Finite Group Databasebeta

← #31908809

Subgroups — #31908809

Order 18176.

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

normal subgroups only

This group has 306 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
1#22598924C211yesmaximal, characteristic
2#22590142C211yesmaximal, characteristic
3#335×D71C211yesmaximal, characteristic
4#22586396C211yesmaximal, characteristic, complemented
5#22586394C211yesmaximal, 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—
15C4⋊C8×D71—21no—
16C4×C8×D71C2211yescharacteristic
17C4⋊(C71⋊C16)C2211yescharacteristic
18(C2×D71)⋊C16—21no—
19#17985774—21no—
20#17985752C2211yescharacteristic
21#17985751—21no—
22#494×C71C2211yescharacteristic
23C4⋊C16×C71C2211yescharacteristic
24#335×C71C2211yescharacteristic
25C2×C8×D71—41no—
26C2×C8×D71C2×C411yescharacteristic
27C2×C8×D71D411yescharacteristic
28#14556128—21no—
29#14556119—41no—
30#14556108—41no—
31#14556104—22no—
32Q8×Dic71—22no—
33#14556093—41no—
34#14556086—41no—
35C42×D71D411yescharacteristic
36C2×C71⋊C16—21no—
37#14556062—21no—
38#14556062C2×C412yes—
39C4⋊(C71⋊C8)—21no—
40Dic71⋊C8—41no—
41#14556052—22no—
42#14556052—22no—
43C4×C71⋊C8D411yescharacteristic
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—
62C8×D71—82no—
63C4⋊Dic71—22no—
64C4⋊Dic71—41no—
65C4⋊Dic71—42no—
66Dic71⋊C4—42no—
67(C2×D71)⋊C4—42no—
68C4×Dic71—42no—
69C4×Dic71C22⋊C411yescharacteristic
70C4×Dic71D811yescharacteristic
71C4×Dic71SD1611yescharacteristic
72C71⋊C16—41no—
73C2×Q8×C71—22no—
74Q16×C71—42no—
75Q16×C71—41no—
76Q16×C71—41no—
77C2×C8×C71—41no—
78C2×C8×C71C22×C411yescharacteristic
79C2×C8×C71C2×D411yescharacteristic
80C4⋊C4×C71—41no—
81C4⋊C4×C71—41no—
82C42×C71C2×D411yescharacteristic
83C16×C71—41no—
84C22×D71#10211yescharacteristic
85Q8×C71—42no—
86Q8×C71—42no—
87C2×C4×C71—42no—
88C2×C4×C71C2×C22⋊C411yescharacteristic
89C2×C4×C71C2×D811yescharacteristic
90C2×C4×C71C2×SD1611yescharacteristic
91C2×Dic71—42no—
92C2×Dic71#10211yescharacteristic
93C2×Dic71#10211yescharacteristic
94C2×Dic71#10211yescharacteristic
95C4×D71—22no—
96C4×D71—21no—
97C4×D71—21no—
98C4×D71—21no—
99C4×D71—21no—
100Dic142—81no—
101Dic142—81no—
102C8×C71—21no—
103C8×C71—81no—
104C8×C71#13511yescharacteristic
105C8×C71#13511yescharacteristic
106C71⋊C8—21no—
107C71⋊C8—21no—
108C71⋊C8—81no—
109C22×C71C2×#10211yescharacteristic
110C2×D71—21no—
111C2×D71—21no—
112C2×D71#36212yes—
113C4×C71—21no—
114C4×C71—82no—
115C4×C71#41311yescharacteristic
116C4×C71#41611yescharacteristic
117C4×C71#50911yescharacteristic
118C4×C71#51011yescharacteristic
119Dic71—21no—
120Dic71—21no—
121Dic71—21no—
122Dic71—82no—
123#11954—711nomaximal, Sylow
124C2×C71#154011yescharacteristic
125C2×C71#213911yescharacteristic
126C2×C71C2×#36211yescharacteristic
127D71—22no—
128#3170—711no—
129#2163—711no—
130C2×#335—711no—
131#1364—711no—
132#1364—711no—
133#1362—711no—
134#1362D7111yescharacteristic, complemented
135C71#1195411yescharacteristic, complemented, Sylow
136#533—1421no—
137#500—711no—
138#494C2×D7111yescharacteristic
139#487—1421no—
140#467—712no—
141#467—1421no—
142C4×Q16—712no—
143C4×Q16—1421no—
144C2×C4⋊C8—1421no—
145C2×C4×C8—711no—
146C4⋊C16—711no—
147C4⋊C16C2×D7111yescharacteristic
148C22⋊C16—1421no—
149#346—1421no—
150#335—711no—
151#335—1421no—
152#335C2×D7111yescharacteristic
153C2×Q16—21no—
154C2×Q16C4×D7112yes—
155C22×C8—2841no—
156C22×C8—711no—
157C22×C8—711no—
158C4⋊Q8—21no—
159C4⋊Q8—1421no—
160#125—2841no—
161C22⋊Q8—2841no—
162C22⋊Q8—1422no—
163C4×Q8—2841no—
164C4×Q8—1422no—
165#117—2841no—
166C2×C42—711no—
167C2×C16—21no—
168C2×C16—1421no—
169#107—712no—
170#107—1421no—
171C4⋊C8—21no—
172C4⋊C8—2841no—
173C4⋊C8—1421no—
174#103—1422no—
175#103—1422no—
176C4×C8—711no—
177C4×C8—711no—
178C4×C8—711no—
179C4×C8C22×D7111yescharacteristic
180C2×Q8—22no—
181C2×Q8—2841no—
182C22×C4—711no—
183C22×C4—711no—
184C22×C4—711no—
185Q16—42no—
186Q16—41no—
187Q16—41no—
188C2×C8—41no—
189C2×C8—2841no—
190C2×C8—2841no—
191C2×C8—5682no—
192C2×C8—711no—
193C2×C8—711no—
194C2×C8—1421no—
195C2×C8—1421no—
196C2×C8C2×C4×D7111yescharacteristic
197C2×C8D4×D7111yescharacteristic
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—2842no—
201C4⋊C4—2842no—
202C4⋊C4—2841no—
203C4⋊C4—1422no—
204C22⋊C4—2842no—
205C42—2842no—
206C42—711no—
207C42—711no—
208C42—711no—
209C42D4×D7111yescharacteristic
210C16—41no—
211C16—2841no—
212C23—711no—
213Q8—42no—
214Q8—42no—
215Q8—5681no—
216Q8—5681no—
217C2×C4—42no—
218C2×C4—2842no—
219C2×C4—711no—
220C2×C4—711no—
221C2×C4—711no—
222C2×C4—1422no—
223C2×C4—1421no—
224C2×C4—1421no—
225C2×C4—1421no—
226C2×C4—1421no—
227C2×C4C22⋊C4×D7111yescharacteristic
228C2×C4D8×D7111yescharacteristic
229C2×C4SD16×D7111yescharacteristic
230C8—21no—
231C8—81no—
232C8—5681no—
233C8—1421no—
234C8—1421no—
235C8#1455614711yescharacteristic
236C8#1455614911yescharacteristic
237C22—712no—
238C22—1421no—
239C22—1421no—
240C22#102×D7111yescharacteristic
241C4—21no—
242C4—82no—
243C4—5682no—
244C4—1421no—
245C4—1421no—
246C4—1421no—
247C4#1798598011yescharacteristic
248C4#1798599211yescharacteristic
249C4#1798643611yescharacteristic
250C4#1798644111yescharacteristic
251C2—1422no—
252C2#2258698611yescharacteristic
253C2#2259005111yescharacteristic
254C2#362×D7111yescharacteristic
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#11954
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. #22586396C2semidirect
5. #22586394C2semidirect
6. #22586389C2semidirect
7. C71×#1362C2semidirect
134. #1362D71semidirect
135. C71#11954semidirect

Maximal subgroups

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

subgroupindex
#225989242
#225901422
#335×D712
#225863962
#225863942
#225863892
C71×#13622
#1195471