Finite Group Databasebeta

← #27187267

Subgroups — #27187267

Order 12928.

360 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 74 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#20266182C211yesmaximal, characteristic, complemented
2C2×#15692320C212yesmaximal, complemented
3C2×#15692319C212yesmaximal, complemented
4C42×#87938C211yesmaximal, characteristic, complemented
5#20265726C211yesmaximal, characteristic
6C2×#13108084C2212yes—
7C2×C4×#87938—22no—
8C2×C4×#87938C2212yes—
9#15692322—22no—
10#15692320—24no—
11#15692319—24no—
12C2×#13108075C2211yescharacteristic
13C2×#13108073C2211yescharacteristic
14#15692253C411yescharacteristic, complemented
15#15692251C411yescharacteristic, complemented
16#15692243C411yescharacteristic, complemented
17#15692228—22no—
18#15692215—22no—
19C42×D101C2211yescharacteristic
20#124×C101C411yescharacteristic, complemented
21C22×#87938—22no—
22#13108084—22no—
23#13108084—22no—
24C2×#11717581—22no—
25C4×#87938—28no—
26C4×#87938—22no—
27C4×#87938D414yescomplemented
28#13108075—22no—
29#13108075—22no—
30#13108073—22no—
31#13108073—22no—
32C2×Dic202C2×C411yescharacteristic
33C2×D404—21no—
34C2×D404C2×C411yescharacteristic
35C2×C22⋊D101—21no—
36C2×C22⋊D101—22no—
37C22×Dic101—21no—
38C2×C4×D101—21no—
39C2×C4×D101D412yes—
40C2×C4×D101C2311yescharacteristic
41#13108053—22no—
42#13108051—22no—
43C4×Dic101C2×C411yescharacteristic
44C4×Dic101D412yes—
45C2×Q8×C101C2×C411yescharacteristic
46C2×D4×C101C2×C411yescharacteristic
47C22⋊C4×C101—22no—
48C42×C101C2×C411yescharacteristic
49C23×C101—21no—
50C22×D101—21no—
51C22×D101C2×D411yescharacteristic
52C2×#87938—28no—
53C2×#87938—24no—
54Q8×C101—22no—
55D4×C101—22no—
56C2×C4×C101—21no—
57C2×C4×C101C22⋊C412yes—
58C2×C4×C101C22×C411yescharacteristic
59C22⋊D101—42no—
60C22⋊D101—44no—
61C2×Dic101—21no—
62C2×Dic101—41no—
63C2×Dic101—41no—
64C2×Dic101C22⋊C412yes—
65C2×Dic101C2×D411yescharacteristic
66D404—22no—
67D404—42no—
68C4×D101—24no—
69C4×D101—41no—
70C4×D101—41no—
71C4×D101C2×D412yes—
72C4×D101#4112yes—
73Dic202—22no—
74#11717581—42no—
75C22×C101—41no—
76C22×C101—42no—
77C22×C101C2×C22⋊C411yescharacteristic
78C2×D101—22no—
79C2×D101—41no—
80C2×D101—42no—
81C2×D101#12112yes—
82#87938—28no—
83C4×C101—22no—
84C4×C101—41no—
85C4×C101C4×D412yes—
86Dic101—41no—
87Dic101#10414yescomplemented
88Dic101#12411yescharacteristic
89Dic101#12711yescharacteristic
90C2×C101—81no—
91C2×C101#39011yescharacteristic
92C2×C101C2×#10412yes—
93D101—22no—
94D101—81no—
95#1962—1011nomaximal, Sylow
96C101#196211yescharacteristic, complemented, Sylow
97#532—1011no—
98#423—1011no—
99C2×#104—1012no—
100C2×#104—1012no—
101C43—1011no—
102C2×#41—1011no—
103C2×#41—1011no—
104C2×M16—1012no—
105C4⋊Q8—1011no—
106#127—1011no—
107#124—1011no—
108#124#8793811yescharacteristic, complemented
109#121—2022no—
110C4×D4—2022no—
111C2×C42—1011no—
112C2×C42—1012no—
113C2×C42—2022no—
114C4⋊C8—2022no—
115#104—2024no—
116#104—2024no—
117#41—2022no—
118#41—2022no—
119#41—2022no—
120#41—2022no—
121C2×Q8—1011no—
122C2×Q8C2×#8793811yescharacteristic
123C2×D4—1011no—
124C2×D4—2021no—
125C2×D4—2022no—
126C2×D4—2021no—
127C2×D4C2×#8793811yescharacteristic
128C22×C4—1011no—
129C22×C4—1012no—
130C22×C4—2021no—
131C22×C4—2021no—
132C22×C4—2022no—
133M16—2022no—
134M16—2022no—
135C2×C8—2022no—
136C4⋊C4—2022no—
137C22⋊C4—22no—
138C22⋊C4—2022no—
139C42—1012no—
140C42—1011no—
141C42—1014no—
142C42—2022no—
143C42—2028no—
144C42C2×#8793811yescharacteristic
145C23—21no—
146C23—1011no—
147C23—2021no—
148Q8—22no—
149Q8—2022no—
150D4—22no—
151D4—4042no—
152D4—4044no—
153D4—4042no—
154D4—2022no—
155C2×C4—21no—
156C2×C4—1012no—
157C2×C4—1011no—
158C2×C4—1012no—
159C2×C4—1012no—
160C2×C4—4041no—
161C2×C4—4041no—
162C2×C4—4041no—
163C2×C4—4041no—
164C2×C4—2024no—
165C2×C4—2021no—
166C2×C4—2024no—
167C2×C4—2028no—
168C2×C4C22⋊#8793812yes—
169C2×C4C22×#8793811yescharacteristic
170C8—4042no—
171C22—41no—
172C22—42no—
173C22—1012no—
174C22—4041no—
175C22—4042no—
176C22—2022no—
177C22C2×C22⋊#8793811yescharacteristic
178C4—22no—
179C4—41no—
180C4—1011no—
181C4—1011no—
182C4—1014no—
183C4—4041no—
184C4—2028no—
185C4D4×#8793812yes—
186C2—81no—
187C2—8081no—
188C2—2022no—
189C2#2026612911yescharacteristic
190C2D101⋊#10412yes—
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#124×C101C4
socleC22×C101C2×C22⋊C4
prime radical#20265726C2
cosocleC2×C4×D101C23
hypercenter#124#87938
hypocenterC101#1962
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #20266182C2semidirect
2. C2×#15692320C2semidirect
3. C2×#15692319C2semidirect
4. C42×#87938C2semidirect
14. #15692253C4semidirect
15. #15692251C4semidirect
16. #15692243C4semidirect
20. #124×C101C4semidirect
27. C4×#87938D4semidirect
87. Dic101#104semidirect
96. C101#1962semidirect
108. #124#87938semidirect

Maximal subgroups

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

subgroupindex
#202661822
C2×#156923202
C2×#156923192
C42×#879382
#202657262
#1962101