Finite Group Databasebeta

← #31901094

Subgroups — #31901094

Order 18176.

376 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 97 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×#17986281C211yesmaximal, characteristic
2#22588333C211yesmaximal, characteristic
3#22587115C211yesmaximal, characteristic
4#22587021C211yesmaximal, characteristic
5#22586734C211yesmaximal, characteristic
6#22586235C211yesmaximal, characteristic
7C71×#1549C211yesmaximal, characteristic
8C2×C4×Dic142C2211yescharacteristic
9#17986281—22no—
10#17986281—22no—
11#17986052C2211yescharacteristic
12C2×#14556059—21no—
13#17986023C2211yescharacteristic
14C2×C4⋊(C71⋊C8)C2211yescharacteristic
15C2×#14556051—21no—
16C2×#14556046—21no—
17#17985767—21no—
18#17985752—22no—
19#17985748—22no—
20#17985729—22no—
21C2×C4⋊Q8×C71C2211yescharacteristic
22#431×C71C2211yescharacteristic
23#423×C71C2211yescharacteristic
24#332×C71—22no—
25C22×Dic142D411yescharacteristic
26C2×#11765849—21no—
27C22×C71⋊C8—21no—
28C4×Dic142—21no—
29C4×Dic142—21no—
30C4×Dic142—21no—
31C4×Dic142—21no—
32C2×C4⋊Dic71D411yescharacteristic
33C2×Dic71⋊C4—21no—
34C2×C4×Dic71—21no—
35#14556059—24no—
36C4⋊(C71⋊C8)—21no—
37C4⋊(C71⋊C8)—21no—
38C4⋊(C71⋊C8)—21no—
39C4⋊(C71⋊C8)—21no—
40C4⋊(C71⋊C8)D412yes—
41#14556051—24no—
42#14556046—42no—
43C22⋊(C71⋊C8)—21no—
44C4×C71⋊C8—21no—
45C22×Q8×C71—21no—
46C2×M16×C71—21no—
47C4⋊Q8×C71—21no—
48C4⋊Q8×C71—21no—
49C4⋊Q8×C71—21no—
50C4⋊Q8×C71D412yes—
51C2×C4⋊C4×C71—21no—
52C2×C4⋊C4×C71—21no—
53C2×C42×C71C2311yescharacteristic
54C4⋊C8×C71—21no—
55C4⋊C8×C71—21no—
56C4⋊C8×C71D412yes—
57C22⋊C8×C71—21no—
58C4×C8×C71—21no—
59C2×Dic142—21no—
60C2×Dic142—21no—
61C2×Dic142—21no—
62C2×Dic142—21no—
63C2×Dic142—21no—
64C2×Dic142Q1612yes—
65C22×Dic71—21no—
66C22×Dic71—21no—
67#11765849—41no—
68#11765849—41no—
69C2×C71⋊C8—22no—
70C2×C71⋊C8—41no—
71C2×C71⋊C8—41no—
72C2×C71⋊C8—41no—
73C2×C71⋊C8—41no—
74C4⋊Dic71—21no—
75C4⋊Dic71—21no—
76Dic71⋊C4—41no—
77Dic71⋊C4—41no—
78C4×Dic71—41no—
79C4×Dic71—41no—
80C2×Q8×C71—22no—
81C2×Q8×C71—41no—
82C2×Q8×C71—82no—
83C22×C4×C71—41no—
84C22×C4×C71C2×D411yescharacteristic
85C22×C4×C71#4111yescharacteristic
86C22×C4×C71#4111yescharacteristic
87M16×C71—41no—
88M16×C71—41no—
89C2×C8×C71—22no—
90C2×C8×C71—41no—
91C4⋊C4×C71—22no—
92C4⋊C4×C71—22no—
93C4⋊C4×C71—41no—
94C4⋊C4×C71—41no—
95C42×C71C2×D411yescharacteristic
96C42×C71C2×D411yescharacteristic
97C42×C71C2×D411yescharacteristic
98C42×C71#4111yescharacteristic
99C23×C71#12311yescharacteristic
100Q8×C71—82no—
101Q8×C71—82no—
102C2×C4×C71—22no—
103C2×C4×C71—41no—
104C2×C4×C71—42no—
105C2×C4×C71—82no—
106C2×C4×C71#12011yescharacteristic
107C2×C4×C71#12111yescharacteristic
108C2×C4×C71#12111yescharacteristic
109C2×C4×C71#12111yescharacteristic
110C2×C4×C71#12311yescharacteristic
111C2×C4×C71#12411yescharacteristic
112C2×C4×C71C2×Q1611yescharacteristic
113C2×C4×C71#13511yescharacteristic
114C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
115C2×C4×C71#13711yescharacteristic
116C2×Dic71—22no—
117C2×Dic71—41no—
118C2×Dic71—41no—
119C2×Dic71—41no—
120C2×Dic71—81no—
121Dic142—22no—
122Dic142—22no—
123Dic142—41no—
124Dic142—41no—
125C8×C71—42no—
126C8×C71—81no—
127C71⋊C8—42no—
128C71⋊C8—81no—
129C71⋊C8—81no—
130C22×C71—21no—
131C22×C71—21no—
132C22×C71#39411yescharacteristic
133C22×C71#48411yescharacteristic
134C22×C71#48511yescharacteristic
135C4×C71—21no—
136C4×C71—21no—
137C4×C71—21no—
138C4×C71—82no—
139C4×C71#44911yescharacteristic
140C4×C71#45111yescharacteristic
141C4×C71#46711yescharacteristic
142C4×C71#46811yescharacteristic
143C4×C71#47112yes—
144Dic71—42no—
145Dic71—81no—
146#10451—711nomaximal, Sylow
147C2×C71—21no—
148C2×C71—21no—
149C2×C71#203611yescharacteristic
150C2×C71#204611yescharacteristic
151C2×C71#204911yescharacteristic
152C2×#475—711no—
153#1801—711no—
154#1571—711no—
155#1549D7111yescharacteristic
156#1548—711no—
157#1479—711no—
158#1301—711no—
159C71#1045111yescharacteristic, complemented, Sylow
160C2×C4⋊Q8C2×D7111yescharacteristic
161C2×C4×Q8—711no—
162#475—1422no—
163#475—1422no—
164#431—711no—
165#431C2×D7111yescharacteristic
166C2×#106—1421no—
167#423—711no—
168#423C2×D7111yescharacteristic
169C2×C4⋊C8—711no—
170C2×#103—1421no—
171C2×#101—1421no—
172#343—1421no—
173#335—1422no—
174#333—1422no—
175#332—22no—
176Q8⋊C8—1422no—
177C22×Q8—21no—
178C22×Q8—711no—
179C2×M16—21no—
180C2×M16—1421no—
181C22×C8—1421no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8D28412yes—
186C4×Q8—1421no—
187C4×Q8—1421no—
188C4×Q8—1421no—
189C4×Q8—1421no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—711no—
193C2×C4⋊C4—1421no—
194C2×C42—1421no—
195C2×C42C22×D7111yescharacteristic
196#106—1424no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—712no—
200C4⋊C8—1421no—
201C4⋊C8—1421no—
202C4⋊C8—1421no—
203C4⋊C8—1421no—
204C4⋊C8C22⋊D7112yes—
205#103—1424no—
206#101—2842no—
207C22⋊C8—21no—
208C22⋊C8—1421no—
209C4×C8—21no—
210C4×C8—1421no—
211C2×Q8—22no—
212C2×Q8—41no—
213C2×Q8—82no—
214C2×Q8—712no—
215C2×Q8—1421no—
216C2×Q8—1421no—
217C2×Q8—1421no—
218C2×Q8—1421no—
219C2×Q8—1421no—
220C22×C4—41no—
221C22×C4—1421no—
222C22×C4—1421no—
223C22×C4D4×D7111yescharacteristic
224C22×C4#1176586411yescharacteristic
225C22×C4#1176586511yescharacteristic
226M16—41no—
227M16—41no—
228M16—2841no—
229M16—2841no—
230C2×C8—22no—
231C2×C8—41no—
232C2×C8—2841no—
233C2×C8—2841no—
234C2×C8—2841no—
235C2×C8—2841no—
236C2×C8—1422no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—2841no—
242C4⋊C4—2841no—
243C4⋊C4—1421no—
244C4⋊C4—1421no—
245C42—2841no—
246C42—2841no—
247C42D4×D7111yescharacteristic
248C42C2×C22⋊D7111yescharacteristic
249C42C2×D28411yescharacteristic
250C42#1176586411yescharacteristic
251C23#1455611011yescharacteristic
252Q8—82no—
253Q8—82no—
254Q8—2841no—
255Q8—2841no—
256Q8—1422no—
257Q8—1422no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—82no—
262C2×C4—2841no—
263C2×C4—2841no—
264C2×C4—2841no—
265C2×C4—5681no—
266C2×C4—1422no—
267C2×C4#1455609811yescharacteristic
268C2×C4#1455609911yescharacteristic
269C2×C4#1455610011yescharacteristic
270C2×C4#1455610211yescharacteristic
271C2×C4#1455611211yescharacteristic
272C2×C4#1455611411yescharacteristic
273C2×C4Q16×D7111yescharacteristic
274C2×C4#1455615111yescharacteristic
275C2×C4#1455615511yescharacteristic
276C2×C4#1455615711yescharacteristic
277C8—42no—
278C8—81no—
279C8—2842no—
280C8—5681no—
281C8—5681no—
282C22—21no—
283C22—21no—
284C22#1798591511yescharacteristic
285C22#1798633311yescharacteristic
286C22#1798633911yescharacteristic
287C4—21no—
288C4—21no—
289C4—21no—
290C4—82no—
291C4—2842no—
292C4—5681no—
293C4#1798614711yescharacteristic
294C4#1798615611yescharacteristic
295C4#1798623711yescharacteristic
296C4#1798624211yescharacteristic
297C4#1798626012yes—
298C2—21no—
299C2—21no—
300C2#2258954211yescharacteristic
301C2#2258960211yescharacteristic
302C2#2258961911yescharacteristic
subgroupG/H
centerC22#17985915
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1549C2
socleC22×C71#394
prime radicalC23×C71#123
hypercenter#1549D71
hypocenterC71#10451
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
159. C71#10451semidirect

Maximal subgroups

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

subgroupindex
C2×#179862812
#225883332
#225871152
#225870212
#225867342
#225862352
C71×#15492
#1045171