Finite Group Databasebeta

← #49160456

Subgroups — #49160456

Order 37504.

270 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 65 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#34036693C211yesmaximal, characteristic
2#34036103C211yesmaximal, characteristic, complemented
3#34035921C214yesmaximal, complemented
4#410×C293C211yesmaximal, characteristic, complemented
5C22×D1172C411yescharacteristic, complemented
6#22627620C2212yes—
7#22627613C412yescomplemented
8#22627598—21no—
9#22627598—21no—
10#22627592—21no—
11#22627592—21no—
12#22627583—21no—
13#22627583C411yescharacteristic, complemented
14#22627539—21no—
15#22627539C2212yescomplemented
16C2×#41×C293C2211yescharacteristic
17#117×C293—21no—
18#99×C293—21no—
19#99×C293C2212yescomplemented
20C23×D293—21no—
21C2×D1172—21no—
22C2×D1172—41no—
23C2×D1172—41no—
24C2×D1172C2×C412yes—
25C2×C22⋊D293—22no—
26C2×C22⋊D293—22no—
27C4⋊Dic293—21no—
28C4⋊Dic293—41no—
29C22⋊Dic293—21no—
30C22⋊Dic293—22no—
31#17989779—22no—
32#17989779—22no—
33C4×Dic293—41no—
34C4×Dic293C2×C412yes—
35#41×C293—41no—
36#41×C293—41no—
37C2×Q8×C293C2×C411yescharacteristic
38C2×D4×C293D411yescharacteristic
39C2×D4×C293D411yescharacteristic
40C2×D4×C293C2311yescharacteristic
41C22×C4×C293C2×C411yescharacteristic
42C22×C4×C293D411yescharacteristic
43C22×C4×C293D411yescharacteristic
44C4⋊C4×C293—41no—
45C22⋊C4×C293—22no—
46C42×C293—41no—
47C22×D293—22no—
48C22×D293—42no—
49C22×D293—82no—
50D1172—42no—
51D1172—41no—
52D1172—41no—
53C22⋊D293—82no—
54C22⋊D293—82no—
55C2×Dic293—22no—
56C2×Dic293—42no—
57C23×C293C22⋊C411yescharacteristic
58C23×C293C2×D411yescharacteristic
59C23×C293C2×D411yescharacteristic
60Q8×C293—41no—
61D4×C293—41no—
62D4×C293—41no—
63D4×C293—41no—
64C2×C4×C293—21no—
65C2×C4×C293—21no—
66C2×C4×C293—21no—
67C2×C4×C293—21no—
68C2×C4×C293—21no—
69C2×C4×C293—21no—
70C2×C4×C293—42no—
71C2×C4×C293C22⋊C411yescharacteristic
72C2×C4×C293C22⋊C411yescharacteristic
73C2×C4×C293C22⋊C411yescharacteristic
74C2×C4×C293C22×C411yescharacteristic
75C22×C293—21no—
76C22×C293—21no—
77C22×C293—21no—
78C22×C293—41no—
79C22×C293—41no—
80C22×C293—41no—
81C22×C293C2×C22⋊C411yescharacteristic
82C2×D293—42no—
83C2×D293—42no—
84C2×D293—42no—
85C2×D293—84no—
86C4×C293—21no—
87C4×C293—21no—
88C4×C293—21no—
89C4×C293—82no—
90C4×C293#9911yescharacteristic
91C4×C293#9911yescharacteristic
92Dic293—42no—
93Dic293—82no—
94C2×C293—21no—
95C2×C293—41no—
96C2×C293—41no—
97C2×C293—41no—
98C2×C293C2×#9911yescharacteristic
99D293—82no—
100C293#212811yescharacteristic, complemented, Sylow
101#2128—2931nomaximal, Sylow
102#535—2931no—
103#410—2931no—
104#410D29311yescharacteristic, complemented
105#353—2934no—
106C2×#41C2×D29311yescharacteristic
107C22×D4—2931no—
108#127—2932no—
109#124—2932no—
110#121—5861no—
111#121—5861no—
112#120—5861no—
113#120—5861no—
114#117—21no—
115#117—2931no—
116#117—5861no—
117#99—21no—
118#99—2932no—
119#99—5861no—
120#99C2×D29312yescomplemented
121C24—5861no—
122#41—41no—
123#41—41no—
124C2×Q8C4×D29311yescharacteristic
125C2×D4—2932no—
126C2×D4—5862no—
127C2×D4—5862no—
128C2×D4—5861no—
129C2×D4—11721no—
130C2×D4—11721no—
131C2×D4C22⋊D29311yescharacteristic
132C2×D4D117211yescharacteristic
133C2×D4C22×D29311yescharacteristic
134C22×C4C4×D29311yescharacteristic
135C22×C4C22⋊D29311yescharacteristic
136C22×C4D117211yescharacteristic
137C4⋊C4—41no—
138C4⋊C4—5861no—
139C4⋊C4—11721no—
140C22⋊C4—22no—
141C22⋊C4—5861no—
142C22⋊C4—5862no—
143C22⋊C4—5862no—
144C22⋊C4—5862no—
145C42—41no—
146C42—2932no—
147C42—11721no—
148C23—23442no—
149C23—5862no—
150C23—11722no—
151C23#1798977911yescharacteristic
152C23C2×C22⋊D29311yescharacteristic
153C23C2×D117211yescharacteristic
154Q8—41no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—23442no—
159D4—23442no—
160D4—11722no—
161D4—11721no—
162D4—11721no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—42no—
170C2×C4—5862no—
171C2×C4—11722no—
172C2×C4#1798977911yescharacteristic
173C2×C4#1798977911yescharacteristic
174C2×C4#1798977911yescharacteristic
175C2×C4C2×C4×D29311yescharacteristic
176C22—21no—
177C22—21no—
178C22—21no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—23444no—
183C22—11722no—
184C22—11722no—
185C22—11722no—
186C22C2×#1798977911yescharacteristic
187C4—21no—
188C4—21no—
189C4—21no—
190C4—82no—
191C4—23442no—
192C4—11722no—
193C4#2262753811yescharacteristic
194C4#2262753811yescharacteristic
195C2—21no—
196C2—41no—
197C2—41no—
198C2—41no—
199C2—23442no—
200C2C2×#2262753811yescharacteristic
subgroupG/H
centerC2C2×#22627538
commutatorC2×C4×C293C22×C4
FrattiniC2×D4C22×D293
Fitting#410×C293C2
socleC2×C293C2×#99
prime radical#34036693C2
cosocleC2×D4×C293C23
hypercenter#410D293
hypocenterC293#2128
LCSG ▷ C2×C4×C293 ▷ C22×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #34036103C2semidirect
3. #34035921C2semidirect
4. #410×C293C2semidirect
5. C22×D1172C4semidirect
7. #22627613C4semidirect
13. #22627583C4semidirect
15. #22627539C22semidirect
19. #99×C293C22semidirect
100. C293#2128semidirect
104. #410D293semidirect
120. #99C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340366932
#340361032
#340359212
#410×C2932
#2128293