Finite Group Databasebeta

← #22589649

Subgroups — #22589649

Order 9088.

360 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C22×#11765850C211yesmaximal, characteristic
2C2×Dic71⋊C8C211yesmaximal, characteristic, complemented
3C2×#14556050C211yesmaximal, characteristic, complemented
4#17985902C211yesmaximal, characteristic, complemented
5#17985900C211yesmaximal, characteristic, complemented
6#17985761C211yesmaximal, characteristic, complemented
7C2×#102×C71C211yesmaximal, characteristic, complemented
8C22×D284C2211yescharacteristic
9C2×#11765850—22no—
10C2×#11765850—22no—
11C2×#11765850—22no—
12C22×C71⋊C8C2211yescharacteristic
13C2×C4⋊Dic71C2211yescharacteristic
14C2×C22⋊Dic71—21no—
15C2×C22⋊Dic71—21no—
16C2×(C2×D71)⋊C4—21no—
17C2×(C2×D71)⋊C4—21no—
18C2×C4×Dic71C2211yescharacteristic
19Dic71⋊C8—22no—
20#14556050—22no—
21C22×D4×C71C2211yescharacteristic
22C22×C8×C71C2211yescharacteristic
23C2×C4⋊C4×C71C2211yescharacteristic
24#102×C71—22no—
25C23×D71—21no—
26C2×D284—21no—
27C2×D284—21no—
28C2×D284—21no—
29C2×D284—21no—
30C2×D284—21no—
31C2×D284D412yes—
32C22×Dic71—21no—
33C22×Dic71D411yescharacteristic
34C22×Dic71D411yescharacteristic
35#11765850—28no—
36C2×C71⋊C8—21no—
37C2×C71⋊C8—21no—
38C2×C71⋊C8D412yes—
39C4⋊Dic71—21no—
40C4⋊Dic71—21no—
41C22⋊Dic71—42no—
42C22⋊Dic71—42no—
43(C2×D71)⋊C4—42no—
44(C2×D71)⋊C4—42no—
45C4×Dic71—21no—
46C4×Dic71—21no—
47C24×C71—21no—
48C2×D4×C71—21no—
49C2×D4×C71—21no—
50C2×D4×C71—21no—
51C2×D4×C71—21no—
52C2×D4×C71—21no—
53C2×D4×C71D412yes—
54C22×C4×C71—21no—
55C22×C4×C71C2311yescharacteristic
56C2×C8×C71—21no—
57C2×C8×C71—21no—
58C2×C8×C71—21no—
59C4⋊C4×C71—21no—
60C4⋊C4×C71—21no—
61C23×C71—22no—
62C23×C71—41no—
63C23×C71—41no—
64C23×C71—41no—
65C23×C71—41no—
66C23×C71—41no—
67C23×C71—41no—
68C23×C71C2×D411yescharacteristic
69C22×D71—22no—
70C22×D71—41no—
71C22×D71—41no—
72C22×D71—41no—
73C22×D71—41no—
74C22×D71—41no—
75C22×D71—41no—
76D4×C71—22no—
77D4×C71—22no—
78D4×C71—41no—
79D4×C71—41no—
80C2×C4×C71—41no—
81C2×C4×C71—41no—
82C2×C4×C71—41no—
83C2×C4×C71C2×D411yescharacteristic
84C2×C4×C71C2×D411yescharacteristic
85C2×C4×C71C2×D411yescharacteristic
86C2×C4×C71#4111yescharacteristic
87C2×C4×C71#4111yescharacteristic
88C2×C4×C71#4111yescharacteristic
89C2×Dic71—21no—
90C2×Dic71—21no—
91C2×Dic71—21no—
92C2×Dic71—21no—
93C2×Dic71—21no—
94C2×Dic71—41no—
95C2×Dic71—41no—
96C2×Dic71—41no—
97C2×Dic71D812yescomplemented
98D284—22no—
99D284—22no—
100D284—41no—
101D284—41no—
102C8×C71—41no—
103C71⋊C8—22no—
104C22×C71—42no—
105C22×C71—42no—
106C22×C71—42no—
107C22×C71—81no—
108C22×C71—81no—
109C22×C71—81no—
110C22×C71—81no—
111C22×C71#12111yescharacteristic
112C22×C71#12111yescharacteristic
113C22×C71#12111yescharacteristic
114C22×C71C2×D811yescharacteristic
115C22×C71#13511yescharacteristic
116C22×C71AGL(1,ℤ/8)11yescharacteristic
117C22×C71AGL(1,ℤ/8)11yescharacteristic
118C2×D71—42no—
119C2×D71—42no—
120C2×D71—42no—
121C2×D71—81no—
122C2×D71—81no—
123C2×D71—81no—
124C2×D71—81no—
125C4×C71—81no—
126C4×C71#12011yescharacteristic
127C4×C71#12311yescharacteristic
128C4×C71#12311yescharacteristic
129C4×C71#12411yescharacteristic
130Dic71—22no—
131Dic71—41no—
132Dic71—81no—
133C2×C71—82no—
134C2×C71#39411yescharacteristic
135C2×C71#45911yescharacteristic
136C2×C71#45911yescharacteristic
137C2×C71#46311yescharacteristic
138C2×C71#46311yescharacteristic
139C2×C71#46611yescharacteristic
140C2×C71#46911yescharacteristic
141D71—82no—
142#2054—711nomaximal, Sylow
143C71#205411yescharacteristic, complemented, Sylow
144C22×D8—711no—
145C2×C4⋊C8—711no—
146C2×#102—711no—
147C2×#102D7111yescharacteristic, complemented
148#390—711no—
149#390—711no—
150#340—711no—
151C22×D4—711no—
152C22×D4C2×D7111yescharacteristic
153C2×D8—1422no—
154C2×D8—1422no—
155C2×D8—1422no—
156C22×C8—711no—
157C22×C8C2×D7111yescharacteristic
158C2×C4⋊C4—711no—
159C2×C4⋊C4C2×D7111yescharacteristic
160C2×C22⋊C4—1421no—
161C2×C22⋊C4—1421no—
162C2×C22⋊C4—1421no—
163C2×C22⋊C4—1421no—
164C2×C42—711no—
165C4⋊C8—1422no—
166#102—22no—
167#102—1422no—
168C24—21no—
169C24—1421no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—712no—
176C2×D4—1421no—
177C2×D4—1421no—
178C2×D4—1421no—
179C2×D4—1421no—
180C2×D4—1421no—
181C2×D4C22⋊D7112yes—
182C22×C4—21no—
183C22×C4—711no—
184C22×C4—711no—
185C22×C4—1421no—
186C22×C4C22×D7111yescharacteristic
187D8—1428no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—712no—
192C2×C8—1421no—
193C2×C8—1421no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—1421no—
197C4⋊C4—1421no—
198C22⋊C4—2842no—
199C22⋊C4—2842no—
200C22⋊C4—2842no—
201C22⋊C4—2842no—
202C42—1421no—
203C42—1421no—
204C23—22no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—2841no—
212C23—2841no—
213C23—2841no—
214C23—2841no—
215C23—2841no—
216C23—2841no—
217C23—1422no—
218C23D4×D7111yescharacteristic
219D4—22no—
220D4—22no—
221D4—41no—
222D4—41no—
223D4—2841no—
224D4—2841no—
225D4—1422no—
226D4—1422no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—2841no—
231C2×C4—2841no—
232C2×C4—2841no—
233C2×C4—712no—
234C2×C4—1421no—
235C2×C4—1421no—
236C2×C4—1421no—
237C2×C4—1421no—
238C2×C4—1421no—
239C2×C4D4×D7111yescharacteristic
240C2×C4D4×D7111yescharacteristic
241C2×C4C2×C22⋊D7111yescharacteristic
242C2×C4#1176586411yescharacteristic
243C2×C4#1176586511yescharacteristic
244C2×C4#1176586511yescharacteristic
245C8—41no—
246C8—1422no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—81no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22—2842no—
255C22—2842no—
256C22—2842no—
257C22—5681no—
258C22—5681no—
259C22—5681no—
260C22—5681no—
261C22#1455610011yescharacteristic
262C22#1455610111yescharacteristic
263C22#1455610111yescharacteristic
264C22D8×D7111yescharacteristic
265C22#1455614911yescharacteristic
266C22#1455615211yescharacteristic
267C22#1455615511yescharacteristic
268C4—81no—
269C4—2841no—
270C4—5681no—
271C4—1422no—
272C4#1455609611yescharacteristic
273C4#1455611011yescharacteristic
274C4#1455611111yescharacteristic
275C4#1455611411yescharacteristic
276C2—82no—
277C2—5682no—
278C2#1798591411yescharacteristic
279C2#1798619211yescharacteristic
280C2#1798619311yescharacteristic
281C2#1798621311yescharacteristic
282C2#1798621611yescharacteristic
283C2#1798623411yescharacteristic
284C2#1798625111yescharacteristic
subgroupG/H
centerC23D4×D71
commutatorC22×C4×C71C23
FrattiniC22×C4C22×D71
FittingC2×#102×C71C2
socleC23×C71C2×D4
hypercenterC2×#102D71
hypocenterC71#2054
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×Dic71⋊C8C2semidirect
3. C2×#14556050C2semidirect
4. #17985902C2semidirect
5. #17985900C2semidirect
6. #17985761C2semidirect
7. C2×#102×C71C2semidirect
97. C2×Dic71D8semidirect
143. C71#2054semidirect
147. C2×#102D71semidirect