Finite Group Databasebeta

← #30640393

Subgroups — #30640393

Order 16256.

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

normal subgroups only

This group has 258 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#22549850C211yesmaximal, characteristic, complemented
3#22549670C212yesmaximal, complemented
4#22549668C212yesmaximal, complemented
5#410×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C411yescharacteristic, complemented
7#16861207C411yescharacteristic, complemented
8#16861205C2211yescharacteristic
9#16861198C411yescharacteristic, complemented
10#16861196C2211yescharacteristic
11#16861181—21no—
12#16861181—21no—
13C4×C22⋊D127—21no—
14C4×C22⋊D127—21no—
15#16861168—21no—
16C2×C4×Dic127C411yescharacteristic, complemented
17#16861124—21no—
18#16861124C2212yes—
19C2×#41×C127C2211yescharacteristic
20#117×C127—21no—
21#99×C127—21no—
22#99×C127C2212yes—
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—
32C4⋊Dic127—41no—
33Dic127⋊C4—21no—
34Dic127⋊C4—21no—
35C22⋊Dic127—21no—
36C22⋊Dic127—21no—
37C22⋊Dic127—22no—
38(C2×D127)⋊C4—21no—
39(C2×D127)⋊C4—21no—
40C4×Dic127—21no—
41C4×Dic127—41no—
42C4×Dic127C2×C411yescharacteristic
43C4×Dic127C2×C411yescharacteristic
44#41×C127—41no—
45#41×C127—41no—
46C2×Q8×C127C2×C411yescharacteristic
47C2×D4×C127D411yescharacteristic
48C2×D4×C127D411yescharacteristic
49C2×D4×C127C2311yescharacteristic
50C22×C4×C127C2×C411yescharacteristic
51C22×C4×C127D411yescharacteristic
52C22×C4×C127D411yescharacteristic
53C4⋊C4×C127—41no—
54C22⋊C4×C127—22no—
55C42×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—
66C2×Dic127—42no—
67C4×D127—41no—
68C4×D127—41no—
69C23×C127C22⋊C411yescharacteristic
70C23×C127C2×D411yescharacteristic
71C23×C127C2×D411yescharacteristic
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×C127—42no—
83C2×C4×C127C22⋊C411yescharacteristic
84C2×C4×C127C22⋊C411yescharacteristic
85C2×C4×C127C22⋊C411yescharacteristic
86C2×C4×C127C22×C411yescharacteristic
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—82no—
101C4×C127#9911yescharacteristic
102C4×C127#9911yescharacteristic
103Dic127—22no—
104Dic127—41no—
105Dic127—41no—
106Dic127—82no—
107C2×C127—21no—
108C2×C127—41no—
109C2×C127—41no—
110C2×C127—41no—
111C2×C127C2×#9911yescharacteristic
112D127—81no—
113#2126—1271nomaximal, Sylow
114C127#212611yescharacteristic, complemented, Sylow
115#532—1271no—
116#410—1271no—
117#410D12711yescharacteristic, complemented
118#354—1272no—
119#353—1272no—
120C2×#41—1271no—
121C2×#41C2×D12711yescharacteristic
122C4⋊Q8—1271no—
123#127—1271no—
124#124—1271no—
125#124—1271no—
126#121—2541no—
127#121—2541no—
128C4×D4—2541no—
129C4×D4—2541no—
130#117—21no—
131#117—2541no—
132C2×C42—1271no—
133#99—21no—
134#99—1272no—
135#99—2541no—
136#99C2×D12712yes—
137#41—41no—
138#41—41no—
139#41—5081no—
140#41—5081no—
141C2×Q8—1271no—
142C2×Q8C4×D12711yescharacteristic
143C2×D4—1271no—
144C2×D4—2541no—
145C2×D4—2541no—
146C2×D4—2541no—
147C2×D4C22⋊D12711yescharacteristic
148C2×D4D50811yescharacteristic
149C2×D4C22×D12711yescharacteristic
150C22×C4—2541no—
151C22×C4—2541no—
152C22×C4C4×D12711yescharacteristic
153C22×C4C22⋊D12711yescharacteristic
154C22×C4D50811yescharacteristic
155C4⋊C4—41no—
156C4⋊C4—5081no—
157C4⋊C4—2541no—
158C4⋊C4—2541no—
159C22⋊C4—22no—
160C22⋊C4—2542no—
161C22⋊C4—2541no—
162C22⋊C4—2541no—
163C22⋊C4—2541no—
164C22⋊C4—2541no—
165C42—41no—
166C42—1271no—
167C42—1271no—
168C42—5081no—
169C42—2541no—
170C23—2541no—
171C23(C2×D127)⋊C411yescharacteristic
172C23C2×C22⋊D12711yescharacteristic
173C23C2×D50811yescharacteristic
174Q8—41no—
175Q8—5081no—
176D4—41no—
177D4—41no—
178D4—41no—
179D4—5081no—
180D4—5082no—
181D4—5082no—
182D4—5082no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—5082no—
191C2×C4—5082no—
192C2×C4—5081no—
193C2×C4—5081no—
194C2×C4—2541no—
195C2×C4—2541no—
196C2×C4—2541no—
197C2×C4(C2×D127)⋊C411yescharacteristic
198C2×C4(C2×D127)⋊C411yescharacteristic
199C2×C4(C2×D127)⋊C411yescharacteristic
200C2×C4C2×C4×D12711yescharacteristic
201C22—21no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—5081no—
208C22—5081no—
209C22—5081no—
210C22C2×(C2×D127)⋊C411yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—82no—
215C4—10162no—
216C4—5081no—
217C4—5081no—
218C4—2542no—
219C4#1686112311yescharacteristic
220C4#1686112311yescharacteristic
221C2—21no—
222C2—41no—
223C2—41no—
224C2—41no—
225C2—10161no—
226C2C2×#1686112311yescharacteristic
subgroupG/H
centerC4#16861123
commutatorC2×C4×C127C22×C4
FrattiniC2×D4C22×D127
Fitting#410×C127C2
socleC2×C127C2×#99
prime radical#22550422C2
cosocleC2×D4×C127C23
hypercenter#410D127
hypocenterC127#2126
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #22549850C2semidirect
3. #22549670C2semidirect
4. #22549668C2semidirect
5. #410×C127C2semidirect
6. C2×#14523248C4semidirect
7. #16861207C4semidirect
9. #16861198C4semidirect
16. C2×C4×Dic127C4semidirect
114. C127#2126semidirect
117. #410D127semidirect

Maximal subgroups

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

subgroupindex
#225504222
#225498502
#225496702
#225496682
#410×C1272
#2126127