Finite Group Databasebeta

← #30640035

Subgroups — #30640035

Order 16256.

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

normal subgroups only

This group has 264 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#22550674C211yesmaximal, characteristic
2#22549903C211yesmaximal, characteristic, complemented
3#22549872C211yesmaximal, characteristic, complemented
4#22549870C211yesmaximal, characteristic, complemented
5#22549849C211yesmaximal, characteristic, complemented
6#22549632C211yesmaximal, characteristic, complemented
7#410×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861240—21no—
10#16861239—21no—
11#16861235—21no—
12#16861234—21no—
13#16861229—22no—
14C2×#14523232C2211yescharacteristic
15C22×C127⋊C8C2211yescharacteristic
16#16861168C2211yescharacteristic
17C4⋊(C127⋊C8)—22no—
18#16861133—21no—
19#16861132—21no—
20#16861130—21no—
21#16861129—21no—
22#16861123—22no—
23C2×#41×C127C2211yescharacteristic
24#117×C127C2211yescharacteristic
25#117×C127C2211yescharacteristic
26#99×C127—22no—
27#14523248—21no—
28#14523248—21no—
29#14523248—21no—
30#14523248—21no—
31C2×Dic254D411yescharacteristic
32C2×D508D411yescharacteristic
33C2×C22⋊D127—21no—
34C2×C4×D127—21no—
35#14523238—22no—
36#14523236—22no—
37#14523235—22no—
38#14523233—22no—
39#14523232—21no—
40#14523232—21no—
41C2×C127⋊C8—21no—
42C2×C127⋊C8—21no—
43C2×C127⋊C8—21no—
44C2×C127⋊C8D412yes—
45C4⋊Dic127—21no—
46C22⋊Dic127—21no—
47C4×Dic127—21no—
48#41×C127—21no—
49#41×C127—21no—
50#41×C127—21no—
51#41×C127—21no—
52C2×Q8×C127D411yescharacteristic
53C2×D4×C127—21no—
54C2×D4×C127D411yescharacteristic
55C22×C4×C127—21no—
56C22×C4×C127C2311yescharacteristic
57C4⋊C4×C127—21no—
58C4⋊C4×C127—21no—
59C22⋊C4×C127—21no—
60C22⋊C4×C127D412yes—
61C42×C127—21no—
62C42×C127—21no—
63C22×D127—21no—
64Dic254—21no—
65Dic254—21no—
66D508—21no—
67D508—21no—
68C22⋊D127—41no—
69C22⋊D127—41no—
70C2×Dic127—21no—
71C2×Dic127—41no—
72C4×D127—41no—
73C4×D127—41no—
74C127⋊C8—22no—
75C127⋊C8—41no—
76C23×C127—21no—
77C23×C127C2×D411yescharacteristic
78Q8×C127—21no—
79Q8×C127—21no—
80D4×C127—21no—
81D4×C127—21no—
82D4×C127—41no—
83D4×C127—41no—
84C2×C4×C127—22no—
85C2×C4×C127—21no—
86C2×C4×C127—41no—
87C2×C4×C127—41no—
88C2×C4×C127—41no—
89C2×C4×C127C2×D411yescharacteristic
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127#4111yescharacteristic
93C2×C4×C127#4111yescharacteristic
94C2×C4×C127#4111yescharacteristic
95C22×C127—41no—
96C22×C127—41no—
97C22×C127—81no—
98C22×C127#12111yescharacteristic
99C22×C127#12111yescharacteristic
100C22×C127#12111yescharacteristic
101C2×D127—41no—
102C2×D127—81no—
103C4×C127—42no—
104C4×C127—41no—
105C4×C127—81no—
106C4×C127#12011yescharacteristic
107C4×C127#12311yescharacteristic
108C4×C127#12311yescharacteristic
109C4×C127#12411yescharacteristic
110Dic127—41no—
111Dic127—81no—
112C2×C127—21no—
113C2×C127—21no—
114C2×C127—21no—
115C2×C127—81no—
116C2×C127#39411yescharacteristic
117D127—81no—
118#2053—1271nomaximal, Sylow
119C127#205311yescharacteristic, complemented, Sylow
120#575—1271no—
121#424—1271no—
122#416—1271no—
123#416—1271no—
124#410—1271no—
125#410D12711yescharacteristic, complemented
126H(3,ℤ/4)—1271no—
127C2×#41—1271no—
128C2×#41C2×D12711yescharacteristic
129#137—2541no—
130#137—2541no—
131AGL(1,ℤ/8)—2541no—
132AGL(1,ℤ/8)—2541no—
133#135—2542no—
134C2×M16—1271no—
135C22×C8—1271no—
136#117—1271no—
137#117C2×D12711yescharacteristic
138#117C2×D12711yescharacteristic
139C4⋊C8—2542no—
140#103—2541no—
141#103—2541no—
142#102—2541no—
143#102—2541no—
144#99—22no—
145#99—2542no—
146#41—21no—
147#41—21no—
148#41—21no—
149#41—21no—
150#41—2541no—
151#41—2541no—
152#41—2541no—
153#41—2541no—
154C2×Q8—1271no—
155C2×Q8D50811yescharacteristic
156C2×D4—21no—
157C2×D4—1271no—
158C2×D4—2541no—
159C2×D4D50811yescharacteristic
160C22×C4—21no—
161C22×C4—2541no—
162C22×C4C22×D12711yescharacteristic
163Q16—2542no—
164SD16—2542no—
165SD16—2542no—
166D8—2542no—
167M16—2541no—
168M16—2541no—
169C2×C8—1272no—
170C2×C8—2541no—
171C2×C8—2541no—
172C2×C8—2541no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—2541no—
176C22⋊C4—21no—
177C22⋊C4—2541no—
178C22⋊C4C22⋊D12712yes—
179C42—21no—
180C42—21no—
181C42—2541no—
182C23—21no—
183C23—2541no—
184C23C2×C22⋊D12711yescharacteristic
185Q8—21no—
186Q8—21no—
187Q8—2541no—
188Q8—2541no—
189D4—21no—
190D4—21no—
191D4—41no—
192D4—41no—
193D4—5081no—
194D4—5081no—
195D4—2541no—
196D4—2541no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—5081no—
203C2×C4—5081no—
204C2×C4—5081no—
205C2×C4—2541no—
206C2×C4D4×D12711yescharacteristic
207C2×C4D4×D12711yescharacteristic
208C2×C4C2×D50811yescharacteristic
209C2×C4#1452324711yescharacteristic
210C2×C4#1452324711yescharacteristic
211C2×C4#1452324811yescharacteristic
212C8—5081no—
213C8—2542no—
214C22—41no—
215C22—41no—
216C22—81no—
217C22—10161no—
218C22—5081no—
219C22#1686118011yescharacteristic
220C22#1686118111yescharacteristic
221C22#1686118311yescharacteristic
222C4—42no—
223C4—41no—
224C4—81no—
225C4—10161no—
226C4—5081no—
227C4#1686117911yescharacteristic
228C4#1686119111yescharacteristic
229C4#1686119311yescharacteristic
230C4#1686119511yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—81no—
235C2—10161no—
236C2#2254979211yescharacteristic
subgroupG/H
centerC4#16861195
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#410×C127C2
socleC2×C127#394
prime radical#22550674C2
hypercenter#410D127
hypocenterC127#2053
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #22549903C2semidirect
3. #22549872C2semidirect
4. #22549870C2semidirect
5. #22549849C2semidirect
6. #22549632C2semidirect
7. #410×C127C2semidirect
119. C127#2053semidirect
125. #410D127semidirect

Maximal subgroups

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

subgroupindex
#225506742
#225499032
#225498722
#225498702
#225498492
#225496322
#410×C1272
#2053127