Finite Group Databasebeta

← #30640741

Subgroups — #30640741

Order 16256.

226 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 41 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#22550629C211yesmaximal, characteristic, complemented
2#22550066C211yesmaximal, characteristic
3#22550054C211yesmaximal, characteristic
4#22549670C211yesmaximal, characteristic, complemented
5#22549666C211yesmaximal, characteristic, complemented
6#22549665C211yesmaximal, characteristic, complemented
7#458×C127C211yesmaximal, characteristic
8#16861239—21no—
9#16861208—21no—
10#16861202—21no—
11#16861202—21no—
12#16861199—21no—
13#16861196C2211yescharacteristic
14#16861190—21no—
15#16861190C2211yescharacteristic
16#16861188—21no—
17#16861186—21no—
18#16861136—21no—
19#16861126C2211yescharacteristic
20#16861124—21no—
21#16861124C2211yescharacteristic
22H(5,𝔽2)×C127C2211yescharacteristic
23#123×C127—21no—
24#123×C127C2211yescharacteristic
25#99×C127—21no—
26#99×C127C2211yescharacteristic
27C2×Dic254—21no—
28C2×Dic254D411yescharacteristic
29C22×Dic127D411yescharacteristic
30#14523238—41no—
31#14523236—41no—
32#14523232—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127—21no—
35Dic127⋊C4—21no—
36Dic127⋊C4—21no—
37Dic127⋊C4—21no—
38Dic127⋊C4—21no—
39Dic127⋊C4—21no—
40Dic127⋊C4—41no—
41C22⋊Dic127—21no—
42C22⋊Dic127—21no—
43C22⋊Dic127—21no—
44C22⋊Dic127—21no—
45C22⋊Dic127—41no—
46C22⋊Dic127D411yescharacteristic
47C4×Dic127—21no—
48C4×Dic127D411yescharacteristic
49#41×C127—21no—
50#41×C127—41no—
51C2×D4×C127—21no—
52C2×D4×C127—21no—
53C2×D4×C127—41no—
54C2×D4×C127C2311yescharacteristic
55C22×C4×C127D411yescharacteristic
56C4⋊C4×C127—21no—
57C4⋊C4×C127—41no—
58C22⋊C4×C127—21no—
59C22⋊C4×C127—21no—
60C22⋊C4×C127—41no—
61C22⋊C4×C127D411yescharacteristic
62Dic254—41no—
63Dic254—81no—
64C2×Dic127—21no—
65C2×Dic127—21no—
66C2×Dic127—21no—
67C2×Dic127—21no—
68C2×Dic127—41no—
69C2×Dic127—41no—
70C127⋊C8—41no—
71C23×C127—21no—
72C23×C127—21no—
73C23×C127C2×D411yescharacteristic
74C23×C127C2×D411yescharacteristic
75Q8×C127—21no—
76D4×C127—21no—
77D4×C127—41no—
78D4×C127—41no—
79D4×C127—41no—
80D4×C127—41no—
81C2×C4×C127—21no—
82C2×C4×C127—21no—
83C2×C4×C127—21no—
84C2×C4×C127—21no—
85C2×C4×C127—41no—
86C2×C4×C127—41no—
87C2×C4×C127—41no—
88C2×C4×C127C2×D411yescharacteristic
89C22×C127—21no—
90C22×C127—21no—
91C22×C127—41no—
92C22×C127—41no—
93C22×C127—41no—
94C22×C127—81no—
95C22×C127—81no—
96C22×C127#12011yescharacteristic
97C4×C127—21no—
98C4×C127—41no—
99C4×C127—41no—
100C4×C127—41no—
101C4×C127—81no—
102Dic127—41no—
103Dic127—41no—
104Dic127—41no—
105Dic127—41no—
106Dic127—81no—
107C2×C127—21no—
108C2×C127—41no—
109C2×C127—41no—
110C2×C127—81no—
111C2×C127B(4,𝔽2)11yescharacteristic
112#2201—1271nomaximal, Sylow
113C127#220111yescharacteristic, complemented, Sylow
114#563—1271no—
115#458—1271no—
116#458D12711yescharacteristic
117#455—1271no—
118#354—1271no—
119#352—1271no—
120#352—1271no—
121H(5,𝔽2)C2×D12711yescharacteristic
122#137—2541no—
123C4⋊Q8—2541no—
124#126—2541no—
125#126—2541no—
126#125—2541no—
127#124—1271no—
128#123—21no—
129#123—1271no—
130#123—2541no—
131#123C2×D12711yescharacteristic
132C22⋊Q8—2541no—
133C22⋊Q8—2541no—
134#104—2541no—
135#100—1271no—
136#99—21no—
137#99—1271no—
138#99—2541no—
139#99C2×D12711yescharacteristic
140#41—21no—
141#41—41no—
142C2×Q8—1271no—
143C2×Q8—2541no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—41no—
147C2×D4C22×D12711yescharacteristic
148C22×C4—1271no—
149C22×C4C22⋊D12711yescharacteristic
150Q16—5081no—
151SD16—5081no—
152M16—2541no—
153C4⋊C4—21no—
154C4⋊C4—41no—
155C4⋊C4—5081no—
156C4⋊C4—2541no—
157C4⋊C4—2541no—
158C4⋊C4—2541no—
159C4⋊C4—2541no—
160C4⋊C4—2541no—
161C4⋊C4—2541no—
162C4⋊C4—2541no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—41no—
166C22⋊C4—1271no—
167C22⋊C4—5081no—
168C22⋊C4—2541no—
169C22⋊C4—2541no—
170C22⋊C4—2541no—
171C22⋊C4—2541no—
172C22⋊C4C22⋊D12711yescharacteristic
173C42—1271no—
174C42—2541no—
175C23—21no—
176C23—21no—
177C23D4×D12711yescharacteristic
178C23C2×C22⋊D12711yescharacteristic
179Q8—21no—
180Q8—10161no—
181Q8—5081no—
182D4—21no—
183D4—41no—
184D4—41no—
185D4—41no—
186D4—41no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—5081no—
195C2×C4—5081no—
196C2×C4—2541no—
197C2×C4—2541no—
198C2×C4—2541no—
199C2×C4—2541no—
200C2×C4D4×D12711yescharacteristic
201C8—5081no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—81no—
208C22—81no—
209C22#1686117711yescharacteristic
210C4—21no—
211C4—41no—
212C4—41no—
213C4—41no—
214C4—81no—
215C4—10161no—
216C4—5081no—
217C4—5081no—
218C4—5081no—
219C4—5081no—
220C2—21no—
221C2—41no—
222C2—41no—
223C2—81no—
224C2#2255006211yescharacteristic
subgroupG/H
centerC2#22550062
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
Fitting#458×C127C2
socleC2×C127B(4,𝔽2)
prime radicalH(5,𝔽2)×C127C22
hypercenter#458D127
hypocenterC127#2201
LCSG ▷ C2×D4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #458

Complements

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

NH ≅ G/Ntype
1. #22550629C2semidirect
4. #22549670C2semidirect
5. #22549666C2semidirect
6. #22549665C2semidirect
113. C127#2201semidirect

Maximal subgroups

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

subgroupindex
#225506292
#225500662
#225500542
#225496702
#225496662
#225496652
#458×C1272
#2201127