Finite Group Databasebeta

← #27172174

Subgroups — #27172174

Order 12928.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 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×#15692280C212yesmaximal, complemented
2C2×#15692182C212yesmaximal, complemented
3#20265078C211yesmaximal, characteristic, complemented
4#20264946C211yesmaximal, characteristic, complemented
5#537×C101C211yesmaximal, characteristic, complemented
6C22×D404C2211yescharacteristic
7#15692280—24no—
8#15692280—24no—
9C2×#13108061—22no—
10C2×#13108059—22no—
11C2×#13108058C2212yes—
12C2×#13108051—22no—
13C2×C4×Dic101C2211yescharacteristic
14#15692182—24no—
15C2×#41×C101C2212yes—
16C22⋊Q8×C101—22no—
17#121×C101—22no—
18#121×C101—22no—
19C2×C4⋊C4×C101C2211yescharacteristic
20C23×D101—21no—
21C2×D404—22no—
22C2×D404—22no—
23C2×D404—21no—
24C2×D404D412yescomplemented
25C22×Dic101—21no—
26#13108061—28no—
27#13108059—28no—
28#13108058—22no—
29#13108058D414yescomplemented
30C2×#11717579—22no—
31#13108051—44no—
32C4×Dic101—22no—
33#41×C101—22no—
34#41×C101—22no—
35#41×C101—22no—
36#41×C101—22no—
37C2×Q8×C101D412yes—
38C2×D4×C101—22no—
39C2×D4×C101—22no—
40C2×D4×C101D412yes—
41C22×C4×C101—22no—
42C22×C4×C101D412yes—
43C22×C4×C101C2311yescharacteristic
44C4⋊C4×C101—22no—
45C4⋊C4×C101—22no—
46C22⋊C4×C101—24no—
47C23×C101—22no—
48C23×C101C2×D411yescharacteristic
49C22×D101—22no—
50C22×D101—42no—
51C22×D101—42no—
52C22×D101—41no—
53C22×D101—41no—
54Q8×C101—22no—
55Q8×C101—22no—
56D4×C101—22no—
57D4×C101—22no—
58D4×C101—42no—
59D4×C101—42no—
60D4×C101—82no—
61C2×C4×C101—22no—
62C2×C4×C101—22no—
63C2×C4×C101—42no—
64C2×C4×C101—42no—
65C2×C4×C101—42no—
66C2×C4×C101C2×D411yescharacteristic
67C2×C4×C101C2×D412yes—
68C2×C4×C101C2×D412yes—
69C2×C4×C101#4111yescharacteristic
70C2×Dic101—22no—
71C2×Dic101—41no—
72C2×Dic101—41no—
73D404—24no—
74D404—42no—
75#11717579—24no—
76C22×C101—21no—
77C22×C101—21no—
78C22×C101—42no—
79C22×C101—42no—
80C22×C101—42no—
81C22×C101#12011yescharacteristic
82C22×C101#12111yescharacteristic
83C22×C101#12711yescharacteristic
84C2×D101—44no—
85C2×D101—42no—
86C2×D101—82no—
87C2×D101—81no—
88C2×D101—81no—
89C4×C101—42no—
90C4×C101—42no—
91C4×C101#12012yes—
92C4×C101#12112yes—
93Dic101—42no—
94C2×C101—21no—
95C2×C101—21no—
96C2×C101—82no—
97C2×C101#39211yescharacteristic
98C2×C101#45311yescharacteristic
99C2×C101#45411yescharacteristic
100D101—82no—
101#2006—1011nomaximal, Sylow
102C101#200611yescharacteristic, complemented, Sylow
103C2×AGL(1,ℤ/8)—1012no—
104#537D10111yescharacteristic, complemented
105C2×#104—1012no—
106#390—1011no—
107#343—1011no—
108C2×#41C2×D10112yes—
109C22×D4—1011no—
110AGL(1,ℤ/8)—2024no—
111AGL(1,ℤ/8)—2024no—
112C2×SD16—2022no—
113C2×D8—2022no—
114C2×M16—1012no—
115C22⋊Q8—22no—
116#121—22no—
117#121—22no—
118C2×C4⋊C4C2×D10111yescharacteristic
119C2×C22⋊C4—2022no—
120C2×C42—1011no—
121#104—2024no—
122C24—2021no—
123#41—22no—
124#41—22no—
125#41—22no—
126#41—22no—
127C2×Q8C22⋊D10112yes—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—1012no—
131C2×D4—2022no—
132C2×D4—2022no—
133C2×D4—2021no—
134C2×D4C22⋊D10112yes—
135C22×C4—22no—
136C22×C4—2021no—
137C22×C4C22⋊D10112yes—
138C22×C4C22×D10111yescharacteristic
139SD16—2028no—
140D8—2028no—
141M16—1014no—
142M16—2022no—
143C2×C8—2022no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C22⋊C4—24no—
147C22⋊C4—4044no—
148C42—2022no—
149C23—22no—
150C23—4042no—
151C23—4042no—
152C23—4041no—
153C23—4041no—
154C23—2022no—
155C23C2×C22⋊D10111yescharacteristic
156Q8—22no—
157Q8—22no—
158D4—22no—
159D4—22no—
160D4—42no—
161D4—42no—
162D4—82no—
163D4—4042no—
164D4—2024no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—4041no—
171C2×C4—4041no—
172C2×C4—2022no—
173C2×C4D4×D10111yescharacteristic
174C2×C4D4×D10112yes—
175C2×C4C2×C22⋊D10112yes—
176C2×C4#1310807311yescharacteristic
177C8—2024no—
178C22—21no—
179C22—21no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—8082no—
184C22—8081no—
185C22—8081no—
186C22—4044no—
187C22—4042no—
188C22#1569222411yescharacteristic
189C22#1569222611yescharacteristic
190C22#1569225111yescharacteristic
191C4—42no—
192C4—42no—
193C4—4042no—
194C4#1569222312yes—
195C4#1569222712yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—8082no—
200C2#2026508511yescharacteristic
201C2#2026534511yescharacteristic
202C2#2026535011yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#537×C101C2
socleC22×C101#120
hypercenter#537D101
hypocenterC101#2006
LCSG ▷ C22×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#15692280C2semidirect
2. C2×#15692182C2semidirect
3. #20265078C2semidirect
4. #20264946C2semidirect
5. #537×C101C2semidirect
24. C2×D404D4semidirect
29. #13108058D4semidirect
102. C101#2006semidirect
104. #537D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922802
C2×#156921822
#202650782
#202649462
#537×C1012
#2006101