Finite Group Databasebeta

← #22599423

Subgroups — #22599423

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#17986611C211yesmaximal, characteristic, complemented
2#17986394C211yesmaximal, characteristic, complemented
3#17986378C211yesmaximal, characteristic, complemented
4#17986367C211yesmaximal, characteristic, complemented
5#17986353C211yesmaximal, characteristic, complemented
6#17986218C212yesmaximal, complemented
7#17986206C212yesmaximal, complemented
8#17986161C212yesmaximal
9#17986155C212yesmaximal, complemented
10#17986060C211yesmaximal, characteristic, complemented
11#536×C71C211yesmaximal, characteristic
12C2×#11765855C2212yes—
13C2×#11765855C2212yes—
14C2×#11765855C2212yes—
15C2×#11765853C2212yes—
16#14556128C2211yescharacteristic
17#14556121—22no—
18#14556120C2211yescharacteristic
19#14556118—22no—
20#14556107C2212yes—
21C4×Dic142C2212yes—
22C4⋊(C71⋊C8)C2212yes—
23#14556052C2212yes—
24#14556052C2212yes—
25#14556052C2212yes—
26#14556049C2212yes—
27C22⋊(C71⋊C8)C2212yes—
28#14556039C2211yescharacteristic
29C4×C71⋊C8C2211yescharacteristic
30C2×#41×C71C2211yescharacteristic
31C22×Q8×C71C2211yescharacteristic
32C4⋊Q8×C71C2211yescharacteristic
33C4⋊Q8×C71C2211yescharacteristic
34#124×C71C2211yescharacteristic
35#124×C71C2211yescharacteristic
36#123×C71—22no—
37C22⋊Q8×C71—22no—
38#117×C71C2211yescharacteristic
39C2×Dic142C2312yes—
40#11765855—42no—
41#11765855—42no—
42#11765855—42no—
43#11765853—42no—
44C2×C71⋊C8C2312yes—
45C2×C71⋊C8C2312yes—
46C4⋊Dic71—22no—
47C4⋊Dic71C2312yes—
48Dic71⋊C4—22no—
49Dic71⋊C4—22no—
50Dic71⋊C4—22no—
51C22⋊Dic71—22no—
52C4×Dic71—22no—
53#41×C71—42no—
54C2×Q8×C71—21no—
55C2×Q8×C71—42no—
56C2×Q8×C71C2311yescharacteristic
57C2×Q8×C71C2311yescharacteristic
58C2×Q8×C71C2311yescharacteristic
59C2×D4×C71—21no—
60C2×D4×C71C2311yescharacteristic
61C22×C4×C71—21no—
62C22×C4×C71—21no—
63C22×C4×C71C2311yescharacteristic
64C4⋊C4×C71—22no—
65C4⋊C4×C71—22no—
66C4⋊C4×C71D412yes—
67C22⋊C4×C71—22no—
68C22⋊C4×C71—22no—
69C22⋊C4×C71D412yes—
70C42×C71C2311yescharacteristic
71C42×C71C2311yescharacteristic
72C23×C71—21no—
73C23×C71C2×D411yescharacteristic
74Q8×C71—22no—
75Q8×C71—22no—
76Q8×C71—22no—
77Q8×C71—42no—
78D4×C71—22no—
79D4×C71—42no—
80C2×C4×C71—22no—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—42no—
85C2×C4×C71—42no—
86C2×C4×C71C2×D412yes—
87C2×C4×C71C2×D412yes—
88C2×C4×C71C2×D411yescharacteristic
89C2×C4×C71C2411yescharacteristic
90C2×Dic71—22no—
91C2×Dic71—22no—
92Dic142—22no—
93Dic142—22no—
94C71⋊C8—22no—
95C71⋊C8—22no—
96C22×C71—21no—
97C22×C71—22no—
98C22×C71—41no—
99C22×C71—42no—
100C22×C71C22×D411yescharacteristic
101C4×C71—21no—
102C4×C71—22no—
103C4×C71—22no—
104C4×C71—41no—
105C4×C71—41no—
106C4×C71—41no—
107C4×C71H(5,𝔽2)12yes—
108Dic71—42no—
109Dic71—42no—
110C2×C71—41no—
111C2×C71—81no—
112C2×C71#53511yescharacteristic
113C2×C71#57812yes—
114#3223—711nomaximal, Sylow
115C71#322311yescharacteristic, complemented, Sylow
116#541—711no—
117#536D7111yescharacteristic
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×#41C2×D7111yescharacteristic
128C22×Q8C2×D7111yescharacteristic
129C2×Q16—712no—
130C2×Q16—712no—
131C2×Q16—712no—
132C2×SD16—712no—
133C4⋊Q8—711no—
134C4⋊Q8C2×D7111yescharacteristic
135C4⋊Q8C2×D7111yescharacteristic
136#126—1422no—
137#125—711no—
138#125—1422no—
139#124C2×D7111yescharacteristic
140#124C2×D7111yescharacteristic
141#123—22no—
142C22⋊Q8—22no—
143C22⋊Q8—712no—
144C4×Q8—712no—
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—42no—
155C2×Q8—21no—
156C2×Q8—42no—
157C2×Q8—712no—
158C2×Q8C22×D7111yescharacteristic
159C2×Q8C22×D7111yescharacteristic
160C2×Q8C22×D7111yescharacteristic
161C2×D4—21no—
162C2×D4C22×D7111yescharacteristic
163C22×C4—21no—
164C22×C4—21no—
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—712no—
175C4⋊C4—1422no—
176C4⋊C4—1422no—
177C4⋊C4—1422no—
178C4⋊C4—1422no—
179C4⋊C4C22⋊D7112yes—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—1422no—
183C22⋊C4C22⋊D7112yes—
184C42—1422no—
185C42C22×D7111yescharacteristic
186C42C22×D7111yescharacteristic
187C23—21no—
188C23C2×C22⋊D7111yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—1422no—
194Q8—1422no—
195D4—22no—
196D4—42no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—1422no—
204C2×C4—1422no—
205C2×C4C2×C22⋊D7112yes—
206C2×C4C2×C22⋊D7112yes—
207C2×C4C2×C22⋊D7111yescharacteristic
208C2×C4C23×D7111yescharacteristic
209C8—1422no—
210C8—1422no—
211C22—21no—
212C22—22no—
213C22—41no—
214C22—42no—
215C22C22×C22⋊D7111yescharacteristic
216C4—21no—
217C4—22no—
218C4—22no—
219C4—41no—
220C4—41no—
221C4—41no—
222C4—2842no—
223C4—2842no—
224C4#1455617312yes—
225C2—41no—
226C2—81no—
227C2#1798656611yescharacteristic
228C2#1798681412yes—
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#536×C71C2
socleC22×C71C22×D4
hypercenter#536D71
hypocenterC71#3223
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #17986611C2semidirect
2. #17986394C2semidirect
3. #17986378C2semidirect
4. #17986367C2semidirect
5. #17986353C2semidirect
6. #17986218C2semidirect
7. #17986206C2semidirect
9. #17986155C2semidirect
10. #17986060C2semidirect
115. C71#3223semidirect