Finite Group Databasebeta

← #44549321

Subgroups — #44549321

Order 32128.

256 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 53 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#30632313C211yesmaximal, characteristic
2#30631843C212yesmaximal, complemented
3#30631423C212yesmaximal, complemented
4#30631411C211yesmaximal, characteristic
5#393×C251C211yesmaximal, characteristic
6C2×#16860800—22no—
7#22548694—21no—
8#22548694C2212yes—
9#22548692—22no—
10#22548692—22no—
11C2×C22⋊Dic251—21no—
12#22548639—22no—
13C22⋊(C251⋊C8)C2212yes—
14C22×Q8×C251C2211yescharacteristic
15C2×C22⋊C4×C251—21no—
16C2×C22⋊C4×C251C2211yescharacteristic
17#95×C251—21no—
18#95×C251C2211yescharacteristic
19C2×Dic502—21no—
20C2×Dic502D412yes—
21C22×Dic251—21no—
22C22×Dic251—21no—
23#16860800—44no—
24C2×C251⋊C8—22no—
25C4⋊Dic251—21no—
26C4⋊Dic251D412yes—
27Dic251⋊C4—22no—
28Dic251⋊C4—22no—
29C22⋊Dic251—22no—
30C22⋊Dic251—22no—
31C24×C251D411yescharacteristic
32C2×Q8×C251—22no—
33C2×Q8×C251—42no—
34C22×C4×C251—21no—
35C22×C4×C251—21no—
36C22×C4×C251D411yescharacteristic
37C22×C4×C251C2311yescharacteristic
38C22⋊C4×C251—22no—
39C22⋊C4×C251—42no—
40Dic502—42no—
41Dic502—81no—
42C2×Dic251—22no—
43C2×Dic251—22no—
44C2×Dic251—81no—
45C2×Dic251—81no—
46C251⋊C8—42no—
47C23×C251—21no—
48C23×C251—21no—
49C23×C251—41no—
50C23×C251—41no—
51C23×C251Q1612yescomplemented
52C23×C251C2×D411yescharacteristic
53Q8×C251—24no—
54Q8×C251—42no—
55C2×C4×C251—22no—
56C2×C4×C251—21no—
57C2×C4×C251—41no—
58C2×C4×C251—41no—
59C2×C4×C251—41no—
60C2×C4×C251—41no—
61C2×C4×C251—41no—
62C2×C4×C251—41no—
63C2×C4×C251—41no—
64C2×C4×C251C2×D412yes—
65C22×C251—21no—
66C22×C251—21no—
67C22×C251—22no—
68C22×C251—42no—
69C22×C251—41no—
70C22×C251—41no—
71C22×C251—81no—
72C22×C251#12011yescharacteristic
73C22×C251C2×Q1611yescharacteristic
74C22×C251#13711yescharacteristic
75C4×C251—22no—
76C4×C251—41no—
77C4×C251—42no—
78C4×C251—81no—
79Dic251—42no—
80Dic251—42no—
81C2×C251—21no—
82C2×C251—21no—
83C2×C251—42no—
84C2×C251#45111yescharacteristic
85C2×C251#45511yescharacteristic
86C2×C251B(4,𝔽2)11yescharacteristic
87C251#160211yescharacteristic, complemented, Sylow
88#1602—2511nomaximal, Sylow
89#543—2511no—
90#451—2512no—
91#393D25111yescharacteristic
92#328—2512no—
93#323—2511no—
94C22×Q8C2×D25111yescharacteristic
95C2×Q16—5022no—
96C22⋊Q8—5022no—
97C22⋊Q8—5022no—
98C22⋊Q8—5021no—
99C22⋊Q8—2512no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—5021no—
102C2×C22⋊C4C2×D25111yescharacteristic
103#103—5022no—
104C22⋊C8—2512no—
105#95—21no—
106#95C2×D25111yescharacteristic
107C24C22⋊D25111yescharacteristic
108C2×Q8—22no—
109C2×Q8—42no—
110C2×Q8—5021no—
111C2×Q8—2512no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—5021no—
115C22×C4—5021no—
116C22×C4C22⋊D25111yescharacteristic
117C22×C4C22×D25111yescharacteristic
118Q16—10044no—
119C2×C8—5022no—
120C4⋊C4—5022no—
121C4⋊C4—5021no—
122C4⋊C4—5022no—
123C4⋊C4—2512no—
124C22⋊C4—22no—
125C22⋊C4—42no—
126C22⋊C4—5022no—
127C22⋊C4—5022no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23#1686080012yescomplemented
133C23C2×C22⋊D25111yescharacteristic
134Q8—24no—
135Q8—42no—
136Q8—20081no—
137Q8—10042no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—20081no—
148C2×C4—20081no—
149C2×C4—5022no—
150C2×C4—5022no—
151C2×C4D4×D25112yes—
152C8—10042no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#2254868311yescharacteristic
161C22C2×#1686080011yescharacteristic
162C22#2254874511yescharacteristic
163C4—22no—
164C4—41no—
165C4—42no—
166C4—81no—
167C4—10042no—
168C4—10042no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#3063184211yescharacteristic
173C2#3063186411yescharacteristic
174C2#3063187311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#393×C251C2
socleC22×C251#120
prime radicalC24×C251D4
hypercenter#393D251
hypocenterC251#1602
LCSG ▷ C22×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #30631843C2semidirect
3. #30631423C2semidirect
51. C23×C251Q16semidirect
87. C251#1602semidirect
132. C23#16860800semidirect

Maximal subgroups

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

subgroupindex
#306323132
#306318432
#306314232
#306314112
#393×C2512
#1602251