Finite Group Databasebeta

← #22599417

Subgroups — #22599417

Order 9088.

364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 117 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#17986575C211yesmaximal, characteristic
2#17986391C211yesmaximal, characteristic, complemented
3#17986378C211yesmaximal, characteristic, complemented
4#17986369C211yesmaximal, characteristic, complemented
5#17986356C211yesmaximal, characteristic, complemented
6#17986221C212yesmaximal, complemented
7#17986209C212yesmaximal, complemented
8#17986159C212yesmaximal
9#17986154C212yesmaximal, complemented
10#17986060C211yesmaximal, characteristic, complemented
11#541×C71C211yesmaximal, characteristic, complemented
12C2×#11765865C2211yescharacteristic
13C22×Dic142C2211yescharacteristic
14C2×#11765855C2212yes—
15C2×#11765855C2212yes—
16C2×#11765855C2212yes—
17C2×#11765852C2212yes—
18#14556128C2211yescharacteristic
19#14556128C2211yescharacteristic
20#14556116C2211yescharacteristic
21#14556116C2211yescharacteristic
22#14556112—22no—
23#14556108—22no—
24C4⋊(C71⋊C8)C2212yes—
25#14556051C2212yes—
26#14556051C2212yes—
27#14556051C2212yes—
28#14556048C2212yes—
29C22⋊(C71⋊C8)C2212yes—
30#14556039C2211yescharacteristic
31C4×C71⋊C8C2211yescharacteristic
32C4⋊Q8×C71C2211yescharacteristic
33#126×C71—22no—
34#125×C71—22no—
35#125×C71C2211yescharacteristic
36C22⋊Q8×C71C2212yes—
37C4×Q8×C71C2212yes—
38#117×C71C2211yescharacteristic
39#11765865—42no—
40C2×Dic142—21no—
41C2×Dic142—42no—
42C2×Dic142C2311yescharacteristic
43C2×Dic142C2311yescharacteristic
44C2×Dic142C2311yescharacteristic
45C2×D284C2311yescharacteristic
46C2×C22⋊D71—21no—
47C22×Dic71—21no—
48C2×C4×D71—21no—
49#11765855—42no—
50#11765855—42no—
51#11765855—42no—
52#11765852—42no—
53C2×C71⋊C8C2312yes—
54C2×C71⋊C8C2312yes—
55C4⋊Dic71—22no—
56C4⋊Dic71—22no—
57(C2×D71)⋊C4—22no—
58(C2×D71)⋊C4—22no—
59C2×Q8×C71C2312yes—
60C22×C4×C71C2311yescharacteristic
61C4⋊C4×C71—22no—
62C4⋊C4×C71—22no—
63C4⋊C4×C71—22no—
64C4⋊C4×C71—22no—
65C4⋊C4×C71D412yes—
66C4⋊C4×C71C2312yes—
67C22⋊C4×C71—22no—
68C22⋊C4×C71D412yes—
69C42×C71—22no—
70C42×C71C2311yescharacteristic
71C42×C71C2311yescharacteristic
72C23×C71C2×D411yescharacteristic
73C22×D71—21no—
74Q8×C71—22no—
75Q8×C71—22no—
76C2×C4×C71—22no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71C2×D412yes—
80C2×C4×C71C2×D412yes—
81C2×C4×C71C2×D411yescharacteristic
82C2×C4×C71C2411yescharacteristic
83C22⋊D71—42no—
84C2×Dic71—21no—
85C2×Dic71—21no—
86C2×Dic71—21no—
87C2×Dic71—42no—
88D284—22no—
89C4×D71—42no—
90Dic142—22no—
91Dic142—22no—
92Dic142—22no—
93Dic142—42no—
94C71⋊C8—22no—
95C71⋊C8—22no—
96C22×C71—21no—
97C22×C71—22no—
98C22×C71C22×D411yescharacteristic
99C2×D71—41no—
100C2×D71—42no—
101C4×C71—21no—
102C4×C71—22no—
103C4×C71—22no—
104C4×C71—42no—
105C4×C71—42no—
106C4×C71H(5,𝔽2)12yes—
107Dic71—41no—
108Dic71—41no—
109Dic71—41no—
110C2×C71—41no—
111C2×C71#53511yescharacteristic
112C2×C71#57812yes—
113D71—81no—
114#3223—711nomaximal, Sylow
115C71#322311yescharacteristic, complemented, Sylow
116#541D7111yescharacteristic, complemented
117#536—711no—
118#497—711no—
119#494—711no—
120#491—711no—
121#488—711no—
122#464—712no—
123#462—712no—
124#452—712no—
125#451—712no—
126#433—711no—
127C2×#41—711no—
128C22×Q8—711no—
129C2×Q16—712no—
130C2×Q16—712no—
131C2×Q16—712no—
132C2×SD16—712no—
133C4⋊Q8—711no—
134C4⋊Q8—711no—
135C4⋊Q8C2×D7111yescharacteristic
136#126—22no—
137#125—22no—
138#125C2×D7111yescharacteristic
139#124—711no—
140#124—711no—
141#123—1422no—
142C22⋊Q8—1422no—
143C22⋊Q8C2×D7112yes—
144C4×Q8C2×D7112yes—
145#117C2×D7111yescharacteristic
146C4⋊C8—712no—
147#103—712no—
148#103—712no—
149#103—712no—
150#102—712no—
151C22⋊C8—712no—
152#97—711no—
153C4×C8—711no—
154#41—2842no—
155C2×Q8—2842no—
156C2×Q8—711no—
157C2×Q8—711no—
158C2×Q8—711no—
159C2×Q8—1421no—
160C2×Q8C22×D7112yes—
161C2×D4—711no—
162C2×D4—1421no—
163C22×C4—1421no—
164C22×C4—1421no—
165C22×C4C22×D7111yescharacteristic
166Q16—2842no—
167Q16—2842no—
168Q16—2842no—
169SD16—2842no—
170C2×C8—712no—
171C2×C8—712no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—1422no—
177C4⋊C4—1422no—
178C4⋊C4C22⋊D7112yes—
179C4⋊C4C22×D7112yes—
180C22⋊C4—22no—
181C22⋊C4—1422no—
182C22⋊C4—1422no—
183C22⋊C4C22⋊D7112yes—
184C42—22no—
185C42C22×D7111yescharacteristic
186C42C22×D7111yescharacteristic
187C23—1421no—
188C23C2×C22⋊D7111yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—2842no—
192Q8—1422no—
193Q8—1422no—
194Q8—1422no—
195D4—2842no—
196D4—1422no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—2842no—
201C2×C4—2842no—
202C2×C4—1421no—
203C2×C4—1421no—
204C2×C4—1421no—
205C2×C4C2×C22⋊D7112yes—
206C2×C4C2×C22⋊D7112yes—
207C2×C4C2×C22⋊D7111yescharacteristic
208C2×C4C23×D7111yescharacteristic
209C8—1422no—
210C8—1422no—
211C22—21no—
212C22—22no—
213C22—2841no—
214C22—2842no—
215C22C22×C22⋊D7111yescharacteristic
216C4—21no—
217C4—22no—
218C4—22no—
219C4—42no—
220C4—42no—
221C4—2841no—
222C4—2841no—
223C4—2841no—
224C4#1455617412yes—
225C2—41no—
226C2—5681no—
227C2#1798656311yescharacteristic
228C2#1798681412yes—
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#541×C71C2
socleC22×C71C22×D4
hypercenter#541D71
hypocenterC71#3223
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #17986391C2semidirect
3. #17986378C2semidirect
4. #17986369C2semidirect
5. #17986356C2semidirect
6. #17986221C2semidirect
7. #17986209C2semidirect
9. #17986154C2semidirect
10. #17986060C2semidirect
11. #541×C71C2semidirect
115. C71#3223semidirect
116. #541D71semidirect