Finite Group Databasebeta

← #22599469

Subgroups — #22599469

Order 9088.

404 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 125 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#17986550C211yesmaximal, characteristic
2C4⋊Q8×D71C211yesmaximal, characteristic, complemented
3#17986395C211yesmaximal, characteristic, complemented
4#17986347C211yesmaximal, characteristic, complemented
5#17986346C211yesmaximal, characteristic, complemented
6#17986319C214yesmaximal, complemented
7#17986288C214yesmaximal, complemented
8#17986050C211yesmaximal, characteristic, complemented
9#498×C71C211yesmaximal, characteristic, complemented
10C2×#11765866—21no—
11C2×Q8×D71—21no—
12#14556128C2211yescharacteristic
13#14556127—21no—
14#14556127C2212yes—
15#14556126—21no—
16#14556125C2211yescharacteristic
17#14556117—21no—
18#14556103—21no—
19#14556103C2212yes—
20#14556090—21no—
21#14556090C2212yes—
22C4⋊C4×D71—21no—
23C4⋊C4×D71C2212yes—
24C42×D71C2211yescharacteristic
25#14556059C2214yes—
26Dic71⋊C8C2212yes—
27#14556050C2214yes—
28#14556048C2214yes—
29(C2×D71)⋊C8C2212yes—
30C4×C71⋊C8C2211yescharacteristic
31C4⋊Q8×C71C2211yescharacteristic
32C4⋊Q8×C71C2211yescharacteristic
33#106×C71C2214yes—
34C4×C8×C71C2211yescharacteristic
35#11765866—44no—
36C2×Dic142—21no—
37C2×Dic142—22no—
38Q8×D71—44no—
39C2×D284—22no—
40C2×D284—21no—
41C2×D284—21no—
42C2×D284C2312yes—
43C2×C4×D71—22no—
44C2×C4×D71—22no—
45C2×C4×D71D411yescharacteristic
46C2×C4×D71D411yescharacteristic
47C2×C4×D71C2311yescharacteristic
48C2×C71⋊C8C2312yes—
49C4⋊Dic71—21no—
50C4⋊Dic71C2312yes—
51Dic71⋊C4—22no—
52Dic71⋊C4—22no—
53(C2×D71)⋊C4—22no—
54(C2×D71)⋊C4—22no—
55C4×Dic71D411yescharacteristic
56C4×Dic71D411yescharacteristic
57C4×Dic71C2311yescharacteristic
58C2×Q8×C71—21no—
59C2×Q8×C71—21no—
60C2×C8×C71C2312yes—
61C4⋊C4×C71—21no—
62C4⋊C4×C71—21no—
63C4⋊C4×C71C2312yes—
64C4⋊C4×C71C2312yes—
65C42×C71C2311yescharacteristic
66C22×D71—22no—
67C22×D71C2×D411yescharacteristic
68Q8×C71—42no—
69Q8×C71—42no—
70C2×C4×C71—22no—
71C2×C4×C71—22no—
72C2×C4×C71C2×D411yescharacteristic
73C2×C4×C71C2×D411yescharacteristic
74C2×C4×C71C2411yescharacteristic
75C2×Dic71—22no—
76C2×Dic71C2×D411yescharacteristic
77C2×Dic71C2×D411yescharacteristic
78C2×Dic71C2×D411yescharacteristic
79D284—24no—
80D284—44no—
81D284—42no—
82D284—42no—
83C4×D71—22no—
84C4×D71—22no—
85C4×D71—44no—
86C4×D71—44no—
87C4×D71SD1614yes—
88Dic142—44no—
89Dic142—42no—
90C8×C71—22no—
91C71⋊C8—22no—
92C22×C71C22×D411yescharacteristic
93C2×D71—21no—
94C2×D71—21no—
95C2×D71—42no—
96C2×D71—42no—
97C2×D71—42no—
98C2×D71C2×SD1612yes—
99C4×C71—42no—
100C4×C71—42no—
101C4×C71C2×SD1612yes—
102C4×C71AGL(1,ℤ/8)12yes—
103C4×C712−1+412yes—
104Dic71—21no—
105Dic71—21no—
106Dic71—42no—
107Dic71C2×SD1612yes—
108C2×C71#53611yescharacteristic
109C2×C71C22×SD1611yescharacteristic
110C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
111D71—22no—
112D71—82no—
113#3227—711nomaximal, Sylow
114C71#322711yescharacteristic, complemented, Sylow
115#532—711no—
116C2×C4⋊Q8—711no—
117#498—711no—
118#498D7111yescharacteristic, complemented
119#486—711no—
120#486—711no—
121#481—714no—
122#476—714no—
123#431—711no—
124C2×#41—1421no—
125C22×Q8—1421no—
126C4⋊Q8—712no—
127C4⋊Q8—711no—
128C4⋊Q8—1421no—
129C4⋊Q8—1421no—
130C4⋊Q8C2×D7111yescharacteristic
131C4⋊Q8C2×D7111yescharacteristic
132#127—711no—
133#124—1421no—
134#121—712no—
135#121—1421no—
136C4×D4—712no—
137C4×D4—1421no—
138C2×C4⋊C4—712no—
139C2×C4⋊C4—1421no—
140C2×C42—711no—
141#106—714no—
142#106C2×D7114yes—
143C4⋊C8—712no—
144#102—714no—
145#102—714no—
146C22⋊C8—712no—
147C4×C8—711no—
148C4×C8C2×D7111yescharacteristic
149#41—2844no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—2844no—
153C2×Q8—1422no—
154C2×Q8—1421no—
155C2×D4—712no—
156C2×D4—1422no—
157C2×D4—1421no—
158C2×D4—1421no—
159C22×C4—711no—
160C22×C4—711no—
161C22×C4—711no—
162C22×C4—1422no—
163C22×C4—1422no—
164C2×C8—712no—
165C2×C8C22×D7112yes—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—712no—
169C4⋊C4—1422no—
170C4⋊C4—1422no—
171C4⋊C4—1421no—
172C4⋊C4C22×D7112yes—
173C4⋊C4C22×D7112yes—
174C22⋊C4—1422no—
175C22⋊C4—1422no—
176C42—711no—
177C42—711no—
178C42—711no—
179C42C22×D7111yescharacteristic
180C23—711no—
181C23—1422no—
182Q8—42no—
183Q8—42no—
184Q8—2844no—
185Q8—2842no—
186D4—2844no—
187D4—2842no—
188D4—2842no—
189D4—1424no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—2844no—
193C2×C4—2844no—
194C2×C4—711no—
195C2×C4—714no—
196C2×C4—711no—
197C2×C4—711no—
198C2×C4—1422no—
199C2×C4—1422no—
200C2×C4—1422no—
201C2×C4D4×D7111yescharacteristic
202C2×C4D4×D7111yescharacteristic
203C2×C4C23×D7111yescharacteristic
204C8—22no—
205C8—1422no—
206C22—2842no—
207C22—2842no—
208C22—2842no—
209C22—712no—
210C22—1421no—
211C22—1421no—
212C22C2×D4×D7111yescharacteristic
213C4—42no—
214C4—42no—
215C4—2842no—
216C4—712no—
217C4—1421no—
218C4—1421no—
219C4SD16×D7112yes—
220C4#1455615512yes—
221C4#1455617612yes—
222C2—5682no—
223C2—1422no—
224C2#1798657411yescharacteristic
225C2C2×SD16×D7111yescharacteristic
226C2C2×#1455615511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#498×C71C2
socleC22×C71C22×D4
hypercenter#498D71
hypocenterC71#3227
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D71C2semidirect
3. #17986395C2semidirect
4. #17986347C2semidirect
5. #17986346C2semidirect
6. #17986319C2semidirect
7. #17986288C2semidirect
8. #17986050C2semidirect
9. #498×C71C2semidirect
114. C71#3227semidirect
118. #498D71semidirect