Finite Group Databasebeta

← #31908776

Subgroups — #31908776

Order 18176.

354 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 91 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#22598920C211yesmaximal, characteristic
2#22590138C211yesmaximal, characteristic, complemented
3#335×D71C211yesmaximal, characteristic, complemented
4#22586395C211yesmaximal, characteristic, complemented
5#22586387C211yesmaximal, characteristic, complemented
6#22586385C211yesmaximal, characteristic, complemented
7C71×#1364C211yesmaximal, characteristic, complemented
8#17986550—21no—
9#17986376C2211yescharacteristic
10#17986347—21no—
11#17986236—21no—
12#17986236C412yes—
13#17986081—21no—
14#17986081C412yes—
15C4⋊C8×D71—21no—
16C4×C8×D71C2211yescharacteristic
17C4⋊(C71⋊C16)C2211yescharacteristic
18(C2×D71)⋊C16—21no—
19C16×Dic71—21no—
20#17985752C2211yescharacteristic
21#17985751—21no—
22#500×C71C2211yescharacteristic
23C4⋊C16×C71C2211yescharacteristic
24#335×C71C2211yescharacteristic
25C2×#11765866—41no—
26C2×D568—21no—
27C2×D568C2×C412yes—
28C2×C8×D71—41no—
29C2×C8×D71C2×C411yescharacteristic
30C2×C8×D71D411yescharacteristic
31#14556125—21no—
32#14556117—41no—
33#14556103—22no—
34#14556103—41no—
35#14556090—22no—
36#14556090—41no—
37C42×D71D411yescharacteristic
38C2×C71⋊C16—21no—
39C4⋊(C71⋊C8)—21no—
40Dic71⋊C8—41no—
41#14556048—22no—
42#14556048—22no—
43C4×C71⋊C8D411yescharacteristic
44C8×Dic71C2×C411yescharacteristic
45C8×Dic71D411yescharacteristic
46C4⋊Q8×C71—21no—
47C2×C16×C71—21no—
48#107×C71—21no—
49#107×C71C2×C412yes—
50C4⋊C8×C71—21no—
51C4×C8×C71C2311yescharacteristic
52#11765866—82no—
53#11765866—82no—
54C2×D284—22no—
55C2×D284—42no—
56C2×D284—41no—
57C2×D284—41no—
58C2×C4×D71—42no—
59C2×C4×D71C22⋊C411yescharacteristic
60C2×C4×D71D811yescharacteristic
61C2×C4×D71SD1611yescharacteristic
62D568—42no—
63D568—41no—
64D568—41no—
65C2×C71⋊C8—41no—
66C2×C71⋊C8C22⋊C411yescharacteristic
67C2×C71⋊C8C22⋊C411yescharacteristic
68C8×D71—21no—
69C8×D71—21no—
70C8×D71—41no—
71C8×D71—82no—
72(C2×D71)⋊C4—42no—
73(C2×D71)⋊C4—42no—
74C4×Dic71C22⋊C411yescharacteristic
75C4×Dic71D811yescharacteristic
76C4×Dic71SD1611yescharacteristic
77C71⋊C16—41no—
78C2×Q8×C71—41no—
79C2×C8×C71—41no—
80C2×C8×C71C22×C411yescharacteristic
81C2×C8×C71C2×D411yescharacteristic
82C4⋊C4×C71—22no—
83C4⋊C4×C71—41no—
84C42×C71C2×D411yescharacteristic
85C16×C71—41no—
86C22×D71—42no—
87C22×D71#10211yescharacteristic
88Q8×C71—81no—
89Q8×C71—81no—
90C2×C4×C71—42no—
91C2×C4×C71C2×C22⋊C411yescharacteristic
92C2×C4×C71C2×D811yescharacteristic
93C2×C4×C71C2×SD1611yescharacteristic
94C2×Dic71#10211yescharacteristic
95C2×Dic71#10211yescharacteristic
96C2×Dic71#10211yescharacteristic
97D284—42no—
98D284—42no—
99D284—82no—
100D284—84no—
101D284—81no—
102D284—81no—
103C4×D71—21no—
104C4×D71—21no—
105C4×D71—41no—
106C4×D71—84no—
107C4×D71D1612yes—
108C4×D71SD3212yes—
109C8×C71—21no—
110C8×C71—81no—
111C8×C71#13511yescharacteristic
112C8×C71#13511yescharacteristic
113C71⋊C8—21no—
114C71⋊C8—21no—
115C71⋊C8—81no—
116C22×C71C2×#10211yescharacteristic
117C2×D71—21no—
118C2×D71—21no—
119C2×D71—82no—
120C2×D71—82no—
121C2×D71—82no—
122C2×D71#35712yes—
123C4×C71—21no—
124C4×C71—82no—
125C4×C71#41311yescharacteristic
126C4×C71#41611yescharacteristic
127C4×C71C2×D1611yescharacteristic
128C4×C71C2×SD3211yescharacteristic
129Dic71—21no—
130Dic71—21no—
131Dic71#35712yes—
132#11947—711nomaximal, Sylow
133C2×C71#154011yescharacteristic
134C2×C71C2×#35711yescharacteristic
135C2×C71#214811yescharacteristic
136D71—22no—
137D71—162no—
138#3169—711no—
139#2162—711no—
140C2×#335—711no—
141#1364—711no—
142#1364D7111yescharacteristic, complemented
143#1361—711no—
144#1361—711no—
145C71#1194711yescharacteristic, complemented, Sylow
146#532—1421no—
147#500C2×D7111yescharacteristic
148#493—711no—
149#486—1421no—
150#466—712no—
151#466—1421no—
152C4×D8—712no—
153C4×D8—1421no—
154C2×C4⋊C8—1421no—
155C2×C4×C8—711no—
156C4⋊C16—711no—
157C4⋊C16C2×D7111yescharacteristic
158C22⋊C16—1421no—
159C4×C16—1421no—
160#335—711no—
161#335—1421no—
162#335C2×D7111yescharacteristic
163C2×#41—2841no—
164C2×D8—712no—
165C2×D8—1421no—
166C22×C8—2841no—
167C22×C8—711no—
168C22×C8—711no—
169C4⋊Q8—21no—
170#127—1421no—
171#124—2841no—
172#121—2841no—
173#121—1422no—
174C4×D4—2841no—
175C4×D4—1422no—
176C2×C42—711no—
177C2×C16—21no—
178C2×C16—1421no—
179#107—21no—
180#107C4×D7112yes—
181C4⋊C8—21no—
182C4⋊C8—2841no—
183C4⋊C8—1421no—
184#102—1422no—
185#102—1422no—
186C4×C8—711no—
187C4×C8—711no—
188C4×C8—711no—
189C4×C8C22×D7111yescharacteristic
190#41—5682no—
191#41—5682no—
192C2×Q8—41no—
193C2×D4—2842no—
194C2×D4—2841no—
195C2×D4—2841no—
196C2×D4—1422no—
197C22×C4—2842no—
198C22×C4—711no—
199C22×C4—711no—
200C22×C4—711no—
201D8—2842no—
202D8—2841no—
203D8—2841no—
204C2×C8—41no—
205C2×C8—2841no—
206C2×C8—2841no—
207C2×C8—5682no—
208C2×C8—711no—
209C2×C8—711no—
210C2×C8—1421no—
211C2×C8—1421no—
212C2×C8C2×C4×D7111yescharacteristic
213C2×C8D4×D7111yescharacteristic
214C4⋊C4—22no—
215C4⋊C4—41no—
216C22⋊C4—2842no—
217C22⋊C4—2842no—
218C42—711no—
219C42—711no—
220C42—711no—
221C42D4×D7111yescharacteristic
222C16—41no—
223C16—2841no—
224C23—2842no—
225C23—711no—
226Q8—81no—
227Q8—81no—
228D4—2842no—
229D4—2842no—
230D4—5682no—
231D4—5684no—
232D4—5681no—
233D4—5681no—
234C2×C4—42no—
235C2×C4—2841no—
236C2×C4—5684no—
237C2×C4—711no—
238C2×C4—712no—
239C2×C4—712no—
240C2×C4—711no—
241C2×C4—711no—
242C2×C4—1421no—
243C2×C4—1421no—
244C2×C4C22⋊C4×D7111yescharacteristic
245C2×C4D8×D7111yescharacteristic
246C2×C4SD16×D7111yescharacteristic
247C8—21no—
248C8—81no—
249C8—5681no—
250C8—1421no—
251C8—1421no—
252C8#1455614911yescharacteristic
253C8#1455615111yescharacteristic
254C22—5682no—
255C22—5682no—
256C22—5682no—
257C22—712no—
258C22—1421no—
259C22—1421no—
260C22#102×D7111yescharacteristic
261C4—21no—
262C4—82no—
263C4—712no—
264C4—1421no—
265C4—1421no—
266C4#1798598211yescharacteristic
267C4#1798599411yescharacteristic
268C4D16×D7111yescharacteristic
269C4SD32×D7111yescharacteristic
270C2—11362no—
271C2—1422no—
272C2#2258698811yescharacteristic
273C2#357×D7111yescharacteristic
274C2#2259008811yescharacteristic
subgroupG/H
centerC22#102×D71
commutatorC2×C8×C71C22×C4
FrattiniC4×C8C22×D71
FittingC71×#1364C2
socleC22×C71C2×#102
prime radical#22598920C2
cosocleC4×C8×C71C23
hypercenter#1364D71
hypocenterC71#11947
LCSG ▷ C2×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1364

Complements

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

NH ≅ G/Ntype
2. #22590138C2semidirect
3. #335×D71C2semidirect
4. #22586395C2semidirect
5. #22586387C2semidirect
6. #22586385C2semidirect
7. C71×#1364C2semidirect
142. #1364D71semidirect
145. C71#11947semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225989202
#225901382
#335×D712
#225863952
#225863872
#225863852
C71×#13642
#1194771