Finite Group Databasebeta

← #27172235

Subgroups — #27172235

Order 12928.

350 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 71 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#20265975C212yesmaximal, complemented
2#20265145C211yesmaximal, characteristic, complemented
3#20265144C212yesmaximal, complemented
4#20264935C211yesmaximal, characteristic, complemented
5#575×C101C211yesmaximal, characteristic, complemented
6C2×#13108074C2211yescharacteristic
7#15692286—22no—
8#15692285—22no—
9#15692281—22no—
10#15692280—22no—
11#15692275—24no—
12C2×#13108058C2212yes—
13C22×#11717579C2211yescharacteristic
14C22⋊#11717579—24no—
15#15692167—22no—
16C2×#41×C101C2212yes—
17#137×C101—22no—
18AGL(1,ℤ/8)×C101—22no—
19#135×C101—22no—
20C2×M16×C101C2211yescharacteristic
21#13108074—22no—
22#13108074—22no—
23C2×Dic202D411yescharacteristic
24C2×D404D411yescharacteristic
25C2×C22⋊D101—21no—
26C2×C4×D101—21no—
27#13108064—24no—
28#13108062—24no—
29#13108061—24no—
30#13108059—24no—
31#13108058—22no—
32#13108058—22no—
33C2×#11717579—21no—
34C2×#11717579—22no—
35C2×#11717579D414yes—
36#41×C101—22no—
37#41×C101—22no—
38#41×C101—22no—
39#41×C101—22no—
40C2×Q8×C101D412yes—
41C2×D4×C101—22no—
42C2×D4×C101D412yes—
43C22×C4×C101—22no—
44C22×C4×C101C2311yescharacteristic
45Q16×C101—22no—
46SD16×C101—24no—
47D8×C101—22no—
48M16×C101—22no—
49C2×C8×C101D412yes—
50C23×C101—22no—
51C23×C101#4111yescharacteristic
52C22×D101—21no—
53Q8×C101—22no—
54Q8×C101—22no—
55D4×C101—22no—
56D4×C101—22no—
57D4×C101—42no—
58D4×C101—42no—
59C2×C4×C101—22no—
60C2×C4×C101—42no—
61C2×C4×C101—42no—
62C2×C4×C101C2×D411yescharacteristic
63C2×C4×C101C2×D412yes—
64C2×C4×C101C2×D412yes—
65C2×C4×C101C2×D411yescharacteristic
66C22⋊D101—42no—
67C2×Dic101—21no—
68D404—22no—
69C4×D101—41no—
70C4×D101—41no—
71Dic202—22no—
72C8×C101—22no—
73#11717579—24no—
74#11717579—41no—
75C22×C101—41no—
76C22×C101—44no—
77C22×C101—42no—
78C22×C101#12111yescharacteristic
79C22×C101#12112yes—
80C2×D101—42no—
81C2×D101—41no—
82C4×C101—42no—
83C4×C101#12012yes—
84C4×C101#12011yescharacteristic
85C4×C101#12711yescharacteristic
86Dic101—41no—
87C2×C101—21no—
88C2×C101—22no—
89C2×C101—82no—
90C2×C101#39211yescharacteristic
91D101—81no—
92#2017—1011nomaximal, Sylow
93C101#201711yescharacteristic, complemented, Sylow
94#575—1012no—
95#575D10111yescharacteristic, complemented
96#408—1012no—
97#408—1011no—
98#338—1011no—
99C2×#41—1011no—
100C2×#41C2×D10112yes—
101#137—22no—
102#137—2022no—
103#137—2022no—
104AGL(1,ℤ/8)—22no—
105AGL(1,ℤ/8)—2022no—
106AGL(1,ℤ/8)—2022no—
107#135—22no—
108#135—2024no—
109C2×M16—1012no—
110C2×M16C2×D10111yescharacteristic
111C22×C8—1011no—
112C22⋊C8—2024no—
113C22⋊C8—2022no—
114#41—22no—
115#41—22no—
116#41—22no—
117#41—22no—
118#41—2022no—
119#41—2022no—
120C2×Q8—1011no—
121C2×Q8C22⋊D10112yes—
122C2×D4—22no—
123C2×D4—1011no—
124C2×D4—2021no—
125C2×D4C22⋊D10112yes—
126C22×C4—22no—
127C22×C4—2021no—
128C22×C4C22×D10111yescharacteristic
129Q16—22no—
130Q16—2024no—
131SD16—24no—
132SD16—2024no—
133SD16—2024no—
134D8—22no—
135D8—2024no—
136M16—22no—
137M16—2022no—
138M16—2022no—
139C2×C8—1014no—
140C2×C8—2022no—
141C2×C8—2021no—
142C2×C8C22⋊D10112yes—
143C23—22no—
144C23—2021no—
145C23#1310807311yescharacteristic
146Q8—22no—
147Q8—22no—
148Q8—2022no—
149D4—22no—
150D4—22no—
151D4—42no—
152D4—42no—
153D4—4042no—
154D4—2022no—
155C2×C4—22no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—4041no—
159C2×C4—4041no—
160C2×C4—2021no—
161C2×C4D4×D10111yescharacteristic
162C2×C4D4×D10112yes—
163C2×C4C2×C22⋊D10112yes—
164C2×C4C2×C22⋊D10111yescharacteristic
165C8—22no—
166C8—4041no—
167C8—2024no—
168C22—41no—
169C22—44no—
170C22—42no—
171C22—4042no—
172C22—4041no—
173C22#1569222611yescharacteristic
174C22#1569222712yes—
175C4—42no—
176C4—4041no—
177C4#1569222312yes—
178C4#1569222411yescharacteristic
179C4#1569225111yescharacteristic
180C2—22no—
181C2—21no—
182C2—82no—
183C2—8081no—
184C2#2026508511yescharacteristic
subgroupG/H
centerC4#15692251
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#575×C101C2
socleC2×C101#392
hypercenter#575D101
hypocenterC101#2017
LCSG ▷ C22×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #20265975C2semidirect
2. #20265145C2semidirect
3. #20265144C2semidirect
4. #20264935C2semidirect
5. #575×C101C2semidirect
93. C101#2017semidirect
95. #575D101semidirect

Maximal subgroups

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

subgroupindex
#202659752
#202651452
#202651442
#202649352
#575×C1012
#2017101