Finite Group Databasebeta

← #30637221

Subgroups — #30637221

Order 16256.

304 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 85 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×C22⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
2C4×C22⋊Dic127C211yesmaximal, characteristic, complemented
3#22549613C212yesmaximal, complemented
4#22549610C212yesmaximal, complemented
5#525×C127C211yesmaximal, characteristic
6C22×C127⋊C8—21no—
7C2×C22⋊Dic127—21no—
8C2×C4×Dic127—21no—
9C22⋊(C127⋊C8)—21no—
10C22⋊(C127⋊C8)C2212yes—
11C127⋊#95C2212yes—
12C2×#41×C127C411yescharacteristic, complemented
13C23×C4×C127C2211yescharacteristic
14#123×C127—21no—
15C22⋊Q8×C127C411yescharacteristic, complemented
16C22⋊Q8×C127C2211yescharacteristic
17#121×C127C411yescharacteristic, complemented
18#121×C127C2211yescharacteristic
19#120×C127—21no—
20C4×D4×C127—21no—
21C4×D4×C127—21no—
22#117×C127C411yescharacteristic, complemented
23C22×Dic127—22no—
24C2×C127⋊C8—22no—
25C2×C127⋊C8—42no—
26C22⋊Dic127—24no—
27C4×Dic127—42no—
28C24×C127D411yescharacteristic
29#41×C127—42no—
30C2×Q8×C127C2×C411yescharacteristic
31C2×D4×C127—21no—
32C2×D4×C127—21no—
33C2×D4×C127—21no—
34C2×D4×C127C2×C411yescharacteristic
35C22×C4×C127—21no—
36C22×C4×C127—42no—
37C22×C4×C127C2×C411yescharacteristic
38C22×C4×C127C2×C411yescharacteristic
39C22×C4×C127D411yescharacteristic
40C22×C4×C127D411yescharacteristic
41C22×C4×C127D411yescharacteristic
42C22×C4×C127C2311yescharacteristic
43C4⋊C4×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127C2×C411yescharacteristic
46C4⋊C4×C127C2×C411yescharacteristic
47C22⋊C4×C127—21no—
48C22⋊C4×C127—21no—
49C22⋊C4×C127—21no—
50C22⋊C4×C127—21no—
51C22⋊C4×C127—21no—
52C42×C127—21no—
53C2×Dic127—24no—
54C2×Dic127—42no—
55C2×Dic127—42no—
56C127⋊C8—42no—
57C23×C127—22no—
58C23×C127—22no—
59C23×C127—21no—
60C23×C127—41no—
61C23×C127C22⋊C411yescharacteristic
62C23×C127C22⋊C411yescharacteristic
63C23×C127C2×D411yescharacteristic
64Q8×C127—41no—
65D4×C127—41no—
66D4×C127—42no—
67D4×C127—42no—
68D4×C127—42no—
69C2×C4×C127—22no—
70C2×C4×C127—22no—
71C2×C4×C127—22no—
72C2×C4×C127—21no—
73C2×C4×C127—21no—
74C2×C4×C127—21no—
75C2×C4×C127—41no—
76C2×C4×C127—41no—
77C2×C4×C127—41no—
78C2×C4×C127—42no—
79C2×C4×C127C22⋊C411yescharacteristic
80C2×C4×C127C22⋊C411yescharacteristic
81C2×C4×C127C22×C411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C22×C127—21no—
84C22×C127—21no—
85C22×C127—21no—
86C22×C127—21no—
87C22×C127—21no—
88C22×C127—21no—
89C22×C127—21no—
90C22×C127—44no—
91C22×C127—41no—
92C22×C127—41no—
93C22×C127—41no—
94C22×C127#10412yes—
95C22×C127C2×C22⋊C411yescharacteristic
96C22×C127#13512yes—
97C4×C127—21no—
98C4×C127—21no—
99C4×C127—21no—
100C4×C127—41no—
101C4×C127—41no—
102C4×C127—41no—
103C4×C127#9912yes—
104Dic127—44no—
105C2×C127—22no—
106C2×C127—22no—
107C2×C127—41no—
108C2×C127—81no—
109C2×C127C2×#9911yescharacteristic
110C2×C127#41611yescharacteristic
111C2×C127C2×#10411yescharacteristic
112#1501—1271nomaximal, Sylow
113C127#150111yescharacteristic, complemented, Sylow
114#525D12711yescharacteristic
115C2×C22⋊C8—1271no—
116C4×C22⋊C4—1271no—
117#328—1272no—
118#327—1272no—
119C2×#41Dic12711yescharacteristic, complemented
120C23×C4C2×D12711yescharacteristic
121C22×C8—2541no—
122#123—21no—
123C22⋊Q8Dic12711yescharacteristic, complemented
124C22⋊Q8C2×D12711yescharacteristic
125#121Dic12711yescharacteristic, complemented
126#121C2×D12711yescharacteristic
127#120—21no—
128C4×D4—21no—
129C4×D4—21no—
130#117Dic12711yescharacteristic, complemented
131C2×C22⋊C4—2541no—
132C2×C42—2541no—
133C22⋊C8—1272no—
134C22⋊C8—2541no—
135#95—1272no—
136C24C22⋊D12711yescharacteristic
137#41—42no—
138C2×Q8C2×Dic12711yescharacteristic
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4C2×Dic12711yescharacteristic
143C22×C4—21no—
144C22×C4—42no—
145C22×C4—2542no—
146C22×C4C2×Dic12711yescharacteristic
147C22×C4C2×Dic12711yescharacteristic
148C22×C4C22⋊D12711yescharacteristic
149C22×C4C22⋊D12711yescharacteristic
150C22×C4C22⋊D12711yescharacteristic
151C22×C4C22×D12711yescharacteristic
152C2×C8—5082no—
153C2×C8—2542no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4C2×Dic12711yescharacteristic
157C4⋊C4C2×Dic12711yescharacteristic
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—21no—
163C22⋊C4—2544no—
164C42—21no—
165C42—5082no—
166C23—22no—
167C23—22no—
168C23—21no—
169C23—41no—
170C23C22⋊Dic12711yescharacteristic
171C23C22⋊Dic12711yescharacteristic
172C23C2×C22⋊D12711yescharacteristic
173Q8—41no—
174D4—41no—
175D4—42no—
176D4—42no—
177D4—42no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—5082no—
189C2×C4—5082no—
190C2×C4—2544no—
191C2×C4C22⋊Dic12711yescharacteristic
192C2×C4C22⋊Dic12711yescharacteristic
193C2×C4C22×Dic12711yescharacteristic
194C2×C4C2×C22⋊D12711yescharacteristic
195C8—5082no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—44no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22#1686113612yes—
208C22C2×C22⋊Dic12711yescharacteristic
209C22#1686122912yes—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—41no—
214C4—41no—
215C4—41no—
216C4—5084no—
217C4#1686112412yes—
218C2—22no—
219C2—22no—
220C2—41no—
221C2—81no—
222C2C2×#1686112411yescharacteristic
223C2#2254987211yescharacteristic
224C2C2×#1686113611yescharacteristic
subgroupG/H
centerC2×C4C22⋊Dic127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting, prime radical#525×C127C2
socleC22×C127C2×C22⋊C4
cosocleC22×C4×C127C23
hypercenter#525D127
hypocenterC127#1501
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊(C127⋊C8)C2semidirect
2. C4×C22⋊Dic127C2semidirect
3. #22549613C2semidirect
4. #22549610C2semidirect
12. C2×#41×C127C4semidirect
15. C22⋊Q8×C127C4semidirect
17. #121×C127C4semidirect
22. #117×C127C4semidirect
113. C127#1501semidirect
119. C2×#41Dic127semidirect
123. C22⋊Q8Dic127semidirect
125. #121Dic127semidirect
130. #117Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊(C127⋊C8)2
C4×C22⋊Dic1272
#225496132
#225496102
#525×C1272
#1501127