Finite Group Databasebeta

← #30647056

Subgroups — #30647056

Order 16256.

688 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 177 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×#16861190C212yesmaximal, complemented
2C22×C4×Dic127C211yesmaximal, characteristic, complemented
3#22549779C212yesmaximal, complemented
4#22549772C212yesmaximal, complemented
5#22549764C214yesmaximal, complemented
6#22549756C212yesmaximal, complemented
7#22549736C211yesmaximal, characteristic, complemented
8C2×#121×C127C211yesmaximal, characteristic
9C23×Dic127C2211yescharacteristic
10C23×Dic127C2211yescharacteristic
11#16861190—28no—
12C2×C4⋊Dic127C2212yes—
13C2×Dic127⋊C4C2214yes—
14C2×C22⋊Dic127C2214yes—
15C2×C22⋊Dic127C2214yes—
16C2×C22⋊Dic127C2212yes—
17C2×C4×Dic127—24no—
18C2×C4×Dic127C2212yes—
19C2×C4×Dic127C2212yes—
20C127⋊#95C2212yes—
21C127⋊#95C2214yes—
22C22×D4×C127C2211yescharacteristic
23C22×D4×C127C2212yes—
24C23×C4×C127C2211yescharacteristic
25#121×C127—24no—
26C2×C4⋊C4×C127C2211yescharacteristic
27C2×C22⋊C4×C127C2212yes—
28C22×Dic127—24no—
29C22×Dic127—22no—
30C22×Dic127—22no—
31C22×Dic127—22no—
32C22×Dic127D414yescomplemented
33C22×Dic127C2314yes—
34C22×Dic127C2312yes—
35C22×Dic127C2312yes—
36C4⋊Dic127—24no—
37Dic127⋊C4—28no—
38C22⋊Dic127—28no—
39C22⋊Dic127—28no—
40C22⋊Dic127—24no—
41C4×Dic127—24no—
42C4×Dic127—24no—
43C24×C127C2312yes—
44C24×C127C2311yescharacteristic
45C2×D4×C127—24no—
46C2×D4×C127—24no—
47C2×D4×C127—24no—
48C2×D4×C127—22no—
49C2×D4×C127—24no—
50C22×C4×C127—24no—
51C22×C4×C127D414yes—
52C22×C4×C127C2312yes—
53C22×C4×C127C2311yescharacteristic
54C22×C4×C127C2311yescharacteristic
55C4⋊C4×C127—22no—
56C22⋊C4×C127—24no—
57C2×Dic127—28no—
58C2×Dic127—28no—
59C2×Dic127—22no—
60C2×Dic127—22no—
61C2×Dic127—22no—
62C2×Dic127—22no—
63C2×Dic127—24no—
64C2×Dic127—24no—
65C2×Dic127—24no—
66C2×Dic127C2×D414yes—
67C2×Dic127#4114yes—
68C23×C127—22no—
69C23×C127—22no—
70C23×C127—22no—
71C23×C127—22no—
72C23×C127—22no—
73C23×C127—24no—
74C23×C127—24no—
75C23×C127C2×D412yes—
76C23×C127C2×D412yes—
77C23×C127#4112yes—
78C23×C127C2411yescharacteristic
79D4×C127—48no—
80D4×C127—44no—
81C2×C4×C127—22no—
82C2×C4×C127—28no—
83C2×C4×C127—22no—
84C2×C4×C127—22no—
85C2×C4×C127—22no—
86C2×C4×C127C2×D412yes—
87C2×C4×C127C2×D412yes—
88C2×C4×C127#4114yes—
89C22×C127—22no—
90C22×C127—22no—
91C22×C127—24no—
92C22×C127—24no—
93C22×C127—42no—
94C22×C127—42no—
95C22×C127—42no—
96C22×C127—44no—
97C22×C127—44no—
98C22×C127#12114yes—
99C22×C127C22×D411yescharacteristic
100C22×C127C22×D411yescharacteristic
101C22×C127C2×#4112yes—
102C22×C127C2×#4111yescharacteristic
103C22×C127C2×#4112yes—
104C4×C127—22no—
105C4×C127—42no—
106C4×C127#12314yescomplemented
107Dic127—24no—
108Dic127—24no—
109Dic127—44no—
110C2×C127—24no—
111C2×C127—82no—
112C2×C127C2×#12111yescharacteristic
113C2×C127C2×#12311yescharacteristic
114C2×C127#52511yescharacteristic
115C2×C127#52912yes—
116C2×C127#53212yes—
117#2854—1271nomaximal, Sylow
118C127#285411yescharacteristic, complemented, Sylow
119C2×#123—1272no—
120C2×#121D12711yescharacteristic
121C22×C42—1271no—
122#390—1272no—
123#389—1272no—
124#388—1274no—
125#386—1272no—
126#380—1271no—
127C22×D4C2×D12712yes—
128C22×D4C2×D12711yescharacteristic
129C23×C4—1271no—
130C23×C4—1271no—
131C23×C4C2×D12711yescharacteristic
132#123—2548no—
133#121—24no—
134C2×C4⋊C4—1274no—
135C2×C4⋊C4—1272no—
136C2×C4⋊C4C2×D12711yescharacteristic
137C2×C22⋊C4—1272no—
138C2×C22⋊C4—1274no—
139C2×C22⋊C4—1274no—
140C2×C22⋊C4C2×D12712yes—
141C2×C42—1272no—
142C2×C42—1272no—
143C2×C42—2544no—
144#95—1272no—
145#95—1274no—
146C24C22×D12712yes—
147C24C22×D12711yescharacteristic
148C2×D4—24no—
149C2×D4—24no—
150C2×D4—24no—
151C2×D4—22no—
152C2×D4—24no—
153C22×C4—24no—
154C22×C4—1274no—
155C22×C4—1272no—
156C22×C4—1272no—
157C22×C4—1274no—
158C22×C4—2544no—
159C22×C4—2542no—
160C22×C4—2542no—
161C22×C4—2542no—
162C22×C4C22⋊D12714yes—
163C22×C4C22×D12712yes—
164C22×C4C22×D12711yescharacteristic
165C22×C4C22×D12711yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—2548no—
168C4⋊C4—2544no—
169C22⋊C4—24no—
170C22⋊C4—2548no—
171C22⋊C4—2548no—
172C22⋊C4—2544no—
173C42—2544no—
174C42—2544no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—24no—
180C23—22no—
181C23—24no—
182C23D4×D12712yes—
183C23C2×C22⋊D12712yes—
184C23#1452324712yes—
185C23C23×D12711yescharacteristic
186D4—48no—
187D4—44no—
188C2×C4—22no—
189C2×C4—28no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—1274no—
194C2×C4—1274no—
195C2×C4—2544no—
196C2×C4—2544no—
197C2×C4—2544no—
198C2×C4—2548no—
199C2×C4—2542no—
200C2×C4—2542no—
201C2×C4—2542no—
202C2×C4—2542no—
203C2×C4—2548no—
204C2×C4C2×C22⋊D12712yes—
205C2×C4C2×C22⋊D12712yes—
206C2×C4#1452324714yes—
207C22—22no—
208C22—22no—
209C22—24no—
210C22—24no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22—44no—
215C22—44no—
216C22#1686118014yes—
217C22C2×D4×D12711yescharacteristic
218C22C22×C22⋊D12711yescharacteristic
219C22#41×D12712yes—
220C22C2×#1452324711yescharacteristic
221C22C2×#1452324712yes—
222C4—22no—
223C4—42no—
224C4—5084no—
225C4—2544no—
226C4—2544no—
227C4#1686119014yescomplemented
228C2—24no—
229C2—82no—
230C2C2×#1686118011yescharacteristic
231C2C2×#1686119011yescharacteristic
232C2#2255038311yescharacteristic
233C2#2255040712yes—
234C2#2255042112yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting, prime radicalC2×#121×C127C2
hypercenterC2×#121D127
hypocenterC127#2854
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16861190C2semidirect
2. C22×C4×Dic127C2semidirect
3. #22549779C2semidirect
4. #22549772C2semidirect
5. #22549764C2semidirect
6. #22549756C2semidirect
7. #22549736C2semidirect
32. C22×Dic127D4semidirect
106. C4×C127#123semidirect
118. C127#2854semidirect
227. C4#16861190semidirect

Maximal subgroups

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

subgroupindex
C2×#168611902
C22×C4×Dic1272
#225497792
#225497722
#225497642
#225497562
#225497362
C2×#121×C1272
#2854127