Finite Group Databasebeta

← #22611902

Subgroups — #22611902

Order 9344.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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#17989027C211yesmaximal, characteristic, complemented
2C2×#14556535C211yesmaximal, characteristic, complemented
3#17988862C211yesmaximal, characteristic, complemented
4#17988400C211yesmaximal, characteristic, complemented
5#17988394C211yesmaximal, characteristic, complemented
6#17988371C211yesmaximal, characteristic, complemented
7#576×C73C211yesmaximal, characteristic
8#14556621—21no—
9C2×Dic292C2211yescharacteristic, complemented
10#14556537—21no—
11#14556535—22no—
12#14556535—21no—
13#14556531C2211yescharacteristic, complemented
14C2×#11924003C2211yescharacteristic, complemented
15#14556527C2211yescharacteristic, complemented
16#14556509—21no—
17H(5,𝔽2)×C73—21no—
18AGL(1,ℤ/8)×C73—22no—
19AGL(1,ℤ/8)×C73—21no—
20#135×C73—21no—
21#135×C73C2211yescharacteristic
22C2×D8×C73—21no—
23C2×D8×C73C2211yescharacteristic
24#131×C73C2211yescharacteristic
25C2×Dic146—21no—
26Dic292—21no—
27Dic292—21no—
28#11924018—41no—
29#11924016—41no—
30#11924012—21no—
31#11924003—21no—
32#11924003—21no—
33#41×C73—22no—
34#41×C73—21no—
35#41×C73—41no—
36#41×C73D411yescharacteristic
37C2×D4×C73—22no—
38C2×D4×C73—21no—
39C2×D4×C73—42no—
40C2×D4×C73—41no—
41Q16×C73D411yescharacteristic
42SD16×C73—22no—
43SD16×C73—21no—
44D8×C73—22no—
45D8×C73—21no—
46D8×C73D411yescharacteristic
47D8×C73D412yes—
48M16×C73—21no—
49M16×C73D411yescharacteristic
50C2×C8×C73—21no—
51C2×C8×C73C2311yescharacteristic
52C23×C73—42no—
53C23×C73—41no—
54Q8×C73—21no—
55Q8×C73D811yescharacteristic
56D4×C73—21no—
57D4×C73—21no—
58D4×C73—22no—
59D4×C73—42no—
60D4×C73—42no—
61D4×C73—41no—
62D4×C73—41no—
63D4×C73D811yescharacteristic
64C2×C4×C73—21no—
65C2×C4×C73—42no—
66C2×C4×C73—41no—
67C2×C4×C73C2×D411yescharacteristic
68C2×Dic73—41no—
69Dic146—41no—
70Dic146—41no—
71C8×C73—21no—
72C8×C73C2×D411yescharacteristic
73C8×C73C2×D411yescharacteristic
74#10596831—41no—
75C22×C73—21no—
76C22×C73—42no—
77C22×C73—41no—
78C22×C73—161no—
79C22×C73—162no—
80C22×C73AGL(1,ℤ/8)11yescharacteristic
81C4×C73—21no—
82C4×C73—41no—
83C4×C73#12011yescharacteristic
84C4×C73C2×D811yescharacteristic
85Dic73—81no—
86C2×C73—21no—
87C2×C73—41no—
88C2×C73—82no—
89C2×C73—81no—
90C2×C73#44711yescharacteristic
91#2194—731nomaximal, Sylow
92C73#219411yescharacteristic, complemented, Sylow
93#576D7311yescharacteristic
94#510—731no—
95C2×SD32—731no—
96#473—731no—
97#361—731no—
98#359—731no—
99#350—731no—
100H(5,𝔽2)—21no—
101#137—1461no—
102AGL(1,ℤ/8)—22no—
103AGL(1,ℤ/8)—21no—
104#135—21no—
105#135C2×D7311yescharacteristic
106C2×Q16—731no—
107C2×D8—21no—
108C2×D8C2×D7311yescharacteristic
109#131C2×D7311yescharacteristic
110Q32—1461no—
111SD32—1462no—
112SD32—1461no—
113M32—731no—
114C2×C16—731no—
115#108—731no—
116#101—1461no—
117#41—22no—
118#41—21no—
119#41—41no—
120#41C22⋊D7311yescharacteristic
121C2×Q8—1461no—
122C2×D4—22no—
123C2×D4—21no—
124C2×D4—42no—
125C2×D4—41no—
126Q16—2921no—
127Q16—1461no—
128Q16—1461no—
129Q16C22⋊D7311yescharacteristic
130SD16—22no—
131SD16—21no—
132SD16—2921no—
133D8—22no—
134D8—21no—
135D8C22⋊D7311yescharacteristic
136D8C22⋊D7312yes—
137M16—21no—
138M16—1461no—
139M16C22⋊D7311yescharacteristic
140C2×C8—21no—
141C2×C8C22×D7311yescharacteristic
142C16—1461no—
143C16—1461no—
144C23—42no—
145C23—41no—
146Q8—21no—
147Q8—2921no—
148Q8—2921no—
149Q8#1192401311yescharacteristic
150D4—21no—
151D4—21no—
152D4—22no—
153D4—42no—
154D4—42no—
155D4—41no—
156D4—41no—
157D4#1192401311yescharacteristic
158C2×C4—21no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—2921no—
162C2×C4C2×C22⋊D7311yescharacteristic
163C8—21no—
164C8—2921no—
165C8C2×C22⋊D7311yescharacteristic
166C8C2×C22⋊D7311yescharacteristic
167C22—21no—
168C22—42no—
169C22—41no—
170C22—161no—
171C22—162no—
172C22#1455661711yescharacteristic
173C4—21no—
174C4—41no—
175C4—5841no—
176C4#1455656011yescharacteristic
177C4C2×#1192401311yescharacteristic
178C2—21no—
179C2—41no—
180C2—82no—
181C2—81no—
182C2#1798872111yescharacteristic
subgroupG/H
centerC2#17988721
commutatorC2×C8×C73C23
FrattiniC2×C8C22×D73
Fitting#576×C73C2
socleC2×C73#447
hypercenter#576D73
hypocenterC73#2194
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #17989027C2semidirect
2. C2×#14556535C2semidirect
3. #17988862C2semidirect
4. #17988400C2semidirect
5. #17988394C2semidirect
6. #17988371C2semidirect
9. C2×Dic292C22semidirect
13. #14556531C22semidirect
14. C2×#11924003C22semidirect
15. #14556527C22semidirect
92. C73#2194semidirect