Finite Group Databasebeta

← #22586864

Subgroups — #22586864

Order 9088.

242 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 63 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#17986535C211yesmaximal, characteristic
2#17986053C211yesmaximal, characteristic, complemented
3#17985850C211yesmaximal, characteristic, complemented
4#17985736C211yesmaximal, characteristic, complemented
5#17985734C211yesmaximal, characteristic
6#17985732C211yesmaximal, characteristic
7#328×C71C211yesmaximal, characteristic, complemented
8#14556123—21no—
9#14556119C411yescharacteristic, complemented
10#14556117C411yescharacteristic, complemented
11#14556112—21no—
12#14556102C2211yescharacteristic
13Q8×Dic71C411yescharacteristic, complemented
14#14556090C411yescharacteristic, complemented
15(C22⋊D71)⋊C4—21no—
16#14556086—21no—
17C2×Dic71⋊C4—21no—
18C2×C4×Dic71—21no—
19C2×C4×Dic71C2211yescharacteristic
20Dic71⋊C8—21no—
21C22⋊(C71⋊C8)C2211yescharacteristic
22#14556038—21no—
23C71⋊#95C2211yescharacteristic
24C22⋊Q8×C71C2211yescharacteristic
25C22⋊C8×C71C2211yescharacteristic
26#95×C71C2211yescharacteristic
27C2×D284C2×C411yescharacteristic
28C2×C22⋊D71—21no—
29C22×Dic71—21no—
30C22×Dic71D411yescharacteristic
31C22×Dic71D411yescharacteristic
32C2×C4×D71—21no—
33C2×C71⋊C8—21no—
34C4⋊Dic71—21no—
35C4⋊Dic71—21no—
36C4⋊Dic71C2×C411yescharacteristic
37Dic71⋊C4—24no—
38Dic71⋊C4—21no—
39(C2×D71)⋊C4—21no—
40(C2×D71)⋊C4—21no—
41(C2×D71)⋊C4—21no—
42(C2×D71)⋊C4—21no—
43C4×Dic71—21no—
44C4×Dic71—42no—
45C4×Dic71C2×C411yescharacteristic
46C4×Dic71C2×C411yescharacteristic
47C4×Dic71D411yescharacteristic
48C4×Dic71D411yescharacteristic
49C2×Q8×C71C2×C411yescharacteristic
50C22×C4×C71—21no—
51C22×C4×C71C2311yescharacteristic
52C2×C8×C71—21no—
53C4⋊C4×C71—21no—
54C4⋊C4×C71C2×C411yescharacteristic
55C22⋊C4×C71—21no—
56C23×C71C2×D411yescharacteristic
57C22×D71—21no—
58Q8×C71—41no—
59C2×C4×C71—22no—
60C2×C4×C71—21no—
61C2×C4×C71—21no—
62C2×C4×C71—41no—
63C2×C4×C71—41no—
64C2×C4×C71—41no—
65C2×C4×C71C22×C411yescharacteristic
66C2×C4×C71C2×D411yescharacteristic
67C22⋊D71—41no—
68C22⋊D71—41no—
69C2×Dic71—21no—
70C2×Dic71—21no—
71C2×Dic71—22no—
72C2×Dic71—22no—
73C2×Dic71—21no—
74C2×Dic71—41no—
75C2×Dic71—41no—
76C2×Dic71C22⋊C411yescharacteristic
77C2×Dic71C22⋊C411yescharacteristic
78C2×Dic71C22⋊C411yescharacteristic
79C2×Dic71C22⋊C411yescharacteristic
80D284—41no—
81C4×D71—42no—
82C8×C71—41no—
83C71⋊C8—41no—
84C22×C71—21no—
85C22×C71—21no—
86C22×C71C2×C22⋊C411yescharacteristic
87C22×C71AGL(1,ℤ/8)11yescharacteristic
88C22×C71#13711yescharacteristic
89C2×D71—41no—
90C2×D71—41no—
91C2×D71—41no—
92C4×C71—21no—
93C4×C71—21no—
94C4×C71—42no—
95C4×C71—41no—
96C4×C71—41no—
97Dic71—21no—
98Dic71—21no—
99Dic71—21no—
100Dic71—42no—
101Dic71—41no—
102Dic71#10412yes—
103C2×C71—21no—
104C2×C71—21no—
105C2×C71#41011yescharacteristic
106C2×C71#41711yescharacteristic
107C2×C71C2×#10411yescharacteristic
108D71—81no—
109#1509—711nomaximal, Sylow
110C71#150911yescharacteristic, complemented, Sylow
111#529—711no—
112#432—711no—
113C4×C4⋊C4—711no—
114#328—711no—
115#328D7111yescharacteristic, complemented
116#327—711no—
117#327—711no—
118#126—1421no—
119#125—711no—
120#124—711no—
121#123—1421no—
122C22⋊Q8C2×D7111yescharacteristic
123#121—711no—
124C4×Q8—711no—
125C4×D4—711no—
126C4×D4—1421no—
127#117—1421no—
128C2×C4⋊C4—1421no—
129C2×C42—711no—
130C2×C42—1421no—
131C4⋊C8—1421no—
132C22⋊C8—711no—
133C22⋊C8C2×D7111yescharacteristic
134#97—1421no—
135#95—711no—
136#95C2×D7111yescharacteristic
137C2×Q8C4×D7111yescharacteristic
138C2×D4—711no—
139C2×D4—1421no—
140C22×C4—21no—
141C22×C4—711no—
142C22×C4—711no—
143C22×C4—1421no—
144C22×C4—1421no—
145C22×C4C22×D7111yescharacteristic
146C2×C8—21no—
147C2×C8—1421no—
148C4⋊C4—21no—
149C4⋊C4—711no—
150C4⋊C4—1421no—
151C4⋊C4—1421no—
152C4⋊C4—1421no—
153C4⋊C4—1424no—
154C4⋊C4C4×D7111yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—1421no—
157C22⋊C4—1421no—
158C22⋊C4—1421no—
159C22⋊C4—1421no—
160C42—2842no—
161C42—711no—
162C42—711no—
163C42—711no—
164C42—711no—
165C42—1421no—
166C23—1421no—
167C23D4×D7111yescharacteristic
168Q8—41no—
169D4—2841no—
170D4—2841no—
171D4—2841no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—2842no—
179C2×C4—2841no—
180C2×C4—2841no—
181C2×C4—711no—
182C2×C4—711no—
183C2×C4—711no—
184C2×C4—711no—
185C2×C4—1422no—
186C2×C4—1421no—
187C2×C4—1422no—
188C2×C4—1421no—
189C2×C4—1421no—
190C2×C4C2×C4×D7111yescharacteristic
191C2×C4D4×D7111yescharacteristic
192C8—41no—
193C8—2841no—
194C22—21no—
195C22—21no—
196C22—2841no—
197C22—2841no—
198C22—2841no—
199C22C22⋊C4×D7111yescharacteristic
200C22#1455615511yescharacteristic
201C22#1455616011yescharacteristic
202C4—21no—
203C4—21no—
204C4—42no—
205C4—41no—
206C4—41no—
207C4—2842no—
208C4—2841no—
209C4—712no—
210C4—1421no—
211C4—1421no—
212C4—1421no—
213C2—21no—
214C2—21no—
215C2—5681no—
216C2#1798597111yescharacteristic
217C2#1798599911yescharacteristic
218C2#104×D7111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D71
commutatorC2×C4×C71C22×C4
FrattiniC22×C4C22×D71
Fitting#328×C71C2
socleC22×C71C2×C22⋊C4
hypercenter#328D71
hypocenterC71#1509
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #17986053C2semidirect
3. #17985850C2semidirect
4. #17985736C2semidirect
7. #328×C71C2semidirect
9. #14556119C4semidirect
10. #14556117C4semidirect
13. Q8×Dic71C4semidirect
14. #14556090C4semidirect
110. C71#1509semidirect
115. #328D71semidirect