Finite Group Databasebeta

← #31900457

Subgroups — #31900457

Order 18176.

500 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 107 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#22594310C211yesmaximal, characteristic
2#22588430C212yesmaximal
3#22586219C212yesmaximal
4#22586183C211yesmaximal, characteristic
5C71×#1445C211yesmaximal, characteristic
6C2×C22⋊(C71⋊C8)C2212yes—
7#17985941—22no—
8#17985933—22no—
9#17985872—22no—
10#17985859C2212yes—
11#17985761—22no—
12#17985761—22no—
13C22×C4⋊C4×C71C2211yescharacteristic
14C2×#95×C71—22no—
15C2×#95×C71C2211yescharacteristic
16C2×#95×C71C2211yescharacteristic
17C22×C71⋊C8—22no—
18C2×C4⋊Dic71D412yes—
19C2×C4⋊Dic71D412yes—
20C2×Dic71⋊C4—22no—
21C2×Dic71⋊C4—22no—
22C2×C22⋊Dic71—22no—
23C22⋊(C71⋊C8)—22no—
24C22⋊(C71⋊C8)—22no—
25C71⋊#95—22no—
26C71⋊#95—22no—
27C71⋊#95—22no—
28C23×C4×C71—21no—
29C23×C4×C71—21no—
30C23×C4×C71D412yes—
31C23×C4×C71C2311yescharacteristic
32C2×C4⋊C4×C71—22no—
33C2×C4⋊C4×C71—22no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71Q812yes—
36#95×C71—22no—
37#95×C71—22no—
38#95×C71—44no—
39C22×Dic71—22no—
40C22×Dic71—22no—
41C2×C71⋊C8—42no—
42C2×C71⋊C8—42no—
43C2×C71⋊C8—42no—
44C4⋊Dic71—42no—
45C4⋊Dic71—42no—
46Dic71⋊C4—82no—
47Dic71⋊C4—82no—
48C22⋊Dic71—42no—
49C22⋊Dic71—42no—
50C24×C71C2×D411yescharacteristic
51C22×C4×C71—22no—
52C22×C4×C71—22no—
53C22×C4×C71—22no—
54C22×C4×C71—22no—
55C22×C4×C71—22no—
56C22×C4×C71—42no—
57C22×C4×C71—42no—
58C22×C4×C71—42no—
59C22×C4×C71—42no—
60C22×C4×C71—41no—
61C22×C4×C71—42no—
62C22×C4×C71—42no—
63C22×C4×C71—41no—
64C22×C4×C71—41no—
65C22×C4×C71—41no—
66C22×C4×C71—42no—
67C22×C4×C71C2×D412yes—
68C22×C4×C71C2×Q811yescharacteristic
69C22×C4×C71#4112yes—
70C22×C4×C71#4111yescharacteristic
71C4⋊C4×C71—24no—
72C4⋊C4×C71—42no—
73C23×C71—21no—
74C23×C71—21no—
75C23×C71—22no—
76C23×C71#12011yescharacteristic
77C23×C71C22⋊Q811yescharacteristic
78C23×C71#12411yescharacteristic
79C23×C71#13512yes—
80C23×C71#13712yes—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71—41no—
84C2×C4×C71—42no—
85C2×C4×C71—41no—
86C2×C4×C71—44no—
87C2×C4×C71—42no—
88C2×C4×C71—44no—
89C2×C4×C71—44no—
90C2×C4×C71—42no—
91C2×C4×C71—42no—
92C2×C4×C71—42no—
93C2×C4×C71—42no—
94C2×C4×C71—82no—
95C2×C4×C71—81no—
96C2×C4×C71—82no—
97C2×C4×C71—81no—
98C2×C4×C71—81no—
99C2×C4×C71—81no—
100C2×C4×C71—82no—
101C2×C4×C71C22⋊Q812yes—
102C2×C4×C71#12412yes—
103C2×Dic71—42no—
104C2×Dic71—42no—
105C2×Dic71—42no—
106C2×Dic71—42no—
107C2×Dic71—42no—
108C2×Dic71—42no—
109C71⋊C8—82no—
110C22×C71—22no—
111C22×C71—22no—
112C22×C71—22no—
113C22×C71—22no—
114C22×C71—21no—
115C22×C71—21no—
116C22×C71—21no—
117C22×C71—21no—
118C22×C71#39311yescharacteristic
119C22×C71#45212yes—
120C22×C71#45311yescharacteristic
121C22×C71#45411yescharacteristic
122C22×C71#45812yes—
123C22×C71#47912yes—
124C22×C71#48811yescharacteristic
125C22×C71#49111yescharacteristic
126C4×C71—22no—
127C4×C71—22no—
128C4×C71—82no—
129C4×C71—82no—
130C4×C71—82no—
131Dic71—82no—
132Dic71—82no—
133#10347—711nomaximal, Sylow
134C2×C71—21no—
135C2×C71—22no—
136C2×C71—21no—
137C2×C71#161512yes—
138C2×C71#161812yes—
139C2×C71#202611yescharacteristic
140C2×C71#202711yescharacteristic
141C2×C71#203011yescharacteristic
142#2646—711no—
143#1824—712no—
144#1445D7111yescharacteristic
145#1294—712no—
146#1280—711no—
147C71#1034711yescharacteristic, complemented, Sylow
148C22×C4⋊C4C2×D7111yescharacteristic
149C2×C22⋊C8—712no—
150#400—1422no—
151#398—1422no—
152#385—1422no—
153#380—712no—
154C2×#95—22no—
155C2×#95C2×D7111yescharacteristic
156C2×#95C2×D7111yescharacteristic
157#340—1422no—
158#340—1422no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4C22⋊D7112yes—
162C23×C4C22×D7111yescharacteristic
163C22×C8—1422no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—712no—
168C2×C4⋊C4—712no—
169C2×C4⋊C4—1422no—
170C2×C4⋊C4—1422no—
171C2×C4⋊C4Dic14212yes—
172C2×C22⋊C4—1422no—
173C22⋊C8—1422no—
174C22⋊C8—1422no—
175#95—22no—
176#95—22no—
177#95—44no—
178#95—1422no—
179#95—1422no—
180#95—1422no—
181C24C2×C22⋊D7111yescharacteristic
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—42no—
198C22×C4—1422no—
199C22×C4—1422no—
200C22×C4D4×D7112yes—
201C22×C4C2×Dic14211yescharacteristic
202C22×C4#1176586412yes—
203C22×C4#1176586511yescharacteristic
204C2×C8—2842no—
205C2×C8—2842no—
206C2×C8—2842no—
207C4⋊C4—24no—
208C4⋊C4—42no—
209C4⋊C4—2842no—
210C4⋊C4—2842no—
211C4⋊C4—5682no—
212C4⋊C4—5682no—
213C22⋊C4—2842no—
214C22⋊C4—2842no—
215C23—21no—
216C23—21no—
217C23—22no—
218C23#1455609611yescharacteristic
219C23#1455610711yescharacteristic
220C23#1455611511yescharacteristic
221C23#1455614812yes—
222C23#1455615912yes—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—81no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—82no—
243C2×C4—2842no—
244C2×C4—2842no—
245C2×C4—2842no—
246C2×C4—2842no—
247C2×C4—2842no—
248C2×C4—2842no—
249C2×C4#1455610512yes—
250C2×C4#1455611412yes—
251C8—5682no—
252C22—21no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—22no—
260C22#1798591111yescharacteristic
261C22#1798615912yes—
262C22#1798616611yescharacteristic
263C22#1798617111yescharacteristic
264C22#1798618812yes—
265C22#1798630612yes—
266C22#1798635311yescharacteristic
267C22#1798636711yescharacteristic
268C4—22no—
269C4—22no—
270C4—82no—
271C4—82no—
272C4—82no—
273C4—5682no—
274C4—5682no—
275C2—21no—
276C2—22no—
277C2—21no—
278C2#2258736912yes—
279C2#2258738612yes—
280C2#2258949111yescharacteristic
281C2#2258949611yescharacteristic
282C2#2258950711yescharacteristic
subgroupG/H
centerC23#14556096
commutator, cosocleC23×C4×C71C23
FrattiniC23×C4C22×D71
FittingC71×#1445C2
socleC23×C71#120
prime radicalC24×C71C2×D4
hypercenter#1445D71
hypocenterC71#10347
LCSG ▷ C23×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1445

Complements

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

NH ≅ G/Ntype
147. C71#10347semidirect

Maximal subgroups

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

subgroupindex
#225943102
#225884302
#225862192
#225861832
C71×#14452
#1034771