Finite Group Databasebeta

← #31900447

Subgroups — #31900447

Order 18176.

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

normal subgroups only

This group has 640 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#22588275C212yesmaximal
3#22586219C212yesmaximal
4#22586183C211yesmaximal, characteristic
5C71×#1446C211yesmaximal, characteristic
6C2×#14556052—22no—
7C2×#14556052—22no—
8C2×C22⋊(C71⋊C8)C2212yes—
9#17985941—22no—
10#17985933—22no—
11#17985872—22no—
12#17985859C2212yes—
13C23×Q8×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—
23#14556052—44no—
24#14556052—44no—
25C22⋊(C71⋊C8)—22no—
26C22⋊(C71⋊C8)—22no—
27C71⋊#95—22no—
28C71⋊#95—22no—
29C71⋊#95—22no—
30C22×Q8×C71—21no—
31C22×Q8×C71—22no—
32C22×Q8×C71—22no—
33C22×Q8×C71—44no—
34C22×Q8×C71Q812yes—
35C23×C4×C71—21no—
36C23×C4×C71—21no—
37C23×C4×C71D412yes—
38C23×C4×C71C2311yescharacteristic
39#95×C71—22no—
40#95×C71—22no—
41#95×C71—44no—
42C22×Dic71—22no—
43C22×Dic71—22no—
44C2×C71⋊C8—42no—
45C2×C71⋊C8—42no—
46C2×C71⋊C8—42no—
47C4⋊Dic71—42no—
48C4⋊Dic71—42no—
49Dic71⋊C4—82no—
50Dic71⋊C4—82no—
51C22⋊Dic71—42no—
52C22⋊Dic71—42no—
53C24×C71C2×D411yescharacteristic
54C2×Q8×C71—24no—
55C2×Q8×C71—24no—
56C2×Q8×C71—24no—
57C2×Q8×C71—44no—
58C2×Q8×C71—42no—
59C2×Q8×C71—42no—
60C2×Q8×C71—44no—
61C2×Q8×C71—48no—
62C2×Q8×C71—42no—
63C22×C4×C71—24no—
64C22×C4×C71—22no—
65C22×C4×C71—22no—
66C22×C4×C71—22no—
67C22×C4×C71—22no—
68C22×C4×C71—22no—
69C22×C4×C71—42no—
70C22×C4×C71—42no—
71C22×C4×C71—41no—
72C22×C4×C71—41no—
73C22×C4×C71—42no—
74C22×C4×C71—42no—
75C22×C4×C71—42no—
76C22×C4×C71—41no—
77C22×C4×C71—41no—
78C22×C4×C71—42no—
79C22×C4×C71C2×D412yes—
80C22×C4×C71C2×Q811yescharacteristic
81C22×C4×C71#4112yes—
82C22×C4×C71#4111yescharacteristic
83C23×C71—21no—
84C23×C71—21no—
85C23×C71—22no—
86C23×C71#12011yescharacteristic
87C23×C71C22⋊Q811yescharacteristic
88C23×C71#12411yescharacteristic
89C23×C71#13512yes—
90C23×C71#13712yes—
91Q8×C71—28no—
92Q8×C71—44no—
93Q8×C71—44no—
94Q8×C71—44no—
95C2×C4×C71—22no—
96C2×C4×C71—22no—
97C2×C4×C71—24no—
98C2×C4×C71—41no—
99C2×C4×C71—42no—
100C2×C4×C71—41no—
101C2×C4×C71—42no—
102C2×C4×C71—44no—
103C2×C4×C71—42no—
104C2×C4×C71—44no—
105C2×C4×C71—44no—
106C2×C4×C71—42no—
107C2×C4×C71—44no—
108C2×C4×C71—42no—
109C2×C4×C71—42no—
110C2×C4×C71—42no—
111C2×C4×C71—82no—
112C2×C4×C71—81no—
113C2×C4×C71—81no—
114C2×C4×C71—82no—
115C2×C4×C71C22⋊Q812yes—
116C2×C4×C71#12412yes—
117C2×Dic71—42no—
118C2×Dic71—42no—
119C2×Dic71—42no—
120C2×Dic71—42no—
121C2×Dic71—42no—
122C2×Dic71—42no—
123C71⋊C8—82no—
124C22×C71—22no—
125C22×C71—22no—
126C22×C71—22no—
127C22×C71—22no—
128C22×C71—21no—
129C22×C71—21no—
130C22×C71—21no—
131C22×C71—21no—
132C22×C71#39311yescharacteristic
133C22×C71#45212yes—
134C22×C71#45511yescharacteristic
135C22×C71#45611yescharacteristic
136C22×C71B(4,𝔽2)12yes—
137C22×C71#47912yes—
138C22×C71#48811yescharacteristic
139C22×C71#49111yescharacteristic
140C4×C71—22no—
141C4×C71—22no—
142C4×C71—44no—
143C4×C71—82no—
144C4×C71—82no—
145Dic71—82no—
146Dic71—82no—
147#10345—711nomaximal, Sylow
148C2×C71—21no—
149C2×C71—22no—
150C2×C71—21no—
151C2×C71#160312yes—
152C2×C71#160612yes—
153C2×C71#202611yescharacteristic
154C2×C71#202811yescharacteristic
155C2×C71#202911yescharacteristic
156#2646—711no—
157#1787—712no—
158#1446D7111yescharacteristic
159#1294—712no—
160#1280—711no—
161C71#1034511yescharacteristic, complemented, Sylow
162C23×Q8C2×D7111yescharacteristic
163C2×#103—1422no—
164C2×#103—1422no—
165C2×C22⋊C8—712no—
166#400—1422no—
167#398—1422no—
168#385—1422no—
169#380—712no—
170C2×#95—22no—
171C2×#95C2×D7111yescharacteristic
172C2×#95C2×D7111yescharacteristic
173C22×Q8—21no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—44no—
177C22×Q8Dic14212yes—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4C22⋊D7112yes—
181C23×C4C22×D7111yescharacteristic
182C22×C8—1422no—
183C2×C4⋊C4—712no—
184C2×C4⋊C4—712no—
185C2×C4⋊C4—1422no—
186C2×C4⋊C4—1422no—
187C2×C22⋊C4—1422no—
188#103—2844no—
189#103—2844no—
190C22⋊C8—1422no—
191C22⋊C8—1422no—
192#95—22no—
193#95—22no—
194#95—44no—
195#95—1422no—
196#95—1422no—
197#95—1422no—
198C24C2×C22⋊D7111yescharacteristic
199C2×Q8—24no—
200C2×Q8—24no—
201C2×Q8—24no—
202C2×Q8—44no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—44no—
206C2×Q8—48no—
207C2×Q8—42no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—1422no—
225C22×C4—1422no—
226C22×C4D4×D7112yes—
227C22×C4C2×Dic14211yescharacteristic
228C22×C4#1176586412yes—
229C22×C4#1176586511yescharacteristic
230C2×C8—2842no—
231C2×C8—2842no—
232C2×C8—2842no—
233C4⋊C4—2842no—
234C4⋊C4—2842no—
235C4⋊C4—5682no—
236C4⋊C4—5682no—
237C22⋊C4—2842no—
238C22⋊C4—2842no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23#1455609611yescharacteristic
243C23#1455610711yescharacteristic
244C23#1455611511yescharacteristic
245C23#1455614812yes—
246C23#1455615912yes—
247Q8—28no—
248Q8—44no—
249Q8—44no—
250Q8—44no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4—82no—
271C2×C4—2842no—
272C2×C4—2842no—
273C2×C4—2842no—
274C2×C4—2842no—
275C2×C4—2842no—
276C2×C4—2842no—
277C2×C4#1455610512yes—
278C2×C4#1455611412yes—
279C8—5682no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22#1798591111yescharacteristic
289C22#1798615912yes—
290C22#1798617611yescharacteristic
291C22#1798618111yescharacteristic
292C22#1798618512yes—
293C22#1798630612yes—
294C22#1798635311yescharacteristic
295C22#1798636711yescharacteristic
296C4—22no—
297C4—22no—
298C4—44no—
299C4—82no—
300C4—82no—
301C4—5682no—
302C4—5682no—
303C2—21no—
304C2—22no—
305C2—21no—
306C2#2258730112yes—
307C2#2258731812yes—
308C2#2258949111yescharacteristic
309C2#2258950111yescharacteristic
310C2#2258950411yescharacteristic
subgroupG/H
centerC23#14556096
commutator, cosocleC23×C4×C71C23
FrattiniC23×C4C22×D71
FittingC71×#1446C2
socleC23×C71#120
prime radicalC24×C71C2×D4
hypercenter#1446D71
hypocenterC71#10345
LCSG ▷ C23×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1446

Complements

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

NH ≅ G/Ntype
161. C71#10345semidirect

Maximal subgroups

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

subgroupindex
#225943102
#225882752
#225862192
#225861832
C71×#14462
#1034571