Finite Group Databasebeta

← #27182566

Subgroups — #27182566

Order 12928.

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#20265780C212yesmaximal, complemented
2#20265480C212yesmaximal, complemented
3#20265475C212yesmaximal, complemented
4#20265428C212yesmaximal, complemented
5#20265413C211yesmaximal, characteristic, complemented
6#20265371C212yesmaximal, complemented
7#20265370C212yesmaximal, complemented
8#20265203C211yesmaximal, characteristic, complemented
9#466×C101C211yesmaximal, characteristic, complemented
10C2×D808C2211yescharacteristic
11C2×#13108059C2212yes—
12C2×#13108058C2211yescharacteristic
13#15692251C2212yes—
14#15692246C2212yes—
15#15692239—22no—
16#15692237—22no—
17#15692229C2212yes—
18#15692227—22no—
19#15692226—22no—
20D4×Dic101C2212yes—
21#15692217C2212yes—
22#15692214C2211yescharacteristic
23#15692189C2211yescharacteristic
24#15692186C2212yes—
25#15692184C2212yes—
26#15692184C2212yes—
27#15692177C2212yes—
28#15692176C2212yes—
29#15692175C2212yes—
30C2×D8×C101C2211yescharacteristic
31C22×C8×C101C2211yescharacteristic
32#121×C101C2212yes—
33#107×C101C2211yescharacteristic
34#102×C101C2212yes—
35C2×D404C2312yes—
36C2×C22⋊D101—22no—
37C2×C22⋊D101—22no—
38C2×C22⋊D101—22no—
39C22×Dic101—22no—
40C2×C4×D101—22no—
41D808—41no—
42#13108059—42no—
43#13108058—41no—
44C2×#11717579C2312yes—
45C4⋊Dic101—22no—
46C4⋊Dic101D411yescharacteristic
47C4⋊Dic101D411yescharacteristic
48C4⋊Dic101C2312yes—
49#13108053—22no—
50C22⋊Dic101—22no—
51C22⋊Dic101D412yes—
52#13108051—22no—
53#13108051—22no—
54C4×Dic101C2312yes—
55C2×D4×C101—22no—
56C2×D4×C101C2312yes—
57C22×C4×C101C2311yescharacteristic
58D8×C101—41no—
59C2×C8×C101—41no—
60C2×C8×C101C2311yescharacteristic
61C2×C8×C101C2311yescharacteristic
62C4⋊C4×C101C2312yes—
63C22⋊C4×C101—22no—
64C23×C101—22no—
65C23×C101C2×D411yescharacteristic
66C22×D101—22no—
67D4×C101—22no—
68D4×C101—22no—
69D4×C101—82no—
70C2×C4×C101—21no—
71C2×C4×C101—21no—
72C2×C4×C101—22no—
73C2×C4×C101C2×D411yescharacteristic
74C2×C4×C101C2411yescharacteristic
75C22⋊D101—82no—
76C22⋊D101—82no—
77C22⋊D101—82no—
78C2×Dic101—22no—
79C2×Dic101—82no—
80C2×Dic101C2×D412yes—
81C2×Dic101C2×D412yes—
82D404—22no—
83D404—22no—
84C4×D101—82no—
85C8×C101—21no—
86C8×C101—21no—
87#11717579—22no—
88C22×C101—21no—
89C22×C101—21no—
90C22×C101—21no—
91C22×C101—42no—
92C22×C101—42no—
93C22×C101—42no—
94C22×C101C22×D411yescharacteristic
95C2×D101—42no—
96C2×D101—42no—
97C2×D101—42no—
98C4×C101—21no—
99C4×C101—42no—
100C4×C101H(5,𝔽2)11yescharacteristic
101C4×C1012−1+411yescharacteristic
102Dic101—22no—
103Dic101—22no—
104Dic101—42no—
105C2×C101—41no—
106C2×C101—82no—
107C2×C101#53711yescharacteristic
108C2×C101#57611yescharacteristic
109C2×C101#57611yescharacteristic
110D101—82no—
111#3259—1011nomaximal, Sylow
112C101#325911yescharacteristic, complemented, Sylow
113#540—1012no—
114#478—1012no—
115#478—1012no—
116#469—1012no—
117#466—1011no—
118#466D10111yescharacteristic, complemented
119#459—1012no—
120#459—1012no—
121#424—1011no—
122C2×D8—1012no—
123C2×D8—1011no—
124C2×D8C2×D10111yescharacteristic
125C2×M16—1011no—
126C22×C8C2×D10111yescharacteristic
127#127—1012no—
128#125—1012no—
129#123—2022no—
130#123—2022no—
131#121—1012no—
132#121—2022no—
133#121—2022no—
134#121C2×D10112yes—
135C4×D4—1012no—
136C4×D4—1012no—
137#117—1011no—
138#107—1011no—
139#107C2×D10111yescharacteristic
140#106—1012no—
141C4⋊C8—1012no—
142C4⋊C8—1012no—
143#102—1012no—
144#102—1012no—
145#102—1012no—
146#102C2×D10112yes—
147C2×D4—22no—
148C2×D4—1012no—
149C2×D4—2022no—
150C2×D4—2022no—
151C2×D4—2022no—
152C2×D4C22×D10112yes—
153C22×C4—2022no—
154C22×C4—2022no—
155C22×C4C22×D10111yescharacteristic
156D8—41no—
157D8—4041no—
158D8—4042no—
159M16—4041no—
160C2×C8—41no—
161C2×C8—1012no—
162C2×C8C22×D10111yescharacteristic
163C2×C8C22×D10111yescharacteristic
164C4⋊C4—1011no—
165C4⋊C4—1011no—
166C4⋊C4—1012no—
167C4⋊C4—2022no—
168C4⋊C4—2022no—
169C4⋊C4C22×D10112yes—
170C22⋊C4—22no—
171C22⋊C4—1012no—
172C22⋊C4—2022no—
173C22⋊C4—2022no—
174C22⋊C4—2022no—
175C42—1012no—
176C23—22no—
177C23—2022no—
178C23D4×D10111yescharacteristic
179D4—22no—
180D4—22no—
181D4—82no—
182D4—8082no—
183D4—8082no—
184D4—8082no—
185D4—2022no—
186D4—2022no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—8082no—
191C2×C4—8082no—
192C2×C4—1012no—
193C2×C4—1012no—
194C2×C4—2022no—
195C2×C4D4×D10111yescharacteristic
196C2×C4C23×D10111yescharacteristic
197C8—21no—
198C8—21no—
199C8—2022no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22—4042no—
207C22—4042no—
208C22—4042no—
209C22C2×D4×D10111yescharacteristic
210C4—21no—
211C4—42no—
212C4—4042no—
213C4—2022no—
214C4—2022no—
215C4#1569230011yescharacteristic
216C4#1569230211yescharacteristic
217C2—41no—
218C2—82no—
219C2—8082no—
220C2#2026575411yescharacteristic
221C2#2026597811yescharacteristic
222C2#2026597911yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#466×C101C2
socleC22×C101C22×D4
hypercenter#466D101
hypocenterC101#3259
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #20265780C2semidirect
2. #20265480C2semidirect
3. #20265475C2semidirect
4. #20265428C2semidirect
5. #20265413C2semidirect
6. #20265371C2semidirect
7. #20265370C2semidirect
8. #20265203C2semidirect
9. #466×C101C2semidirect
112. C101#3259semidirect
118. #466D101semidirect

Maximal subgroups

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

subgroupindex
#202657802
#202654802
#202654752
#202654282
#202654132
#202653712
#202653702
#202652032
#466×C1012
#3259101