Finite Group Databasebeta

← #31908804

Subgroups — #31908804

Order 18176.

296 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 83 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#22598924C211yesmaximal, characteristic
2#22590142C211yesmaximal, characteristic
3#22587087C211yesmaximal, characteristic
4#22586399C211yesmaximal, characteristic, complemented
5#22586397C211yesmaximal, characteristic, complemented
6#22586389C211yesmaximal, characteristic, complemented
7C71×#1362C211yesmaximal, characteristic, complemented
8#17986554—21no—
9#17986404C2211yescharacteristic
10#17986349—21no—
11#17986237—21no—
12#17986237C412yes—
13Q16×Dic71—21no—
14Q16×Dic71C412yes—
15#17986051—21no—
16C4×C8×D71C2211yescharacteristic
17C4⋊(C71⋊C16)C2211yescharacteristic
18(C2×D71)⋊C16—21no—
19#17985774—21no—
20#17985752C2211yescharacteristic
21#17985749—21no—
22#494×C71C2211yescharacteristic
23C4⋊C16×C71C2211yescharacteristic
24#335×C71C2211yescharacteristic
25C2×C8×D71C2×C411yescharacteristic
26C2×C8×D71D411yescharacteristic
27#14556128—21no—
28#14556119—41no—
29#14556108—41no—
30#14556104—22no—
31Q8×Dic71—22no—
32#14556093—41no—
33#14556086—41no—
34C42×D71D411yescharacteristic
35C2×C71⋊C16—21no—
36#14556062—21no—
37#14556062C2×C412yes—
38C4⋊(C71⋊C8)—21no—
39#14556052—22no—
40#14556052—22no—
41(C2×D71)⋊C8—41no—
42C4×C71⋊C8D411yescharacteristic
43C8×Dic71—41no—
44C8×Dic71C2×C411yescharacteristic
45C8×Dic71D411yescharacteristic
46C2×Q16×C71—21no—
47C2×Q16×C71C2×C412yes—
48C4⋊Q8×C71—21no—
49C2×C16×C71—21no—
50C4⋊C8×C71—21no—
51C4×C8×C71C2311yescharacteristic
52C2×Dic142—41no—
53C2×C4×D71C22⋊C411yescharacteristic
54C2×C4×D71D811yescharacteristic
55C2×C4×D71SD1611yescharacteristic
56C2×C71⋊C8—41no—
57C2×C71⋊C8C22⋊C411yescharacteristic
58C2×C71⋊C8C22⋊C411yescharacteristic
59C8×D71—21no—
60C8×D71—21no—
61C8×D71—41no—
62C4⋊Dic71—22no—
63C4⋊Dic71—41no—
64C4⋊Dic71—42no—
65Dic71⋊C4—42no—
66(C2×D71)⋊C4—42no—
67C4×Dic71—42no—
68C4×Dic71C22⋊C411yescharacteristic
69C4×Dic71D811yescharacteristic
70C4×Dic71SD1611yescharacteristic
71C71⋊C16—41no—
72C2×Q8×C71—22no—
73Q16×C71—42no—
74Q16×C71—41no—
75Q16×C71—41no—
76C2×C8×C71—41no—
77C2×C8×C71C22×C411yescharacteristic
78C2×C8×C71C2×D411yescharacteristic
79C4⋊C4×C71—41no—
80C4⋊C4×C71—41no—
81C42×C71C2×D411yescharacteristic
82C16×C71—41no—
83C22×D71#10211yescharacteristic
84Q8×C71—42no—
85Q8×C71—42no—
86C2×C4×C71—42no—
87C2×C4×C71C2×C22⋊C411yescharacteristic
88C2×C4×C71C2×D811yescharacteristic
89C2×C4×C71C2×SD1611yescharacteristic
90C2×Dic71—42no—
91C2×Dic71#10211yescharacteristic
92C2×Dic71#10211yescharacteristic
93C2×Dic71#10211yescharacteristic
94C4×D71—21no—
95C4×D71—21no—
96C4×D71—21no—
97C4×D71—21no—
98C4×D71—41no—
99Dic142—82no—
100C8×C71—21no—
101C8×C71—81no—
102C8×C71#13511yescharacteristic
103C8×C71#13511yescharacteristic
104C71⋊C8—21no—
105C71⋊C8—21no—
106C71⋊C8—81no—
107C22×C71C2×#10211yescharacteristic
108C2×D71—21no—
109C2×D71—21no—
110C2×D71—21no—
111C4×C71—21no—
112C4×C71—82no—
113C4×C71#41311yescharacteristic
114C4×C71#41611yescharacteristic
115C4×C71#51011yescharacteristic
116C4×C71#51011yescharacteristic
117Dic71—21no—
118Dic71—21no—
119Dic71—21no—
120Dic71—82no—
121#11953—711nomaximal, Sylow
122C2×C71#154011yescharacteristic
123C2×C71#214111yescharacteristic
124C2×C71#214811yescharacteristic
125D71—41no—
126#3170—711no—
127#2163—711no—
128#1564—711no—
129#1365—711no—
130#1365—711no—
131#1362—711no—
132#1362D7111yescharacteristic, complemented
133C71#1195311yescharacteristic, complemented, Sylow
134#533—1421no—
135#500—711no—
136#494C2×D7111yescharacteristic
137#487—1421no—
138#467—712no—
139#467—1421no—
140C4×Q16—712no—
141C4×Q16—1421no—
142#431—1421no—
143C2×C4×C8—711no—
144C4⋊C16—711no—
145C4⋊C16C2×D7111yescharacteristic
146C22⋊C16—1421no—
147#346—1421no—
148#335—711no—
149#335C2×D7111yescharacteristic
150#334—1421no—
151C2×Q16—21no—
152C2×Q16C4×D7112yes—
153C22×C8—711no—
154C22×C8—711no—
155C4⋊Q8—21no—
156C4⋊Q8—1421no—
157#125—2841no—
158C22⋊Q8—2841no—
159C22⋊Q8—1422no—
160C4×Q8—2841no—
161C4×Q8—1422no—
162#117—2841no—
163C2×C42—711no—
164C2×C16—21no—
165C2×C16—1421no—
166#107—712no—
167#107—1421no—
168C4⋊C8—21no—
169C4⋊C8—1421no—
170#103—1422no—
171#103—1422no—
172C22⋊C8—2841no—
173C4×C8—2841no—
174C4×C8—711no—
175C4×C8—711no—
176C4×C8—711no—
177C4×C8C22×D7111yescharacteristic
178C2×Q8—22no—
179C2×Q8—2841no—
180C22×C4—711no—
181C22×C4—711no—
182C22×C4—711no—
183Q16—42no—
184Q16—41no—
185Q16—41no—
186C2×C8—41no—
187C2×C8—2841no—
188C2×C8—2841no—
189C2×C8—711no—
190C2×C8—711no—
191C2×C8—1421no—
192C2×C8—1421no—
193C2×C8C2×C4×D7111yescharacteristic
194C2×C8D4×D7111yescharacteristic
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—2842no—
198C4⋊C4—2842no—
199C4⋊C4—2841no—
200C4⋊C4—1422no—
201C22⋊C4—2842no—
202C42—2842no—
203C42—711no—
204C42—711no—
205C42—711no—
206C42D4×D7111yescharacteristic
207C16—41no—
208C16—2841no—
209C23—711no—
210Q8—42no—
211Q8—42no—
212Q8—5682no—
213C2×C4—42no—
214C2×C4—2841no—
215C2×C4—2842no—
216C2×C4—711no—
217C2×C4—711no—
218C2×C4—711no—
219C2×C4—1421no—
220C2×C4—1421no—
221C2×C4—1421no—
222C2×C4—1421no—
223C2×C4C22⋊C4×D7111yescharacteristic
224C2×C4D8×D7111yescharacteristic
225C2×C4SD16×D7111yescharacteristic
226C8—21no—
227C8—81no—
228C8—5681no—
229C8—1421no—
230C8—1421no—
231C8#1455614711yescharacteristic
232C8#1455614911yescharacteristic
233C22—1421no—
234C22—1421no—
235C22—1421no—
236C22#102×D7111yescharacteristic
237C4—21no—
238C4—82no—
239C4—5682no—
240C4—1421no—
241C4—1421no—
242C4—1421no—
243C4#1798598011yescharacteristic
244C4#1798599211yescharacteristic
245C4#1798643811yescharacteristic
246C4#1798644111yescharacteristic
247C2—2841no—
248C2#2258698611yescharacteristic
249C2#2259005811yescharacteristic
250C2#2259008811yescharacteristic
subgroupG/H
centerC22#102×D71
commutatorC2×C8×C71C22×C4
FrattiniC4×C8C22×D71
FittingC71×#1362C2
socleC22×C71C2×#102
prime radicalC22×D71#102
cosocleC4×C8×C71C23
hypercenter#1362D71
hypocenterC71#11953
LCSG ▷ C2×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
4. #22586399C2semidirect
5. #22586397C2semidirect
6. #22586389C2semidirect
7. C71×#1362C2semidirect
132. #1362D71semidirect
133. C71#11953semidirect

Maximal subgroups

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

subgroupindex
#225989242
#225901422
#225870872
#225863992
#225863972
#225863892
C71×#13622
#1195371