Finite Group Databasebeta

← #49132697

Subgroups — #49132697

Order 36224.

370 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 109 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#31827068C212yesmaximal, complemented
2#31826767C212yesmaximal, complemented
3#31826764C212yesmaximal, complemented
4#31826718C211yesmaximal, characteristic, complemented
5#31826699C212yesmaximal, complemented
6#31826659C212yesmaximal, complemented
7#31826658C212yesmaximal, complemented
8#31826493C211yesmaximal, characteristic, complemented
9#469×C283C211yesmaximal, characteristic, complemented
10C2×D2264C2211yescharacteristic
11C2×#17985377C2212yes—
12C22×C283⋊C8C2211yescharacteristic
13#22583606C2212yes—
14#22583601C2212yes—
15#22583594—22no—
16#22583592—22no—
17#22583584C2212yes—
18#22583582—22no—
19#22583581—22no—
20D4×Dic283C2212yes—
21#22583572C2212yes—
22#22583569C2211yescharacteristic
23#22583543C2212yes—
24#22583542C2211yescharacteristic
25Dic283⋊C8C2212yes—
26Dic283⋊C8C2212yes—
27#22583532C2212yes—
28#22583531C2212yes—
29#22583530C2212yes—
30C2×D8×C283C2211yescharacteristic
31C2×M16×C283C2211yescharacteristic
32#121×C283C2212yes—
33#106×C283C2211yescharacteristic
34#102×C283C2212yes—
35C2×D1132C2312yes—
36C2×C22⋊D283—22no—
37C2×C22⋊D283—22no—
38C2×C22⋊D283—22no—
39C22×Dic283—22no—
40C2×C4×D283—22no—
41D2264—41no—
42#17985377—42no—
43C2×C283⋊C8—41no—
44C2×C283⋊C8C2312yes—
45C4⋊Dic283—22no—
46C4⋊Dic283D412yes—
47C4⋊Dic283C2312yes—
48Dic283⋊C4—22no—
49C22⋊Dic283—22no—
50C22⋊Dic283D411yescharacteristic
51C22⋊Dic283D411yescharacteristic
52(C2×D283)⋊C4—22no—
53(C2×D283)⋊C4—22no—
54C4×Dic283C2312yes—
55C2×D4×C283—22no—
56C2×D4×C283C2312yes—
57C22×C4×C283C2311yescharacteristic
58D8×C283—41no—
59M16×C283—41no—
60C2×C8×C283C2311yescharacteristic
61C2×C8×C283C2311yescharacteristic
62C4⋊C4×C283C2312yes—
63C22⋊C4×C283—22no—
64C22×D283—22no—
65D1132—22no—
66D1132—22no—
67C22⋊D283—82no—
68C22⋊D283—82no—
69C22⋊D283—82no—
70C2×Dic283—22no—
71C2×Dic283—82no—
72C2×Dic283C2×D412yes—
73C2×Dic283C2×D412yes—
74C4×D283—82no—
75C283⋊C8—22no—
76C23×C283—22no—
77C23×C283C2×D411yescharacteristic
78D4×C283—22no—
79D4×C283—22no—
80D4×C283—82no—
81C2×C4×C283—21no—
82C2×C4×C283—21no—
83C2×C4×C283—22no—
84C2×C4×C283C2×D411yescharacteristic
85C2×C4×C283C2411yescharacteristic
86C8×C283—21no—
87C8×C283—21no—
88C22×C283—21no—
89C22×C283—22no—
90C22×C283—42no—
91C22×C283—42no—
92C22×C283—42no—
93C22×C283C22×D411yescharacteristic
94C2×D283—42no—
95C2×D283—42no—
96C2×D283—42no—
97C4×C283—21no—
98C4×C283—42no—
99C4×C283H(5,𝔽2)11yescharacteristic
100C4×C2832−1+411yescharacteristic
101Dic283—22no—
102Dic283—22no—
103Dic283—42no—
104C2×C283—41no—
105C2×C283—82no—
106C2×C283#53711yescharacteristic
107C2×C283#57612yes—
108D283—82no—
109C283#325911yescharacteristic, complemented, Sylow
110#3259—2831nomaximal, Sylow
111#540—2832no—
112#478—2832no—
113#478—2832no—
114#469—2831no—
115#469D28311yescharacteristic, complemented
116#466—2832no—
117#459—2832no—
118#459—2832no—
119#424—2831no—
120C2×D8—2832no—
121C2×D8—2831no—
122C2×D8C2×D28311yescharacteristic
123C2×M16C2×D28311yescharacteristic
124C22×C8—2831no—
125#127—2832no—
126#125—2832no—
127#123—5662no—
128#123—5662no—
129#121—2832no—
130#121—5662no—
131#121—5662no—
132#121C2×D28312yes—
133C4×D4—2832no—
134C4×D4—2832no—
135#117—2831no—
136#107—2832no—
137#106—2831no—
138#106C2×D28311yescharacteristic
139C4⋊C8—2832no—
140C4⋊C8—2832no—
141#102—2832no—
142#102—2832no—
143#102—2832no—
144#102C2×D28312yes—
145C2×D4—22no—
146C2×D4—2832no—
147C2×D4—5662no—
148C2×D4—5662no—
149C2×D4—5662no—
150C2×D4C22×D28312yes—
151C22×C4—5662no—
152C22×C4—5662no—
153C22×C4C22×D28311yescharacteristic
154D8—41no—
155D8—11321no—
156D8—11322no—
157M16—41no—
158C2×C8—2832no—
159C2×C8—11321no—
160C2×C8C22×D28311yescharacteristic
161C2×C8C22×D28311yescharacteristic
162C4⋊C4—2832no—
163C4⋊C4—2832no—
164C4⋊C4—5662no—
165C4⋊C4—5662no—
166C4⋊C4C22×D28312yes—
167C22⋊C4—22no—
168C22⋊C4—2831no—
169C22⋊C4—2831no—
170C22⋊C4—5662no—
171C22⋊C4—5662no—
172C22⋊C4—5662no—
173C42—2832no—
174C23—22no—
175C23—5662no—
176C23D4×D28311yescharacteristic
177D4—22no—
178D4—22no—
179D4—82no—
180D4—5662no—
181D4—5662no—
182D4—22642no—
183D4—22642no—
184D4—22642no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—2832no—
189C2×C4—2832no—
190C2×C4—5662no—
191C2×C4—22642no—
192C2×C4—22642no—
193C2×C4D4×D28311yescharacteristic
194C2×C4C23×D28311yescharacteristic
195C8—21no—
196C8—21no—
197C8—5662no—
198C22—22no—
199C22—21no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—11322no—
204C22—11322no—
205C22—11322no—
206C22C2×D4×D28311yescharacteristic
207C4—21no—
208C4—42no—
209C4—5662no—
210C4—5662no—
211C4—11322no—
212C4#2258365511yescharacteristic
213C4#2258365711yescharacteristic
214C2—41no—
215C2—82no—
216C2—22642no—
217C2#3182704211yescharacteristic
218C2#3182726912yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#469×C283C2
socleC22×C283C22×D4
hypercenter#469D283
hypocenterC283#3259
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #31827068C2semidirect
2. #31826767C2semidirect
3. #31826764C2semidirect
4. #31826718C2semidirect
5. #31826699C2semidirect
6. #31826659C2semidirect
7. #31826658C2semidirect
8. #31826493C2semidirect
9. #469×C283C2semidirect
109. C283#3259semidirect
115. #469D283semidirect

Maximal subgroups

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

subgroupindex
#318270682
#318267672
#318267642
#318267182
#318266992
#318266592
#318266582
#318264932
#469×C2832
#3259283