Finite Group Databasebeta

← #44398436

Subgroups — #44398436

Order 29056.

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

normal subgroups only

This group has 242 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#28409580C211yesmaximal, characteristic
2#28409088C211yesmaximal, characteristic, complemented
3#28409084C211yesmaximal, characteristic, complemented
4#28408779C211yesmaximal, characteristic, complemented
5#28408777C211yesmaximal, characteristic, complemented
6#28408776C211yesmaximal, characteristic, complemented
7#331×C227C211yesmaximal, characteristic, complemented
8C2×#16813883—21no—
9#20300288C2211yescharacteristic
10#20300275—21no—
11#20300259—21no—
12#20300259C412yes—
13D4×Dic227—21no—
14D4×Dic227C412yes—
15C42×D227C2211yescharacteristic
16C4⋊(C227⋊C8)C2211yescharacteristic
17C4⋊(C227⋊C8)C2211yescharacteristic
18(C2×D227)⋊C8—21no—
19(C2×D227)⋊C8—21no—
20#20300198—21no—
21C8×Dic227—21no—
22#127×C227C2211yescharacteristic
23C4⋊C8×C227C2211yescharacteristic
24C4⋊C8×C227C2211yescharacteristic
25#16813883—42no—
26#16813883—42no—
27C2×Dic454—21no—
28C2×C22⋊D227—22no—
29C22×Dic227—22no—
30C2×C4×D227C2×C411yescharacteristic
31C2×C4×D227D411yescharacteristic
32C2×C4×D227D411yescharacteristic
33C2×C227⋊C8—21no—
34C2×C227⋊C8—21no—
35C4⋊Dic227—21no—
36C4⋊Dic227C2×C412yes—
37C22⋊Dic227—22no—
38C22⋊Dic227—22no—
39C4×Dic227C2×C411yescharacteristic
40C4×Dic227D411yescharacteristic
41C4×Dic227D411yescharacteristic
42C2×D4×C227—21no—
43C2×D4×C227—21no—
44C2×D4×C227C2×C412yes—
45C2×C8×C227—21no—
46C2×C8×C227—21no—
47C42×C227C2311yescharacteristic
48C22×D227C22⋊C411yescharacteristic
49Dic454—41no—
50Dic454—41no—
51C22⋊D227—44no—
52C2×Dic227—22no—
53C2×Dic227—44no—
54C2×Dic227C22⋊C411yescharacteristic
55C2×Dic227C22⋊C411yescharacteristic
56C2×Dic227C22⋊C411yescharacteristic
57C4×D227—21no—
58C4×D227—21no—
59C4×D227—21no—
60C4×D227—21no—
61C4×D227—41no—
62C227⋊C8—41no—
63C227⋊C8—41no—
64C23×C227—22no—
65D4×C227—42no—
66D4×C227—41no—
67D4×C227—41no—
68D4×C227—41no—
69D4×C227—41no—
70C2×C4×C227C22×C411yescharacteristic
71C2×C4×C227C2×D411yescharacteristic
72C2×C4×C227C2×D411yescharacteristic
73C8×C227—41no—
74C8×C227—41no—
75C22×C227—42no—
76C22×C227—42no—
77C22×C227—42no—
78C22×C227C2×C22⋊C411yescharacteristic
79C2×D227—21no—
80C2×D227—21no—
81C2×D227—21no—
82C4×C227—21no—
83C4×C227#13511yescharacteristic
84C4×C227#13511yescharacteristic
85C4×C227AGL(1,ℤ/8)11yescharacteristic
86C4×C227AGL(1,ℤ/8)11yescharacteristic
87Dic227—21no—
88Dic227—21no—
89Dic227—21no—
90Dic227—42no—
91C2×C227—82no—
92C2×C227#41311yescharacteristic
93C2×C227#41611yescharacteristic
94C2×C227#41811yescharacteristic
95D227—41no—
96C227#155311yescharacteristic, complemented, Sylow
97#1553—2271nomaximal, Sylow
98#532—2271no—
99#432—2271no—
100#431—2271no—
101#332—2271no—
102#332—2271no—
103#331—2271no—
104#331D22711yescharacteristic, complemented
105C2×#41—4541no—
106C4⋊Q8—2271no—
107#127C2×D22711yescharacteristic
108#124—4541no—
109#121—4541no—
110#121—2272no—
111C4×D4—4541no—
112C4×D4—2272no—
113C2×C42—2271no—
114C4⋊C8—2271no—
115C4⋊C8—2271no—
116C4⋊C8C2×D22711yescharacteristic
117C4⋊C8C2×D22711yescharacteristic
118C22⋊C8—4541no—
119C22⋊C8—4541no—
120#97—4541no—
121C4×C8—4541no—
122#41—9082no—
123#41—9082no—
124C2×Q8—4541no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—4542no—
128C2×D4C4×D22712yes—
129C22×C4—4542no—
130C22×C4—2271no—
131C22×C4—2271no—
132C22×C4—2271no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—4541no—
136C2×C8—4541no—
137C4⋊C4—4541no—
138C4⋊C4—2272no—
139C22⋊C4—4542no—
140C22⋊C4—4542no—
141C42—2271no—
142C42—2271no—
143C42—2271no—
144C42C22×D22711yescharacteristic
145C23—22no—
146C23—2271no—
147Q8—9081no—
148Q8—9081no—
149D4—42no—
150D4—41no—
151D4—41no—
152D4—41no—
153D4—41no—
154D4—9084no—
155C2×C4—9081no—
156C2×C4—9084no—
157C2×C4—4542no—
158C2×C4—4541no—
159C2×C4—4541no—
160C2×C4—4541no—
161C2×C4—4541no—
162C2×C4—2271no—
163C2×C4—2271no—
164C2×C4—2271no—
165C2×C4C2×C4×D22711yescharacteristic
166C2×C4D4×D22711yescharacteristic
167C2×C4D4×D22711yescharacteristic
168C8—41no—
169C8—41no—
170C8—9081no—
171C8—9081no—
172C22—42no—
173C22—42no—
174C22—42no—
175C22—4541no—
176C22—4541no—
177C22—4541no—
178C22C22⋊C4×D22711yescharacteristic
179C4—21no—
180C4—9082no—
181C4—4541no—
182C4—4541no—
183C4—4541no—
184C4#2030030711yescharacteristic
185C4#2030030911yescharacteristic
186C4#2030031211yescharacteristic
187C4#2030031511yescharacteristic
188C2—82no—
189C2—9081no—
190C2#2840901311yescharacteristic
191C2#2840902511yescharacteristic
192C2#2840903511yescharacteristic
subgroupG/H
centerC22C22⋊C4×D227
commutatorC2×C4×C227C22×C4
FrattiniC42C22×D227
Fitting#331×C227C2
socleC22×C227C2×C22⋊C4
prime radical#28409580C2
cosocleC42×C227C23
hypercenter#331D227
hypocenterC227#1553
LCSG ▷ C2×C4×C227 ▷ C22×C227 ▷ C227
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #28409088C2semidirect
3. #28409084C2semidirect
4. #28408779C2semidirect
5. #28408777C2semidirect
6. #28408776C2semidirect
7. #331×C227C2semidirect
96. C227#1553semidirect
104. #331D227semidirect

Maximal subgroups

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

subgroupindex
#284095802
#284090882
#284090842
#284087792
#284087772
#284087762
#331×C2272
#1553227