Finite Group Databasebeta

← #30640727

Subgroups — #30640727

Order 16256.

246 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 41 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#22550610C211yesmaximal, characteristic, complemented
2#22550067C211yesmaximal, characteristic, complemented
3#22550065C211yesmaximal, characteristic, complemented
4#22550046C211yesmaximal, characteristic
5#22549668C211yesmaximal, characteristic, complemented
6#22549667C211yesmaximal, characteristic, complemented
7#352×C127C211yesmaximal, characteristic, complemented
8#16861254C2211yescharacteristic
9#16861237—21no—
10#16861205C2211yescharacteristic
11#16861195—21no—
12#16861195—21no—
13#16861194—21no—
14#16861193—21no—
15#16861192—21no—
16#16861190C2211yescharacteristic
17#16861174—21no—
18C4×C22⋊D127—21no—
19(C22⋊D127)⋊C4—21no—
20#16861135—21no—
21#16861124C2211yescharacteristic
22#16861123—21no—
23#16861123—21no—
24#123×C127C2211yescharacteristic
25#100×C127C2211yescharacteristic
26#99×C127C2211yescharacteristic
27#14523248—21no—
28#14523247—41no—
29C2×Dic254—21no—
30C2×D508D411yescharacteristic
31C2×C22⋊D127—21no—
32C2×C22⋊D127—21no—
33C2×C22⋊D127—21no—
34C2×C22⋊D127—21no—
35D4×D127—41no—
36C22×Dic127D411yescharacteristic
37C2×C4×D127—21no—
38#14523237—41no—
39D1016—41no—
40C4⋊Dic127—41no—
41Dic127⋊C4—21no—
42Dic127⋊C4—21no—
43Dic127⋊C4—41no—
44C22⋊Dic127—21no—
45C22⋊Dic127—21no—
46C22⋊Dic127D411yescharacteristic
47(C2×D127)⋊C4—21no—
48(C2×D127)⋊C4—21no—
49(C2×D127)⋊C4—21no—
50(C2×D127)⋊C4—41no—
51(C2×D127)⋊C4—41no—
52C4×Dic127—21no—
53C4×Dic127—21no—
54C4×Dic127D411yescharacteristic
55C2×D4×C127C2311yescharacteristic
56C22×C4×C127D411yescharacteristic
57M16×C127—21no—
58C4⋊C4×C127—21no—
59C22⋊C4×C127—21no—
60C22⋊C4×C127—21no—
61C22⋊C4×C127D411yescharacteristic
62C22×D127—21no—
63C22×D127—21no—
64C22×D127—21no—
65Dic254—21no—
66Dic254—42no—
67D508—21no—
68D508—41no—
69C22⋊D127—42no—
70C22⋊D127—42no—
71C22⋊D127—41no—
72C22⋊D127—41no—
73C22⋊D127—41no—
74C2×Dic127—21no—
75C2×Dic127—21no—
76C2×Dic127—21no—
77C2×Dic127—21no—
78C2×Dic127—21no—
79C2×Dic127—41no—
80C2×Dic127—41no—
81C4×D127—41no—
82C4×D127—81no—
83C23×C127C2×D411yescharacteristic
84C23×C127C2×D411yescharacteristic
85D4×C127—41no—
86C2×C4×C127—21no—
87C2×C4×C127—21no—
88C2×C4×C127—41no—
89C2×C4×C127—41no—
90C2×C4×C127C2×D411yescharacteristic
91C8×C127—41no—
92C22×C127—21no—
93C22×C127—21no—
94C22×C127—41no—
95C22×C127—41no—
96C22×C127#12011yescharacteristic
97C2×D127—41no—
98C2×D127—41no—
99C2×D127—41no—
100C2×D127—41no—
101C2×D127—81no—
102C2×D127—81no—
103C4×C127—21no—
104C4×C127—41no—
105C4×C127—41no—
106C4×C127—81no—
107Dic127—22no—
108Dic127—41no—
109Dic127—41no—
110Dic127—41no—
111Dic127—81no—
112C2×C127—21no—
113C2×C127—41no—
114C2×C127—41no—
115C2×C127B(4,𝔽2)11yescharacteristic
116D127—81no—
117D127—81no—
118#2198—1271nomaximal, Sylow
119C127#219811yescharacteristic, complemented, Sylow
120#559—1271no—
121#458—1271no—
122#458—1271no—
123#453—1271no—
124#353—1271no—
125#352—1271no—
126#352D12711yescharacteristic, complemented
127H(5,𝔽2)—1271no—
128AGL(1,ℤ/8)—2541no—
129#127—1271no—
130#124—2541no—
131#124—2541no—
132#123—1271no—
133#123—2541no—
134#123—2541no—
135#123—2541no—
136#123C2×D12711yescharacteristic
137C4×Q8—2541no—
138C4×D4—2541no—
139C4×D4—2541no—
140#104—2541no—
141#100C2×D12711yescharacteristic
142#99—1271no—
143#99—2541no—
144#99—2541no—
145#99C2×D12711yescharacteristic
146#41—5081no—
147#41—2541no—
148C2×Q8—2541no—
149C2×D4—1271no—
150C2×D4—5081no—
151C2×D4—2541no—
152C2×D4—2541no—
153C2×D4—2541no—
154C2×D4—2541no—
155C2×D4C22×D12711yescharacteristic
156C22×C4—1271no—
157C22×C4—2541no—
158C22×C4D50811yescharacteristic
159SD16—5081no—
160D8—5081no—
161M16—21no—
162C4⋊C4—21no—
163C4⋊C4—5081no—
164C4⋊C4—5081no—
165C4⋊C4—2541no—
166C4⋊C4—2541no—
167C22⋊C4—21no—
168C22⋊C4—21no—
169C22⋊C4—1271no—
170C22⋊C4—5081no—
171C22⋊C4—5081no—
172C22⋊C4—2541no—
173C22⋊C4—2541no—
174C22⋊C4—2541no—
175C22⋊C4—2541no—
176C22⋊C4—2541no—
177C22⋊C4D50811yescharacteristic
178C42—1271no—
179C42—2541no—
180C42—2541no—
181C23—2541no—
182C23—2541no—
183C23—2541no—
184C23D4×D12711yescharacteristic
185C23C2×D50811yescharacteristic
186Q8—5082no—
187Q8—2541no—
188D4—41no—
189D4—5081no—
190D4—5081no—
191D4—5081no—
192D4—5081no—
193D4—5082no—
194D4—5082no—
195D4—2541no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—10161no—
201C2×C4—5081no—
202C2×C4—5081no—
203C2×C4—5081no—
204C2×C4—2541no—
205C2×C4—2541no—
206C2×C4—2541no—
207C2×C4—2541no—
208C2×C4—2541no—
209C2×C4D4×D12711yescharacteristic
210C8—41no—
211C22—21no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—10161no—
216C22—10161no—
217C22—5081no—
218C22—5081no—
219C22—5081no—
220C22—5081no—
221C22#1686117911yescharacteristic
222C4—21no—
223C4—41no—
224C4—41no—
225C4—81no—
226C4—10161no—
227C4—5081no—
228C4—5081no—
229C4—5081no—
230C4—2542no—
231C2—21no—
232C2—41no—
233C2—41no—
234C2—10161no—
235C2—10161no—
236C2#2255006411yescharacteristic
subgroupG/H
centerC2#22550064
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
Fitting#352×C127C2
socleC2×C127B(4,𝔽2)
prime radical#22550046C2
hypercenter#352D127
hypocenterC127#2198
LCSG ▷ C2×D4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
1. #22550610C2semidirect
2. #22550067C2semidirect
3. #22550065C2semidirect
5. #22549668C2semidirect
6. #22549667C2semidirect
7. #352×C127C2semidirect
119. C127#2198semidirect
126. #352D127semidirect

Maximal subgroups

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

subgroupindex
#225506102
#225500672
#225500652
#225500462
#225496682
#225496672
#352×C1272
#2198127