Finite Group Databasebeta

← #31884092

Subgroups — #31884092

Order 18176.

224 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 53 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#22588517C211yesmaximal, characteristic
2#22586264C211yesmaximal, characteristic
3C71×#1314C211yesmaximal, characteristic
4C23×C71⋊C8C411yescharacteristic
5C2×Dic71⋊C8—21no—
6C4×C22⋊Dic71C411yescharacteristic
7C22⋊(C71⋊C16)—21no—
8#17985753—21no—
9C2×C22⋊C8×C71C2211yescharacteristic
10C22⋊C16×C71—21no—
11C22×C71⋊C8—21no—
12C22×C71⋊C8—41no—
13C22×C71⋊C8—42no—
14C2×C22⋊Dic71C811yescharacteristic
15C2×C22⋊Dic71C811yescharacteristic
16C2×C4×Dic71—21no—
17C2×C71⋊C16—41no—
18Dic71⋊C8—42no—
19C71⋊#95—21no—
20C23×C4×C71C2×C411yescharacteristic
21C22×C8×C71D411yescharacteristic
22C22×C8×C71D411yescharacteristic
23C2×C16×C71—41no—
24C22⋊C8×C71—41no—
25C22×Dic71—21no—
26C22×Dic71—21no—
27C2×C71⋊C8—22no—
28C2×C71⋊C8—41no—
29C2×C71⋊C8—42no—
30C2×C71⋊C8—41no—
31C2×C71⋊C8—42no—
32C22⋊Dic71—21no—
33C22⋊Dic71—41no—
34C22⋊Dic71C1612yescomplemented
35C4×Dic71—41no—
36C4×Dic71—41no—
37C71⋊C16—81no—
38C24×C71C2×C811yescharacteristic
39C22×C4×C71—21no—
40C22×C4×C71—21no—
41C22×C4×C71—41no—
42C22×C4×C71—41no—
43C22×C4×C71C22⋊C411yescharacteristic
44C22×C4×C71M1611yescharacteristic
45C2×C8×C71—21no—
46C2×C8×C71—21no—
47C2×C8×C71—41no—
48C2×C8×C71—41no—
49C16×C71—81no—
50C23×C71—21no—
51C23×C71—21no—
52C23×C71—41no—
53C23×C71—41no—
54C23×C71C22⋊C811yescharacteristic
55C23×C71C2×C1611yescharacteristic
56C23×C71M3211yescharacteristic
57C2×C4×C71—21no—
58C2×C4×C71—21no—
59C2×C4×C71—21no—
60C2×C4×C71—42no—
61C2×C4×C71—41no—
62C2×C4×C71—41no—
63C2×C4×C71—41no—
64C2×C4×C71#9911yescharacteristic
65C2×C4×C71#10111yescharacteristic
66C2×Dic71—22no—
67C2×Dic71—41no—
68C2×Dic71—41no—
69C2×Dic71—41no—
70C2×Dic71—41no—
71C2×Dic71—41no—
72C8×C71—41no—
73C8×C71—41no—
74C71⋊C8—42no—
75C22×C71—21no—
76C22×C71—21no—
77C22×C71—21no—
78C22×C71—21no—
79C22×C71—41no—
80C22×C71—41no—
81C22×C71—42no—
82C22×C71—41no—
83C22×C71—41no—
84C22×C71#32411yescharacteristic
85C22×C71C22⋊C1611yescharacteristic
86C22×C71#34911yescharacteristic
87C4×C71—21no—
88C4×C71—41no—
89C4×C71AΣL(1,𝔽16)11yescharacteristic
90C4×C71#35211yescharacteristic
91Dic71—42no—
92Dic71—81no—
93#7064—711nomaximal, Sylow
94C2×C71—21no—
95C2×C71—21no—
96C2×C71—41no—
97C2×C71—41no—
98C2×C71#132211yescharacteristic
99C2×C71#132611yescharacteristic
100C2×C71#132811yescharacteristic
101#1850—711no—
102#1314—711no—
103#1314D7111yescharacteristic
104C71#706411yescharacteristic, complemented, Sylow
105C23×C8—711no—
106C2×C4⋊C8—1421no—
107C2×C22⋊C8C2×D7111yescharacteristic
108C4×C22⋊C4—711no—
109C22⋊C16—21no—
110C22⋊C16—1421no—
111#336—1421no—
112C23×C4C4×D7111yescharacteristic
113C22×C8—2841no—
114C22×C8—2842no—
115C22×C8—1421no—
116C22×C8D28411yescharacteristic
117C22×C8C22⋊D7111yescharacteristic
118C2×C22⋊C4—711no—
119C2×C22⋊C4—711no—
120C2×C42—1421no—
121C2×C16—41no—
122C2×C16—2841no—
123C4⋊C8—2842no—
124C22⋊C8—41no—
125#95—1421no—
126C24C8×D7111yescharacteristic
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—1421no—
132C22×C4—1421no—
133C22×C4(C2×D71)⋊C411yescharacteristic
134C22×C4#1176584811yescharacteristic
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—2841no—
140C2×C8—2842no—
141C2×C8—2841no—
142C2×C8—2842no—
143C2×C8—1422no—
144C22⋊C4—2841no—
145C22⋊C4—712no—
146C22⋊C4—1421no—
147C42—2841no—
148C42—2841no—
149C16—81no—
150C16—5681no—
151C23—21no—
152C23—21no—
153C23—41no—
154C23—41no—
155C23(C2×D71)⋊C811yescharacteristic
156C23C16×D7111yescharacteristic
157C23#1455606711yescharacteristic
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—2841no—
166C2×C4—2841no—
167C2×C4—2841no—
168C2×C4—2841no—
169C2×C4—2841no—
170C2×C4—1422no—
171C2×C4#1455604211yescharacteristic
172C2×C4#1455604611yescharacteristic
173C8—41no—
174C8—41no—
175C8—2842no—
176C22—21no—
177C22—21no—
178C22—21no—
179C22—21no—
180C22—41no—
181C22—41no—
182C22—42no—
183C22—41no—
184C22—41no—
185C22#1798572411yescharacteristic
186C22(C2×D71)⋊C1611yescharacteristic
187C22#1798578011yescharacteristic
188C4—21no—
189C4—41no—
190C4—2842no—
191C4—5681no—
192C4#1798578711yescharacteristic
193C4#1798578811yescharacteristic
194C2—21no—
195C2—21no—
196C2—41no—
197C2—41no—
198C2#2258627911yescharacteristic
199C2#2258628911yescharacteristic
200C2#2258629211yescharacteristic
subgroupG/H
centerC2×C4#14556042
commutatorC23×C71C2×C16
FrattiniC2×C22⋊C8C2×D71
FittingC71×#1314C2
socleC22×C71#324
prime radicalC24×C71C2×C8
cosocleC2×C22⋊C8×C71C22
hypercenter#1314D71
hypocenterC71#7064
LCSG ▷ C23×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
34. C22⋊Dic71C16semidirect
104. C71#7064semidirect

Maximal subgroups

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

subgroupindex
#225885172
#225862642
C71×#13142
#706471