Finite Group Databasebeta

← #44549299

Subgroups — #44549299

Order 32128.

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

normal subgroups only

This group has 310 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#30632268C211yesmaximal, characteristic, complemented
2#30631821C212yesmaximal, complemented
3#30631420C212yesmaximal, complemented
4#30631413C211yesmaximal, characteristic, complemented
5#394×C251C211yesmaximal, characteristic, complemented
6C2×#16860810—21no—
7C2×#16860795—22no—
8#22548695—22no—
9#22548690—22no—
10#22548689—21no—
11#22548689C2212yes—
12#22548636—22no—
13C22⋊(C251⋊C8)C2212yes—
14C22×D4×C251C2211yescharacteristic
15C2×C4⋊C4×C251C2211yescharacteristic
16C2×C22⋊C4×C251—21no—
17C2×C22⋊C4×C251—21no—
18#95×C251C2211yescharacteristic
19#16860810—42no—
20#16860810—42no—
21C2×Dic502—21no—
22C2×D1004—21no—
23C2×D1004—21no—
24C2×D1004D412yes—
25C2×C22⋊D251—22no—
26C2×C4×D251—22no—
27#16860795—44no—
28C2×C251⋊C8—22no—
29C4⋊Dic251—21no—
30C4⋊Dic251D412yes—
31(C2×D251)⋊C4—22no—
32(C2×D251)⋊C4—22no—
33C24×C251—21no—
34C2×D4×C251—22no—
35C2×D4×C251—42no—
36C22×C4×C251—21no—
37C22×C4×C251D411yescharacteristic
38C22×C4×C251D411yescharacteristic
39C22×C4×C251C2311yescharacteristic
40C4⋊C4×C251—22no—
41C22⋊C4×C251—42no—
42C22⋊C4×C251—42no—
43C22×D251—22no—
44Dic502—42no—
45D1004—42no—
46D1004—42no—
47D1004—81no—
48C22⋊D251—42no—
49C22⋊D251—42no—
50C2×Dic251—22no—
51C4×D251—44no—
52C251⋊C8—42no—
53C23×C251—22no—
54C23×C251—41no—
55C23×C251—41no—
56C23×C251—41no—
57C23×C251—41no—
58C23×C251—41no—
59C23×C251—41no—
60C23×C251C2×D411yescharacteristic
61D4×C251—24no—
62D4×C251—42no—
63C2×C4×C251—21no—
64C2×C4×C251—21no—
65C2×C4×C251—21no—
66C2×C4×C251—41no—
67C2×C4×C251—41no—
68C2×C4×C251—41no—
69C2×C4×C251—41no—
70C2×C4×C251—41no—
71C2×C4×C251D812yescomplemented
72C2×C4×C251C2×D412yes—
73C22×C251—21no—
74C22×C251—21no—
75C22×C251—42no—
76C22×C251—42no—
77C22×C251—42no—
78C22×C251—81no—
79C22×C251—81no—
80C22×C251—81no—
81C22×C251—81no—
82C22×C251#12011yescharacteristic
83C22×C251C2×D811yescharacteristic
84C22×C251AGL(1,ℤ/8)11yescharacteristic
85C2×D251—42no—
86C2×D251—42no—
87C2×D251—42no—
88C4×C251—22no—
89C4×C251—22no—
90C4×C251—41no—
91C4×C251—81no—
92Dic251—42no—
93C2×C251—21no—
94C2×C251—21no—
95C2×C251—82no—
96C2×C251#44711yescharacteristic
97C2×C251#45411yescharacteristic
98C2×C251B(4,𝔽2)11yescharacteristic
99D251—82no—
100C251#159811yescharacteristic, complemented, Sylow
101#1598—2511nomaximal, Sylow
102#537—2511no—
103#447—2512no—
104#394D25111yescharacteristic, complemented
105#327—2512no—
106#324—2511no—
107C2×#41—5021no—
108C22×D4C2×D25111yescharacteristic
109C2×D8—5022no—
110C22⋊Q8—5022no—
111#121—5022no—
112#121—5021no—
113#121—2512no—
114C2×C4⋊C4C2×D25111yescharacteristic
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117#102—5022no—
118C22⋊C8—2512no—
119#95C2×D25111yescharacteristic
120C24—21no—
121#41—10042no—
122#41—10042no—
123C2×Q8—5021no—
124C2×D4—22no—
125C2×D4—42no—
126C2×D4—5021no—
127C2×D4—5021no—
128C2×D4—5022no—
129C2×D4—2512no—
130C22×C4—21no—
131C22×C4—5022no—
132C22×C4C22⋊D25111yescharacteristic
133C22×C4C22⋊D25111yescharacteristic
134C22×C4C22×D25111yescharacteristic
135D8—10044no—
136C2×C8—5022no—
137C4⋊C4—22no—
138C4⋊C4—5021no—
139C4⋊C4—2512no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C22⋊C4—5022no—
143C22⋊C4—5022no—
144C23—22no—
145C23—41no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—5022no—
152C23C2×C22⋊D25111yescharacteristic
153Q8—10042no—
154D4—24no—
155D4—42no—
156D4—20081no—
157D4—10042no—
158D4—10042no—
159D4—10042no—
160D4—10042no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—10044no—
170C2×C4—5022no—
171C2×C4#1686079512yescomplemented
172C2×C4D4×D25112yes—
173C8—10042no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—81no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22—10042no—
184C22—10042no—
185C22—10042no—
186C22#2254868311yescharacteristic
187C22C2×#1686079511yescharacteristic
188C22#2254874011yescharacteristic
189C4—22no—
190C4—22no—
191C4—41no—
192C4—81no—
193C4—10042no—
194C2—21no—
195C2—21no—
196C2—82no—
197C2—20082no—
198C2#3063182011yescharacteristic
199C2#3063185911yescharacteristic
200C2#3063187311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#394×C251C2
socleC22×C251#120
hypercenter#394D251
hypocenterC251#1598
LCSG ▷ C22×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #30632268C2semidirect
2. #30631821C2semidirect
3. #30631420C2semidirect
4. #30631413C2semidirect
5. #394×C251C2semidirect
71. C2×C4×C251D8semidirect
100. C251#1598semidirect
104. #394D251semidirect
171. C2×C4#16860795semidirect

Maximal subgroups

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

subgroupindex
#306322682
#306318212
#306314202
#306314132
#394×C2512
#1598251