Finite Group Databasebeta

← #44552027

Subgroups — #44552027

Order 32128.

258 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 55 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#30632235C211yesmaximal, characteristic
2#30631659C212yesmaximal, complemented
3#30631481C212yesmaximal, complemented
4#30631480C211yesmaximal, characteristic, complemented
5#353×C251C211yesmaximal, characteristic, complemented
6C2×#16860809C411yescharacteristic, complemented
7C2×#16860809C2211yescharacteristic
8#22548715C411yescharacteristic, complemented
9#22548702C411yescharacteristic, complemented
10#22548702C2211yescharacteristic
11#22548686—22no—
12D4×Dic251—22no—
13#22548671—22no—
14C42×D251C411yescharacteristic, complemented
15#22548630C2212yes—
16#22548629—22no—
17#127×C251C2211yescharacteristic
18#99×C251C2212yes—
19#16860809—41no—
20#16860809—41no—
21#16860809—41no—
22#16860809—41no—
23C2×Dic502C2×C411yescharacteristic
24C2×Dic502C2×C411yescharacteristic
25C2×C22⋊D251—21no—
26C2×C22⋊D251D412yes—
27C22×Dic251—21no—
28C22×Dic251D412yes—
29C2×C4×D251—21no—
30C2×C4×D251C2×C411yescharacteristic
31C4⋊Dic251—22no—
32Dic251⋊C4—42no—
33C22⋊Dic251—22no—
34C22⋊Dic251—22no—
35C22⋊Dic251—22no—
36C4×Dic251—21no—
37C4×Dic251—42no—
38C4×Dic251C2×C411yescharacteristic
39C2×D4×C251—22no—
40C2×D4×C251C2×C411yescharacteristic
41C2×D4×C251C2311yescharacteristic
42C22⋊C4×C251—22no—
43C42×C251C2×C411yescharacteristic
44C22×D251C22⋊C411yescharacteristic
45Dic502—41no—
46Dic502—41no—
47C22⋊D251—42no—
48C22⋊D251—42no—
49C2×Dic251—22no—
50C2×Dic251—22no—
51C2×Dic251—21no—
52C2×Dic251—21no—
53C2×Dic251—41no—
54C2×Dic251—41no—
55C2×Dic251—42no—
56C2×Dic251C22⋊C412yes—
57C2×Dic251C22⋊C411yescharacteristic
58C4×D251—21no—
59C4×D251—21no—
60C4×D251—42no—
61C23×C251—21no—
62C23×C251C2×D412yes—
63D4×C251—41no—
64D4×C251—41no—
65D4×C251—44no—
66C2×C4×C251—21no—
67C2×C4×C251—42no—
68C2×C4×C251C22×C411yescharacteristic
69C22×C251—22no—
70C22×C251—42no—
71C22×C251—42no—
72C22×C251—41no—
73C22×C251C2×C22⋊C411yescharacteristic
74C2×D251—21no—
75C2×D251—41no—
76C4×C251—22no—
77C4×C251—21no—
78C4×C251—82no—
79Dic251—22no—
80Dic251—41no—
81Dic251—41no—
82Dic251—82no—
83Dic251#9911yescharacteristic
84Dic251#9911yescharacteristic
85C2×C251—21no—
86C2×C251—42no—
87C2×C251—81no—
88C2×C251C2×#9911yescharacteristic
89D251—41no—
90C251#212611yescharacteristic, complemented, Sylow
91#2126—2511nomaximal, Sylow
92#532—2511no—
93#410—2512no—
94#354—2512no—
95#353—2511no—
96#353D25111yescharacteristic, complemented
97C2×#41—2511no—
98C2×#41—2511no—
99C4⋊Q8—2511no—
100#127C2×D25111yescharacteristic
101#124—2511no—
102#124—2511no—
103#121—5022no—
104C4×D4—5022no—
105#117—5022no—
106C2×C42—2511no—
107#99—5022no—
108#99—2512no—
109#99C2×D25112yes—
110#41—10041no—
111#41—10041no—
112#41—10041no—
113#41—10041no—
114C2×Q8—2511no—
115C2×Q8—2511no—
116C2×D4—22no—
117C2×D4—5021no—
118C2×D4—2512no—
119C2×D4C4×D25111yescharacteristic
120C2×D4C22×D25111yescharacteristic
121C22×C4—5021no—
122C22×C4—5021no—
123C22×C4—2512no—
124C22×C4—2511no—
125C4⋊C4—10042no—
126C4⋊C4—5022no—
127C22⋊C4—22no—
128C22⋊C4—5022no—
129C22⋊C4—5022no—
130C22⋊C4—5022no—
131C42—10042no—
132C42—5021no—
133C42—2511no—
134C42C4×D25111yescharacteristic
135C23—21no—
136C23—2511no—
137C23D4×D25112yes—
138Q8—10041no—
139Q8—10041no—
140D4—41no—
141D4—41no—
142D4—44no—
143D4—10042no—
144D4—10042no—
145C2×C4—21no—
146C2×C4—42no—
147C2×C4—10042no—
148C2×C4—10042no—
149C2×C4—10041no—
150C2×C4—10041no—
151C2×C4—5021no—
152C2×C4—5021no—
153C2×C4—5021no—
154C2×C4—5021no—
155C2×C4—5022no—
156C2×C4—5022no—
157C2×C4—2512no—
158C2×C4—2511no—
159C2×C4C2×C4×D25111yescharacteristic
160C22—22no—
161C22—42no—
162C22—42no—
163C22—41no—
164C22—10041no—
165C22—5021no—
166C22C22⋊C4×D25111yescharacteristic
167C4—22no—
168C4—21no—
169C4—82no—
170C4—20082no—
171C4—10041no—
172C4—10041no—
173C4—5022no—
174C4—2511no—
175C4—2511no—
176C2—21no—
177C2—42no—
178C2—81no—
179C2—10041no—
180C2#99×D25111yescharacteristic
subgroupG/H
centerC2#99×D251
commutatorC2×C4×C251C22×C4
FrattiniC2×D4C22×D251
Fitting#353×C251C2
socleC2×C251C2×#99
prime radical#30632235C2
cosocleC2×D4×C251C23
hypercenter#353D251
hypocenterC251#2126
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #30631659C2semidirect
3. #30631481C2semidirect
4. #30631480C2semidirect
5. #353×C251C2semidirect
6. C2×#16860809C4semidirect
8. #22548715C4semidirect
9. #22548702C4semidirect
14. C42×D251C4semidirect
90. C251#2126semidirect
96. #353D251semidirect

Maximal subgroups

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

subgroupindex
#306322352
#306316592
#306314812
#306314802
#353×C2512
#2126251