Finite Group Databasebeta

← #396×D293 #49162004

Subgroups — #396×D293 #49162004

Order 37504.

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×D293C213yesmaximal, complemented
2C2×C22⋊C4×D293C213yesmaximal
3#34036053C211yesmaximal, characteristic, complemented
4#34036052C213yesmaximal, complemented
5#34036051C213yesmaximal, complemented
6#95×D293C211yesmaximal, characteristic, complemented
7#396×C293C211yesmaximal, characteristic, complemented
8C24×D293C2211yescharacteristic
9C22×C4×D293C2213yes—
10C22×C4×D293C2213yes—
11C2×C4⋊Dic293C2213yescomplemented
12C2×#17989781C2213yescomplemented
13C2×#17989781C2213yescomplemented
14C4⋊C4×D293—212no—
15C2×C22⋊Dic293C2213yes—
16C2×#17989779C2213yes—
17C2×#17989779C2213yes—
18C22⋊C4×D293—212no—
19#22627531C2213yescomplemented
20C2×C4⋊C4×C293C2213yescomplemented
21C2×C22⋊C4×C293C2213yes—
22#95×C293C2211yescharacteristic, complemented
23C23×D293—26no—
24C23×D293—26no—
25C23×D293Q814yes—
26C23×D293C2311yescharacteristic
27C23×D293C2311yescharacteristic
28C22×Dic293C2313yes—
29C22×Dic293C2313yes—
30C2×C4×D293—26no—
31C2×C4×D293—26no—
32C2×C4×D293—26no—
33C2×C4×D293—26no—
34C2×C4×D293—26no—
35C2×C4×D293D4112yes—
36C4⋊Dic293—26no—
37#17989781—26no—
38#17989781—26no—
39C22⋊Dic293—26no—
40#17989779—26no—
41#17989779—26no—
42C24×C293C2311yescharacteristic
43C22×C4×C293C2313yes—
44C22×C4×C293C2313yes—
45C4⋊C4×C293—26no—
46C22⋊C4×C293—26no—
47C22×D293—23no—
48C22×D293—23no—
49C22×D293—224no—
50C22×D293—412no—
51C22×D293—44no—
52C22×D293C2×D416yes—
53C22×D293C2×Q812yes—
54C22×D293C2×Q812yes—
55C22×D293#4116yes—
56C2×Dic293—23no—
57C2×Dic293—23no—
58C2×Dic293—23no—
59C2×Dic293—23no—
60C2×Dic293—23no—
61C2×Dic293C2×D416yes—
62C4×D293—224no—
63C4×D293—412no—
64C23×C293—23no—
65C23×C293—23no—
66C23×C293C2×Q812yes—
67C23×C293C2411yescharacteristic
68C2×C4×C293—23no—
69C2×C4×C293—23no—
70C2×C4×C293—23no—
71C2×C4×C293—23no—
72C2×C4×C293—23no—
73C2×C4×C293C2×D416yes—
74C22×C293—26no—
75C22×C293—43no—
76C22×C293—41no—
77C22×C293C22×D413yes—
78C22×C293C22×Q811yescharacteristic
79C22×C293C2×#4113yes—
80C2×D293—26no—
81C2×D293—43no—
82C2×D293—41no—
83C2×D293—416no—
84C2×D293#121112yes—
85C2×D293C22⋊Q8112yes—
86C2×D293#12614yes—
87C4×C293—26no—
88C4×C293—43no—
89Dic293—26no—
90Dic293—43no—
91C2×C293—42no—
92C2×C293C2×#12113yes—
93C2×C293C2×C22⋊Q813yes—
94C2×C293C2×#12611yescharacteristic
95D293—42no—
96D293#39618yesdirect product factor
97C293C2×#39611yescharacteristic, complemented, Sylow
98C2×#396—2931nomaximal, Sylow
99C22×C4⋊C4—2933no—
100C22×C22⋊C4—2933no—
101#396—2933no—
102#396—2933no—
103#396—2931no—
104#396D29311yescharacteristic, direct product factor
105C2×#95—2931no—
106C25—2931no—
107C23×C4—2933no—
108C23×C4—2933no—
109C2×C4⋊C4—2933no—
110C2×C4⋊C4—2933no—
111C2×C4⋊C4—2933no—
112C2×C4⋊C4—58612no—
113C2×C4⋊C4C2×D29313yescomplemented
114C2×C22⋊C4—2933no—
115C2×C22⋊C4—2933no—
116C2×C22⋊C4—2933no—
117C2×C22⋊C4—58612no—
118C2×C22⋊C4C2×D29313yes—
119#95—2933no—
120#95C2×D29311yescharacteristic, complemented
121C24—2931no—
122C24—2934no—
123C24—2931no—
124C24—5866no—
125C24—5866no—
126C24C22×D29311yescharacteristic
127C22×C4—2933no—
128C22×C4—29312no—
129C22×C4—2933no—
130C22×C4—5866no—
131C22×C4—5866no—
132C22×C4—5866no—
133C22×C4—5866no—
134C22×C4—5866no—
135C22×C4C22×D29313yes—
136C22×C4C22×D29313yes—
137C4⋊C4—26no—
138C4⋊C4—5866no—
139C4⋊C4—5866no—
140C4⋊C4—5866no—
141C22⋊C4—26no—
142C22⋊C4—5866no—
143C22⋊C4—5866no—
144C22⋊C4—5866no—
145C23—23no—
146C23—23no—
147C23—2936no—
148C23—2932no—
149C23—2936no—
150C23—2932no—
151C23—58624no—
152C23—5863no—
153C23—5863no—
154C23—117212no—
155C23—11724no—
156C23Q8×D29312yes—
157C23C23×D29311yescharacteristic
158C2×C4—23no—
159C2×C4—23no—
160C2×C4—23no—
161C2×C4—23no—
162C2×C4—23no—
163C2×C4—2936no—
164C2×C4—58624no—
165C2×C4—5863no—
166C2×C4—5863no—
167C2×C4—5863no—
168C2×C4—5863no—
169C2×C4—5863no—
170C2×C4—117212no—
171C2×C4D4×D29316yes—
172C22—26no—
173C22—43no—
174C22—41no—
175C22—29312no—
176C22—29312no—
177C22—2934no—
178C22—5866no—
179C22—117216no—
180C22—11723no—
181C22—11721no—
182C22C2×D4×D29313yes—
183C22C2×Q8×D29311yescharacteristic
184C22#41×D29313yes—
185C4—26no—
186C4—43no—
187C4—5866no—
188C4—11723no—
189C2—42no—
190C2—2938no—
191C2—11722no—
192C2#121×D29313yes—
193C2C22⋊Q8×D29313yes—
194C2#126×D29311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D293
commutator, socle, cosocleC23×C293C24
Fitting#396×C293C2
prime radicalC24×D293C22
hypercenter#396D293
hypocenterC293C2×#396
LCSG ▷ C23×C293 ▷ C293
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×D293C2semidirect
3. #34036053C2semidirect
4. #34036052C2semidirect
5. #34036051C2semidirect
6. #95×D293C2semidirect
7. #396×C293C2semidirect
11. C2×C4⋊Dic293C22semidirect
12. C2×#17989781C22semidirect
13. C2×#17989781C22semidirect
19. #22627531C22semidirect
20. C2×C4⋊C4×C293C22semidirect
22. #95×C293C22semidirect
96. D293#396direct (N × H)
97. C293C2×#396semidirect
104. #396D293direct (N × H)
113. C2×C4⋊C4C2×D293semidirect
120. #95C2×D293semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D2932
C2×C22⋊C4×D2932
#340360532
#340360522
#340360512
#95×D2932
#396×C2932
C2×#396293