Finite Group Databasebeta

← #22599532

Subgroups — #22599532

Order 9088.

368 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 121 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#17986549C211yesmaximal, characteristic
2C2×#14556120C211yesmaximal, characteristic, complemented
3#17986408C212yesmaximal, complemented
4#17986364C212yesmaximal, complemented
5#17986318C212yesmaximal, complemented
6#17986313C212yesmaximal, complemented
7#17986300C214yesmaximal, complemented
8#432×C71C211yesmaximal, characteristic, complemented
9C2×#11765865—21no—
10#14556128C2211yescharacteristic
11#14556125C2211yescharacteristic
12#14556120—21no—
13#14556120—21no—
14#14556120C2212yes—
15#14556120C2212yes—
16#14556116—21no—
17#14556102—21no—
18#14556102C2212yes—
19C4×D284—21no—
20C4×D284C2212yes—
21C2×C4⋊Dic71C2211yescharacteristic
22C2×C4⋊Dic71C2211yescharacteristic
23C2×Dic71⋊C4—21no—
24C2×Dic71⋊C4—21no—
25#14556062C2214yes—
26#14556060C2214yes—
27#14556050C2214yes—
28#14556050C2214yes—
29C2×C42×C71C2211yescharacteristic
30C4⋊C8×C71C2212yes—
31C22⋊C8×C71C2212yes—
32#97×C71C2212yes—
33#11765865—44no—
34C2×Dic142—21no—
35C2×D284—21no—
36C2×D284—21no—
37C2×D284C2312yes—
38C2×C22⋊D71—22no—
39C22×Dic71—21no—
40C22×Dic71—21no—
41C2×C4×D71—22no—
42C4⋊Dic71—21no—
43C4⋊Dic71—21no—
44C4⋊Dic71—21no—
45C4⋊Dic71C2312yes—
46C4⋊Dic71C2312yes—
47C4⋊Dic71C2312yes—
48Dic71⋊C4—22no—
49Dic71⋊C4—22no—
50Dic71⋊C4—22no—
51Dic71⋊C4—22no—
52(C2×D71)⋊C4—22no—
53(C2×D71)⋊C4—22no—
54C22×C4×C71D411yescharacteristic
55C22×C4×C71D411yescharacteristic
56C22×C4×C71C2311yescharacteristic
57C2×C8×C71C2314yes—
58C42×C71D411yescharacteristic
59C42×C71D411yescharacteristic
60C42×C71C2311yescharacteristic
61C42×C71C2311yescharacteristic
62C23×C71C2×D411yescharacteristic
63C22×D71—22no—
64C2×C4×C71—22no—
65C2×C4×C71—22no—
66C2×C4×C71—41no—
67C2×C4×C71C2×D411yescharacteristic
68C2×C4×C71C2×D411yescharacteristic
69C2×C4×C71C2×D411yescharacteristic
70C2×C4×C71C2×D411yescharacteristic
71C2×C4×C71C2×D411yescharacteristic
72C2×C4×C71C2411yescharacteristic
73C22⋊D71—44no—
74C2×Dic71—22no—
75C2×Dic71—22no—
76C2×Dic71—22no—
77C2×Dic71—42no—
78C2×Dic71—42no—
79D284—24no—
80D284—42no—
81D284—42no—
82C4×D71—44no—
83Dic142—42no—
84C8×C71—24no—
85C22×C71—21no—
86C22×C71—21no—
87C22×C71AGL(1,ℤ/8)12yes—
88C22×C71C22×D411yescharacteristic
89C2×D71—42no—
90C2×D71—42no—
91C2×D71—42no—
92C4×C71—21no—
93C4×C71—21no—
94C4×C71—21no—
95C4×C71—21no—
96C4×C71—21no—
97C4×C712−1+412yes—
98Dic71—42no—
99Dic71—42no—
100Dic71—42no—
101C2×C71—22no—
102C2×C71#53611yescharacteristic
103C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
104C2×C71#57511yescharacteristic
105D71—82no—
106#3234—711nomaximal, Sylow
107C71#323411yescharacteristic, complemented, Sylow
108#532—711no—
109C2×#125—711no—
110#501—712no—
111#490—712no—
112#481—712no—
113#480—712no—
114#478—714no—
115#432D7111yescharacteristic, complemented
116C2×#41—1421no—
117C4⋊Q8—711no—
118#127—711no—
119#125—712no—
120#125—712no—
121#125—1421no—
122#125—1421no—
123#124—1421no—
124#121—712no—
125#121—1421no—
126C4×D4—712no—
127C4×D4—1421no—
128C2×C4⋊C4—711no—
129C2×C4⋊C4—711no—
130C2×C4⋊C4—1421no—
131C2×C4⋊C4—1421no—
132C2×C42C2×D7111yescharacteristic
133#107—714no—
134#106—714no—
135C4⋊C8C2×D7112yes—
136#102—714no—
137#102—714no—
138C22⋊C8C2×D7112yes—
139#97C2×D7112yes—
140#41—2844no—
141C2×Q8—1421no—
142C2×D4—712no—
143C2×D4—1422no—
144C2×D4—1421no—
145C2×D4—1421no—
146C22×C4—1421no—
147C22×C4—1421no—
148C22×C4—1422no—
149C22×C4D28411yescharacteristic
150C22×C4D28411yescharacteristic
151C22×C4C22×D7111yescharacteristic
152C2×C8C22×D7114yes—
153C4⋊C4—712no—
154C4⋊C4—712no—
155C4⋊C4—712no—
156C4⋊C4—1422no—
157C4⋊C4—1422no—
158C4⋊C4—1422no—
159C4⋊C4—1422no—
160C4⋊C4—1421no—
161C4⋊C4—1421no—
162C4⋊C4—1421no—
163C22⋊C4—1422no—
164C22⋊C4—1422no—
165C42D28411yescharacteristic
166C42D28411yescharacteristic
167C42C22×D7111yescharacteristic
168C42C22×D7111yescharacteristic
169C23—1422no—
170C23C2×D28411yescharacteristic
171Q8—2842no—
172D4—2842no—
173D4—2842no—
174D4—2844no—
175D4—1424no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—2844no—
180C2×C4—2842no—
181C2×C4—2842no—
182C2×C4—1422no—
183C2×C4—1422no—
184C2×C4—1422no—
185C2×C4C2×D28411yescharacteristic
186C2×C4C2×D28411yescharacteristic
187C2×C4C2×D28411yescharacteristic
188C2×C4C2×D28411yescharacteristic
189C2×C4C2×D28411yescharacteristic
190C2×C4C23×D7111yescharacteristic
191C8—24no—
192C22—21no—
193C22—21no—
194C22—2842no—
195C22—2842no—
196C22—2842no—
197C22#1455615612yes—
198C22C22×D28411yescharacteristic
199C4—21no—
200C4—21no—
201C4—21no—
202C4—21no—
203C4—21no—
204C4—2842no—
205C4—2842no—
206C4—2842no—
207C4#1455617512yes—
208C2—22no—
209C2—5682no—
210C2#1798657511yescharacteristic
211C2C2×#1455615611yescharacteristic
212C2#1798679911yescharacteristic
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#432×C71C2
socleC22×C71C22×D4
hypercenter#432D71
hypocenterC71#3234
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. C2×#14556120C2semidirect
3. #17986408C2semidirect
4. #17986364C2semidirect
5. #17986318C2semidirect
6. #17986313C2semidirect
7. #17986300C2semidirect
8. #432×C71C2semidirect
107. C71#3234semidirect
115. #432D71semidirect