Finite Group Databasebeta

← #22601197

Subgroups — #22601197

Order 9088.

370 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 133 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#17986729C211yesmaximal, characteristic, complemented
2#17986664C211yesmaximal, characteristic
3#17986396C212yesmaximal, complemented
4#17986288C212yesmaximal
5#17986286C211yesmaximal, characteristic
6#17986283C212yesmaximal, complemented
7#17986281C211yesmaximal, characteristic, complemented
8C8×Dic142C211yesmaximal, characteristic, complemented
9C4×#11765854C211yesmaximal, characteristic
10#17986086C212yesmaximal
11#498×C71C211yesmaximal, characteristic, complemented
12C2×Q8×D71—21no—
13C2×#11765854C2211yescharacteristic
14#14556127—21no—
15#14556127C2212yes—
16#14556127C2212yes—
17#14556126—22no—
18#14556106—22no—
19#14556104—21no—
20C4×Dic142C2211yescharacteristic
21C4×Dic142C2211yescharacteristic
22#14556093—21no—
23#14556093C2212yes—
24C4×D284C2211yescharacteristic
25#14556090C2212yes—
26#14556060C2211yescharacteristic
27#14556059C2212yes—
28#14556059C2212yes—
29C4⋊(C71⋊C8)C2211yescharacteristic
30Dic71⋊C8C2212yes—
31#14556053C2211yescharacteristic
32#14556051C2212yes—
33#14556050C2211yescharacteristic
34#14556048C2212yes—
35C8×Dic71C2212yes—
36C4⋊Q8×C71C2211yescharacteristic
37C4⋊Q8×C71C2211yescharacteristic
38#106×C71C2212yes—
39#106×C71C2212yes—
40C4×C8×C71C2211yescharacteristic
41C2×Dic142—22no—
42C2×Dic142—22no—
43C2×Dic142D411yescharacteristic
44C2×Dic142C2311yescharacteristic
45Q8×D71—44no—
46C2×D284C2311yescharacteristic
47C2×C4×D71—21no—
48C2×C4×D71—22no—
49#11765854Q814yescomplemented
50C2×C71⋊C8C2312yes—
51C4⋊Dic71D411yescharacteristic
52C4⋊Dic71C2311yescharacteristic
53C4⋊Dic71C2311yescharacteristic
54Dic71⋊C4—21no—
55Dic71⋊C4—22no—
56Dic71⋊C4—22no—
57Dic71⋊C4—22no—
58Dic71⋊C4D412yes—
59(C2×D71)⋊C4—21no—
60(C2×D71)⋊C4—22no—
61C4×Dic71—21no—
62C4×Dic71—22no—
63C4×Dic71C2312yes—
64C2×Q8×C71—21no—
65C2×Q8×C71—21no—
66C2×C8×C71C2311yescharacteristic
67C2×C8×C71C2311yescharacteristic
68C4⋊C4×C71—21no—
69C4⋊C4×C71—21no—
70C4⋊C4×C71C2312yes—
71C4⋊C4×C71C2312yes—
72C42×C71C2311yescharacteristic
73C22×D71—21no—
74Q8×C71—42no—
75Q8×C71—81no—
76C2×C4×C71—22no—
77C2×C4×C71—22no—
78C2×C4×C71C2×D411yescharacteristic
79C2×C4×C71C2×D411yescharacteristic
80C2×C4×C71C2411yescharacteristic
81C2×Dic71—21no—
82C2×Dic71—21no—
83C2×Dic71—21no—
84C2×Dic71C2×D412yes—
85C2×Dic71C2×D412yes—
86D284—21no—
87D284C2×Q812yes—
88C4×D71—42no—
89C4×D71—44no—
90Dic142—21no—
91Dic142—44no—
92Dic142—82no—
93Dic142SD1614yescomplemented
94Dic142C2×Q812yescomplemented
95C8×C71—21no—
96C8×C71C2×Q812yescomplemented
97C71⋊C8—22no—
98C22×C71C22×D411yescharacteristic
99C2×D71—22no—
100C2×D71—41no—
101C2×D71—41no—
102C4×C71—21no—
103C4×C71—42no—
104C4×C71—42no—
105C4×C71C2×SD1612yes—
106C4×C71C22×Q811yescharacteristic
107C4×C712−1+411yescharacteristic
108Dic71—22no—
109Dic71—22no—
110Dic71—41no—
111Dic71—41no—
112Dic71C2×SD1614yes—
113C2×C71D4×Q811yescharacteristic
114C2×C71C22×SD1611yescharacteristic
115C2×C71#57711yescharacteristic
116D71—42no—
117Q8×SD16—711nomaximal, Sylow
118C71Q8×SD1611yescharacteristic, complemented, Sylow
119Q82—711no—
120D4×Q8—711no—
121#498—712no—
122#498D7111yescharacteristic, complemented
123#476—711no—
124#476—712no—
125#475—711no—
126#475—712no—
127C8×Q8—711no—
128C4×SD16—712no—
129C4×SD16—711no—
130C22×Q8—1421no—
131C2×SD16—711no—
132C4⋊Q8—712no—
133C4⋊Q8—712no—
134C4⋊Q8—1421no—
135C4⋊Q8—1422no—
136C4⋊Q8C2×D7111yescharacteristic
137C4⋊Q8C2×D7111yescharacteristic
138C22⋊Q8—1422no—
139C22⋊Q8—1421no—
140C4×Q8—711no—
141C4×Q8—712no—
142C4×Q8—711no—
143C4×Q8—1421no—
144C4×D4—712no—
145C4×D4—711no—
146#106—711no—
147#106—712no—
148#106—712no—
149#106C2×D7112yes—
150#106C2×D7112yes—
151C4⋊C8—712no—
152C4⋊C8—711no—
153#103—711no—
154#103—712no—
155#102—711no—
156#102—712no—
157C4×C8—712no—
158C4×C8C2×D7111yescharacteristic
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—2844no—
162C2×Q8—711no—
163C2×Q8—711no—
164C2×Q8—1422no—
165C2×Q8—1422no—
166C2×D4—711no—
167C22×C4—1422no—
168C22×C4—1421no—
169SD16—714no—
170C2×C8—712no—
171C2×C8C22×D7111yescharacteristic
172C2×C8C22×D7111yescharacteristic
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—712no—
176C4⋊C4—711no—
177C4⋊C4—711no—
178C4⋊C4—711no—
179C4⋊C4—1422no—
180C4⋊C4—1422no—
181C4⋊C4—1422no—
182C4⋊C4—1421no—
183C4⋊C4C22×D7112yes—
184C4⋊C4C22×D7112yes—
185C22⋊C4—1422no—
186C22⋊C4—1421no—
187C42—712no—
188C42—1422no—
189C42—1421no—
190C42C22×D7111yescharacteristic
191C23—1421no—
192Q8—42no—
193Q8—81no—
194Q8—2844no—
195Q8—5682no—
196Q8—714no—
197Q8—712no—
198Q8—1421no—
199D4—712no—
200D4—1421no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—2844no—
204C2×C4—2842no—
205C2×C4—712no—
206C2×C4—712no—
207C2×C4—1421no—
208C2×C4—1421no—
209C2×C4—1421no—
210C2×C4D4×D7111yescharacteristic
211C2×C4D4×D7111yescharacteristic
212C2×C4C23×D7111yescharacteristic
213C8—21no—
214C8—1422no—
215C8Q8×D7112yescomplemented
216C22—2841no—
217C22—2841no—
218C22—1422no—
219C22C2×D4×D7111yescharacteristic
220C4—21no—
221C4—42no—
222C4—42no—
223C4—2841no—
224C4—2841no—
225C4—714no—
226C4—1422no—
227C4—1422no—
228C4SD16×D7112yes—
229C4C2×Q8×D7111yescharacteristic
230C4#1455617611yescharacteristic
231C2—2842no—
232C2#1798666411yescharacteristic
233C2C2×SD16×D7111yescharacteristic
234C2#1798680511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#498×C71C2
socleC22×C71C22×D4
hypercenter#498D71
hypocenterC71Q8×SD16
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
1. #17986729C2semidirect
3. #17986396C2semidirect
6. #17986283C2semidirect
7. #17986281C2semidirect
8. C8×Dic142C2semidirect
11. #498×C71C2semidirect
49. #11765854Q8semidirect
93. Dic142SD16semidirect
94. Dic142C2×Q8semidirect
96. C8×C71C2×Q8semidirect
118. C71Q8×SD16semidirect
122. #498D71semidirect
215. C8Q8×D71semidirect