Finite Group Databasebeta

← #27184223

Subgroups — #27184223

Order 12928.

380 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 113 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#20265909C212yesmaximal, complemented
2#20265563C212yesmaximal, complemented
3#20265401C212yesmaximal, complemented
4#20265389C212yesmaximal, complemented
5#20265370C212yesmaximal, complemented
6#20265307C211yesmaximal, characteristic, complemented
7#20265278C212yesmaximal, complemented
8C4×D808C211yesmaximal, characteristic, complemented
9#495×C101C211yesmaximal, characteristic, complemented
10C2×#13108061C2212yes—
11C2×D808C2211yescharacteristic
12C2×#13108059C2212yes—
13#15692251C2212yes—
14#15692244C2212yes—
15#15692241—22no—
16#15692239—22no—
17#15692237—22no—
18Q8×Dic101C2212yes—
19C4×Dic202C2211yescharacteristic
20C4×D404C2212yes—
21D4×Dic101C2212yes—
22#15692215—22no—
23#15692189C2211yescharacteristic
24C4⋊#11717579C2211yescharacteristic
25#15692184C2212yes—
26#15692179C2212yes—
27#15692177C2212yes—
28#15692176C2212yes—
29#15692165C2212yes—
30C2×Q16×C101C2211yescharacteristic
31C2×SD16×C101C2212yes—
32C2×D8×C101C2211yescharacteristic
33#124×C101C2212yes—
34C4×C8×C101C2211yescharacteristic
35C2×Dic202D411yescharacteristic
36C2×D404C2312yes—
37C2×C22⋊D101—22no—
38C2×C22⋊D101—22no—
39C22×Dic101—22no—
40C2×C4×D101—22no—
41#13108061—42no—
42D808—22no—
43#13108059—42no—
44C2×#11717579C2312yes—
45C4⋊Dic101—22no—
46C4⋊Dic101D411yescharacteristic
47C4⋊Dic101C2312yes—
48#13108053D412yes—
49C22⋊Dic101—22no—
50#13108051—22no—
51#13108051—22no—
52#13108051—22no—
53C4×Dic101—22no—
54C4×Dic101C2312yes—
55C2×Q8×C101C2312yes—
56C2×D4×C101C2312yes—
57Q16×C101—41no—
58SD16×C101—24no—
59D8×C101—41no—
60C2×C8×C101C2311yescharacteristic
61C2×C8×C101C2311yescharacteristic
62C22⋊C4×C101—22no—
63C22⋊C4×C101—22no—
64C42×C101C2311yescharacteristic
65C23×C101—22no—
66C22×D101—22no—
67Q8×C101—22no—
68Q8×C101—22no—
69D4×C101—22no—
70D4×C101—22no—
71C2×C4×C101—22no—
72C2×C4×C101C2×D411yescharacteristic
73C2×C4×C101C2×D411yescharacteristic
74C2×C4×C101C2411yescharacteristic
75C22⋊D101—44no—
76C22⋊D101—82no—
77C2×Dic101—22no—
78C2×Dic101—82no—
79C2×Dic101C2×D412yes—
80C2×Dic101C2×D412yes—
81D404—22no—
82D404—22no—
83C4×D101—82no—
84Dic202—41no—
85C8×C101—21no—
86C8×C101#4112yes—
87#11717579—22no—
88C22×C101—42no—
89C22×C101—42no—
90C22×C101—42no—
91C22×C101C22×D411yescharacteristic
92C2×D101—42no—
93C2×D101—42no—
94C2×D101—42no—
95C4×C101—21no—
96C4×C101—21no—
97C4×C101—42no—
98C4×C101C2×#4111yescharacteristic
99C4×C101H(5,𝔽2)11yescharacteristic
100Dic101—22no—
101Dic101—22no—
102Dic101—42no—
103C2×C101—82no—
104C2×C101#55011yescharacteristic
105C2×C101#57511yescharacteristic
106C2×C101#57611yescharacteristic
107D101—82no—
108#3403—1011nomaximal, Sylow
109C101#340311yescharacteristic, complemented, Sylow
110#559—1012no—
111#496—1012no—
112#495D10111yescharacteristic, complemented
113#464—1012no—
114#463—1012no—
115#459—1012no—
116#446—1011no—
117#440—1012no—
118C4×D8—1011no—
119C2×Q16C2×D10111yescharacteristic
120C2×SD16—1012no—
121C2×SD16C2×D10112yes—
122C2×D8—1012no—
123C2×D8—1011no—
124C2×D8C2×D10111yescharacteristic
125#127—1012no—
126#124—1012no—
127#124—2022no—
128#124C2×D10112yes—
129#123—2022no—
130#123—2022no—
131C4×Q8—1011no—
132C4×Q8—1012no—
133C4×D4—1012no—
134C4×D4—1012no—
135C4×D4—2022no—
136#107—1011no—
137C4⋊C8—1012no—
138C4⋊C8—1011no—
139#103—1012no—
140#102—1012no—
141#102—1012no—
142#97—1012no—
143C4×C8C2×D10111yescharacteristic
144C2×Q8—1011no—
145C2×Q8C22×D10112yes—
146C2×D4—1012no—
147C2×D4—2022no—
148C2×D4—2022no—
149C2×D4C22×D10112yes—
150C22×C4—2022no—
151C22×C4—2022no—
152Q16—41no—
153SD16—24no—
154SD16—4042no—
155D8—41no—
156D8—4042no—
157D8—2022no—
158C2×C8—1012no—
159C2×C8C22×D10111yescharacteristic
160C2×C8C22×D10111yescharacteristic
161C4⋊C4—1012no—
162C4⋊C4—1011no—
163C4⋊C4—1012no—
164C4⋊C4—2022no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C22⋊C4—2022no—
168C22⋊C4—2022no—
169C22⋊C4—2022no—
170C22⋊C4—2022no—
171C42—1012no—
172C42—2022no—
173C42C22×D10111yescharacteristic
174C23—22no—
175C23—2022no—
176Q8—22no—
177Q8—22no—
178Q8—4041no—
179D4—22no—
180D4—22no—
181D4—8082no—
182D4—4044no—
183D4—2022no—
184D4—2022no—
185C2×C4—22no—
186C2×C4—8082no—
187C2×C4—8082no—
188C2×C4—1012no—
189C2×C4—1012no—
190C2×C4—2022no—
191C2×C4D4×D10111yescharacteristic
192C2×C4D4×D10111yescharacteristic
193C2×C4C23×D10111yescharacteristic
194C8—21no—
195C8—2022no—
196C8#1310807312yes—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—4042no—
201C22—4042no—
202C22—4042no—
203C22C2×D4×D10111yescharacteristic
204C4—21no—
205C4—21no—
206C4—42no—
207C4—4042no—
208C4—2022no—
209C4—2022no—
210C4C2×#1310807311yescharacteristic
211C4#1569230011yescharacteristic
212C2—82no—
213C2—8082no—
214C2#2026584711yescharacteristic
215C2#2026597411yescharacteristic
216C2#2026597911yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#495×C101C2
socleC22×C101C22×D4
hypercenter#495D101
hypocenterC101#3403
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #20265909C2semidirect
2. #20265563C2semidirect
3. #20265401C2semidirect
4. #20265389C2semidirect
5. #20265370C2semidirect
6. #20265307C2semidirect
7. #20265278C2semidirect
8. C4×D808C2semidirect
9. #495×C101C2semidirect
109. C101#3403semidirect
112. #495D101semidirect

Maximal subgroups

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

subgroupindex
#202659092
#202655632
#202654012
#202653892
#202653702
#202653072
#202652782
C4×D8082
#495×C1012
#3403101