Finite Group Databasebeta

← #44549392

Subgroups — #44549392

Order 32128.

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

normal subgroups only

This group has 236 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#30632023C212yesmaximal, complemented
3#30631623C211yesmaximal, characteristic
4#30631425C212yesmaximal, complemented
5#323×C251C211yesmaximal, characteristic
6#22548694—21no—
7#22548694C2212yes—
8#22548692—22no—
9#22548692—22no—
10C2×C4⋊Dic251C2211yescharacteristic
11C2×Dic251⋊C4—21no—
12C2×C22⋊Dic251—21no—
13C2×C22⋊Dic251—21no—
14#22548649—22no—
15#22548640—22no—
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—
30Dic251⋊C4—42no—
31C22⋊Dic251—22no—
32C22⋊Dic251—22no—
33C22⋊Dic251—42no—
34C24×C251D411yescharacteristic
35C22×C4×C251D411yescharacteristic
36C22×C4×C251C2311yescharacteristic
37C2×C8×C251—22no—
38C22⋊C4×C251—22no—
39Dic502—42no—
40Dic502—81no—
41C2×Dic251—22no—
42C2×Dic251—22no—
43C2×Dic251—22no—
44C2×Dic251—41no—
45C2×Dic251—41no—
46C2×Dic251—41no—
47C2×Dic251—41no—
48C2×Dic251—41no—
49C2×Dic251—81no—
50C2×Dic251—81no—
51C23×C251—21no—
52C23×C251—21no—
53C23×C251—41no—
54C23×C251—41no—
55C23×C251Q1612yes—
56C23×C251C2×D411yescharacteristic
57C2×C4×C251—21no—
58C2×C4×C251—41no—
59C2×C4×C251—41no—
60C2×C4×C251C2×D412yes—
61C8×C251—42no—
62C22×C251—21no—
63C22×C251—21no—
64C22×C251—22no—
65C22×C251—42no—
66C22×C251—41no—
67C22×C251—41no—
68C22×C251—81no—
69C22×C251#12011yescharacteristic
70C22×C251C2×Q1611yescharacteristic
71C22×C251#13711yescharacteristic
72C4×C251—22no—
73C4×C251—41no—
74Dic251—42no—
75Dic251—42no—
76Dic251—42no—
77Dic251—81no—
78C2×C251—21no—
79C2×C251—21no—
80C2×C251—42no—
81C2×C251#45111yescharacteristic
82C2×C251#45311yescharacteristic
83C2×C251#45811yescharacteristic
84C251#161411yescharacteristic, complemented, Sylow
85#1614—2511nomaximal, Sylow
86#543—2511no—
87#484—2512no—
88#399—2511no—
89#328—2512no—
90#323D25111yescharacteristic
91C22⋊Q8—5022no—
92C22⋊Q8—5022no—
93C22⋊Q8—5021no—
94C22⋊Q8—2512no—
95C2×C4⋊C4—5021no—
96C2×C4⋊C4—2511no—
97C2×C22⋊C4—5021no—
98C2×C22⋊C4—5021no—
99C2×C22⋊C4C2×D25111yescharacteristic
100#107—5022no—
101#103—5022no—
102C22⋊C8C2×D25112yes—
103#95—2511no—
104C24D100411yescharacteristic
105C2×Q8—5021no—
106C2×Q8—2512no—
107C22×C4—5021no—
108C22×C4—5021no—
109C22×C4—5021no—
110C22×C4—5021no—
111C22×C4D100411yescharacteristic
112C22×C4C22×D25111yescharacteristic
113C2×C8—22no—
114C4⋊C4—10042no—
115C4⋊C4—5022no—
116C4⋊C4—5021no—
117C4⋊C4—5022no—
118C4⋊C4—5022no—
119C4⋊C4—2512no—
120C22⋊C4—22no—
121C22⋊C4—10042no—
122C22⋊C4—5022no—
123C22⋊C4—5022no—
124C23—21no—
125C23—21no—
126C23—41no—
127C23—41no—
128C23Dic100412yes—
129C23C2×D100411yescharacteristic
130Q8—20081no—
131Q8—10042no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—20081no—
136C2×C4—20081no—
137C2×C4—10041no—
138C2×C4—10041no—
139C2×C4—10041no—
140C2×C4—10041no—
141C2×C4—10041no—
142C2×C4—5022no—
143C2×C4—5022no—
144C2×C4—5022no—
145C2×C4D4×D25112yes—
146C8—42no—
147C22—21no—
148C22—21no—
149C22—22no—
150C22—42no—
151C22—41no—
152C22—41no—
153C22—81no—
154C22#2254868511yescharacteristic
155C22C2×Dic100411yescharacteristic
156C22#2254874811yescharacteristic
157C4—22no—
158C4—41no—
159C4—20081no—
160C4—10042no—
161C4—10042no—
162C4—10042no—
163C2—21no—
164C2—21no—
165C2—42no—
166C2#3063184511yescharacteristic
167C2#3063185711yescharacteristic
168C2#3063187811yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#323×C251C2
socleC22×C251#120
prime radicalC24×C251D4
hypercenter#323D251
hypocenterC251#1614
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. #30632023C2semidirect
4. #30631425C2semidirect
84. C251#1614semidirect

Maximal subgroups

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

subgroupindex
#306323132
#306320232
#306316232
#306314252
#323×C2512
#1614251