Finite Group Databasebeta

← #44549358

Subgroups — #44549358

Order 32128.

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

normal subgroups only

This group has 228 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#30632018C212yesmaximal, complemented
3#30631612C211yesmaximal, characteristic
4#30631425C212yesmaximal, complemented
5#323×C251C211yesmaximal, characteristic
6#22548694—21no—
7#22548694C2212yes—
8#22548692—22no—
9#22548692—22no—
10C2×C4⋊Dic251C2211yescharacteristic
11C2×C22⋊Dic251—21no—
12C2×C22⋊Dic251—21no—
13#22548647—22no—
14#22548640—22no—
15C251⋊#95—21no—
16C251⋊#95C2211yescharacteristic
17C2×C22⋊C4×C251C2211yescharacteristic
18C22⋊C8×C251C2212yes—
19C2×Dic502—21no—
20C2×Dic502D412yes—
21C22×Dic251—21no—
22C22×Dic251—21no—
23C22×Dic251—21no—
24C22×Dic251—21no—
25C4⋊Dic251—21no—
26C4⋊Dic251—22no—
27C4⋊Dic251D412yes—
28Dic251⋊C4—22no—
29Dic251⋊C4—22no—
30C22⋊Dic251—22no—
31C22⋊Dic251—22no—
32C22⋊Dic251—42no—
33C24×C251D411yescharacteristic
34C22×C4×C251D411yescharacteristic
35C22×C4×C251C2311yescharacteristic
36C2×C8×C251—22no—
37C22⋊C4×C251—22no—
38Dic502—42no—
39Dic502—81no—
40C2×Dic251—22no—
41C2×Dic251—22no—
42C2×Dic251—41no—
43C2×Dic251—41no—
44C2×Dic251—41no—
45C2×Dic251—41no—
46C2×Dic251—41no—
47C2×Dic251—41no—
48C2×Dic251—81no—
49C2×Dic251—81no—
50C23×C251—21no—
51C23×C251—21no—
52C23×C251—41no—
53C23×C251—41no—
54C23×C251SD1612yes—
55C23×C251C2×D411yescharacteristic
56C2×C4×C251—21no—
57C2×C4×C251—41no—
58C2×C4×C251—41no—
59C2×C4×C251C2×D412yes—
60C8×C251—42no—
61C22×C251—21no—
62C22×C251—21no—
63C22×C251—22no—
64C22×C251—42no—
65C22×C251—41no—
66C22×C251—41no—
67C22×C251—81no—
68C22×C251#12011yescharacteristic
69C22×C251C2×SD1611yescharacteristic
70C22×C251#13711yescharacteristic
71C4×C251—22no—
72C4×C251—41no—
73Dic251—42no—
74Dic251—42no—
75Dic251—81no—
76Dic251—81no—
77C2×C251—21no—
78C2×C251—21no—
79C2×C251—42no—
80C2×C251#44811yescharacteristic
81C2×C251#45511yescharacteristic
82C2×C251#45811yescharacteristic
83C251#160811yescharacteristic, complemented, Sylow
84#1608—2511nomaximal, Sylow
85#543—2511no—
86#483—2512no—
87#397—2511no—
88#328—2512no—
89#323D25111yescharacteristic
90C22⋊Q8—5022no—
91C22⋊Q8—5022no—
92C22⋊Q8—5021no—
93C22⋊Q8—2512no—
94C2×C4⋊C4—2511no—
95C2×C22⋊C4—5021no—
96C2×C22⋊C4—5021no—
97C2×C22⋊C4C2×D25111yescharacteristic
98#106—5022no—
99#103—5022no—
100C22⋊C8C2×D25112yes—
101#95—5021no—
102#95—2511no—
103C24D100411yescharacteristic
104C2×Q8—5021no—
105C2×Q8—2512no—
106C22×C4—5021no—
107C22×C4—5021no—
108C22×C4—5021no—
109C22×C4—5021no—
110C22×C4D100411yescharacteristic
111C22×C4C22×D25111yescharacteristic
112C2×C8—22no—
113C4⋊C4—5022no—
114C4⋊C4—5021no—
115C4⋊C4—5022no—
116C4⋊C4—5022no—
117C4⋊C4—2512no—
118C22⋊C4—22no—
119C22⋊C4—10042no—
120C22⋊C4—5022no—
121C22⋊C4—5022no—
122C23—21no—
123C23—21no—
124C23—41no—
125C23—41no—
126C23#1686079912yes—
127C23C2×D100411yescharacteristic
128Q8—20081no—
129Q8—10042no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—20081no—
134C2×C4—20081no—
135C2×C4—10041no—
136C2×C4—10041no—
137C2×C4—10041no—
138C2×C4—10041no—
139C2×C4—10041no—
140C2×C4—10041no—
141C2×C4—5022no—
142C2×C4—5022no—
143C2×C4D4×D25112yes—
144C8—42no—
145C22—21no—
146C22—21no—
147C22—22no—
148C22—42no—
149C22—41no—
150C22—41no—
151C22—81no—
152C22#2254868511yescharacteristic
153C22C2×#1686079911yescharacteristic
154C22#2254874811yescharacteristic
155C4—22no—
156C4—41no—
157C4—20081no—
158C4—20081no—
159C4—10042no—
160C4—10042no—
161C2—21no—
162C2—21no—
163C2—42no—
164C2#3063182811yescharacteristic
165C2#3063186711yescharacteristic
166C2#3063187811yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#323×C251C2
socleC22×C251#120
prime radicalC24×C251D4
hypercenter#323D251
hypocenterC251#1608
LCSG ▷ C22×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #30632018C2semidirect
4. #30631425C2semidirect
83. C251#1608semidirect

Maximal subgroups

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

subgroupindex
#306323132
#306320182
#306316122
#306314252
#323×C2512
#1608251