Finite Group Databasebeta

← #44548732

Subgroups — #44548732

Order 32128.

214 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 69 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
1C2×#22548709C211yesmaximal, characteristic
2#30631744C211yesmaximal, characteristic, complemented
3#30631412C212yesmaximal, complemented
4#30631410C212yesmaximal
5#431×C251C211yesmaximal, characteristic, complemented
6#22548709—21no—
7#22548709—21no—
8#22548709—21no—
9#22548709—21no—
10C2×Dic251⋊C4C411yescharacteristic
11C2×Dic251⋊C4C411yescharacteristic
12C2×Dic251⋊C4C2211yescharacteristic
13C2×(C2×D251)⋊C4C411yescharacteristic
14C2×(C2×D251)⋊C4C411yescharacteristic
15C2×(C2×D251)⋊C4C2211yescharacteristic
16C4⋊(C251⋊C8)—21no—
17C22⋊(C251⋊C8)C2212yes—
18#22548626—21no—
19C2×C42×C251C2211yescharacteristic
20C4⋊C8×C251—21no—
21C22⋊C8×C251C2212yes—
22C4×C8×C251—21no—
23C23×D251C2×C411yescharacteristic
24C22×Dic251C2×C411yescharacteristic
25C22×Dic251C2×C411yescharacteristic
26C22×Dic251C2×C411yescharacteristic
27C2×C251⋊C8—22no—
28Dic251⋊C4—21no—
29Dic251⋊C4—21no—
30Dic251⋊C4—21no—
31Dic251⋊C4—21no—
32Dic251⋊C4—21no—
33Dic251⋊C4—21no—
34(C2×D251)⋊C4—21no—
35(C2×D251)⋊C4—21no—
36(C2×D251)⋊C4—21no—
37(C2×D251)⋊C4—21no—
38(C2×D251)⋊C4—21no—
39(C2×D251)⋊C4—21no—
40C22×C4×C251C2×C411yescharacteristic
41C22×C4×C251C2×C411yescharacteristic
42C22×C4×C251C2311yescharacteristic
43C2×C8×C251—22no—
44C42×C251D411yescharacteristic
45C42×C251D411yescharacteristic
46C42×C251D411yescharacteristic
47C42×C251D411yescharacteristic
48C22×D251—21no—
49C22×D251—22no—
50C22×D251—41no—
51C22×D251—41no—
52C2×Dic251—21no—
53C2×Dic251—21no—
54C2×Dic251—21no—
55C2×Dic251—22no—
56C2×Dic251—41no—
57C2×Dic251—41no—
58C251⋊C8—42no—
59C23×C251C22×C411yescharacteristic
60C2×C4×C251—22no—
61C2×C4×C251—41no—
62C2×C4×C251—41no—
63C2×C4×C251C22⋊C411yescharacteristic
64C2×C4×C251C22⋊C411yescharacteristic
65C2×C4×C251C22⋊C411yescharacteristic
66C2×C4×C251C22⋊C411yescharacteristic
67C2×C4×C251C2×D411yescharacteristic
68C2×C4×C251C2×D411yescharacteristic
69C8×C251—42no—
70C22×C251—21no—
71C22×C251—21no—
72C22×C251C2×C22⋊C411yescharacteristic
73C22×C251#13112yes—
74C2×D251—41no—
75C2×D251—41no—
76C2×D251—41no—
77C2×D251—82no—
78C4×C251—21no—
79C4×C251—21no—
80C4×C251—21no—
81C4×C251—21no—
82C4×C251—21no—
83C4×C251—21no—
84Dic251—41no—
85Dic251—41no—
86Dic251—41no—
87C2×C251—22no—
88C2×C251#40811yescharacteristic
89C2×C251#41011yescharacteristic
90C2×C251#41311yescharacteristic
91D251—81no—
92C251#147311yescharacteristic, complemented, Sylow
93#1473—2511nomaximal, Sylow
94C2×#126—2511no—
95#432—2511no—
96#431D25111yescharacteristic, complemented
97#324—2512no—
98#323—2512no—
99#126—5021no—
100#126—5021no—
101#126—5021no—
102#126—5021no—
103C2×C4⋊C4—2511no—
104C2×C4⋊C4—2511no—
105C2×C4⋊C4—2511no—
106C2×C22⋊C4—2511no—
107C2×C22⋊C4—2511no—
108C2×C22⋊C4—2511no—
109C2×C42C2×D25111yescharacteristic
110C4⋊C8—21no—
111C4⋊C8—5021no—
112C22⋊C8—2512no—
113C22⋊C8C2×D25112yes—
114#97—5021no—
115C4×C8—21no—
116C24—2511no—
117C22×C4—2511no—
118C22×C4—2511no—
119C22×C4—2511no—
120C22×C4C4×D25111yescharacteristic
121C22×C4C4×D25111yescharacteristic
122C22×C4C22×D25111yescharacteristic
123C2×C8—22no—
124C2×C8—5022no—
125C4⋊C4—5021no—
126C4⋊C4—5021no—
127C4⋊C4—5021no—
128C4⋊C4—5021no—
129C4⋊C4—5021no—
130C4⋊C4—5021no—
131C22⋊C4—5021no—
132C22⋊C4—5021no—
133C22⋊C4—5021no—
134C22⋊C4—5021no—
135C22⋊C4—5021no—
136C22⋊C4—5021no—
137C42C22⋊D25111yescharacteristic
138C42C22⋊D25111yescharacteristic
139C42D100411yescharacteristic
140C42D100411yescharacteristic
141C23—10041no—
142C23—10041no—
143C23—5022no—
144C23—5021no—
145C23C2×C4×D25111yescharacteristic
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—10041no—
150C2×C4—10041no—
151C2×C4—5021no—
152C2×C4—5021no—
153C2×C4—5021no—
154C2×C4—5022no—
155C2×C4(C2×D251)⋊C411yescharacteristic
156C2×C4(C2×D251)⋊C411yescharacteristic
157C2×C4(C2×D251)⋊C411yescharacteristic
158C2×C4(C2×D251)⋊C411yescharacteristic
159C2×C4C2×C22⋊D25111yescharacteristic
160C2×C4C2×D100411yescharacteristic
161C8—42no—
162C8—10042no—
163C22—21no—
164C22—21no—
165C22—20082no—
166C22—10041no—
167C22—10041no—
168C22—10041no—
169C22C2×(C2×D251)⋊C411yescharacteristic
170C22#2254872112yes—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—10041no—
178C4—10041no—
179C4—10041no—
180C2—22no—
181C2—20081no—
182C2#3063165311yescharacteristic
183C2#3063166011yescharacteristic
184C2#3063166911yescharacteristic
subgroupG/H
centerC22C2×(C2×D251)⋊C4
commutatorC23×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#431×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC23×D251C2×C4
cosocleC22×C4×C251C23
hypercenter#431D251
hypocenterC251#1473
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
2. #30631744C2semidirect
3. #30631412C2semidirect
5. #431×C251C2semidirect
92. C251#1473semidirect
96. #431D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487092
#306317442
#306314122
#306314102
#431×C2512
#1473251