Finite Group Databasebeta

← #22599418

Subgroups — #22599418

Order 9088.

364 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 109 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#17986610C211yesmaximal, characteristic, complemented
2#17986569C211yesmaximal, characteristic
3#17986378C211yesmaximal, characteristic, complemented
4#17986368C211yesmaximal, characteristic, complemented
5#17986353C211yesmaximal, characteristic, complemented
6#17986219C212yesmaximal, complemented
7#17986207C212yesmaximal, complemented
8#17986158C212yesmaximal
9#17986153C212yesmaximal, complemented
10#17986061C211yesmaximal, characteristic, complemented
11#497×C71C211yesmaximal, characteristic, complemented
12C2×#11765864C2211yescharacteristic
13C2×Q8×D71C2211yescharacteristic
14C2×#11765855C2212yes—
15C2×#11765855C2212yes—
16#14556128C2211yescharacteristic
17#14556126C2211yescharacteristic
18#14556123—22no—
19#14556120C2211yescharacteristic
20#14556118—22no—
21#14556115C2211yescharacteristic
22#14556110—22no—
23#14556105—22no—
24#14556104C2212yes—
25Q8×Dic71C2212yes—
26#14556085C2211yescharacteristic
27Dic71⋊C8C2212yes—
28#14556053C2212yes—
29#14556053C2212yes—
30#14556052C2212yes—
31#14556049C2212yes—
32(C2×D71)⋊C8C2212yes—
33C4×C71⋊C8C2211yescharacteristic
34C2×Q16×C71C2212yes—
35C2×SD16×C71C2212yes—
36C4⋊Q8×C71C2211yescharacteristic
37#124×C71C2211yescharacteristic
38#97×C71C2211yescharacteristic
39#11765864—42no—
40C2×Dic142—21no—
41C2×Dic142C2311yescharacteristic
42C2×Dic142C2311yescharacteristic
43Q8×D71—42no—
44C2×C22⋊D71—21no—
45C22×Dic71—21no—
46C2×C4×D71—21no—
47C2×C4×D71C2311yescharacteristic
48#11765855—42no—
49#11765855—42no—
50C2×C71⋊C8C2312yes—
51C4⋊Dic71—22no—
52C4⋊Dic71—22no—
53C4⋊Dic71C2312yes—
54Dic71⋊C4—22no—
55Dic71⋊C4—22no—
56Dic71⋊C4—22no—
57Dic71⋊C4D412yes—
58C22⋊Dic71—22no—
59(C2×D71)⋊C4—22no—
60(C2×D71)⋊C4D412yes—
61C4×Dic71—22no—
62C4×Dic71C2311yescharacteristic
63C2×Q8×C71C2311yescharacteristic
64C2×Q8×C71C2312yes—
65C2×D4×C71C2311yescharacteristic
66Q16×C71—42no—
67SD16×C71—42no—
68C2×C8×C71C2312yes—
69C4⋊C4×C71—22no—
70C22⋊C4×C71—22no—
71C42×C71C2311yescharacteristic
72C23×C71—21no—
73C22×D71C2×D411yescharacteristic
74Q8×C71—22no—
75Q8×C71—22no—
76Q8×C71—22no—
77D4×C71—22no—
78C2×C4×C71—21no—
79C2×C4×C71—22no—
80C2×C4×C71C2×D412yes—
81C2×C4×C71C2411yescharacteristic
82C22⋊D71—42no—
83C2×Dic71—22no—
84C2×Dic71—21no—
85C2×Dic71—21no—
86C2×Dic71—42no—
87C2×Dic71C2×D412yes—
88C2×Dic71C2×D411yescharacteristic
89C4×D71—22no—
90C4×D71—42no—
91Dic142—22no—
92Dic142—22no—
93Dic142—42no—
94C8×C71—22no—
95C71⋊C8—22no—
96C22×C71—41no—
97C22×C71—42no—
98C22×C71C22×D411yescharacteristic
99C2×D71—21no—
100C2×D71—22no—
101C4×C71—22no—
102C4×C71—41no—
103C4×C71—42no—
104C4×C71H(5,𝔽2)12yes—
105Dic71—21no—
106Dic71—22no—
107Dic71—41no—
108Dic71—41no—
109Dic71—42no—
110C2×C71—81no—
111C2×C71#53511yescharacteristic
112C2×C71#57812yes—
113D71—41no—
114#3223—711nomaximal, Sylow
115C71#322311yescharacteristic, complemented, Sylow
116#541—711no—
117#536—711no—
118#497D7111yescharacteristic, complemented
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×Q16C2×D7112yes—
132C2×SD16C2×D7112yes—
133C4⋊Q8—711no—
134C4⋊Q8—711no—
135C4⋊Q8C2×D7111yescharacteristic
136#126—1422no—
137#125—711no—
138#125—1422no—
139#124—711no—
140#124C2×D7111yescharacteristic
141#123—1422no—
142C22⋊Q8—712no—
143C22⋊Q8—1422no—
144C4×Q8—712no—
145#117—711no—
146C4⋊C8—712no—
147#103—712no—
148#103—712no—
149#103—712no—
150#102—712no—
151C22⋊C8—712no—
152#97C2×D7111yescharacteristic
153C4×C8—711no—
154#41—2842no—
155C2×Q8—2842no—
156C2×Q8—711no—
157C2×Q8—711no—
158C2×Q8—1421no—
159C2×Q8C22×D7111yescharacteristic
160C2×Q8C22×D7112yes—
161C2×D4—1421no—
162C2×D4C22×D7111yescharacteristic
163C22×C4—711no—
164C22×C4—1421no—
165C22×C4—1421no—
166Q16—42no—
167Q16—2842no—
168Q16—2842no—
169SD16—42no—
170C2×C8—712no—
171C2×C8C22×D7112yes—
172C4⋊C4—22no—
173C4⋊C4—712no—
174C4⋊C4—712no—
175C4⋊C4—1422no—
176C4⋊C4—1422no—
177C4⋊C4—1422no—
178C4⋊C4—1422no—
179C4⋊C4—1422no—
180C22⋊C4—22no—
181C22⋊C4—712no—
182C22⋊C4—1422no—
183C22⋊C4—1422no—
184C42—711no—
185C42—1422no—
186C42C22×D7111yescharacteristic
187C23—21no—
188C23—711no—
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—2842no—
193Q8—1422no—
194Q8—1422no—
195D4—22no—
196D4—2842no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—2842no—
200C2×C4—2842no—
201C2×C4—712no—
202C2×C4—711no—
203C2×C4—1422no—
204C2×C4—1421no—
205C2×C4—1421no—
206C2×C4—1422no—
207C2×C4D4×D7112yes—
208C2×C4C23×D7111yescharacteristic
209C8—22no—
210C8—1422no—
211C22—41no—
212C22—42no—
213C22—1421no—
214C22—1422no—
215C22C2×D4×D7111yescharacteristic
216C4—22no—
217C4—41no—
218C4—42no—
219C4—2841no—
220C4—2841no—
221C4—2842no—
222C4—1421no—
223C4—1422no—
224C4#1455617312yes—
225C2—81no—
226C2—2841no—
227C2#1798656211yescharacteristic
228C2#1798681512yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#497×C71C2
socleC22×C71C22×D4
hypercenter#497D71
hypocenterC71#3223
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #17986610C2semidirect
3. #17986378C2semidirect
4. #17986368C2semidirect
5. #17986353C2semidirect
6. #17986219C2semidirect
7. #17986207C2semidirect
9. #17986153C2semidirect
10. #17986061C2semidirect
11. #497×C71C2semidirect
115. C71#3223semidirect
118. #497D71semidirect