Finite Group Databasebeta

← #30637778

Subgroups — #30637778

Order 16256.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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#22550209C211yesmaximal, characteristic, complemented
2#22550192C211yesmaximal, characteristic, complemented
3#22549812C211yesmaximal, characteristic, complemented
4#22549613C211yesmaximal, characteristic, complemented
5#22549610C211yesmaximal, characteristic, complemented
6#22549600C211yesmaximal, characteristic, complemented
7#538×C127C211yesmaximal, characteristic
8C2×C4⋊Dic127C2211yescharacteristic
9C2×Dic127⋊C4—21no—
10C2×C22⋊Dic127—21no—
11#16861142—21no—
12#16861140—21no—
13#16861133—21no—
14#16861130—21no—
15C22⋊(C127⋊C8)C2211yescharacteristic
16C22⋊(C127⋊C8)C2211yescharacteristic
17C127⋊#95C2211yescharacteristic
18#126×C127—21no—
19#124×C127—21no—
20#123×C127—21no—
21C22⋊Q8×C127C2211yescharacteristic
22#121×C127—21no—
23#121×C127C2211yescharacteristic
24#120×C127—21no—
25C4×D4×C127—21no—
26C2×C22⋊C4×C127C2211yescharacteristic
27C22×Dic127—21no—
28C22×Dic127—21no—
29C2×C127⋊C8—21no—
30C2×C127⋊C8—21no—
31C4⋊Dic127—21no—
32C4⋊Dic127—21no—
33Dic127⋊C4—42no—
34C22⋊Dic127—42no—
35C24×C127D411yescharacteristic
36C2×Q8×C127D411yescharacteristic
37C2×D4×C127—21no—
38C2×D4×C127—21no—
39C2×D4×C127—21no—
40C2×D4×C127D411yescharacteristic
41C22×C4×C127—21no—
42C22×C4×C127D411yescharacteristic
43C22×C4×C127C2311yescharacteristic
44C4⋊C4×C127—21no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127D411yescharacteristic
47C4⋊C4×C127D411yescharacteristic
48C22⋊C4×C127—21no—
49C22⋊C4×C127—21no—
50C22⋊C4×C127—21no—
51C22⋊C4×C127—21no—
52C22⋊C4×C127—21no—
53C22⋊C4×C127—21no—
54C22⋊C4×C127—21no—
55C42×C127—21no—
56C2×Dic127—22no—
57C2×Dic127—41no—
58C2×Dic127—41no—
59C2×Dic127—41no—
60C2×Dic127—41no—
61C2×Dic127—41no—
62C127⋊C8—41no—
63C127⋊C8—41no—
64C23×C127—21no—
65C23×C127—21no—
66C23×C127—21no—
67C23×C127—21no—
68C23×C127—41no—
69C23×C127—41no—
70C23×C127C2×D411yescharacteristic
71Q8×C127—41no—
72D4×C127—41no—
73D4×C127—41no—
74D4×C127—41no—
75D4×C127—81no—
76D4×C127—81no—
77C2×C4×C127—21no—
78C2×C4×C127—21no—
79C2×C4×C127—21no—
80C2×C4×C127—21no—
81C2×C4×C127—41no—
82C2×C4×C127—41no—
83C2×C4×C127—81no—
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C22×C127—21no—
87C22×C127—21no—
88C22×C127—42no—
89C22×C127—41no—
90C22×C127—41no—
91C22×C127—41no—
92C22×C127—41no—
93C22×C127—41no—
94C22×C127—41no—
95C22×C127—81no—
96C22×C127#12011yescharacteristic
97C22×C127#13511yescharacteristic
98C22×C127#13711yescharacteristic
99C4×C127—21no—
100C4×C127—21no—
101C4×C127—41no—
102C4×C127—41no—
103C4×C127—41no—
104C4×C127—41no—
105Dic127—42no—
106Dic127—81no—
107C2×C127—21no—
108C2×C127—21no—
109C2×C127—81no—
110C2×C127—81no—
111C2×C127#45211yescharacteristic
112C2×C127#45311yescharacteristic
113C2×C127#45811yescharacteristic
114#1618—1271nomaximal, Sylow
115C127#161811yescharacteristic, complemented, Sylow
116#538D12711yescharacteristic
117#484—1271no—
118#481—1271no—
119#399—1271no—
120#328—1271no—
121#327—1271no—
122#323—1271no—
123#126—21no—
124#124—21no—
125#123—21no—
126C22⋊Q8C2×D12711yescharacteristic
127#121—21no—
128#121C2×D12711yescharacteristic
129#120—21no—
130C4×D4—21no—
131C2×C4⋊C4—1271no—
132C2×C4⋊C4—2541no—
133C2×C22⋊C4—2541no—
134C2×C22⋊C4C2×D12711yescharacteristic
135#107—2541no—
136#106—2541no—
137#103—2541no—
138#102—2541no—
139C22⋊C8—1271no—
140C22⋊C8—1271no—
141#95—1271no—
142C24C22⋊D12711yescharacteristic
143C2×Q8C22⋊D12711yescharacteristic
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4C22⋊D12711yescharacteristic
148C22×C4—21no—
149C22×C4—2541no—
150C22×C4—2541no—
151C22×C4C22⋊D12711yescharacteristic
152C22×C4C22×D12711yescharacteristic
153C2×C8—2541no—
154C2×C8—2541no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—5082no—
158C4⋊C4—2541no—
159C4⋊C4—2541no—
160C4⋊C4C22⋊D12711yescharacteristic
161C4⋊C4C22⋊D12711yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—21no—
166C22⋊C4—21no—
167C22⋊C4—21no—
168C22⋊C4—21no—
169C22⋊C4—5082no—
170C42—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—41no—
176C23—41no—
177C23C2×C22⋊D12711yescharacteristic
178Q8—41no—
179D4—41no—
180D4—41no—
181D4—41no—
182D4—81no—
183D4—81no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—81no—
191C2×C4—5081no—
192C2×C4—5081no—
193C2×C4—5081no—
194C2×C4—5081no—
195C2×C4—5081no—
196C2×C4—2542no—
197C2×C4C2×C22⋊D12711yescharacteristic
198C2×C4C2×C22⋊D12711yescharacteristic
199C8—5081no—
200C8—5081no—
201C22—21no—
202C22—21no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—81no—
211C22#1686117811yescharacteristic
212C22#1686122911yescharacteristic
213C22#1686123911yescharacteristic
214C4—21no—
215C4—21no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—10161no—
221C4—5082no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—81no—
226C2#2255003811yescharacteristic
227C2#2255004411yescharacteristic
228C2#2255006611yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting, prime radical#538×C127C2
socleC22×C127#120
hypercenter#538D127
hypocenterC127#1618
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #538

Complements

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

NH ≅ G/Ntype
1. #22550209C2semidirect
2. #22550192C2semidirect
3. #22549812C2semidirect
4. #22549613C2semidirect
5. #22549610C2semidirect
6. #22549600C2semidirect
115. C127#1618semidirect

Maximal subgroups

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

subgroupindex
#225502092
#225501922
#225498122
#225496132
#225496102
#225496002
#538×C1272
#1618127