Finite Group Databasebeta

← #22599289

Subgroups — #22599289

Order 9088.

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

normal subgroups only

This group has 384 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#17986554C211yesmaximal, characteristic
2C4⋊Q8×D71C211yesmaximal, characteristic
3#17986379C211yesmaximal, characteristic, complemented
4#17986378C211yesmaximal, characteristic, complemented
5#17986351C211yesmaximal, characteristic, complemented
6#17986208C214yesmaximal, complemented
7#17986156C214yesmaximal
8#17986050C211yesmaximal, characteristic
9#487×C71C211yesmaximal, characteristic, complemented
10C2×Q8×D71C2212yes—
11C2×Dic284C2214yes—
12C2×#11765855C2214yes—
13#14556128C2211yescharacteristic
14#14556128C2211yescharacteristic
15#14556127—21no—
16#14556127—21no—
17#14556126C2212yes—
18#14556119—21no—
19#14556108C2212yes—
20#14556104—21no—
21Q8×Dic71—21no—
22#14556093C2212yes—
23#14556086—21no—
24C4⋊C4×D71—21no—
25C4⋊C4×D71—21no—
26C42×D71C2211yescharacteristic
27Dic71⋊C8C2212yes—
28#14556051C2214yes—
29(C2×D71)⋊C8C2212yes—
30C4×C71⋊C8C2211yescharacteristic
31C4⋊Q8×C71C2211yescharacteristic
32C4⋊Q8×C71C2211yescharacteristic
33#103×C71C2214yes—
34C4×C8×C71C2211yescharacteristic
35C2×Dic142—22no—
36C2×Dic142C2312yes—
37C2×Dic142C2312yes—
38Q8×D71—44no—
39C2×C4×D71—22no—
40C2×C4×D71D411yescharacteristic
41C2×C4×D71D411yescharacteristic
42C2×C4×D71C2311yescharacteristic
43Dic284—44no—
44#11765855—44no—
45C2×C71⋊C8C2312yes—
46C4⋊Dic71—21no—
47C4⋊Dic71—21no—
48C4⋊Dic71—21no—
49C4⋊Dic71—21no—
50C4⋊Dic71—22no—
51Dic71⋊C4—22no—
52Dic71⋊C4—22no—
53Dic71⋊C4—22no—
54(C2×D71)⋊C4—22no—
55C4×Dic71—22no—
56C4×Dic71D411yescharacteristic
57C4×Dic71D411yescharacteristic
58C4×Dic71C2311yescharacteristic
59C2×Q8×C71—21no—
60C2×Q8×C71C2312yes—
61C2×C8×C71C2312yes—
62C4⋊C4×C71—21no—
63C4⋊C4×C71—21no—
64C4⋊C4×C71—21no—
65C4⋊C4×C71C2312yes—
66C42×C71C2311yescharacteristic
67C22×D71C2×D411yescharacteristic
68Q8×C71—24no—
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×Dic71—22no—
77C2×Dic71C2×D411yescharacteristic
78C2×Dic71C2×D411yescharacteristic
79C2×Dic71C2×D411yescharacteristic
80C4×D71—22no—
81C4×D71—22no—
82C4×D71—44no—
83C4×D71Q1614yescomplemented
84Dic142—24no—
85Dic142—24no—
86Dic142—44no—
87C8×C71—22no—
88C71⋊C8—22no—
89C22×C71C22×D411yescharacteristic
90C2×D71—21no—
91C2×D71—21no—
92C2×D71C2×Q1612yes—
93C4×C71—42no—
94C4×C71—42no—
95C4×C71C2×Q1612yes—
96C4×C71#13712yes—
97C4×C71H(5,𝔽2)12yes—
98Dic71—21no—
99Dic71—21no—
100Dic71—42no—
101Dic71—42no—
102Dic71C2×Q1612yes—
103C2×C71#53511yescharacteristic
104C2×C71C22×Q1611yescharacteristic
105C2×C71C2×#13711yescharacteristic
106D71—22no—
107#3209—711nomaximal, Sylow
108C71#320911yescharacteristic, complemented, Sylow
109#533—711no—
110C2×C4⋊Q8—711no—
111#494—711no—
112#494—711no—
113#487—711no—
114#487D7111yescharacteristic, complemented
115#462—714no—
116#451—714no—
117#431—711no—
118C22×Q8—712no—
119C2×Q16—714no—
120C2×Q16—714no—
121C4⋊Q8—712no—
122C4⋊Q8—711no—
123C4⋊Q8—711no—
124C4⋊Q8—1421no—
125C4⋊Q8—1421no—
126C4⋊Q8C2×D7111yescharacteristic
127C4⋊Q8C2×D7111yescharacteristic
128#125—1421no—
129C22⋊Q8—712no—
130C22⋊Q8—1421no—
131C4×Q8—712no—
132C4×Q8—1421no—
133#117—1421no—
134C2×C4⋊C4—1421no—
135C2×C4⋊C4—1421no—
136C2×C42—711no—
137C4⋊C8—712no—
138#103—714no—
139#103C2×D7114yes—
140C22⋊C8—712no—
141C4×C8—711no—
142C4×C8C2×D7111yescharacteristic
143C2×Q8—21no—
144C2×Q8—2844no—
145C2×Q8—712no—
146C2×Q8—712no—
147C2×Q8—1422no—
148C2×Q8C22×D7112yes—
149C22×C4—711no—
150C22×C4—711no—
151C22×C4—711no—
152C22×C4—1422no—
153Q16—2844no—
154Q16—2844no—
155C2×C8—712no—
156C2×C8C22×D7112yes—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—1422no—
161C4⋊C4—1422no—
162C4⋊C4—1422no—
163C4⋊C4—1422no—
164C4⋊C4—1421no—
165C4⋊C4—1421no—
166C4⋊C4—1421no—
167C4⋊C4—1421no—
168C4⋊C4C22×D7112yes—
169C22⋊C4—1422no—
170C42—711no—
171C42—711no—
172C42—711no—
173C42—1422no—
174C42C22×D7111yescharacteristic
175C23—711no—
176Q8—24no—
177Q8—42no—
178Q8—2844no—
179Q8—1424no—
180Q8—1424no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—2844no—
184C2×C4—711no—
185C2×C4—714no—
186C2×C4—711no—
187C2×C4—711no—
188C2×C4—1422no—
189C2×C4—1422no—
190C2×C4—1422no—
191C2×C4—1422no—
192C2×C4D4×D7111yescharacteristic
193C2×C4D4×D7111yescharacteristic
194C2×C4C23×D7111yescharacteristic
195C8—22no—
196C8—1422no—
197C22—712no—
198C22—1421no—
199C22—1421no—
200C22C2×D4×D7111yescharacteristic
201C4—42no—
202C4—42no—
203C4—2842no—
204C4—2842no—
205C4—712no—
206C4—1421no—
207C4—1421no—
208C4Q16×D7112yes—
209C4#1455615712yes—
210C4#1455617412yes—
211C2—1422no—
212C2#1798656711yescharacteristic
213C2C2×Q16×D7111yescharacteristic
214C2C2×#1455615711yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#487×C71C2
socleC22×C71C22×D4
hypercenter#487D71
hypocenterC71#3209
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. #17986379C2semidirect
4. #17986378C2semidirect
5. #17986351C2semidirect
6. #17986208C2semidirect
9. #487×C71C2semidirect
83. C4×D71Q16semidirect
108. C71#3209semidirect
114. #487D71semidirect