Finite Group Databasebeta

← #22598178

Subgroups — #22598178

Order 9088.

362 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 141 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#17986630C211yesmaximal, characteristic
2#17986477C211yesmaximal, characteristic
3#17986308C212yesmaximal, complemented
4#17986304C212yesmaximal, complemented
5#17986292C212yesmaximal, complemented
6#17986282C212yesmaximal, complemented
7#17986039C211yesmaximal, characteristic
8#17986037C211yesmaximal, characteristic
9#17986025C211yesmaximal, characteristic
10#17986008C211yesmaximal, characteristic
11C2×#103×C71C211yesmaximal, characteristic
12C2×#11765849C2211yescharacteristic
13#14556127—21no—
14#14556127C2212yes—
15#14556119—21no—
16#14556119C2212yes—
17#14556107—22no—
18Q8×Dic71—21no—
19Q8×Dic71—21no—
20Q8×Dic71C2212yes—
21Q8×Dic71C2212yes—
22#14556087—22no—
23#14556087C2211yescharacteristic
24#14556087C2211yescharacteristic
25#14556087C2211yescharacteristic
26#14556062C2211yescharacteristic
27#14556062C2211yescharacteristic
28#14556061C2212yes—
29#14556060C2211yescharacteristic
30#14556060C2211yescharacteristic
31#14556059C2212yes—
32Dic71⋊C8C2212yes—
33Dic71⋊C8C2212yes—
34#14556052C2212yes—
35#14556052C2212yes—
36C22×Q8×C71C2211yescharacteristic
37C22×C8×C71C2211yescharacteristic
38C2×C4⋊C4×C71C2211yescharacteristic
39#103×C71C2212yes—
40#103×C71C2212yes—
41C2×Dic142—22no—
42#11765849—22no—
43C2×C71⋊C8C2312yes—
44C4⋊Dic71—22no—
45C4⋊Dic71—22no—
46C4⋊Dic71—22no—
47C4⋊Dic71D411yescharacteristic
48C4⋊Dic71D411yescharacteristic
49C4⋊Dic71C2311yescharacteristic
50C4⋊Dic71C2311yescharacteristic
51C4⋊Dic71C2311yescharacteristic
52C4⋊Dic71C2311yescharacteristic
53Dic71⋊C4—22no—
54Dic71⋊C4—22no—
55C22⋊Dic71—21no—
56C22⋊Dic71—21no—
57C22⋊Dic71D412yes—
58C4×Dic71—21no—
59C4×Dic71—21no—
60C4×Dic71—22no—
61C4×Dic71—22no—
62C4×Dic71C2312yes—
63C2×Q8×C71—21no—
64C2×Q8×C71—22no—
65C2×Q8×C71Q814yescomplemented
66C2×Q8×C71C2312yes—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71C2311yescharacteristic
70C2×C8×C71—22no—
71C2×C8×C71C2311yescharacteristic
72C2×C8×C71C2311yescharacteristic
73C4⋊C4×C71—21no—
74C4⋊C4×C71C2312yes—
75C23×C71C2×D411yescharacteristic
76Q8×C71—24no—
77Q8×C71—22no—
78Q8×C71C2×Q814yes—
79C2×C4×C71—24no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71—81no—
83C2×C4×C71C2×D411yescharacteristic
84C2×C4×C71C2×Q812yes—
85C2×C4×C71#4112yes—
86C2×C4×C71C2411yescharacteristic
87C2×Dic71—21no—
88C2×Dic71—21no—
89C2×Dic71—21no—
90C2×Dic71—21no—
91C2×Dic71C2×D412yes—
92C2×Dic71C2×D412yes—
93Dic142—82no—
94C8×C71—21no—
95C8×C71—21no—
96C71⋊C8—22no—
97C22×C71—21no—
98C22×C71—21no—
99C22×C71C22⋊Q812yes—
100C22×C71C22×D411yescharacteristic
101C4×C71—24no—
102C4×C71—42no—
103C4×C71C22⋊Q812yes—
104C4×C71C22×Q811yescharacteristic
105C4×C71C2×#4111yescharacteristic
106Dic71—22no—
107Dic71—22no—
108Dic71—41no—
109Dic71—41no—
110Dic71—41no—
111Dic71—41no—
112C2×C71—22no—
113C2×C71C2×C22⋊Q811yescharacteristic
114C2×C71#57711yescharacteristic
115C2×C71#57811yescharacteristic
116#3082—711nomaximal, Sylow
117C71#308211yescharacteristic, complemented, Sylow
118#544—711no—
119#519—711no—
120#479—712no—
121#479—712no—
122#477—712no—
123#475—712no—
124#428—711no—
125#427—711no—
126#424—711no—
127#419—711no—
128C2×#103D7111yescharacteristic
129C22×Q8C2×D7111yescharacteristic
130C2×M16—711no—
131C22×C8C2×D7111yescharacteristic
132C4⋊Q8—712no—
133C4⋊Q8—1421no—
134#125—712no—
135#125—1421no—
136C22⋊Q8—1422no—
137C4×Q8—712no—
138C4×Q8—712no—
139C4×Q8—1421no—
140C4×Q8—1421no—
141#117—711no—
142#117—711no—
143#117—711no—
144#117—1422no—
145C2×C4⋊C4C2×D7111yescharacteristic
146#107—711no—
147#107—711no—
148#107—712no—
149#106—711no—
150#106—711no—
151#106—712no—
152C4⋊C8—712no—
153C4⋊C8—712no—
154#103—712no—
155#103—712no—
156#103C2×D7112yes—
157#103C2×D7112yes—
158C2×Q8—21no—
159C2×Q8—22no—
160C2×Q8—1422no—
161C2×Q8Dic14214yescomplemented
162C2×Q8C22×D7112yes—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4C22×D7111yescharacteristic
166M16—1422no—
167C2×C8—22no—
168C2×C8—712no—
169C2×C8C22×D7111yescharacteristic
170C2×C8C22×D7111yescharacteristic
171C4⋊C4—21no—
172C4⋊C4—711no—
173C4⋊C4—711no—
174C4⋊C4—711no—
175C4⋊C4—711no—
176C4⋊C4—711no—
177C4⋊C4—711no—
178C4⋊C4—1422no—
179C4⋊C4—1422no—
180C4⋊C4—1422no—
181C4⋊C4—1422no—
182C4⋊C4—1422no—
183C4⋊C4C22×D7112yes—
184C22⋊C4—712no—
185C22⋊C4—1421no—
186C22⋊C4—1421no—
187C42—712no—
188C42—1422no—
189C42—1422no—
190C42—1421no—
191C42—1421no—
192C23D4×D7111yescharacteristic
193Q8—22no—
194Q8—24no—
195Q8—5682no—
196Q8C2×Dic14214yes—
197C2×C4—22no—
198C2×C4—24no—
199C2×C4—22no—
200C2×C4—81no—
201C2×C4—712no—
202C2×C4—712no—
203C2×C4—1421no—
204C2×C4—1421no—
205C2×C4—1421no—
206C2×C4—1421no—
207C2×C4D4×D7111yescharacteristic
208C2×C4C2×Dic14212yes—
209C2×C4#1176586412yes—
210C2×C4C23×D7111yescharacteristic
211C8—21no—
212C8—21no—
213C8—1422no—
214C22—21no—
215C22—21no—
216C22#1455610512yes—
217C22C2×D4×D7111yescharacteristic
218C4—24no—
219C4—42no—
220C4—2841no—
221C4—2841no—
222C4—2841no—
223C4—2841no—
224C4—1422no—
225C4—1422no—
226C4#1455610512yes—
227C4C22×Dic14211yescharacteristic
228C4C2×#1176586411yescharacteristic
229C2—22no—
230C2C2×#1455610511yescharacteristic
231C2#1798680511yescharacteristic
232C2#1798681211yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×#103×C71C2
socleC22×C71C22×D4
hypercenterC2×#103D71
hypocenterC71#3082
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. #17986308C2semidirect
4. #17986304C2semidirect
5. #17986292C2semidirect
6. #17986282C2semidirect
65. C2×Q8×C71Q8semidirect
117. C71#3082semidirect
161. C2×Q8Dic142semidirect