Finite Group Databasebeta

← #27187412

Subgroups — #27187412

Order 12928.

270 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 52 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#20266158C212yesmaximal, complemented
2#20266082C212yesmaximal, complemented
3#20266081C212yesmaximal, complemented
4#20265745C211yesmaximal, characteristic
5#15692334—22no—
6#15692314—22no—
7#15692314C2214yescomplemented
8C2×#13108075C2211yescharacteristic
9C2×D4×D101C411yescharacteristic, complemented
10#15692252C2211yescharacteristic
11#15692251C2211yescharacteristic
12#15692244C411yescharacteristic, complemented
13#15692228—22no—
14#15692225—22no—
15#15692212C411yescharacteristic, complemented
16#124×C101C411yescharacteristic, complemented
17C4⋊#87938—42no—
18C22⋊#87938—24no—
19C4×#87938—42no—
20C23×D101—21no—
21#13108075—42no—
22C2×D404—22no—
23C2×D404C2×C411yescharacteristic
24C2×D404D412yes—
25C2×D404C2311yescharacteristic
26C2×C22⋊D101—22no—
27C2×C22⋊D101—21no—
28D4×D101—42no—
29C2×C4×D101C2×C411yescharacteristic
30C2×C4×D101D412yes—
31#13108053—21no—
32#13108051—21no—
33#13108051—22no—
34C4×Dic101C2×C411yescharacteristic
35C2×Q8×C101C2×C411yescharacteristic
36C2×D4×C101C2×C411yescharacteristic
37C22⋊C4×C101—22no—
38C42×C101C2×C411yescharacteristic
39C23×C101—21no—
40C22×D101—21no—
41C22×D101—42no—
42C22×D101—82no—
43C22×D101C22⋊C411yescharacteristic
44C22×D101C2×D412yes—
45C2×#87938—44no—
46Q8×C101—41no—
47D4×C101—41no—
48C2×C4×C101—21no—
49C2×C4×C101C22⋊C412yes—
50C2×C4×C101C22×C411yescharacteristic
51C22⋊D101—42no—
52C22⋊D101—82no—
53C2×Dic101—21no—
54C2×Dic101C22⋊C411yescharacteristic
55D404—42no—
56D404—41no—
57D404—41no—
58D404—82no—
59C4×D101—22no—
60C4×D101—22no—
61C4×D101—22no—
62C22×C101—42no—
63C22×C101—41no—
64C22×C101C2×C22⋊C411yescharacteristic
65C2×D101—22no—
66C2×D101—21no—
67C2×D101—42no—
68C2×D101—41no—
69C2×D101—42no—
70C2×D101—41no—
71C2×D101—84no—
72#87938—84no—
73C4×C101—21no—
74C4×C101—22no—
75C4×C101—41no—
76Dic101—41no—
77Dic101#9912yes—
78C2×C101—21no—
79C2×C101—81no—
80C2×C101C2×#9911yescharacteristic
81D101—42no—
82D101—41no—
83D101—81no—
84#2128—1011nomaximal, Sylow
85C101#212811yescharacteristic, complemented, Sylow
86#535—1011no—
87#410—1012no—
88#353—1012no—
89#353—1012no—
90C2×#41—1011no—
91C22×D4—1011no—
92#127—1011no—
93#127—1011no—
94#124—1011no—
95#124#8793811yescharacteristic, complemented
96#121—2022no—
97#120—2022no—
98#117—1011no—
99#117—2022no—
100#99—1014no—
101#99—2022no—
102C24—2021no—
103#41—4042no—
104C2×Q8C2×#8793811yescharacteristic
105C2×D4—1012no—
106C2×D4—1011no—
107C2×D4—1011no—
108C2×D4—4042no—
109C2×D4—2022no—
110C2×D4—2021no—
111C2×D4—2022no—
112C2×D4C2×#8793811yescharacteristic
113C22×C4—1012no—
114C22×C4—1011no—
115C4⋊C4—4042no—
116C4⋊C4—2021no—
117C22⋊C4—22no—
118C22⋊C4—2021no—
119C22⋊C4—2024no—
120C22⋊C4—2022no—
121C42—1011no—
122C42—4042no—
123C42C2×#8793811yescharacteristic
124C23—21no—
125C23—8082no—
126C23—1012no—
127C23—1011no—
128C23—4042no—
129C23—2021no—
130Q8—41no—
131D4—41no—
132D4—8082no—
133D4—8082no—
134D4—4041no—
135D4—4042no—
136D4—4041no—
137D4—4042no—
138C2×C4—21no—
139C2×C4—1011no—
140C2×C4—4044no—
141C2×C4—2022no—
142C2×C4—2022no—
143C2×C4—2022no—
144C2×C4—2021no—
145C2×C4C22⋊#8793812yes—
146C2×C4C22×#8793811yescharacteristic
147C22—42no—
148C22—41no—
149C22—8084no—
150C22—4042no—
151C22—4042no—
152C22—4041no—
153C22—4041no—
154C22—2022no—
155C22—2021no—
156C22C2×C22⋊#8793811yescharacteristic
157C4—22no—
158C4—21no—
159C4—41no—
160C4—8084no—
161C4—1012no—
162C4—4041no—
163C2—21no—
164C2—81no—
165C2—8081no—
166C2—4042no—
167C2—4041no—
168C2D101⋊#9911yescharacteristic
subgroupG/H
centerC2D101⋊#99
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#124×C101C4
socleC2×C101C2×#99
prime radical#20265745C2
cosocleC2×D404C23
hypercenter#124#87938
hypocenterC101#2128
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #20266158C2semidirect
2. #20266082C2semidirect
3. #20266081C2semidirect
7. #15692314C22semidirect
9. C2×D4×D101C4semidirect
12. #15692244C4semidirect
15. #15692212C4semidirect
16. #124×C101C4semidirect
85. C101#2128semidirect
95. #124#87938semidirect

Maximal subgroups

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

subgroupindex
#202661582
#202660822
#202660812
#202657452
#2128101