Finite Group Databasebeta

← #32024216

Subgroups — #32024216

Order 18176.

626 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 201 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#22591638C211yesmaximal, characteristic
2#22589055C213yesmaximal, complemented
3#22588945C213yesmaximal
4#22588894C213yesmaximal, complemented
5#22588521C213yesmaximal
6#22588485C211yesmaximal, characteristic, complemented
7C71×#1843C211yesmaximal, characteristic, complemented
8C22×#11765848C2211yescharacteristic
9C2×Dic71⋊C8C2213yes—
10C2×(C2×D71)⋊C8C2213yes—
11C2×C8×Dic71C2213yes—
12C71⋊#401C411yescharacteristic
13#17985940C413yes—
14#17985928C413yes—
15#17985923C411yescharacteristic
16C4×Dic71⋊C4C2213yes—
17C4×(C2×D71)⋊C4C2213yes—
18#17985754C2213yes—
19#17985754C2213yes—
20#17985753C2213yes—
21#17985753C2213yes—
22C2×C4⋊C8×C71C2213yes—
23C2×C4⋊C8×C71C2213yes—
24#376×C71C2211yescharacteristic
25C22×C4×D71C2311yescharacteristic
26C2×#11765848—43no—
27C22×C71⋊C8C2313yes—
28C22×C71⋊C8C2311yescharacteristic
29C2×Dic71⋊C4C2×C413yes—
30C2×Dic71⋊C4C2×C413yes—
31C2×(C2×D71)⋊C4C2×C413yes—
32C2×(C2×D71)⋊C4C2×C413yes—
33C2×C4×Dic71C2313yes—
34Dic71⋊C8—43no—
35(C2×D71)⋊C8—43no—
36C8×Dic71—43no—
37C71⋊#95C2×C413yes—
38C71⋊#95C2×C413yes—
39C71⋊#95C2×C413yes—
40C71⋊#95C2×C413yes—
41C22×C8×C71C2313yes—
42C22×C8×C71C2311yescharacteristic
43C2×C42×C71C2313yes—
44C4⋊C8×C71—43no—
45C4⋊C8×C71—43no—
46#95×C71C2×C411yescharacteristic
47#95×C71C2×C413yes—
48C23×D71C22×C411yescharacteristic
49C22×Dic71C22×C411yescharacteristic
50C22×Dic71C22×C413yes—
51C22×Dic71C22×C413yes—
52C2×C4×D71—23no—
53C2×C4×D71—23no—
54#11765848—44no—
55C2×C71⋊C8—23no—
56C2×C71⋊C8—23no—
57C2×C71⋊C8—26no—
58Dic71⋊C4—26no—
59Dic71⋊C4—26no—
60(C2×D71)⋊C4—26no—
61(C2×D71)⋊C4—43no—
62C4×Dic71—23no—
63C4×Dic71—23no—
64C22×C4×C71C22×C413yes—
65C22×C4×C71C22×C413yes—
66C22×C4×C71C2411yescharacteristic
67C2×C8×C71—23no—
68C2×C8×C71—23no—
69C2×C8×C71—23no—
70C2×C8×C71—23no—
71C42×C71—23no—
72C42×C71—23no—
73C23×C71C23×C411yescharacteristic
74C22×D71—23no—
75C22×D71—23no—
76C22×D71—21no—
77C2×C4×C71—23no—
78C2×C4×C71—23no—
79C2×C4×C71—23no—
80C2×C4×C71—23no—
81C2×C4×C71—23no—
82C2×C4×C71—23no—
83C2×C4×C71H(5,𝔽2)13yes—
84C2×C4×C712−1+413yes—
85C2×Dic71—23no—
86C2×Dic71—23no—
87C2×Dic71—23no—
88C2×Dic71—23no—
89C2×Dic71—23no—
90C2×Dic71—23no—
91C2×Dic71#13116yes—
92C4×D71—41no—
93C4×D71—43no—
94C8×C71—43no—
95C8×C71—41no—
96C71⋊C8—41no—
97C71⋊C8—43no—
98C22×C71#52013yes—
99C22×C71C2×#13113yes—
100C22×C71#56811yescharacteristic
101C2×D71—41no—
102C2×D71—43no—
103C2×D71—43no—
104C4×C71—43no—
105C4×C71—43no—
106C4×C71#56011yescharacteristic
107C4×C71#56313yes—
108Dic71—26no—
109Dic71—41no—
110Dic71—43no—
111#23381—711nomaximal, Sylow
112C2×C71#238311yescharacteristic
113C2×C71#299313yes—
114C2×C71#299413yes—
115D71—81no—
116#2401—711no—
117#1939—713no—
118#1923—713no—
119#1916—713no—
120#1851—713no—
121#1843D7111yescharacteristic, complemented
122#1840—711no—
123C71#2338111yescharacteristic, complemented, Sylow
124C22×M16—711no—
125C2×C4⋊C8—713no—
126C2×C4⋊C8C2×D7113yes—
127C2×C4⋊C8C2×D7113yes—
128C2×C22⋊C8—713no—
129C2×C4×C8—713no—
130#401—711no—
131#400—713no—
132#397—713no—
133#396—711no—
134C4×C4⋊C4—713no—
135C4×C22⋊C4—713no—
136#376C2×D7111yescharacteristic
137#336—713no—
138#336—713no—
139#336—713no—
140#336—713no—
141C23×C4—711no—
142C2×M16—2843no—
143C22×C8—711no—
144C22×C8—713no—
145C22×C8C22×D7111yescharacteristic
146C22×C8C22×D7113yes—
147C2×C4⋊C4—713no—
148C2×C4⋊C4—713no—
149C2×C22⋊C4—713no—
150C2×C22⋊C4—713no—
151C2×C42—713no—
152C2×C42C22×D7113yes—
153C4⋊C8—43no—
154C4⋊C8—43no—
155C4⋊C8—2843no—
156C22⋊C8—2843no—
157C4×C8—2843no—
158#95—713no—
159#95—713no—
160#95—713no—
161#95—713no—
162#95C4×D7111yescharacteristic
163#95C4×D7113yes—
164C24—711no—
165C22×C4—713no—
166C22×C4—711no—
167C22×C4—713no—
168C22×C4—1423no—
169C22×C4—1423no—
170C22×C4C2×C4×D7113yes—
171C22×C4C2×C4×D7113yes—
172C22×C4C23×D7111yescharacteristic
173M16—2844no—
174C2×C8—23no—
175C2×C8—23no—
176C2×C8—23no—
177C2×C8—23no—
178C2×C8—1423no—
179C2×C8—1423no—
180C2×C8—1426no—
181C4⋊C4—1426no—
182C4⋊C4—1426no—
183C22⋊C4—2843no—
184C22⋊C4—1426no—
185C42—23no—
186C42—23no—
187C42—1423no—
188C42—1423no—
189C23—1423no—
190C23—1423no—
191C23—1421no—
192C23C22×C4×D7111yescharacteristic
193C2×C4—23no—
194C2×C4—23no—
195C2×C4—23no—
196C2×C4—23no—
197C2×C4—23no—
198C2×C4—23no—
199C2×C4—2841no—
200C2×C4—2843no—
201C2×C4—716no—
202C2×C4—1423no—
203C2×C4—1423no—
204C2×C4—1423no—
205C2×C4—1423no—
206C2×C4—1423no—
207C2×C4—1423no—
208C2×C4#1455617313yes—
209C2×C4#1455617613yes—
210C8—41no—
211C8—43no—
212C8—2841no—
213C8—2843no—
214C22—2841no—
215C22—2843no—
216C22—2843no—
217C22#1798648013yes—
218C22#131×D7113yes—
219C22#1798676411yescharacteristic
220C4—43no—
221C4—43no—
222C4—2841no—
223C4—2843no—
224C4—1426no—
225C4#1798673611yescharacteristic
226C4#1798674813yes—
227C2—5681no—
228C2#2259153311yescharacteristic
229C2#2259748713yes—
230C2#2259749813yes—
subgroupG/H
center, FrattiniC22×C4C23×D71
commutator, socleC23×C71C23×C4
FittingC71×#1843C2
prime radicalC23×D71C22×C4
cosocleC22×C4×C71C24
hypercenter#1843D71
hypocenterC71#23381
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ #1843

Complements

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

NH ≅ G/Ntype
2. #22589055C2semidirect
4. #22588894C2semidirect
6. #22588485C2semidirect
7. C71×#1843C2semidirect
121. #1843D71semidirect
123. C71#23381semidirect

Maximal subgroups

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

subgroupindex
#225916382
#225890552
#225889452
#225888942
#225885212
#225884852
C71×#18432
#2338171