Finite Group Databasebeta

← #30640413

Subgroups — #30640413

Order 16256.

250 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 65 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#22550422C211yesmaximal, characteristic
2#22549860C211yesmaximal, characteristic, complemented
3#22549674C212yesmaximal, complemented
4#22549672C212yesmaximal, complemented
5#413×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C411yescharacteristic
7C2×#14523232—21no—
8#16861207C2211yescharacteristic
9#16861205C411yescharacteristic
10#16861198C2211yescharacteristic
11#16861196C411yescharacteristic
12#16861181—21no—
13#16861181—21no—
14C4×C22⋊D127—21no—
15C4×C22⋊D127—21no—
16C2×C4×Dic127C411yescharacteristic
17#16861128C2212yes—
18#16861126—21no—
19C2×#41×C127C2211yescharacteristic
20C2×M16×C127—21no—
21#101×C127C2212yes—
22#100×C127—21no—
23#14523248—41no—
24#14523248—41no—
25C2×Dic254C2×C411yescharacteristic
26C2×D508C2×C411yescharacteristic
27C2×C22⋊D127—21no—
28C2×C22⋊D127—21no—
29C2×C22⋊D127—21no—
30C22×Dic127—21no—
31C2×C4×D127—21no—
32#14523232—22no—
33#14523232—41no—
34C2×C127⋊C8—41no—
35Dic127⋊C4—21no—
36Dic127⋊C4—21no—
37C22⋊Dic127—21no—
38C22⋊Dic127—21no—
39(C2×D127)⋊C4—21no—
40(C2×D127)⋊C4—21no—
41C4×Dic127—21no—
42C4×Dic127C2×C411yescharacteristic
43C4×Dic127C2×C411yescharacteristic
44#41×C127—41no—
45#41×C127—41no—
46C2×Q8×C127C2311yescharacteristic
47C2×D4×C127C2×C411yescharacteristic
48C2×D4×C127D411yescharacteristic
49C2×D4×C127D411yescharacteristic
50C22×C4×C127C2×C411yescharacteristic
51C22×C4×C127D411yescharacteristic
52C22×C4×C127D411yescharacteristic
53M16×C127—22no—
54M16×C127—41no—
55C2×C8×C127—41no—
56C22×D127—21no—
57Dic254—41no—
58D508—41no—
59C22⋊D127—42no—
60C22⋊D127—42no—
61C22⋊D127—42no—
62C2×Dic127—21no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—42no—
66C4×D127—41no—
67C4×D127—41no—
68C127⋊C8—42no—
69C23×C127C22⋊C411yescharacteristic
70C23×C127C22⋊C411yescharacteristic
71C23×C127C22⋊C411yescharacteristic
72Q8×C127—41no—
73D4×C127—41no—
74D4×C127—41no—
75D4×C127—41no—
76C2×C4×C127—21no—
77C2×C4×C127—21no—
78C2×C4×C127—21no—
79C2×C4×C127—21no—
80C2×C4×C127—21no—
81C2×C4×C127—21no—
82C2×C4×C127C22⋊C411yescharacteristic
83C2×C4×C127C22×C411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C8×C127—42no—
87C22×C127—21no—
88C22×C127—21no—
89C22×C127—21no—
90C22×C127—41no—
91C22×C127—41no—
92C22×C127—41no—
93C22×C127C2×C22⋊C411yescharacteristic
94C2×D127—41no—
95C2×D127—41no—
96C2×D127—41no—
97C4×C127—21no—
98C4×C127—21no—
99C4×C127—21no—
100C4×C127#9911yescharacteristic
101C4×C127#9911yescharacteristic
102Dic127—22no—
103Dic127—41no—
104Dic127—41no—
105C2×C127—21no—
106C2×C127—41no—
107C2×C127—41no—
108C2×C127—41no—
109C2×C127C2×#9911yescharacteristic
110D127—81no—
111#2132—1271nomaximal, Sylow
112C127#213211yescharacteristic, complemented, Sylow
113#532—1271no—
114#413—1271no—
115#413D12711yescharacteristic, complemented
116#356—1272no—
117#355—1272no—
118C2×#41—1271no—
119C2×#41C2×D12711yescharacteristic
120C2×M16—21no—
121C2×M16—2541no—
122C4⋊Q8—1271no—
123#127—1271no—
124#124—1271no—
125#124—1271no—
126#121—2541no—
127#121—2541no—
128C4×D4—2541no—
129C4×D4—2541no—
130C2×C42—1271no—
131#101—1272no—
132#101C2×D12712yes—
133#100—21no—
134#100—2541no—
135#41—41no—
136#41—41no—
137#41—5081no—
138#41—5081no—
139C2×Q8—1271no—
140C2×Q8C22×D12711yescharacteristic
141C2×D4—1271no—
142C2×D4—2541no—
143C2×D4—2541no—
144C2×D4—2541no—
145C2×D4C4×D12711yescharacteristic
146C2×D4C22⋊D12711yescharacteristic
147C2×D4D50811yescharacteristic
148C22×C4—2541no—
149C22×C4—2541no—
150C22×C4C4×D12711yescharacteristic
151C22×C4C22⋊D12711yescharacteristic
152C22×C4D50811yescharacteristic
153M16—22no—
154M16—41no—
155M16—5081no—
156M16—2542no—
157C2×C8—41no—
158C2×C8—5081no—
159C4⋊C4—2541no—
160C4⋊C4—2541no—
161C22⋊C4—2541no—
162C22⋊C4—2541no—
163C22⋊C4—2541no—
164C22⋊C4—2541no—
165C42—1271no—
166C42—1271no—
167C42—2541no—
168C23—2541no—
169C23(C2×D127)⋊C411yescharacteristic
170C23(C2×D127)⋊C411yescharacteristic
171C23(C2×D127)⋊C411yescharacteristic
172Q8—41no—
173Q8—5081no—
174D4—41no—
175D4—41no—
176D4—41no—
177D4—5081no—
178D4—5082no—
179D4—5082no—
180D4—5082no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—5082no—
188C2×C4—5081no—
189C2×C4—5081no—
190C2×C4—2541no—
191C2×C4—2541no—
192C2×C4—2541no—
193C2×C4(C2×D127)⋊C411yescharacteristic
194C2×C4C2×C4×D12711yescharacteristic
195C2×C4C2×C22⋊D12711yescharacteristic
196C2×C4C2×D50811yescharacteristic
197C8—42no—
198C8—5082no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—5081no—
206C22—5081no—
207C22—5081no—
208C22C2×(C2×D127)⋊C411yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—5081no—
213C4—5081no—
214C4—2542no—
215C4#1686112311yescharacteristic
216C4#1686112311yescharacteristic
217C2—21no—
218C2—41no—
219C2—41no—
220C2—41no—
221C2—10161no—
222C2C2×#1686112311yescharacteristic
subgroupG/H
centerC4#16861123
commutatorC2×C4×C127C22×C4
FrattiniC2×Q8C22×D127
Fitting#413×C127C2
socleC2×C127C2×#99
prime radical#22550422C2
cosocleC2×Q8×C127C23
hypercenter#413D127
hypocenterC127#2132
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #22549860C2semidirect
3. #22549674C2semidirect
4. #22549672C2semidirect
5. #413×C127C2semidirect
112. C127#2132semidirect
115. #413D127semidirect

Maximal subgroups

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

subgroupindex
#225504222
#225498602
#225496742
#225496722
#413×C1272
#2132127