Finite Group Databasebeta

← #31898047

Subgroups — #31898047

Order 18176.

632 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C2×#17986329C212yesmaximal
2#22594454C211yesmaximal, characteristic
3C2×#17985735C212yesmaximal
4#22586183C211yesmaximal, characteristic
5C71×#1445C211yesmaximal, characteristic
6C2×#14556107C2212yes—
7#17986329—24no—
8#17986329—24no—
9C2×#14556059—22no—
10C2×#14556051—22no—
11C2×C22⋊(C71⋊C8)C2212yes—
12#17985931—22no—
13#17985920—22no—
14#17985859—22no—
15#17985735—24no—
16C22×C4⋊C4×C71C2211yescharacteristic
17C2×#95×C71—22no—
18C2×#95×C71C2211yescharacteristic
19C2×#95×C71C2211yescharacteristic
20C22×Dic142D412yes—
21C22×C71⋊C8—22no—
22#14556107—22no—
23#14556107—22no—
24#14556107—22no—
25#14556107—22no—
26C2×C4⋊Dic71—22no—
27C2×C4⋊Dic71—22no—
28C2×C4⋊Dic71D412yes—
29C2×Dic71⋊C4—22no—
30C2×C22⋊Dic71—22no—
31#14556059—28no—
32#14556051—28no—
33C22⋊(C71⋊C8)—22no—
34C22⋊(C71⋊C8)D414yes—
35C71⋊#95—24no—
36C23×C4×C71—21no—
37C23×C4×C71—21no—
38C23×C4×C71D412yes—
39C23×C4×C71C2311yescharacteristic
40C2×C4⋊C4×C71—22no—
41C2×C4⋊C4×C71—22no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71D412yes—
44#95×C71—22no—
45#95×C71—22no—
46#95×C71—44no—
47C2×Dic142—22no—
48C2×Dic142—22no—
49C2×Dic142—44no—
50C22×Dic71—22no—
51C22×Dic71—22no—
52C2×C71⋊C8—24no—
53C2×C71⋊C8—44no—
54C4⋊Dic71—22no—
55C4⋊Dic71—22no—
56C4⋊Dic71—82no—
57C4⋊Dic71—82no—
58Dic71⋊C4—42no—
59Dic71⋊C4—42no—
60C22⋊Dic71—42no—
61C22⋊Dic71—42no—
62C24×C71C2×D411yescharacteristic
63C22×C4×C71—22no—
64C22×C4×C71—22no—
65C22×C4×C71—22no—
66C22×C4×C71—22no—
67C22×C4×C71—42no—
68C22×C4×C71—42no—
69C22×C4×C71—42no—
70C22×C4×C71—41no—
71C22×C4×C71—42no—
72C22×C4×C71—41no—
73C22×C4×C71—41no—
74C22×C4×C71—41no—
75C22×C4×C71—42no—
76C22×C4×C71—42no—
77C22×C4×C71—42no—
78C22×C4×C71SD1614yes—
79C22×C4×C71C2×D412yes—
80C22×C4×C71C2×D412yes—
81C22×C4×C71C2×D411yescharacteristic
82C22×C4×C71#4111yescharacteristic
83C4⋊C4×C71—24no—
84C4⋊C4×C71—42no—
85C23×C71—21no—
86C23×C71—21no—
87C23×C71—22no—
88C23×C71#12011yescharacteristic
89C23×C71#12111yescharacteristic
90C23×C71#12711yescharacteristic
91C23×C71C2×SD1612yes—
92C23×C71#13712yes—
93C2×C4×C71—24no—
94C2×C4×C71—24no—
95C2×C4×C71—42no—
96C2×C4×C71—41no—
97C2×C4×C71—41no—
98C2×C4×C71—42no—
99C2×C4×C71—42no—
100C2×C4×C71—44no—
101C2×C4×C71—44no—
102C2×C4×C71—42no—
103C2×C4×C71—42no—
104C2×C4×C71—44no—
105C2×C4×C71—82no—
106C2×C4×C71—82no—
107C2×C4×C71—81no—
108C2×C4×C71—82no—
109C2×C4×C71—81no—
110C2×C4×C71—81no—
111C2×C4×C71—81no—
112C2×C4×C71#12012yes—
113C2×C4×C71#12112yes—
114C2×Dic71—42no—
115C2×Dic71—42no—
116C2×Dic71—42no—
117C2×Dic71—42no—
118C2×Dic71—44no—
119Dic142—44no—
120Dic142—82no—
121C71⋊C8—44no—
122C22×C71—22no—
123C22×C71—22no—
124C22×C71—22no—
125C22×C71—22no—
126C22×C71—21no—
127C22×C71—21no—
128C22×C71—21no—
129C22×C71—21no—
130C22×C71#39211yescharacteristic
131C22×C71#44812yes—
132C22×C71#45511yescharacteristic
133C22×C71#45611yescharacteristic
134C22×C71#45812yes—
135C22×C71#46112yes—
136C22×C71#49211yescharacteristic
137C22×C71#49711yescharacteristic
138C4×C71—22no—
139C4×C71—22no—
140C4×C71—44no—
141C4×C71—82no—
142C4×C71—82no—
143Dic71—82no—
144Dic71—82no—
145#9973—711nomaximal, Sylow
146C2×C71—21no—
147C2×C71—22no—
148C2×C71—21no—
149C2×C71#160812yes—
150C2×C71#161112yes—
151C2×C71#200711yescharacteristic
152C2×C71#201311yescharacteristic
153C2×C71#202211yescharacteristic
154C2×#483—712no—
155#2659—711no—
156C2×#328—712no—
157#1445D7111yescharacteristic
158#1280—711no—
159C71#997311yescharacteristic, complemented, Sylow
160C2×C22⋊Q8—712no—
161C22×C4⋊C4C2×D7111yescharacteristic
162#483—1424no—
163#483—1424no—
164C2×#106—1422no—
165C2×#103—1422no—
166C2×C22⋊C8—712no—
167#398—1422no—
168#395—1422no—
169#380—1422no—
170C2×#95—22no—
171C2×#95C2×D7111yescharacteristic
172C2×#95C2×D7111yescharacteristic
173#328—1424no—
174C22×Q8—712no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4C22⋊D7112yes—
178C23×C4C22×D7111yescharacteristic
179C22×C8—1422no—
180C22⋊Q8—1422no—
181C22⋊Q8—1422no—
182C22⋊Q8—1422no—
183C22⋊Q8—1422no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—712no—
188C2×C4⋊C4—1422no—
189C2×C4⋊C4—1422no—
190C2×C4⋊C4—1422no—
191C2×C4⋊C4D28412yes—
192C2×C22⋊C4—1422no—
193#106—1428no—
194#103—1428no—
195C22⋊C8—714no—
196C22⋊C8—1422no—
197#95—22no—
198#95—22no—
199#95—44no—
200#95—1424no—
201C24C2×C22⋊D7111yescharacteristic
202C2×Q8—2844no—
203C2×Q8—1422no—
204C2×Q8—1422no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—1422no—
221C22×C4—1422no—
222C22×C4#1176585314yes—
223C22×C4D4×D7112yes—
224C22×C4D4×D7112yes—
225C22×C4C2×D28411yescharacteristic
226C22×C4#1176586511yescharacteristic
227C2×C8—2844no—
228C2×C8—1424no—
229C4⋊C4—24no—
230C4⋊C4—42no—
231C4⋊C4—2842no—
232C4⋊C4—2842no—
233C4⋊C4—5682no—
234C4⋊C4—5682no—
235C4⋊C4—1422no—
236C4⋊C4—1422no—
237C22⋊C4—2842no—
238C22⋊C4—2842no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23#1455609611yescharacteristic
243C23#1455610211yescharacteristic
244C23#1455612411yescharacteristic
245C23C2×#1176585312yes—
246C23#1455615812yes—
247Q8—2844no—
248Q8—5682no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—82no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—82no—
267C2×C4—81no—
268C2×C4—2842no—
269C2×C4—2842no—
270C2×C4—2842no—
271C2×C4—2842no—
272C2×C4—2844no—
273C2×C4#1455609812yes—
274C2×C4#1455610112yes—
275C8—2844no—
276C22—21no—
277C22—21no—
278C22—22no—
279C22—22no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—22no—
284C22#1798590811yescharacteristic
285C22#1798613712yes—
286C22#1798617611yescharacteristic
287C22#1798618111yescharacteristic
288C22#1798618812yes—
289C22#1798620412yes—
290C22#1798637111yescharacteristic
291C22#1798639411yescharacteristic
292C4—22no—
293C4—22no—
294C4—44no—
295C4—82no—
296C4—82no—
297C4—5682no—
298C4—5682no—
299C2—21no—
300C2—21no—
301C2—22no—
302C2#2258733012yes—
303C2#2258734712yes—
304C2#2258938611yescharacteristic
305C2#2258942211yescharacteristic
306C2#2258947111yescharacteristic
subgroupG/H
centerC23#14556096
commutator, cosocleC23×C4×C71C23
FrattiniC23×C4C22×D71
FittingC71×#1445C2
socleC23×C71#120
prime radicalC24×C71C2×D4
hypercenter#1445D71
hypocenterC71#9973
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
159. C71#9973semidirect

Maximal subgroups

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

subgroupindex
C2×#179863292
#225944542
C2×#179857352
#225861832
C71×#14452
#997371