Finite Group Databasebeta

← #30637702

Subgroups — #30637702

Order 16256.

238 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 51 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#22550505C211yesmaximal, characteristic
2#22550032C211yesmaximal, characteristic
3#22549795C211yesmaximal, characteristic
4#22549614C211yesmaximal, characteristic
5#22549613C211yesmaximal, characteristic
6#22549601C211yesmaximal, characteristic
7#451×C127C211yesmaximal, characteristic
8C2×#14523238—21no—
9#16861208—21no—
10#16861207—21no—
11#16861199—21no—
12#16861199—21no—
13#16861188C2211yescharacteristic
14#16861186—21no—
15#16861165—21no—
16C2×Dic127⋊C4—21no—
17C2×Dic127⋊C4C2211yescharacteristic
18#16861133—21no—
19C22⋊(C127⋊C8)C2211yescharacteristic
20C127⋊#95—21no—
21C127⋊#95C2211yescharacteristic
22C22×Q8×C127C2211yescharacteristic
23C2×Q16×C127—21no—
24C22⋊Q8×C127C2211yescharacteristic
25#103×C127—21no—
26C22⋊C8×C127C2211yescharacteristic
27C2×Dic254—21no—
28C2×Dic254D411yescharacteristic
29C22×Dic127—21no—
30C22×Dic127D411yescharacteristic
31C22×Dic127D411yescharacteristic
32#14523238—42no—
33C2×C127⋊C8—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127D411yescharacteristic
36Dic127⋊C4—21no—
37Dic127⋊C4—21no—
38Dic127⋊C4—21no—
39Dic127⋊C4—21no—
40Dic127⋊C4—21no—
41Dic127⋊C4—21no—
42Dic127⋊C4—21no—
43Dic127⋊C4—42no—
44C22⋊Dic127—21no—
45C4×Dic127—21no—
46C4×Dic127—21no—
47C2×Q8×C127—21no—
48C2×Q8×C127—21no—
49C2×Q8×C127—42no—
50C2×Q8×C127D411yescharacteristic
51C22×C4×C127—21no—
52C22×C4×C127C2311yescharacteristic
53Q16×C127—42no—
54C2×C8×C127—21no—
55C4⋊C4×C127—21no—
56C4⋊C4×C127D411yescharacteristic
57C22⋊C4×C127—21no—
58Dic254—41no—
59Dic254—81no—
60C2×Dic127—21no—
61C2×Dic127—21no—
62C2×Dic127—22no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—21no—
66C2×Dic127—41no—
67C2×Dic127—41no—
68C2×Dic127—41no—
69C2×Dic127Q1612yescomplemented
70C127⋊C8—41no—
71C23×C127C2×D411yescharacteristic
72Q8×C127—22no—
73Q8×C127—22no—
74Q8×C127—41no—
75Q8×C127—41no—
76Q8×C127—41no—
77C2×C4×C127—22no—
78C2×C4×C127—21no—
79C2×C4×C127—21no—
80C2×C4×C127—41no—
81C2×C4×C127—41no—
82C2×C4×C127—41no—
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C8×C127—41no—
86C22×C127—21no—
87C22×C127—21no—
88C22×C127#12011yescharacteristic
89C22×C127C2×Q1611yescharacteristic
90C22×C127#13711yescharacteristic
91C4×C127—21no—
92C4×C127—21no—
93C4×C127—41no—
94C4×C127—42no—
95C4×C127—41no—
96Dic127—22no—
97Dic127—42no—
98Dic127—41no—
99Dic127—41no—
100Dic127—41no—
101C2×C127—21no—
102C2×C127—21no—
103C2×C127#45111yescharacteristic
104C2×C127#45611yescharacteristic
105C2×C127B(4,𝔽2)11yescharacteristic
106#1605—1271nomaximal, Sylow
107C127#160511yescharacteristic, complemented, Sylow
108#544—1271no—
109#451—1271no—
110#451D12711yescharacteristic
111#395—1271no—
112#328—1271no—
113#328—1271no—
114#324—1271no—
115C22×Q8C2×D12711yescharacteristic
116C2×Q16—21no—
117C2×Q16—2541no—
118C4⋊Q8—2541no—
119C4⋊Q8—2541no—
120#125—2541no—
121#125—2541no—
122C22⋊Q8—1271no—
123C22⋊Q8—2541no—
124C22⋊Q8C2×D12711yescharacteristic
125#117—2541no—
126C2×C4⋊C4—1271no—
127C2×C4⋊C4—2541no—
128#103—21no—
129#103—2541no—
130C22⋊C8—1271no—
131C22⋊C8C2×D12711yescharacteristic
132#95—1271no—
133#95—2541no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—42no—
137C2×Q8—1271no—
138C2×Q8—2541no—
139C2×Q8C22⋊D12711yescharacteristic
140C22×C4—21no—
141C22×C4—1271no—
142C22×C4—1271no—
143C22×C4—2541no—
144C22×C4C22×D12711yescharacteristic
145Q16—42no—
146Q16—5082no—
147C2×C8—21no—
148C2×C8—2541no—
149C4⋊C4—21no—
150C4⋊C4—1271no—
151C4⋊C4—5082no—
152C4⋊C4—2541no—
153C4⋊C4—2541no—
154C4⋊C4—2541no—
155C4⋊C4—2541no—
156C4⋊C4—2541no—
157C4⋊C4—2541no—
158C4⋊C4—2541no—
159C4⋊C4—2541no—
160C4⋊C4C22⋊D12711yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—2541no—
163C42—2541no—
164C42—2541no—
165C23D4×D12711yescharacteristic
166Q8—22no—
167Q8—22no—
168Q8—41no—
169Q8—41no—
170Q8—41no—
171Q8—10161no—
172Q8—5081no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—1272no—
180C2×C4—5081no—
181C2×C4—5081no—
182C2×C4—5081no—
183C2×C4—2542no—
184C2×C4—2541no—
185C2×C4—2541no—
186C2×C4—2541no—
187C2×C4—2541no—
188C2×C4—2541no—
189C2×C4D4×D12711yescharacteristic
190C2×C4C2×C22⋊D12711yescharacteristic
191C8—41no—
192C8—5081no—
193C22—21no—
194C22—21no—
195C22#1686117711yescharacteristic
196C22Q16×D12711yescharacteristic
197C22#1686124111yescharacteristic
198C4—21no—
199C4—21no—
200C4—41no—
201C4—42no—
202C4—41no—
203C4—5082no—
204C4—5081no—
205C4—5081no—
206C4—5081no—
207C4—2542no—
208C2—21no—
209C2—21no—
210C2#2255003011yescharacteristic
211C2#2255005711yescharacteristic
212C2#2255006211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#451×C127C2
socleC22×C127#120
prime radicalC23×C127C2×D4
hypercenter#451D127
hypocenterC127#1605
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
69. C2×Dic127Q16semidirect
107. C127#1605semidirect

Maximal subgroups

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

subgroupindex
#225505052
#225500322
#225497952
#225496142
#225496132
#225496012
#451×C1272
#1605127