Finite Group Databasebeta

← #31902353

Subgroups — #31902353

Order 18176.

236 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 69 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#22587111C211yesmaximal, characteristic
2#22587071C211yesmaximal, characteristic
3#22587041C211yesmaximal, characteristic
4#22586814C211yesmaximal, characteristic
5#22586250C211yesmaximal, characteristic
6#22586209C211yesmaximal, characteristic
7C71×#1287C211yesmaximal, characteristic
8#17986556C2211yescharacteristic
9#17986056C2211yescharacteristic
10#17986023C2211yescharacteristic
11C2×C4⋊(C71⋊C8)C2211yescharacteristic
12#17985750—22no—
13#17985747—21no—
14#17985746—21no—
15#17985744—21no—
16#17985744—21no—
17#17985731—22no—
18#432×C71C2211yescharacteristic
19#432×C71C2211yescharacteristic
20#431×C71C2211yescharacteristic
21C2×#11765849—21no—
22C22×C71⋊C8—21no—
23#14556128D411yescharacteristic
24#14556128D411yescharacteristic
25#14556120—21no—
26#14556107—21no—
27#14556107—21no—
28C4×Dic142—21no—
29C4×Dic142—21no—
30#14556087—21no—
31C4⋊(C71⋊C8)—21no—
32C4⋊(C71⋊C8)—21no—
33C4⋊(C71⋊C8)—21no—
34C4⋊(C71⋊C8)—21no—
35C4⋊(C71⋊C8)—21no—
36C22⋊(C71⋊C8)—21no—
37#14556039—21no—
38C2×C42×C71C2311yescharacteristic
39C4⋊C8×C71—21no—
40C4⋊C8×C71—21no—
41C4⋊C8×C71—21no—
42C22⋊C8×C71—21no—
43C22⋊C8×C71—21no—
44C22⋊C8×C71—21no—
45#97×C71—21no—
46#97×C71—21no—
47C4×C8×C71D412yes—
48C2×Dic142—21no—
49C2×Dic142—21no—
50#11765849—81no—
51C2×C71⋊C8—41no—
52C2×C71⋊C8—41no—
53C2×C71⋊C8—41no—
54C2×C71⋊C8—81no—
55C4⋊Dic71—21no—
56C4⋊Dic71—21no—
57C4⋊Dic71—21no—
58C4⋊Dic71—21no—
59Dic71⋊C4—41no—
60Dic71⋊C4—41no—
61C22⋊Dic71—41no—
62C4×Dic71—41no—
63C22×C4×C71#4111yescharacteristic
64C22×C4×C71#4111yescharacteristic
65C22×C4×C71#4111yescharacteristic
66C2×C8×C71—22no—
67C2×C8×C71—41no—
68C2×C8×C71—41no—
69C42×C71C2×D411yescharacteristic
70C42×C71C2×D411yescharacteristic
71C42×C71#4111yescharacteristic
72C42×C71#4111yescharacteristic
73C23×C71#12611yescharacteristic
74C2×C4×C71—21no—
75C2×C4×C71—21no—
76C2×C4×C71—41no—
77C2×C4×C71—41no—
78C2×C4×C71#12111yescharacteristic
79C2×C4×C71#12311yescharacteristic
80C2×C4×C71#12311yescharacteristic
81C2×C4×C71#12311yescharacteristic
82C2×C4×C71#12411yescharacteristic
83C2×C4×C71#12611yescharacteristic
84C2×Dic71—41no—
85C2×Dic71—41no—
86Dic142—41no—
87Dic142—41no—
88Dic142—81no—
89C8×C71—42no—
90C8×C71—81no—
91C8×C71—81no—
92C71⋊C8—81no—
93C71⋊C8—81no—
94C71⋊C8—81no—
95C22×C71—21no—
96C22×C71—21no—
97C22×C71—21no—
98C22×C71#39711yescharacteristic
99C4×C71—21no—
100C4×C71—21no—
101C4×C71—21no—
102C4×C71—21no—
103C4×C71—21no—
104C4×C71#47111yescharacteristic
105C4×C71#47111yescharacteristic
106Dic71—81no—
107Dic71—81no—
108#10648—711nomaximal, Sylow
109C2×C71—41no—
110C2×C71#207911yescharacteristic
111C2×C71#208011yescharacteristic
112C2×C71#208211yescharacteristic
113#1570—711no—
114#1559—711no—
115#1552—711no—
116#1497—711no—
117#1307—711no—
118#1290—711no—
119#1287D7111yescharacteristic
120C71#1064811yescharacteristic, complemented, Sylow
121#533—711no—
122#432—711no—
123#432C2×D7111yescharacteristic
124#432C2×D7111yescharacteristic
125#431C2×D7111yescharacteristic
126#423—711no—
127C2×C4⋊C8—711no—
128#334—1422no—
129#333—1421no—
130#332—1421no—
131#332—1421no—
132#332—1421no—
133Q8⋊C8—1422no—
134C2×M16—1421no—
135C22×C8—1421no—
136C4⋊Q8—711no—
137C4⋊Q8—711no—
138#125—1421no—
139C22⋊Q8—1421no—
140C22⋊Q8—1421no—
141C4×Q8—1421no—
142C4×Q8—1421no—
143#117—1421no—
144C2×C42C22×D7111yescharacteristic
145C4⋊C8—21no—
146C4⋊C8—21no—
147C4⋊C8—21no—
148C4⋊C8—1421no—
149C4⋊C8—1421no—
150C4⋊C8—1421no—
151C4⋊C8—1421no—
152C4⋊C8—1421no—
153C22⋊C8—21no—
154C22⋊C8—21no—
155C22⋊C8—21no—
156C22⋊C8—1421no—
157#97—21no—
158#97—21no—
159#97—1421no—
160C4×C8D28412yes—
161C2×Q8—1421no—
162C2×Q8—1421no—
163C22×C4#1176586411yescharacteristic
164C22×C4#1176586511yescharacteristic
165C22×C4#1176586611yescharacteristic
166M16—5681no—
167C2×C8—22no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—2841no—
171C2×C8—2841no—
172C2×C8—2841no—
173C2×C8—5681no—
174C4⋊C4—2841no—
175C4⋊C4—2841no—
176C4⋊C4—1421no—
177C4⋊C4—1421no—
178C4⋊C4—1421no—
179C4⋊C4—1421no—
180C22⋊C4—2841no—
181C42—2841no—
182C42D4×D7111yescharacteristic
183C42C2×D28411yescharacteristic
184C42#1176586411yescharacteristic
185C42#1176586511yescharacteristic
186C23#1455612311yescharacteristic
187Q8—2841no—
188Q8—2841no—
189Q8—5681no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—2841no—
195C2×C4—2841no—
196C2×C4#1455610311yescharacteristic
197C2×C4#1455611011yescharacteristic
198C2×C4#1455611011yescharacteristic
199C2×C4#1455611211yescharacteristic
200C2×C4#1455611611yescharacteristic
201C2×C4#1455612311yescharacteristic
202C8—42no—
203C8—81no—
204C8—81no—
205C8—5681no—
206C8—5681no—
207C8—5681no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22#1798592611yescharacteristic
212C4—21no—
213C4—21no—
214C4—21no—
215C4—21no—
216C4—21no—
217C4—5681no—
218C4—5681no—
219C4#1798626311yescharacteristic
220C4#1798626311yescharacteristic
221C2—41no—
222C2#2258978711yescharacteristic
223C2#2258979211yescharacteristic
224C2#2258980611yescharacteristic
subgroupG/H
centerC22#17985926
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1287C2
socleC22×C71#397
prime radicalC23×C71#126
hypercenter#1287D71
hypocenterC71#10648
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1287

Complements

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

NH ≅ G/Ntype
120. C71#10648semidirect

Maximal subgroups

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

subgroupindex
#225871112
#225870712
#225870412
#225868142
#225862502
#225862092
C71×#12872
#1064871