Finite Group Databasebeta

← #44550783

Subgroups — #44550783

Order 32128.

342 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 89 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#30632194C211yesmaximal, characteristic
2#30631671C211yesmaximal, characteristic, complemented
3#30631650C212yesmaximal, complemented
4#30631458C212yesmaximal, complemented
5#568×C251C211yesmaximal, characteristic, complemented
6C22×C4×D251C2211yescharacteristic
7C2×#16860794C2212yes—
8#22548696C412yes—
9#22548691—21no—
10#22548686—21no—
11#22548683C412yes—
12C4×C22⋊D251—22no—
13#22548674C2211yescharacteristic
14#22548634—21no—
15#22548632—21no—
16(C2×D251)⋊C8—24no—
17C2×#41×C251C2211yescharacteristic
18#131×C251—24no—
19C2×M16×C251—22no—
20C2×M16×C251C2212yes—
21C23×D251C2×C411yescharacteristic
22C2×C22⋊D251—22no—
23C22×Dic251C2×C411yescharacteristic
24C2×C4×D251—22no—
25C2×C4×D251—21no—
26C2×C4×D251—21no—
27C2×C4×D251—22no—
28#16860794—22no—
29#16860794—22no—
30C2×C251⋊C8—22no—
31C4⋊Dic251—21no—
32Dic251⋊C4—22no—
33C22⋊Dic251—22no—
34C22⋊Dic251C2×C412yes—
35(C2×D251)⋊C4—22no—
36C4×Dic251—21no—
37#41×C251—22no—
38#41×C251—22no—
39C2×Q8×C251D411yescharacteristic
40C2×D4×C251C2×C412yes—
41C2×D4×C251D411yescharacteristic
42C22×C4×C251D412yes—
43C22×C4×C251C2311yescharacteristic
44M16×C251—24no—
45M16×C251—22no—
46M16×C251—22no—
47C2×C8×C251—24no—
48C2×C8×C251D414yescomplemented
49C22×D251—21no—
50C22×D251—22no—
51C22×D251—42no—
52C22⋊D251—44no—
53C2×Dic251—22no—
54C2×Dic251—21no—
55C2×Dic251—22no—
56C4×D251—41no—
57C4×D251—42no—
58C4×D251—41no—
59C251⋊C8—42no—
60C23×C251C22⋊C412yes—
61C23×C251C22×C411yescharacteristic
62Q8×C251—22no—
63D4×C251—24no—
64D4×C251—22no—
65C2×C4×C251—22no—
66C2×C4×C251—22no—
67C2×C4×C251C22⋊C412yes—
68C2×C4×C251C2×D412yes—
69C2×C4×C251C2×D411yescharacteristic
70C2×C4×C251C2×D411yescharacteristic
71C2×C4×C251#4112yes—
72C8×C251—24no—
73C22×C251—22no—
74C22×C251—41no—
75C22×C251—42no—
76C22×C251C2×C22⋊C411yescharacteristic
77C22×C251C4×D412yes—
78C2×D251—22no—
79C2×D251—24no—
80C2×D251—41no—
81C2×D251—42no—
82C2×D251—41no—
83C4×C251—22no—
84C4×C251#12011yescharacteristic
85C4×C251#12112yes—
86C4×C251#12311yescharacteristic
87Dic251—42no—
88Dic251—41no—
89C2×C251—21no—
90C2×C251—22no—
91C2×C251—42no—
92C2×C251#38611yescharacteristic
93D251—42no—
94C251#189111yescharacteristic, complemented, Sylow
95#1891—2511nomaximal, Sylow
96#568D25111yescharacteristic, complemented
97#525—2511no—
98#413—2511no—
99#407—2512no—
100#344—2512no—
101C2×#41C2×D25111yescharacteristic
102C23×C4—2511no—
103#131—24no—
104C2×M16—22no—
105C2×M16—2512no—
106C2×M16C2×D25112yes—
107#123—2512no—
108C22⋊Q8—5021no—
109#121—5021no—
110#120—2512no—
111C4×D4—5022no—
112#117—2511no—
113#101—5021no—
114#100—5021no—
115C22⋊C8—5024no—
116C24—2511no—
117#41—22no—
118#41—22no—
119C2×Q8D100411yescharacteristic
120C2×D4—5022no—
121C2×D4C4×D25112yes—
122C2×D4D100411yescharacteristic
123C22×C4—5022no—
124C22×C4—5021no—
125C22×C4—5021no—
126C22×C4—5022no—
127C22×C4—2511no—
128C22×C4C22⋊D25112yes—
129C22×C4C22×D25111yescharacteristic
130M16—24no—
131M16—22no—
132M16—22no—
133M16—5022no—
134M16—5022no—
135C2×C8—24no—
136C2×C8—5022no—
137C2×C8C22⋊D25114yescomplemented
138C4⋊C4—5022no—
139C4⋊C4—5021no—
140C22⋊C4—5022no—
141C22⋊C4—5022no—
142C22⋊C4—2512no—
143C42—5021no—
144C23—10042no—
145C23—5021no—
146C23—5022no—
147C23(C2×D251)⋊C412yes—
148C23C2×C4×D25111yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—10044no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—10041no—
156C2×C4—10042no—
157C2×C4—10041no—
158C2×C4—5022no—
159C2×C4—5021no—
160C2×C4—5022no—
161C2×C4(C2×D251)⋊C412yes—
162C2×C4C2×C22⋊D25112yes—
163C2×C4C2×C22⋊D25111yescharacteristic
164C2×C4C2×D100411yescharacteristic
165C2×C4#1686081012yes—
166C8—24no—
167C8—10042no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22—10041no—
172C22—10042no—
173C22—10041no—
174C22—5022no—
175C22—5024no—
176C22C2×(C2×D251)⋊C411yescharacteristic
177C22C4×C22⋊D25112yes—
178C4—22no—
179C4—10042no—
180C4—10041no—
181C4#2254868411yescharacteristic
182C4#2254868912yes—
183C4#2254869811yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—10042no—
188C2#3063156611yescharacteristic
subgroupG/H
centerC4#22548684
commutatorC23×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#568×C251C2
socleC2×C251#386
prime radical#30632194C2
cosocleC22×C4×C251C23
hypercenter#568D251
hypocenterC251#1891
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #30631671C2semidirect
3. #30631650C2semidirect
4. #30631458C2semidirect
5. #568×C251C2semidirect
48. C2×C8×C251D4semidirect
94. C251#1891semidirect
96. #568D251semidirect
137. C2×C8C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
#306321942
#306316712
#306316502
#306314582
#568×C2512
#1891251