Finite Group Databasebeta

← #22598185

Subgroups — #22598185

Order 9088.

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

normal subgroups only

This group has 362 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#17986627C211yesmaximal, characteristic
2#17986476C211yesmaximal, characteristic
3#17986309C212yesmaximal, complemented
4#17986305C212yesmaximal, complemented
5#17986293C212yesmaximal, complemented
6#17986283C212yesmaximal, complemented
7#17986042C211yesmaximal, characteristic
8#17986025C211yesmaximal, characteristic
9#17986007C211yesmaximal, characteristic
10C2×#14556053C211yesmaximal, characteristic
11#427×C71C211yesmaximal, characteristic
12C22×Dic142C2211yescharacteristic
13C2×#11765849C2211yescharacteristic
14#14556128—21no—
15#14556127C2212yes—
16#14556120—21no—
17#14556119C2212yes—
18#14556105—22no—
19C4×Dic142—21no—
20C4×Dic142—21no—
21#14556093C2212yes—
22#14556093C2212yes—
23#14556087C2211yescharacteristic
24#14556084—22no—
25C2×C4⋊Dic71C2211yescharacteristic
26#14556061C2212yes—
27#14556059C2212yes—
28Dic71⋊C8C2212yes—
29Dic71⋊C8C2212yes—
30#14556053C2212yes—
31#14556053C2212yes—
32#14556051C2212yes—
33#14556051C2212yes—
34C22×C8×C71C2211yescharacteristic
35#117×C71C2211yescharacteristic
36#117×C71C2211yescharacteristic
37#107×C71C2211yescharacteristic
38#107×C71C2211yescharacteristic
39#106×C71C2211yescharacteristic
40#106×C71C2211yescharacteristic
41C2×Dic142—21no—
42C2×Dic142—22no—
43C2×Dic142—22no—
44C2×Dic142Q814yescomplemented
45C2×Dic142C2312yes—
46C22×Dic71—21no—
47C22×Dic71—21no—
48#11765849—22no—
49C2×C71⋊C8C2312yes—
50C4⋊Dic71—21no—
51C4⋊Dic71—22no—
52C4⋊Dic71D411yescharacteristic
53C4⋊Dic71D411yescharacteristic
54C4⋊Dic71C2312yes—
55Dic71⋊C4—22no—
56Dic71⋊C4—22no—
57Dic71⋊C4—22no—
58Dic71⋊C4—22no—
59C22⋊Dic71D412yes—
60C4×Dic71—22no—
61C4×Dic71—22no—
62C4×Dic71C2312yes—
63C22×C4×C71C2311yescharacteristic
64C2×C8×C71—22no—
65C2×C8×C71C2311yescharacteristic
66C2×C8×C71C2311yescharacteristic
67C4⋊C4×C71C2311yescharacteristic
68C4⋊C4×C71C2311yescharacteristic
69C4⋊C4×C71C2311yescharacteristic
70C4⋊C4×C71C2311yescharacteristic
71C22⋊C4×C71—21no—
72C22⋊C4×C71—21no—
73C42×C71—21no—
74C42×C71—21no—
75C23×C71C2×D411yescharacteristic
76C2×C4×C71—21no—
77C2×C4×C71—21no—
78C2×C4×C71—21no—
79C2×C4×C71—21no—
80C2×C4×C71C2×D411yescharacteristic
81C2×C4×C71C2×Q812yes—
82C2×C4×C71#4112yes—
83C2×C4×C71C2411yescharacteristic
84C2×Dic71—24no—
85C2×Dic71—22no—
86C2×Dic71—22no—
87C2×Dic71—81no—
88C2×Dic71C2×D412yes—
89C2×Dic71C2×D412yes—
90Dic142—24no—
91Dic142—22no—
92Dic142—82no—
93Dic142C2×Q814yes—
94C8×C71—21no—
95C8×C71—21no—
96C71⋊C8—22no—
97C22×C71—21no—
98C22×C71—21no—
99C22×C71C22⋊Q812yes—
100C22×C71C22×D411yescharacteristic
101C4×C71—41no—
102C4×C71—41no—
103C4×C71—41no—
104C4×C71—41no—
105C4×C71C22⋊Q812yes—
106C4×C71C22×Q811yescharacteristic
107C4×C71C2×#4111yescharacteristic
108Dic71—24no—
109Dic71—22no—
110Dic71—22no—
111Dic71—42no—
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#427D7111yescharacteristic
126#424—711no—
127#419—711no—
128C2×#103—711no—
129C22×Q8—711no—
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—1422no—
143#117C2×D7111yescharacteristic
144#117C2×D7111yescharacteristic
145C2×C4⋊C4—711no—
146#107—712no—
147#107C2×D7111yescharacteristic
148#107C2×D7111yescharacteristic
149#106—712no—
150#106C2×D7111yescharacteristic
151#106C2×D7111yescharacteristic
152C4⋊C8—712no—
153C4⋊C8—712no—
154#103—712no—
155#103—712no—
156#103—712no—
157#103—712no—
158C2×Q8—714no—
159C2×Q8—712no—
160C2×Q8—1422no—
161C2×Q8—1421no—
162C2×Q8—1422no—
163C22×C4—1421no—
164C22×C4—1421no—
165C22×C4C22×D7111yescharacteristic
166M16—1422no—
167C2×C8—22no—
168C2×C8—712no—
169C2×C8C22×D7111yescharacteristic
170C2×C8C22×D7111yescharacteristic
171C4⋊C4—711no—
172C4⋊C4—711no—
173C4⋊C4—712no—
174C4⋊C4—1422no—
175C4⋊C4—1422no—
176C4⋊C4—1422no—
177C4⋊C4—1422no—
178C4⋊C4—1422no—
179C4⋊C4—1421no—
180C4⋊C4C22×D7111yescharacteristic
181C4⋊C4C22×D7111yescharacteristic
182C4⋊C4C22×D7111yescharacteristic
183C4⋊C4C22×D7111yescharacteristic
184C22⋊C4—21no—
185C22⋊C4—21no—
186C22⋊C4—712no—
187C42—21no—
188C42—21no—
189C42—712no—
190C42—1422no—
191C42—1422no—
192C23D4×D7111yescharacteristic
193Q8—5682no—
194Q8—714no—
195Q8—1422no—
196Q8—1424no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—5681no—
202C2×C4—712no—
203C2×C4—712no—
204C2×C4—1422no—
205C2×C4—1422no—
206C2×C4—1424no—
207C2×C4D4×D7111yescharacteristic
208C2×C4Q8×D7112yes—
209C2×C4#1176586512yes—
210C2×C4C23×D7111yescharacteristic
211C8—21no—
212C8—21no—
213C8—1422no—
214C22—21no—
215C22—21no—
216C22#1455610612yes—
217C22C2×D4×D7111yescharacteristic
218C4—41no—
219C4—41no—
220C4—41no—
221C4—41no—
222C4—2842no—
223C4—1424no—
224C4—1422no—
225C4—1422no—
226C4#1455610612yes—
227C4C2×Q8×D7111yescharacteristic
228C4C2×#1176586511yescharacteristic
229C2—22no—
230C2C2×#1455610611yescharacteristic
231C2#1798680611yescharacteristic
232C2#1798681311yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#427×C71C2
socleC22×C71C22×D4
hypercenter#427D71
hypocenterC71#3082
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
3. #17986309C2semidirect
4. #17986305C2semidirect
5. #17986293C2semidirect
6. #17986283C2semidirect
44. C2×Dic142Q8semidirect
117. C71#3082semidirect