Finite Group Databasebeta

← #31885194

Subgroups — #31885194

Order 18176.

286 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 59 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#22590005C211yesmaximal, characteristic
2#22586932C211yesmaximal, characteristic
3C71×#1884C211yesmaximal, characteristic
4#17985973—21no—
5C4×#11765849—21no—
6#17985794—21no—
7#17985792C412yescomplemented
8#17985739—21no—
9#17985739C412yescomplemented
10#568×C71C411yescharacteristic, complemented
11#532×C71C2211yescharacteristic
12#421×C71—21no—
13#410×C71C411yescharacteristic, complemented
14#339×C71—21no—
15C2×#11765849—41no—
16#14556087—41no—
17#14556043—22no—
18#14556043—41no—
19#14556039—22no—
20C4×C71⋊C8—41no—
21C2×#41×C71C2×C411yescharacteristic
22C2×#41×C71D411yescharacteristic
23#131×C71—42no—
24C2×M16×C71—21no—
25C2×M16×C71—41no—
26C4⋊Q8×C71Q811yescharacteristic
27#127×C71C2×C411yescharacteristic
28#124×C71C2×C411yescharacteristic
29#124×C71D411yescharacteristic
30#121×C71—41no—
31C4×D4×C71—41no—
32#117×C71—21no—
33C2×C42×C71D411yescharacteristic
34#104×C71—42no—
35#99×C71—41no—
36#11765849—44no—
37C2×C71⋊C8—42no—
38C4⋊Dic71—81no—
39C22⋊Dic71—42no—
40C4×Dic71—81no—
41#41×C71—41no—
42#41×C71—41no—
43#41×C71—41no—
44#41×C71—41no—
45C2×Q8×C71C4⋊C411yescharacteristic
46C2×Q8×C71C4⋊C411yescharacteristic
47C2×D4×C71—21no—
48C2×D4×C71—21no—
49C2×D4×C71—41no—
50C2×D4×C71C4211yescharacteristic
51C2×D4×C71C22⋊C411yescharacteristic
52C22×C4×C71—21no—
53C22×C4×C71—21no—
54C22×C4×C71—21no—
55C22×C4×C71C22⋊C411yescharacteristic
56M16×C71—22no—
57M16×C71—42no—
58M16×C71—41no—
59C2×C8×C71—41no—
60C2×C8×C71—42no—
61C4⋊C4×C71—41no—
62C4⋊C4×C71—41no—
63C22⋊C4×C71—41no—
64C22⋊C4×C71—41no—
65C22⋊C4×C71—41no—
66C42×C71—21no—
67C42×C71—41no—
68C42×C71C22⋊C411yescharacteristic
69C42×C71C4⋊C411yescharacteristic
70C23×C71—21no—
71C23×C71—21no—
72C23×C71#9911yescharacteristic
73Q8×C71—41no—
74Q8×C71—41no—
75D4×C71—41no—
76D4×C71—41no—
77D4×C71—42no—
78D4×C71—82no—
79D4×C71—81no—
80C2×C4×C71—21no—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—42no—
87C2×C4×C71—41no—
88C2×C4×C71—41no—
89C2×C4×C71—41no—
90C2×C4×C71—41no—
91C2×C4×C71—81no—
92C2×C4×C71#9511yescharacteristic
93C2×C4×C71#9911yescharacteristic
94C2×Dic71—82no—
95C8×C71—42no—
96C71⋊C8—44no—
97C22×C71—21no—
98C22×C71—41no—
99C22×C71—41no—
100C22×C71—41no—
101C22×C71—81no—
102C22×C71—81no—
103C22×C71#34211yescharacteristic
104C4×C71—22no—
105C4×C71—21no—
106C4×C71—41no—
107C4×C71—41no—
108C4×C71—41no—
109C4×C71—161no—
110C4×C71AΣL(1,𝔽16)11yescharacteristic
111C4×C71#35211yescharacteristic
112Dic71—162no—
113#7486—711nomaximal, Sylow
114C2×C71—21no—
115C2×C71—41no—
116C2×C71—81no—
117C2×C71—81no—
118C2×C71#139011yescharacteristic
119#2126—711no—
120#1884D7111yescharacteristic
121#1525—711no—
122C71#748611yescharacteristic, complemented, Sylow
123#568Dic7111yescharacteristic, complemented
124#532C2×D7111yescharacteristic
125#421—21no—
126#410—1421no—
127#410Dic7111yescharacteristic, complemented
128C4×M16—1421no—
129#354—1421no—
130#353—712no—
131#339—21no—
132#329—712no—
133#329—1421no—
134C2×#41C2×Dic7111yescharacteristic
135C2×#41C22⋊D7111yescharacteristic
136#131—42no—
137C2×M16—21no—
138C2×M16—41no—
139C2×M16—2841no—
140C4⋊Q8Dic14211yescharacteristic
141#127C4×D7111yescharacteristic
142#124C4×D7111yescharacteristic
143#124D28411yescharacteristic
144#121—41no—
145C4×D4—41no—
146#117—21no—
147#117—2841no—
148C2×C42C22⋊D7111yescharacteristic
149#104—42no—
150#99—41no—
151#99—2841no—
152#99—1422no—
153#97—1422no—
154C4×C8—2841no—
155#41—41no—
156#41—41no—
157#41—41no—
158#41—41no—
159C2×Q8Dic71⋊C411yescharacteristic
160C2×Q8C4⋊Dic7111yescharacteristic
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—41no—
164C2×D4C4×Dic7111yescharacteristic
165C2×D4(C2×D71)⋊C411yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4C22⋊Dic7111yescharacteristic
170M16—22no—
171M16—42no—
172M16—41no—
173M16—2844no—
174C2×C8—41no—
175C2×C8—42no—
176C2×C8—2842no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—5681no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—41no—
183C22⋊C4—2842no—
184C42—21no—
185C42—41no—
186C42—5681no—
187C42(C2×D71)⋊C411yescharacteristic
188C42Dic71⋊C411yescharacteristic
189C23—21no—
190C23—21no—
191C23#1455604311yescharacteristic
192Q8—41no—
193Q8—41no—
194D4—41no—
195D4—41no—
196D4—42no—
197D4—82no—
198D4—81no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—81no—
211C2×C4—5682no—
212C2×C4C71⋊#9511yescharacteristic
213C2×C4#1455604311yescharacteristic
214C8—42no—
215C8—2844no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—81no—
221C22—81no—
222C22#1798576611yescharacteristic
223C4—22no—
224C4—21no—
225C4—41no—
226C4—41no—
227C4—41no—
228C4—161no—
229C4—11362no—
230C4#1798578611yescharacteristic
231C4#1798578911yescharacteristic
232C2—21no—
233C2—41no—
234C2—81no—
235C2—81no—
236C2#2258645811yescharacteristic
subgroupG/H
centerC4#17985786
commutatorC2×D4×C71C42
Frattini#532C2×D71
FittingC71×#1884C2
socleC2×C71#1390
prime radical, cosocle#532×C71C22
hypercenter#1884D71
hypocenterC71#7486
LCSG ▷ C2×D4×C71 ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #1884

Complements

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

NH ≅ G/Ntype
7. #17985792C4semidirect
9. #17985739C4semidirect
10. #568×C71C4semidirect
13. #410×C71C4semidirect
122. C71#7486semidirect
123. #568Dic71semidirect
127. #410Dic71semidirect

Maximal subgroups

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

subgroupindex
#225900052
#225869322
C71×#18842
#748671