Finite Group Databasebeta

← #44550746

Subgroups — #44550746

Order 32128.

332 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 85 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#30632454C211yesmaximal, characteristic, complemented
2#30632233C211yesmaximal, characteristic
3#30631705C212yesmaximal, complemented
4#30631446C212yesmaximal, complemented
5#410×C251C211yesmaximal, characteristic, complemented
6C2×#16860810C2211yescharacteristic
7#22548722—24no—
8C2×#16860794—22no—
9C2×#16860794C2212yes—
10#22548713C411yescharacteristic, complemented
11#22548711C411yescharacteristic, complemented
12#22548704C411yescharacteristic, complemented
13#22548702C411yescharacteristic, complemented
14#22548687—22no—
15C4×C22⋊D251—22no—
16C2×C4×Dic251C2211yescharacteristic
17#22548643—24no—
18C2×#41×C251C2211yescharacteristic
19#117×C251C2212yes—
20#99×C251—22no—
21#16860810—22no—
22#16860810—22no—
23C2×Dic502C2×C411yescharacteristic
24C2×D1004C2×C411yescharacteristic
25C2×C22⋊D251—22no—
26C2×C22⋊D251—21no—
27C22×Dic251—21no—
28C2×C4×D251—21no—
29#16860794—24no—
30#16860794—22no—
31#16860794D414yescomplemented
32C2×C251⋊C8—24no—
33C2×C251⋊C8—22no—
34Dic251⋊C4—22no—
35C22⋊Dic251—22no—
36(C2×D251)⋊C4—22no—
37C4×Dic251—21no—
38C4×Dic251C2×C411yescharacteristic
39C4×Dic251C2×C411yescharacteristic
40#41×C251—22no—
41#41×C251—22no—
42C2×Q8×C251C2×C411yescharacteristic
43C2×D4×C251C2×C411yescharacteristic
44C2×D4×C251D412yes—
45C22×C4×C251D412yes—
46C22×C4×C251C2311yescharacteristic
47C4⋊C4×C251—22no—
48C22⋊C4×C251—22no—
49C42×C251—22no—
50C22×D251—21no—
51Dic502—22no—
52D1004—22no—
53C22⋊D251—44no—
54C22⋊D251—42no—
55C2×Dic251—21no—
56C2×Dic251—21no—
57C2×Dic251—21no—
58C2×Dic251—42no—
59C4×D251—41no—
60C4×D251—41no—
61C251⋊C8—24no—
62C23×C251C22⋊C412yes—
63C23×C251C2×D411yescharacteristic
64Q8×C251—22no—
65D4×C251—24no—
66D4×C251—22no—
67C2×C4×C251—22no—
68C2×C4×C251—22no—
69C2×C4×C251—42no—
70C2×C4×C251C22⋊C412yes—
71C2×C4×C251C22×C411yescharacteristic
72C2×C4×C251C2×D412yes—
73C2×C4×C251C2×D411yescharacteristic
74C2×C4×C251#4112yes—
75C22×C251—22no—
76C22×C251—41no—
77C22×C251—42no—
78C22×C251C2×C22⋊C411yescharacteristic
79C22×C251#12112yes—
80C2×D251—42no—
81C2×D251—41no—
82C4×C251—22no—
83C4×C251—82no—
84C4×C251C4×D412yes—
85C4×C251#12011yescharacteristic
86C4×C251#12311yescharacteristic
87Dic251—22no—
88Dic251—41no—
89Dic251—41no—
90C2×C251—21no—
91C2×C251—22no—
92C2×C251—42no—
93C2×C251#38611yescharacteristic
94D251—81no—
95C251#188411yescharacteristic, complemented, Sylow
96#1884—2511nomaximal, Sylow
97#568—2511no—
98#532—2511no—
99#421—2512no—
100#410D25111yescharacteristic, complemented
101#339—2512no—
102C2×#41—2511no—
103C2×#41C2×D25111yescharacteristic
104#131—5024no—
105C2×M16—5022no—
106C2×M16—2512no—
107C4⋊Q8—2511no—
108#127—2511no—
109#124—2511no—
110#124—2511no—
111#121—5022no—
112C4×D4—5022no—
113#117C2×D25112yes—
114C2×C42—2511no—
115#104—5024no—
116#99—22no—
117#41—22no—
118#41—22no—
119#41—5022no—
120#41—5022no—
121C2×Q8—2511no—
122C2×Q8C4×D25111yescharacteristic
123C2×D4—5022no—
124C2×D4—5021no—
125C2×D4—2511no—
126C2×D4C4×D25111yescharacteristic
127C2×D4C22⋊D25112yes—
128C22×C4—5021no—
129C22×C4—5021no—
130C22×C4D100412yes—
131C22×C4C22×D25111yescharacteristic
132M16—5024no—
133M16—5022no—
134M16—2514no—
135C2×C8—5024no—
136C2×C8—5022no—
137C4⋊C4—22no—
138C4⋊C4—5022no—
139C22⋊C4—22no—
140C22⋊C4—5022no—
141C22⋊C4—5022no—
142C42—22no—
143C42—5021no—
144C42—2511no—
145C42—2511no—
146C23—5021no—
147C23(C2×D251)⋊C412yes—
148C23C2×C22⋊D25111yescharacteristic
149Q8—22no—
150Q8—5022no—
151D4—24no—
152D4—22no—
153D4—10042no—
154D4—10044no—
155D4—5022no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—42no—
159C2×C4—10042no—
160C2×C4—10041no—
161C2×C4—10041no—
162C2×C4—5021no—
163C2×C4—5021no—
164C2×C4—5021no—
165C2×C4(C2×D251)⋊C412yes—
166C2×C4C2×C4×D25111yescharacteristic
167C2×C4D4×D25112yes—
168C2×C4C2×D100411yescharacteristic
169C2×C4#1686080912yes—
170C8—5024no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—10042no—
175C22—10041no—
176C22C2×(C2×D251)⋊C411yescharacteristic
177C22#2254868712yes—
178C4—22no—
179C4—82no—
180C4—10041no—
181C4—10041no—
182C4—5022no—
183C4(C22⋊D251)⋊C412yes—
184C4#2254868511yescharacteristic
185C4#2254869911yescharacteristic
186C2—22no—
187C2—21no—
188C2—42no—
189C2—20081no—
190C2#3063156411yescharacteristic
subgroupG/H
centerC4#22548685
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#410×C251C2
socleC2×C251#386
prime radical#30632233C2
cosocleC22×C4×C251C23
hypercenter#410D251
hypocenterC251#1884
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #30632454C2semidirect
3. #30631705C2semidirect
4. #30631446C2semidirect
5. #410×C251C2semidirect
10. #22548713C4semidirect
11. #22548711C4semidirect
12. #22548704C4semidirect
13. #22548702C4semidirect
31. #16860794D4semidirect
95. C251#1884semidirect
100. #410D251semidirect

Maximal subgroups

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

subgroupindex
#306324542
#306322332
#306317052
#306314462
#410×C2512
#1884251