Finite Group Databasebeta

← #31884480

Subgroups — #31884480

Order 18176.

290 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 67 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#22588417C211yesmaximal, characteristic
2#22586674C211yesmaximal, characteristic
3C71×#1750C211yesmaximal, characteristic
4#17986056—21no—
5#17985862—21no—
6#17985766—21no—
7#17985766C412yes—
8#17985725C412yescomplemented
9#17985723—21no—
10C2×#117×C71C2211yescharacteristic
11C2×C22⋊C8×C71—21no—
12C2×#97×C71C411yescharacteristic, complemented
13C2×C4×C8×C71C411yescharacteristic, complemented
14#336×C71—21no—
15C2×C22⋊Dic71—22no—
16C4⋊(C71⋊C8)—41no—
17C22⋊(C71⋊C8)—22no—
18#14556039—41no—
19C71⋊#95—41no—
20C71⋊#95—41no—
21C23×C4×C71D411yescharacteristic
22C22×C8×C71—21no—
23C22×C8×C71—21no—
24#117×C71—41no—
25#117×C71—41no—
26C2×C4⋊C4×C71C2×C411yescharacteristic
27C2×C4⋊C4×C71Q811yescharacteristic
28C2×C22⋊C4×C71C2×C411yescharacteristic
29C2×C22⋊C4×C71D411yescharacteristic
30C2×C42×C71C2×C411yescharacteristic
31C2×C42×C71D411yescharacteristic
32C22⋊C8×C71—24no—
33#97×C71—41no—
34C4×C8×C71—41no—
35C22×Dic71—42no—
36C2×C71⋊C8—42no—
37C22⋊Dic71—42no—
38C22⋊Dic71—42no—
39C24×C71C22⋊C411yescharacteristic
40C22×C4×C71—22no—
41C22×C4×C71—21no—
42C22×C4×C71—21no—
43C22×C4×C71—41no—
44C22×C4×C71C4211yescharacteristic
45C22×C4×C71C22⋊C411yescharacteristic
46C22×C4×C71C22⋊C411yescharacteristic
47C22×C4×C71C4⋊C411yescharacteristic
48C22×C4×C71C4⋊C411yescharacteristic
49C22×C4×C71C4⋊C411yescharacteristic
50C2×C8×C71—22no—
51C2×C8×C71—22no—
52C2×C8×C71—41no—
53C2×C8×C71—41no—
54C2×C8×C71—41no—
55C2×C8×C71—41no—
56C4⋊C4×C71—41no—
57C4⋊C4×C71—41no—
58C22⋊C4×C71—41no—
59C22⋊C4×C71—41no—
60C42×C71—21no—
61C42×C71—21no—
62C42×C71—21no—
63C42×C71—21no—
64C23×C71—21no—
65C23×C71—21no—
66C23×C71—41no—
67C23×C71—41no—
68C23×C71#9511yescharacteristic
69C23×C71#10412yes—
70C2×C4×C71—21no—
71C2×C4×C71—21no—
72C2×C4×C71—21no—
73C2×C4×C71—21no—
74C2×C4×C71—21no—
75C2×C4×C71—21no—
76C2×C4×C71—21no—
77C2×C4×C71—21no—
78C2×C4×C71—21no—
79C2×C4×C71—42no—
80C2×C4×C71—42no—
81C2×C4×C71—41no—
82C2×C4×C71—41no—
83C2×C4×C71—41no—
84C2×C4×C71—41no—
85C2×C4×C71#9911yescharacteristic
86C2×C4×C71#9911yescharacteristic
87C2×Dic71—82no—
88C2×Dic71—82no—
89C2×Dic71—82no—
90C8×C71—42no—
91C8×C71—42no—
92C71⋊C8—82no—
93C22×C71—21no—
94C22×C71—21no—
95C22×C71—22no—
96C22×C71—42no—
97C22×C71—41no—
98C22×C71—41no—
99C22×C71—81no—
100C22×C71H(3,ℤ/4)11yescharacteristic
101C22×C71#33911yescharacteristic
102C22×C71#34211yescharacteristic
103C4×C71—21no—
104C4×C71—21no—
105C4×C71—41no—
106C4×C71—41no—
107C4×C71—41no—
108C4×C71—41no—
109C4×C71—41no—
110Dic71—162no—
111#7214—711nomaximal, Sylow
112C2×C71—21no—
113C2×C71—21no—
114C2×C71—42no—
115C2×C71#128411yescharacteristic
116C2×C71#139111yescharacteristic
117C2×C71#139511yescharacteristic
118#1820—711no—
119#1750D7111yescharacteristic
120#1464—711no—
121C71#721411yescharacteristic, complemented, Sylow
122C2×#117C2×D7111yescharacteristic
123#432—1421no—
124C2×C22⋊C8—21no—
125C2×#97Dic7111yescharacteristic, complemented
126C2×C4×C8Dic7111yescharacteristic, complemented
127#381—1421no—
128#342—712no—
129#342—1421no—
130#336—21no—
131#324—712no—
132#323—1421no—
133C23×C4C22⋊D7111yescharacteristic
134C22×C8—21no—
135C22×C8—21no—
136#117—41no—
137#117—41no—
138C2×C4⋊C4Dic14211yescharacteristic
139C2×C4⋊C4C4×D7111yescharacteristic
140C2×C22⋊C4—1422no—
141C2×C22⋊C4C4×D7111yescharacteristic
142C2×C22⋊C4D28411yescharacteristic
143C2×C42C2×Dic7111yescharacteristic
144C2×C42C22⋊D7111yescharacteristic
145C4⋊C8—2841no—
146C22⋊C8—24no—
147C22⋊C8—1422no—
148#97—41no—
149#97—2841no—
150C4×C8—41no—
151#95—2841no—
152#95—2841no—
153C24(C2×D71)⋊C411yescharacteristic
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—41no—
158C22×C4—2842no—
159C22×C4C4×Dic7111yescharacteristic
160C22×C4(C2×D71)⋊C411yescharacteristic
161C22×C4C22⋊Dic7111yescharacteristic
162C22×C4Dic71⋊C411yescharacteristic
163C22×C4Dic71⋊C411yescharacteristic
164C22×C4C4⋊Dic7111yescharacteristic
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—2842no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C22⋊C4—41no—
175C22⋊C4—41no—
176C22⋊C4—2842no—
177C22⋊C4—2842no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—21no—
182C23—21no—
183C23—21no—
184C23—41no—
185C23—41no—
186C23C71⋊#9511yescharacteristic
187C23#1455605612yes—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—5682no—
204C2×C4—5682no—
205C2×C4—5682no—
206C2×C4#1455604311yescharacteristic
207C2×C4#1455604311yescharacteristic
208C8—42no—
209C8—42no—
210C8—5682no—
211C22—21no—
212C22—21no—
213C22—22no—
214C22—42no—
215C22—41no—
216C22—41no—
217C22—81no—
218C22#1798575511yescharacteristic
219C22#1798575911yescharacteristic
220C22#1798576611yescharacteristic
221C4—21no—
222C4—21no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—41no—
227C4—41no—
228C4—11362no—
229C2—21no—
230C2—21no—
231C2—42no—
232C2#2258619311yescharacteristic
233C2#2258646111yescharacteristic
234C2#2258646911yescharacteristic
subgroupG/H
centerC22#17985766
commutatorC22×C4×C71C42
FrattiniC2×#117C2×D71
FittingC71×#1750C2
socleC22×C71#342
prime radicalC24×C71C22⋊C4
cosocleC2×#117×C71C22
hypercenter#1750D71
hypocenterC71#7214
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1750

Complements

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

NH ≅ G/Ntype
8. #17985725C4semidirect
12. C2×#97×C71C4semidirect
13. C2×C4×C8×C71C4semidirect
121. C71#7214semidirect
125. C2×#97Dic71semidirect
126. C2×C4×C8Dic71semidirect

Maximal subgroups

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

subgroupindex
#225884172
#225866742
C71×#17502
#721471