Finite Group Databasebeta

← #27187409

Subgroups — #27187409

Order 12928.

286 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 54 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×#15692314C212yesmaximal, complemented
2#20266084C212yesmaximal, complemented
3#20266082C212yesmaximal, complemented
4#20265746C211yesmaximal, characteristic
5C2×C22⋊#87938—22no—
6#15692314—22no—
7#15692314C2214yes—
8C2×#13108074C411yescharacteristic, complemented
9C22×D404C2211yescharacteristic
10#15692252C411yescharacteristic, complemented
11#15692252C2211yescharacteristic
12#15692243C411yescharacteristic, complemented
13#15692243C2211yescharacteristic
14#15692230—22no—
15#15692225—22no—
16#117×C101C411yescharacteristic, complemented
17C22×#87938—42no—
18C22⋊#87938—24no—
19C22⋊#87938—42no—
20C23×D101D412yes—
21#13108074—42no—
22C2×Dic202C2×C411yescharacteristic, complemented
23C2×D404—22no—
24C2×D404—22no—
25C2×D404—42no—
26C2×D404C2×C411yescharacteristic
27C2×D404C2×C411yescharacteristic, complemented
28C2×D404D412yes—
29C2×D404C2311yescharacteristic
30C2×C22⋊D101—21no—
31C2×C4×D101—21no—
32#13108051—22no—
33#13108051—22no—
34C22×C4×C101C2×C411yescharacteristic
35C4⋊C4×C101—21no—
36C22⋊C4×C101—21no—
37C42×C101C2×C411yescharacteristic, complemented
38C42×C101C2×C411yescharacteristic, complemented
39C23×C101C22⋊C411yescharacteristic
40C22×D101—22no—
41C22×D101—22no—
42C22×D101—21no—
43C22×D101—82no—
44C22×D101C22⋊C412yes—
45C22×D101C2×D412yes—
46C2×#87938—44no—
47C2×#87938—84no—
48C2×C4×C101—22no—
49C2×C4×C101—21no—
50C2×C4×C101—21no—
51C2×C4×C101C22⋊C411yescharacteristic
52C2×C4×C101C22×C411yescharacteristic
53C22⋊D101—42no—
54C2×Dic101—21no—
55D404—41no—
56D404—42no—
57D404—41no—
58D404—41no—
59D404—82no—
60D404—82no—
61C4×D101—42no—
62Dic202—41no—
63C22×C101—41no—
64C22×C101#9912yes—
65C22×C101C2×C22⋊C411yescharacteristic
66C2×D101—22no—
67C2×D101—22no—
68C2×D101—44no—
69C2×D101—42no—
70C2×D101—42no—
71C2×D101—42no—
72C2×D101—41no—
73C2×D101—82no—
74#87938—84no—
75C4×C101—21no—
76C4×C101—21no—
77C4×C101—41no—
78C4×C101—41no—
79Dic101—41no—
80C2×C101—21no—
81C2×C101—22no—
82C2×C101C2×#9911yescharacteristic
83D101—42no—
84D101—42no—
85D101—81no—
86#2127—1011nomaximal, Sylow
87C101#212711yescharacteristic, complemented, Sylow
88#535—1011no—
89C2×#99—1012no—
90#354—1012no—
91#353—1012no—
92C2×#41—1011no—
93C22×D4—1011no—
94#127—1011no—
95#127—1011no—
96#124—1011no—
97#124—1011no—
98#121—2022no—
99#120—2022no—
100#117#8793811yescharacteristic, complemented
101C2×C22⋊C4—2022no—
102#99—1014no—
103#99—2022no—
104C24—1012no—
105#41—4042no—
106C2×Q8—1011no—
107C2×D4—1012no—
108C2×D4—1011no—
109C2×D4—1011no—
110C2×D4—1011no—
111C2×D4—4042no—
112C2×D4—2022no—
113C2×D4—2022no—
114C2×D4—2021no—
115C22×C4—4042no—
116C22×C4—2021no—
117C22×C4C2×#8793811yescharacteristic
118C4⋊C4—21no—
119C22⋊C4—21no—
120C22⋊C4—4042no—
121C22⋊C4—2022no—
122C22⋊C4—2022no—
123C22⋊C4—2024no—
124C42C2×#8793811yescharacteristic, complemented
125C42C2×#8793811yescharacteristic, complemented
126C23—8082no—
127C23—1012no—
128C23—1012no—
129C23—2022no—
130C23—2021no—
131C23—2022no—
132C23C22⋊#8793811yescharacteristic
133Q8—4041no—
134D4—8082no—
135D4—8082no—
136D4—4041no—
137D4—4042no—
138D4—4041no—
139D4—4041no—
140D4—4042no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—8084no—
145C2×C4—4042no—
146C2×C4—4044no—
147C2×C4—2021no—
148C2×C4C22⋊#8793811yescharacteristic
149C2×C4C22×#8793811yescharacteristic
150C22—41no—
151C22—8082no—
152C22—4042no—
153C22—4042no—
154C22—4042no—
155C22—4044no—
156C22—4041no—
157C22—2022no—
158C22—2022no—
159C22#1569231412yes—
160C22C2×C22⋊#8793811yescharacteristic
161C4—21no—
162C4—21no—
163C4—41no—
164C4—41no—
165C4—8084no—
166C4—4041no—
167C2—21no—
168C2—22no—
169C2—8081no—
170C2—4042no—
171C2—4042no—
172C2C2×#1569231411yescharacteristic
subgroupG/H
centerC2C2×#15692314
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#117×C101C4
socleC2×C101C2×#99
prime radical#20265746C2
cosocleC2×D404C23
hypercenter#117#87938
hypocenterC101#2127
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C2×#15692314C2semidirect
2. #20266084C2semidirect
3. #20266082C2semidirect
8. C2×#13108074C4semidirect
10. #15692252C4semidirect
12. #15692243C4semidirect
16. #117×C101C4semidirect
22. C2×Dic202C2×C4semidirect
27. C2×D404C2×C4semidirect
37. C42×C101C2×C4semidirect
38. C42×C101C2×C4semidirect
87. C101#2127semidirect
100. #117#87938semidirect
124. C42C2×#87938semidirect
125. C42C2×#87938semidirect

Maximal subgroups

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

subgroupindex
C2×#156923142
#202660842
#202660822
#202657462
#2127101