Finite Group Databasebeta

← #396×D101 #27174351

Subgroups — #396×D101 #27174351

Order 12928.

908 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 203 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×C4⋊C4×D101C213yesmaximal, complemented
2C2×C22⋊C4×D101C213yesmaximal
3#20265101C211yesmaximal, characteristic, complemented
4#20265100C213yesmaximal, complemented
5#20265099C213yesmaximal, complemented
6#95×D101C211yesmaximal, characteristic, complemented
7#396×C101C211yesmaximal, characteristic, complemented
8C24×D101C2211yescharacteristic
9C22×C4×D101C2213yes—
10C22×C4×D101C2213yes—
11C2×C4⋊Dic101C2213yescomplemented
12C2×#13108053C2213yescomplemented
13C2×#13108053C2213yescomplemented
14C4⋊C4×D101—212no—
15C2×C22⋊Dic101C2213yes—
16C2×#13108051C2213yes—
17C2×#13108051C2213yes—
18C22⋊C4×D101—212no—
19#15692162C2213yescomplemented
20C2×C4⋊C4×C101C2213yescomplemented
21C2×C22⋊C4×C101C2213yes—
22#95×C101C2211yescharacteristic, complemented
23C23×D101—26no—
24C23×D101—26no—
25C23×D101Q814yes—
26C23×D101C2311yescharacteristic
27C23×D101C2311yescharacteristic
28C22×Dic101C2313yes—
29C22×Dic101C2313yes—
30C2×C4×D101—26no—
31C2×C4×D101—26no—
32C2×C4×D101—26no—
33C2×C4×D101—26no—
34C2×C4×D101—26no—
35C2×C4×D101D4112yes—
36C4⋊Dic101—26no—
37#13108053—26no—
38#13108053—26no—
39C22⋊Dic101—26no—
40#13108051—26no—
41#13108051—26no—
42C24×C101C2311yescharacteristic
43C22×C4×C101C2313yes—
44C22×C4×C101C2313yes—
45C4⋊C4×C101—26no—
46C22⋊C4×C101—26no—
47C23×C101—23no—
48C23×C101—23no—
49C23×C101C2×Q812yes—
50C23×C101C2411yescharacteristic
51C22×D101—23no—
52C22×D101—23no—
53C22×D101—224no—
54C22×D101—412no—
55C22×D101—44no—
56C22×D101C2×D416yes—
57C22×D101C2×Q812yes—
58C22×D101C2×Q812yes—
59C22×D101#4116yes—
60C2×C4×C101—23no—
61C2×C4×C101—23no—
62C2×C4×C101—23no—
63C2×C4×C101—23no—
64C2×C4×C101—23no—
65C2×C4×C101C2×D416yes—
66C2×Dic101—23no—
67C2×Dic101—23no—
68C2×Dic101—23no—
69C2×Dic101—23no—
70C2×Dic101—23no—
71C2×Dic101C2×D416yes—
72C4×D101—224no—
73C4×D101—412no—
74C22×C101—26no—
75C22×C101—43no—
76C22×C101—41no—
77C22×C101C22×D413yes—
78C22×C101C22×Q811yescharacteristic
79C22×C101C2×#4113yes—
80C2×D101—26no—
81C2×D101—43no—
82C2×D101—41no—
83C2×D101—416no—
84C2×D101#121112yes—
85C2×D101C22⋊Q8112yes—
86C2×D101#12614yes—
87C4×C101—26no—
88C4×C101—43no—
89Dic101—26no—
90Dic101—43no—
91C2×C101—42no—
92C2×C101C2×#12113yes—
93C2×C101C2×C22⋊Q813yes—
94C2×C101C2×#12611yescharacteristic
95D101—42no—
96D101#39618yesdirect product factor
97C2×#396—1011nomaximal, Sylow
98C101C2×#39611yescharacteristic, complemented, Sylow
99C22×C4⋊C4—1013no—
100C22×C22⋊C4—1013no—
101#396—1013no—
102#396—1013no—
103#396—1011no—
104#396D10111yescharacteristic, direct product factor
105C2×#95—1011no—
106C25—1011no—
107C23×C4—1013no—
108C23×C4—1013no—
109C2×C4⋊C4—1013no—
110C2×C4⋊C4—1013no—
111C2×C4⋊C4—1013no—
112C2×C4⋊C4—20212no—
113C2×C4⋊C4C2×D10113yescomplemented
114C2×C22⋊C4—1013no—
115C2×C22⋊C4—1013no—
116C2×C22⋊C4—1013no—
117C2×C22⋊C4—20212no—
118C2×C22⋊C4C2×D10113yes—
119#95—1013no—
120#95C2×D10111yescharacteristic, complemented
121C24—1011no—
122C24—1014no—
123C24—1011no—
124C24—2026no—
125C24—2026no—
126C24C22×D10111yescharacteristic
127C22×C4—1013no—
128C22×C4—10112no—
129C22×C4—1013no—
130C22×C4—2026no—
131C22×C4—2026no—
132C22×C4—2026no—
133C22×C4—2026no—
134C22×C4—2026no—
135C22×C4C22×D10113yes—
136C22×C4C22×D10113yes—
137C4⋊C4—26no—
138C4⋊C4—2026no—
139C4⋊C4—2026no—
140C4⋊C4—2026no—
141C22⋊C4—26no—
142C22⋊C4—2026no—
143C22⋊C4—2026no—
144C22⋊C4—2026no—
145C23—23no—
146C23—23no—
147C23—1016no—
148C23—1012no—
149C23—1016no—
150C23—1012no—
151C23—40412no—
152C23—4044no—
153C23—20224no—
154C23—2023no—
155C23—2023no—
156C23Q8×D10112yes—
157C23C23×D10111yescharacteristic
158C2×C4—23no—
159C2×C4—23no—
160C2×C4—23no—
161C2×C4—23no—
162C2×C4—23no—
163C2×C4—1016no—
164C2×C4—40412no—
165C2×C4—20224no—
166C2×C4—2023no—
167C2×C4—2023no—
168C2×C4—2023no—
169C2×C4—2023no—
170C2×C4—2023no—
171C2×C4D4×D10116yes—
172C22—26no—
173C22—43no—
174C22—41no—
175C22—10112no—
176C22—10112no—
177C22—1014no—
178C22—40416no—
179C22—4043no—
180C22—4041no—
181C22—2026no—
182C22C2×D4×D10113yes—
183C22C2×Q8×D10111yescharacteristic
184C22#41×D10113yes—
185C4—26no—
186C4—43no—
187C4—4043no—
188C4—2026no—
189C2—42no—
190C2—1018no—
191C2—4042no—
192C2#121×D10113yes—
193C2C22⋊Q8×D10113yes—
194C2#126×D10111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#396×C101C2
prime radicalC24×D101C22
hypercenter#396D101
hypocenterC101C2×#396
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×D101C2semidirect
3. #20265101C2semidirect
4. #20265100C2semidirect
5. #20265099C2semidirect
6. #95×D101C2semidirect
7. #396×C101C2semidirect
11. C2×C4⋊Dic101C22semidirect
12. C2×#13108053C22semidirect
13. C2×#13108053C22semidirect
19. #15692162C22semidirect
20. C2×C4⋊C4×C101C22semidirect
22. #95×C101C22semidirect
96. D101#396direct (N × H)
98. C101C2×#396semidirect
104. #396D101direct (N × H)
113. C2×C4⋊C4C2×D101semidirect
120. #95C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D1012
C2×C22⋊C4×D1012
#202651012
#202651002
#202650992
#95×D1012
#396×C1012
C2×#396101