Finite Group Databasebeta

← #22599230

Subgroups — #22599230

Order 9088.

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

normal subgroups only

This group has 378 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#17986572C212yesmaximal
2#17986342C212yesmaximal, complemented
3#17986339C212yesmaximal, complemented
4#17986255C212yesmaximal, complemented
5#17986240C211yesmaximal, characteristic, complemented
6#17986156C212yesmaximal, complemented
7#17986153C212yesmaximal, complemented
8#17985965C211yesmaximal, characteristic
9#467×C71C211yesmaximal, characteristic, complemented
10C2×#11765865C2211yescharacteristic
11C2×Q8×D71C2212yes—
12C2×Dic284C2211yescharacteristic
13C2×#11765855C2212yes—
14C2×#11765849C2211yescharacteristic
15#14556127—22no—
16#14556114—22no—
17#14556113—22no—
18#14556110—22no—
19#14556108C2212yes—
20#14556107C2212yes—
21#14556104C2212yes—
22#14556086C2212yes—
23#14556062C2211yescharacteristic
24#14556059C2212yes—
25#14556053C2212yes—
26#14556052C2212yes—
27#14556051C2212yes—
28(C2×D71)⋊C8C2212yes—
29(C2×D71)⋊C8C2212yes—
30C2×Q16×C71C2211yescharacteristic
31C22×C8×C71C2211yescharacteristic
32C22⋊Q8×C71C2212yes—
33#107×C71C2211yescharacteristic
34#103×C71C2212yes—
35#11765865—41no—
36#11765865—41no—
37C2×Dic142—22no—
38C2×Dic142D411yescharacteristic
39C2×Dic142C2312yes—
40Q8×D71—42no—
41Q8×D71—42no—
42C2×D284D411yescharacteristic
43C2×C22⋊D71D412yes—
44C2×C4×D71—22no—
45C2×C4×D71C2312yes—
46Dic284—41no—
47#11765855—42no—
48#11765849—41no—
49C2×C71⋊C8C2312yes—
50C4⋊Dic71—22no—
51C4⋊Dic71C2312yes—
52Dic71⋊C4—22no—
53Dic71⋊C4—22no—
54C22⋊Dic71—22no—
55(C2×D71)⋊C4—22no—
56(C2×D71)⋊C4—22no—
57(C2×D71)⋊C4—22no—
58C4×Dic71—22no—
59C2×Q8×C71C2312yes—
60C22×C4×C71C2311yescharacteristic
61Q16×C71—41no—
62C2×C8×C71—41no—
63C2×C8×C71C2311yescharacteristic
64C2×C8×C71C2311yescharacteristic
65C4⋊C4×C71—22no—
66C4⋊C4×C71C2312yes—
67C22⋊C4×C71—22no—
68C23×C71C2×D411yescharacteristic
69C22×D71C2×D412yes—
70Q8×C71—22no—
71Q8×C71—22no—
72C2×C4×C71—21no—
73C2×C4×C71—21no—
74C2×C4×C71—22no—
75C2×C4×C71—22no—
76C2×C4×C71C2×D411yescharacteristic
77C2×C4×C71C2411yescharacteristic
78C22⋊D71—42no—
79C2×Dic71—22no—
80C2×Dic71—22no—
81C2×Dic71C2×D412yes—
82D284—41no—
83C4×D71—22no—
84C4×D71—22no—
85C4×D71—44no—
86Dic142—22no—
87Dic142—22no—
88Dic142—41no—
89Dic142—44no—
90C8×C71—21no—
91C8×C71—21no—
92C71⋊C8—22no—
93C22×C71—21no—
94C22×C71—21no—
95C22×C71—21no—
96C22×C71C22×D411yescharacteristic
97C2×D71—22no—
98C2×D71—22no—
99C2×D71—22no—
100C4×C71—21no—
101C4×C71—42no—
102C4×C71—42no—
103C4×C71H(5,𝔽2)11yescharacteristic
104C4×C71H(5,𝔽2)11yescharacteristic
105Dic71—22no—
106Dic71—42no—
107Dic71—42no—
108C2×C71—41no—
109C2×C71#53411yescharacteristic
110C2×C71#57811yescharacteristic
111C2×C71#57811yescharacteristic
112D71—42no—
113#3202—711nomaximal, Sylow
114C71#320211yescharacteristic, complemented, Sylow
115#536—712no—
116#485—712no—
117#485—712no—
118#470—712no—
119#467—711no—
120#467D7111yescharacteristic, complemented
121#451—712no—
122#451—712no—
123#408—711no—
124C2×#41—711no—
125C22×Q8—712no—
126C2×Q16—712no—
127C2×Q16—711no—
128C2×Q16C2×D7111yescharacteristic
129C2×M16—711no—
130C22×C8C2×D7111yescharacteristic
131C4⋊Q8—1422no—
132#124—1422no—
133#123—1422no—
134#123—1422no—
135C22⋊Q8—712no—
136C22⋊Q8—712no—
137C22⋊Q8—712no—
138C22⋊Q8C2×D7112yes—
139#117—712no—
140#107—711no—
141#107C2×D7111yescharacteristic
142#106—712no—
143#103—712no—
144#103—712no—
145#103—712no—
146#103C2×D7112yes—
147C22⋊C8—712no—
148C22⋊C8—712no—
149#41—2841no—
150#41—2841no—
151C2×Q8—2842no—
152C2×Q8—2842no—
153C2×Q8—711no—
154C2×Q8—712no—
155C2×Q8—1422no—
156C2×Q8C22×D7112yes—
157C2×D4—712no—
158C2×D4—711no—
159C22×C4—712no—
160C22×C4—1422no—
161C22×C4C22×D7111yescharacteristic
162Q16—41no—
163Q16—2841no—
164Q16—2842no—
165M16—2841no—
166C2×C8—41no—
167C2×C8—712no—
168C2×C8C22×D7111yescharacteristic
169C2×C8C22×D7111yescharacteristic
170C4⋊C4—22no—
171C4⋊C4—712no—
172C4⋊C4—1422no—
173C4⋊C4—1422no—
174C4⋊C4—1422no—
175C4⋊C4C22×D7112yes—
176C22⋊C4—22no—
177C22⋊C4—1422no—
178C22⋊C4—1422no—
179C22⋊C4—1422no—
180C22⋊C4—1422no—
181C42—1422no—
182C23—712no—
183C23D4×D7111yescharacteristic
184Q8—22no—
185Q8—22no—
186Q8—2841no—
187Q8—2844no—
188Q8—1422no—
189Q8—1422no—
190D4—2841no—
191D4—2842no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—2844no—
197C2×C4—712no—
198C2×C4—1422no—
199C2×C4—1422no—
200C2×C4—1422no—
201C2×C4—1422no—
202C2×C4D4×D7111yescharacteristic
203C2×C4C23×D7111yescharacteristic
204C8—21no—
205C8—21no—
206C8—1422no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—1422no—
211C22—1422no—
212C22—1422no—
213C22C2×D4×D7111yescharacteristic
214C4—21no—
215C4—42no—
216C4—42no—
217C4—2842no—
218C4—2842no—
219C4—1422no—
220C4#1455617311yescharacteristic
221C4#1455617411yescharacteristic
222C2—41no—
223C2—2842no—
224C2#1798655711yescharacteristic
225C2#1798681211yescharacteristic
226C2#1798681311yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#467×C71C2
socleC22×C71C22×D4
hypercenter#467D71
hypocenterC71#3202
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
2. #17986342C2semidirect
3. #17986339C2semidirect
4. #17986255C2semidirect
5. #17986240C2semidirect
6. #17986156C2semidirect
7. #17986153C2semidirect
9. #467×C71C2semidirect
114. C71#3202semidirect
120. #467D71semidirect