Finite Group Databasebeta

← #22599367

Subgroups — #22599367

Order 9088.

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

normal subgroups only

This group has 386 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#17986533C211yesmaximal, characteristic
2#17986391C212yesmaximal, complemented
3#17986367C212yesmaximal, complemented
4#17986206C212yesmaximal, complemented
5#17986201C212yesmaximal, complemented
6#17986155C212yesmaximal, complemented
7#17986138C212yesmaximal
8#17986056C211yesmaximal, characteristic, complemented
9C2×C4⋊Q8×C71C211yesmaximal, characteristic, complemented
10C2×#11765855C2214yes—
11C2×#11765852C2214yes—
12#14556122—21no—
13#14556120C2211yescharacteristic
14#14556116C2211yescharacteristic
15#14556111—21no—
16#14556107C2211yescharacteristic
17#14556102C2211yescharacteristic
18C4×Dic142C2211yescharacteristic
19C4×D284C2211yescharacteristic
20C4×C22⋊D71—21no—
21#14556085—21no—
22C4⋊(C71⋊C8)C2212yes—
23#14556052C2214yes—
24#14556052C2214yes—
25C22⋊(C71⋊C8)C2212yes—
26#14556039C2212yes—
27C22×Q8×C71C2212yes—
28C4⋊Q8×C71—21no—
29C4⋊Q8×C71—21no—
30C4⋊Q8×C71C2212yes—
31C4⋊Q8×C71C2212yes—
32C2×C4⋊C4×C71—21no—
33C2×C4⋊C4×C71—21no—
34C2×C42×C71C2211yescharacteristic
35C2×Dic142C2311yescharacteristic
36C2×D284C2311yescharacteristic
37C2×C22⋊D71—21no—
38C2×C4×D71—21no—
39#11765855—44no—
40#11765852—44no—
41C2×C71⋊C8C2314yes—
42C4⋊Dic71C2311yescharacteristic
43C4⋊Dic71C2311yescharacteristic
44Dic71⋊C4—21no—
45Dic71⋊C4—21no—
46Dic71⋊C4—21no—
47Dic71⋊C4—21no—
48C22⋊Dic71—21no—
49(C2×D71)⋊C4—21no—
50(C2×D71)⋊C4—21no—
51(C2×D71)⋊C4—21no—
52(C2×D71)⋊C4—21no—
53C4×Dic71—21no—
54C2×Q8×C71—22no—
55C2×Q8×C71—44no—
56C2×Q8×C71C2314yes—
57C22×C4×C71—22no—
58C22×C4×C71D411yescharacteristic
59C22×C4×C71D411yescharacteristic
60C22×C4×C71C2311yescharacteristic
61C4⋊C4×C71—22no—
62C4⋊C4×C71—22no—
63C4⋊C4×C71—22no—
64C4⋊C4×C71—22no—
65C42×C71D411yescharacteristic
66C42×C71D411yescharacteristic
67C42×C71C2311yescharacteristic
68C42×C71C2311yescharacteristic
69C23×C71C2×D411yescharacteristic
70C22×D71—21no—
71Q8×C71—24no—
72Q8×C71—24no—
73Q8×C71—44no—
74C2×C4×C71—22no—
75C2×C4×C71—22no—
76C2×C4×C71—22no—
77C2×C4×C71—24no—
78C2×C4×C71—44no—
79C2×C4×C71C2×D411yescharacteristic
80C2×C4×C71C2×D411yescharacteristic
81C2×C4×C71C2×D411yescharacteristic
82C2×C4×C71C2×D411yescharacteristic
83C2×C4×C71C2×D411yescharacteristic
84C2×C4×C71C2411yescharacteristic
85C22⋊D71—42no—
86C2×Dic71—21no—
87C2×Dic71—21no—
88C2×Dic71—21no—
89D284—22no—
90C4×D71—42no—
91Dic142—22no—
92C71⋊C8—24no—
93C22×C71—21no—
94C22×C71—21no—
95C22×C71#13712yes—
96C22×C71C22×D411yescharacteristic
97C2×D71—41no—
98C2×D71—41no—
99C2×D71—41no—
100C4×C71—21no—
101C4×C71—21no—
102C4×C71—21no—
103C4×C71—21no—
104C4×C71—44no—
105C4×C71#13712yes—
106C4×C71H(5,𝔽2)12yes—
107Dic71—41no—
108Dic71—41no—
109Dic71—41no—
110C2×C71—22no—
111C2×C71#53511yescharacteristic
112C2×C71C2×#13711yescharacteristic
113C2×C71C2×#13711yescharacteristic
114D71—81no—
115#3218—711nomaximal, Sylow
116C71#321811yescharacteristic, complemented, Sylow
117C2×C4⋊Q8D7111yescharacteristic, complemented
118#529—711no—
119#497—712no—
120#491—712no—
121#462—712no—
122#461—712no—
123#451—712no—
124#448—712no—
125#432—711no—
126C22×Q8C2×D7112yes—
127C2×Q16—714no—
128C2×SD16—714no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8C2×D7112yes—
132C4⋊Q8C2×D7112yes—
133#126—1421no—
134#125—711no—
135#124—711no—
136#123—1421no—
137C22⋊Q8—711no—
138#121—711no—
139C4×Q8—711no—
140C4×D4—711no—
141C4×D4—1421no—
142#117—1421no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C42C2×D7111yescharacteristic
146C4⋊C8—712no—
147#103—714no—
148#103—714no—
149C22⋊C8—712no—
150#97—712no—
151C2×Q8—22no—
152C2×Q8—44no—
153C2×Q8—711no—
154C2×Q8C22×D7114yes—
155C2×D4—711no—
156C2×D4—1421no—
157C22×C4—22no—
158C22×C4—1421no—
159C22×C4C22⋊D7111yescharacteristic
160C22×C4C22⋊D7111yescharacteristic
161C22×C4C22×D7111yescharacteristic
162Q16—2844no—
163SD16—2844no—
164C2×C8—714no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—711no—
170C4⋊C4—711no—
171C4⋊C4—1421no—
172C4⋊C4—1421no—
173C4⋊C4—1421no—
174C4⋊C4—1421no—
175C22⋊C4—1421no—
176C22⋊C4—1421no—
177C22⋊C4—1421no—
178C22⋊C4—1421no—
179C22⋊C4—1421no—
180C42—1421no—
181C42C22⋊D7111yescharacteristic
182C42C22⋊D7111yescharacteristic
183C42C22×D7111yescharacteristic
184C42C22×D7111yescharacteristic
185C23—1421no—
186C23C2×C22⋊D7111yescharacteristic
187Q8—24no—
188Q8—24no—
189Q8—44no—
190Q8—1422no—
191D4—2842no—
192D4—1422no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—44no—
198C2×C4—2842no—
199C2×C4—1421no—
200C2×C4—1421no—
201C2×C4—1421no—
202C2×C4C2×C22⋊D7111yescharacteristic
203C2×C4C2×C22⋊D7111yescharacteristic
204C2×C4C2×C22⋊D7111yescharacteristic
205C2×C4C2×C22⋊D7111yescharacteristic
206C2×C4C2×C22⋊D7111yescharacteristic
207C2×C4C23×D7111yescharacteristic
208C8—1424no—
209C22—21no—
210C22—21no—
211C22—2841no—
212C22—2841no—
213C22—2841no—
214C22#1455615912yes—
215C22C22×C22⋊D7111yescharacteristic
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—44no—
221C4—2841no—
222C4—2841no—
223C4—2841no—
224C4#1455615912yes—
225C4#1455617312yes—
226C2—22no—
227C2—5681no—
228C2#1798656611yescharacteristic
229C2C2×#1455615911yescharacteristic
230C2C2×#1455615911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×C4⋊Q8×C71C2
socleC22×C71C22×D4
hypercenterC2×C4⋊Q8D71
hypocenterC71#3218
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #17986391C2semidirect
3. #17986367C2semidirect
4. #17986206C2semidirect
5. #17986201C2semidirect
6. #17986155C2semidirect
8. #17986056C2semidirect
9. C2×C4⋊Q8×C71C2semidirect
116. C71#3218semidirect
117. C2×C4⋊Q8D71semidirect