Finite Group Databasebeta

← #27172822

Subgroups — #27172822

Order 12928.

262 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 55 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#20265745C211yesmaximal, characteristic
2#20265160C212yesmaximal, complemented
3#20264974C211yesmaximal, characteristic, complemented
4#20264973C212yesmaximal, complemented
5#355×C101C211yesmaximal, characteristic, complemented
6C2×#13108075C2211yescharacteristic
7C2×D4×D101C411yescharacteristic
8M16×D101—22no—
9#15692252C411yescharacteristic
10#15692251C411yescharacteristic
11#15692244C2211yescharacteristic
12#15692228—22no—
13#15692225—22no—
14#15692212C411yescharacteristic
15#15692174C2212yescomplemented
16#15692171—22no—
17#124×C101C2211yescharacteristic
18#101×C101C2212yescomplemented
19C23×D101—21no—
20#13108075—42no—
21C2×D404—22no—
22C2×D404C2×C411yescharacteristic
23C2×D404C2×C411yescharacteristic
24C2×D404D412yes—
25C2×C22⋊D101—22no—
26C2×C22⋊D101—21no—
27D4×D101—42no—
28C2×C4×D101C2×C411yescharacteristic
29C2×C4×D101D412yes—
30#13108058—22no—
31#13108057—42no—
32C8×D101—42no—
33#13108053—21no—
34#13108051—21no—
35#13108051—22no—
36C4×Dic101C2×C411yescharacteristic
37C2×Q8×C101C2311yescharacteristic
38C2×D4×C101C2×C411yescharacteristic
39M16×C101—22no—
40C22⋊C4×C101—22no—
41C42×C101C2×C411yescharacteristic
42C23×C101—21no—
43C22×D101—21no—
44C22×D101—42no—
45C22×D101—82no—
46C22×D101C22⋊C411yescharacteristic
47C22×D101C22⋊C412yes—
48Q8×C101—41no—
49D4×C101—41no—
50C2×C4×C101—21no—
51C2×C4×C101C22×C411yescharacteristic
52C2×C4×C101C2×D412yes—
53C22⋊D101—42no—
54C22⋊D101—82no—
55C2×Dic101—21no—
56C2×Dic101C22⋊C411yescharacteristic
57D404—41no—
58D404—42no—
59D404—41no—
60D404—82no—
61C4×D101—22no—
62C4×D101—22no—
63C4×D101—22no—
64C8×C101—42no—
65#11717579—42no—
66C22×C101—42no—
67C22×C101—41no—
68C22×C101C2×C22⋊C411yescharacteristic
69C2×D101—21no—
70C2×D101—22no—
71C2×D101—42no—
72C2×D101—41no—
73C2×D101—42no—
74C2×D101—41no—
75C2×D101—84no—
76C4×C101—22no—
77C4×C101—21no—
78C4×C101—41no—
79Dic101—41no—
80Dic101#9912yes—
81C2×C101—21no—
82C2×C101—81no—
83C2×C101C2×#9911yescharacteristic
84D101—42no—
85D101—41no—
86D101—81no—
87#2134—1011nomaximal, Sylow
88C101#213411yescharacteristic, complemented, Sylow
89#535—1011no—
90#413—1012no—
91#355—1011no—
92#355—1012no—
93#355D10111yescharacteristic, complemented
94C2×#41—1011no—
95C22×D4—1011no—
96C2×M16—2022no—
97#127—1011no—
98#127—1011no—
99#124—1011no—
100#124C2×D10111yescharacteristic
101#121—2022no—
102#120—2022no—
103#117—1011no—
104#101—1012no—
105#101C2×D10112yescomplemented
106#100—2022no—
107C24—2021no—
108#41—4042no—
109C2×Q8C22×D10111yescharacteristic
110C2×D4—1011no—
111C2×D4—1011no—
112C2×D4—1012no—
113C2×D4—4042no—
114C2×D4—2022no—
115C2×D4—2021no—
116C2×D4—2022no—
117C2×D4C4×D10111yescharacteristic
118C22×C4—1011no—
119C22×C4—1012no—
120M16—22no—
121M16—4042no—
122M16—2022no—
123C2×C8—4042no—
124C4⋊C4—2021no—
125C22⋊C4—22no—
126C22⋊C4—2021no—
127C22⋊C4—2022no—
128C42—1011no—
129C42C4×D10111yescharacteristic
130C23—21no—
131C23—8082no—
132C23—1011no—
133C23—1012no—
134C23—4042no—
135C23—2021no—
136Q8—41no—
137D4—41no—
138D4—8082no—
139D4—8082no—
140D4—4041no—
141D4—4042no—
142D4—4041no—
143D4—4042no—
144C2×C4—21no—
145C2×C4—1011no—
146C2×C4—2022no—
147C2×C4—2022no—
148C2×C4—2022no—
149C2×C4—2021no—
150C2×C4C2×C4×D10111yescharacteristic
151C2×C4D4×D10112yes—
152C8—42no—
153C8—4042no—
154C22—42no—
155C22—41no—
156C22—8084no—
157C22—4042no—
158C22—4042no—
159C22—4041no—
160C22—4041no—
161C22—2021no—
162C22—2022no—
163C22C22⋊C4×D10111yescharacteristic
164C4—22no—
165C4—21no—
166C4—41no—
167C4—1012no—
168C4—4041no—
169C2—21no—
170C2—81no—
171C2—8081no—
172C2—4041no—
173C2—4042no—
174C2#99×D10111yescharacteristic
subgroupG/H
centerC2#99×D101
commutatorC2×C4×C101C22×C4
FrattiniC2×Q8C22×D101
Fitting#355×C101C2
socleC2×C101C2×#99
prime radical#20265745C2
cosocleC2×Q8×C101C23
hypercenter#355D101
hypocenterC101#2134
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #20265160C2semidirect
3. #20264974C2semidirect
4. #20264973C2semidirect
5. #355×C101C2semidirect
15. #15692174C22semidirect
18. #101×C101C22semidirect
88. C101#2134semidirect
93. #355D101semidirect
105. #101C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202657452
#202651602
#202649742
#202649732
#355×C1012
#2134101