Finite Group Databasebeta

← #49160458

Subgroups — #49160458

Order 37504.

238 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 55 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#34036699C211yesmaximal, characteristic
2#34036101C212yesmaximal
3#34035924C211yesmaximal, characteristic, complemented
4#34035923C212yesmaximal, complemented
5#354×C293C211yesmaximal, characteristic, complemented
6C2×#17989801C2211yescharacteristic
7C2×Q8×D293C411yescharacteristic, complemented
8#22627624C411yescharacteristic, complemented
9#22627622C411yescharacteristic, complemented
10#22627611C2211yescharacteristic
11#22627606—22no—
12#22627601—22no—
13#22627581C411yescharacteristic, complemented
14#22627580—22no—
15#22627539C2212yes—
16#22627538—22no—
17#124×C293C2211yescharacteristic
18#99×C293C2212yes—
19#17989801—42no—
20C2×Dic586—21no—
21C2×Dic586C2×C411yescharacteristic
22C2×Dic586C2×C411yescharacteristic
23Q8×D293—42no—
24C2×C22⋊D293D412yes—
25C22×Dic293D412yes—
26C2×C4×D293—21no—
27C2×C4×D293C2×C411yescharacteristic
28C4⋊Dic293—22no—
29#17989781—21no—
30#17989781—22no—
31#17989781—42no—
32C22⋊Dic293—22no—
33C22⋊Dic293—22no—
34#17989779—21no—
35C4×Dic293—42no—
36C4×Dic293C2×C411yescharacteristic
37C2×Q8×C293C2×C411yescharacteristic
38C2×D4×C293C2311yescharacteristic
39C22⋊C4×C293—22no—
40C22⋊C4×C293—22no—
41C42×C293C2×C411yescharacteristic
42C22×D293C22⋊C411yescharacteristic
43Dic586—41no—
44Dic586—42no—
45Dic586—41no—
46C22⋊D293—42no—
47C2×Dic293—22no—
48C2×Dic293—22no—
49C2×Dic293—21no—
50C2×Dic293—21no—
51C2×Dic293—42no—
52C2×Dic293C22⋊C412yes—
53C2×Dic293C22⋊C411yescharacteristic
54C4×D293—22no—
55C4×D293—42no—
56C23×C293C2×D412yes—
57Q8×C293—41no—
58D4×C293—41no—
59C2×C4×C293—21no—
60C2×C4×C293—21no—
61C2×C4×C293—42no—
62C2×C4×C293C22×C411yescharacteristic
63C22×C293—22no—
64C22×C293—42no—
65C22×C293C2×C22⋊C411yescharacteristic
66C2×D293—21no—
67C2×D293—41no—
68C4×C293—21no—
69C4×C293—41no—
70C4×C293—41no—
71C4×C293—82no—
72Dic293—22no—
73Dic293—41no—
74Dic293—41no—
75Dic293—82no—
76Dic293#9912yes—
77C2×C293—21no—
78C2×C293—42no—
79C2×C293C2×#9911yescharacteristic
80D293—41no—
81C293#212911yescharacteristic, complemented, Sylow
82#2129—2931nomaximal, Sylow
83#536—2931no—
84#410—2932no—
85#354—2931no—
86#354—2932no—
87#354D29311yescharacteristic, complemented
88C2×#41—2931no—
89C22×Q8—2931no—
90C4⋊Q8—2931no—
91C4⋊Q8—2931no—
92#124—2931no—
93#124C2×D29311yescharacteristic
94#123—5862no—
95C22⋊Q8—5862no—
96#117—2931no—
97#117—5862no—
98#99—2932no—
99#99—5862no—
100#99C2×D29312yes—
101#41—11722no—
102C2×Q8—2931no—
103C2×Q8—2931no—
104C2×Q8—5861no—
105C2×Q8—11722no—
106C2×Q8C4×D29311yescharacteristic
107C2×D4—2932no—
108C2×D4C22×D29311yescharacteristic
109C22×C4—2932no—
110C22×C4—2931no—
111C22×C4—5861no—
112C4⋊C4—5861no—
113C4⋊C4—5862no—
114C4⋊C4—5862no—
115C4⋊C4—11722no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—5861no—
119C22⋊C4—5862no—
120C22⋊C4—5862no—
121C42—2931no—
122C42—11722no—
123C42C4×D29311yescharacteristic
124C23—2931no—
125C23D4×D29312yes—
126Q8—41no—
127Q8—11721no—
128Q8—11721no—
129Q8—11722no—
130D4—41no—
131D4—11722no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—42no—
135C2×C4—2931no—
136C2×C4—2932no—
137C2×C4—5862no—
138C2×C4—5862no—
139C2×C4—5862no—
140C2×C4—5861no—
141C2×C4—5861no—
142C2×C4—11722no—
143C2×C4—11722no—
144C2×C4C2×C4×D29311yescharacteristic
145C22—22no—
146C22—42no—
147C22—5861no—
148C22—11721no—
149C22C22⋊C4×D29311yescharacteristic
150C4—21no—
151C4—41no—
152C4—41no—
153C4—82no—
154C4—2932no—
155C4—23442no—
156C4—5862no—
157C4—11721no—
158C4—11721no—
159C2—21no—
160C2—42no—
161C2—11721no—
162C2#99×D29311yescharacteristic
subgroupG/H
centerC2#99×D293
commutatorC2×C4×C293C22×C4
FrattiniC2×D4C22×D293
Fitting#354×C293C2
socleC2×C293C2×#99
prime radicalC2×#17989801C22
cosocleC2×D4×C293C23
hypercenter#354D293
hypocenterC293#2129
LCSG ▷ C2×C4×C293 ▷ C22×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #34035924C2semidirect
4. #34035923C2semidirect
5. #354×C293C2semidirect
7. C2×Q8×D293C4semidirect
8. #22627624C4semidirect
9. #22627622C4semidirect
13. #22627581C4semidirect
81. C293#2129semidirect
87. #354D293semidirect

Maximal subgroups

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

subgroupindex
#340366992
#340361012
#340359242
#340359232
#354×C2932
#2129293