Finite Group Databasebeta

← #30640724

Subgroups — #30640724

Order 16256.

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

normal subgroups only

This group has 246 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#22550610C211yesmaximal, characteristic, complemented
2#22550066C211yesmaximal, characteristic, complemented
3#22550044C211yesmaximal, characteristic
4#22549668C211yesmaximal, characteristic, complemented
5#22549666C211yesmaximal, characteristic, complemented
6#22549665C211yesmaximal, characteristic, complemented
7#458×C127C211yesmaximal, characteristic, complemented
8#16861234—21no—
9#16861205C2211yescharacteristic
10#16861195—21no—
11#16861195—21no—
12#16861194—21no—
13#16861190—21no—
14#16861190C2211yescharacteristic
15#16861174—21no—
16C4×C22⋊D127—21no—
17(C22⋊D127)⋊C4—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×D508D411yescharacteristic
29C2×C22⋊D127—21no—
30C2×C22⋊D127—21no—
31C22×Dic127D411yescharacteristic
32C2×C4×D127—21no—
33#14523235—41no—
34#14523233—41no—
35#14523232—21no—
36Dic127⋊C4—21no—
37Dic127⋊C4—21no—
38Dic127⋊C4—41no—
39C22⋊Dic127—21no—
40C22⋊Dic127—21no—
41C22⋊Dic127—41no—
42C22⋊Dic127D411yescharacteristic
43(C2×D127)⋊C4—21no—
44(C2×D127)⋊C4—21no—
45(C2×D127)⋊C4—21no—
46C4×Dic127—21no—
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
62C22×D127—21no—
63Dic254—42no—
64D508—41no—
65C22⋊D127—42no—
66C22⋊D127—42no—
67C2×Dic127—21no—
68C2×Dic127—21no—
69C2×Dic127—21no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C4×D127—81no—
73C127⋊C8—41no—
74C23×C127—21no—
75C23×C127—21no—
76C23×C127C2×D411yescharacteristic
77C23×C127C2×D411yescharacteristic
78Q8×C127—21no—
79D4×C127—21no—
80D4×C127—41no—
81D4×C127—41no—
82D4×C127—41no—
83D4×C127—41no—
84C2×C4×C127—21no—
85C2×C4×C127—21no—
86C2×C4×C127—21no—
87C2×C4×C127—21no—
88C2×C4×C127—41no—
89C2×C4×C127—41no—
90C2×C4×C127—41no—
91C2×C4×C127C2×D411yescharacteristic
92C22×C127—21no—
93C22×C127—21no—
94C22×C127—41no—
95C22×C127—41no—
96C22×C127—41no—
97C22×C127—81no—
98C22×C127—81no—
99C22×C127#12011yescharacteristic
100C2×D127—41no—
101C2×D127—41no—
102C2×D127—41no—
103C4×C127—21no—
104C4×C127—41no—
105C4×C127—41no—
106C4×C127—41no—
107C4×C127—81no—
108Dic127—22no—
109Dic127—41no—
110Dic127—41no—
111Dic127—81no—
112C2×C127—21no—
113C2×C127—41no—
114C2×C127—41no—
115C2×C127—81no—
116C2×C127B(4,𝔽2)11yescharacteristic
117D127—81no—
118#2198—1271nomaximal, Sylow
119C127#219811yescharacteristic, complemented, Sylow
120#559—1271no—
121#458—1271no—
122#458D12711yescharacteristic, complemented
123#453—1271no—
124#353—1271no—
125#352—1271no—
126#352—1271no—
127H(5,𝔽2)C2×D12711yescharacteristic
128AGL(1,ℤ/8)—2541no—
129#127—1271no—
130#124—2541no—
131#124—2541no—
132#123—21no—
133#123—1271no—
134#123—2541no—
135#123—2541no—
136#123C2×D12711yescharacteristic
137C4×Q8—2541no—
138C4×D4—2541no—
139C4×D4—2541no—
140#104—2541no—
141#100—1271no—
142#99—21no—
143#99—1271no—
144#99—2541no—
145#99C2×D12711yescharacteristic
146#41—21no—
147#41—41no—
148C2×Q8—2541no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—41no—
152C2×D4—1271no—
153C2×D4—2541no—
154C2×D4—2541no—
155C2×D4C22×D12711yescharacteristic
156C22×C4—1271no—
157C22×C4—2541no—
158C22×C4C22⋊D12711yescharacteristic
159SD16—5081no—
160D8—5081no—
161M16—2541no—
162C4⋊C4—21no—
163C4⋊C4—41no—
164C4⋊C4—5081no—
165C4⋊C4—2541no—
166C4⋊C4—2541no—
167C22⋊C4—21no—
168C22⋊C4—21no—
169C22⋊C4—41no—
170C22⋊C4—1271no—
171C22⋊C4—5081no—
172C22⋊C4—2541no—
173C22⋊C4—2541no—
174C22⋊C4—2541no—
175C22⋊C4—2541no—
176C22⋊C4—2541no—
177C22⋊C4C22⋊D12711yescharacteristic
178C42—1271no—
179C42—2541no—
180C42—2541no—
181C23—21no—
182C23—21no—
183C23—2541no—
184C23D4×D12711yescharacteristic
185C23C2×C22⋊D12711yescharacteristic
186Q8—21no—
187Q8—5082no—
188D4—21no—
189D4—41no—
190D4—41no—
191D4—41no—
192D4—41no—
193D4—5081no—
194D4—5082no—
195D4—5082no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—10161no—
204C2×C4—5081no—
205C2×C4—5081no—
206C2×C4—2541no—
207C2×C4—2541no—
208C2×C4—2541no—
209C2×C4D4×D12711yescharacteristic
210C8—5081no—
211C22—21no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—81no—
217C22—81no—
218C22—5081no—
219C22—5081no—
220C22—5081no—
221C22#1686117711yescharacteristic
222C4—21no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—81no—
227C4—10161no—
228C4—5081no—
229C4—5081no—
230C4—2542no—
231C2—21no—
232C2—41no—
233C2—41no—
234C2—81no—
235C2—10161no—
236C2#2255006211yescharacteristic
subgroupG/H
centerC2#22550062
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
Fitting#458×C127C2
socleC2×C127B(4,𝔽2)
prime radical#22550044C2
hypercenter#458D127
hypocenterC127#2198
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. #22550610C2semidirect
2. #22550066C2semidirect
4. #22549668C2semidirect
5. #22549666C2semidirect
6. #22549665C2semidirect
7. #458×C127C2semidirect
119. C127#2198semidirect
122. #458D127semidirect

Maximal subgroups

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

subgroupindex
#225506102
#225500662
#225500442
#225496682
#225496662
#225496652
#458×C1272
#2198127