101—1271no—
114C2×#100—1271no—
115C2×#100D12711yescharacteristic
116C4×M16—1271no—
117C22×Q8—1271no—
118C22×D4C2×D12711yescharacteristic
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—2542no—
130#101—2542no—
131#100—22no—
132#100—2542no—
133#97—2541no—
134C4×C8—2541no—
135C24C4×D12712yes—
136C2×Q8—2541no—
137C2×Q8—2541no—
138C2×Q8—2541no—
139C2×Q8—2541no—
140C2×Q8—2541no—
141C2×Q8—2541no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C22×C4—1272no—
149C22×C4—1272no—
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—22no—
169C23—22no—
170C23—22no—
171C23—44no—
172C23C2×C4×D12711yescharacteristic
173Q8—5081no—
174Q8—5081no—
175Q8—5081no—
176Q8—5081no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—41no—
181C2×C4—1272no—
182C2×C4—1272no—
183C2×C4—5082no—
184C2×C4—2542no—
185C2×C4—2542no—
186C2×C4—2542no—
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—42no—
199C22—44no—
200C22—82no—
201C22—82no—
202C22C22⋊C4×D12711yescharacteristic
203C22(C22⋊D127)⋊C411yescharacteristic
204C22C4×C22⋊D12711yescharacteristic
205C4—5082no—
206C4—2542no—
207C4—2542no—
208C4#1686118111yescharacteristic
209C4#1686118211yescharacteristic
210C4#1686119511yescharacteristic
211C4#1686120511yescharacteristic
212C2—21no—
213C2—21no—
214C2—82no—
215C2#2254977611yescharacteristic
216C2#2254985712yes—
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#100×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×D4×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#100D127
hypocenterC127#1967
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. #22549899C2semidirect
3. C2×#16861127C2semidirect
4. C2×#16861127C2semidirect
6. M16×Dic127C2semidirect
109. C127#1967semidirect

Maximal subgroups

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

subgroupindex
C2×#168611962
#225498992
C2×#168611272
C2×#168611262
M16×Dic1272
C2×#100×C1272
#1967127