Finite Group Databasebeta

← #31901097

Subgroups — #31901097

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×#17986285C211yesmaximal, characteristic
2#22588333C211yesmaximal, characteristic
3#22587024C211yesmaximal, characteristic
4#22587022C211yesmaximal, characteristic
5#22586734C211yesmaximal, characteristic
6#22586237C211yesmaximal, characteristic
7C71×#1571C211yesmaximal, characteristic
8C2×#14556128C2211yescharacteristic
9C2×C4×Dic142C2211yescharacteristic
10#17986285—22no—
11#17986285—22no—
12#17986052C2211yescharacteristic
13C2×#14556060—21no—
14#17986023C2211yescharacteristic
15C2×#14556053—21no—
16C2×#14556046—21no—
17#17985767—21no—
18#17985747—22no—
19#17985746—22no—
20#17985729—22no—
21#431×C71C2211yescharacteristic
22#423×C71C2211yescharacteristic
23C2×C4⋊C8×C71C2211yescharacteristic
24#335×C71—22no—
25C22×Dic142—21no—
26C22×Dic142D411yescharacteristic
27C2×#11765849—21no—
28#14556128—21no—
29#14556128—21no—
30#14556128—21no—
31#14556128D412yes—
32C4×Dic142—21no—
33C4×Dic142—21no—
34C4×Dic142—21no—
35C4×Dic142—21no—
36C2×C4⋊Dic71—21no—
37C2×C4⋊Dic71—21no—
38C2×C4⋊Dic71D411yescharacteristic
39C2×Dic71⋊C4—21no—
40C2×C4×Dic71—21no—
41#14556060—24no—
42C4⋊(C71⋊C8)—21no—
43C4⋊(C71⋊C8)—21no—
44C4⋊(C71⋊C8)—21no—
45#14556053—24no—
46#14556046—42no—
47C22⋊(C71⋊C8)—21no—
48C4×C71⋊C8—21no—
49C2×M16×C71—21no—
50C22×C8×C71—21no—
51C2×C42×C71C2311yescharacteristic
52C4⋊C8×C71—21no—
53C4⋊C8×C71—21no—
54C4⋊C8×C71—21no—
55C4⋊C8×C71D412yes—
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×Dic142—22no—
65C2×Dic142—41no—
66C2×Dic142—82no—
67C2×Dic142Q1612yes—
68C22×Dic71—21no—
69C22×Dic71—21no—
70C22×Dic71—41no—
71#11765849—41no—
72#11765849—41no—
73C2×C71⋊C8—41no—
74C2×C71⋊C8—41no—
75C4⋊Dic71—22no—
76C4⋊Dic71—21no—
77C4⋊Dic71—22no—
78C4⋊Dic71—21no—
79C4⋊Dic71—41no—
80C4⋊Dic71—41no—
81Dic71⋊C4—41no—
82Dic71⋊C4—41no—
83C4×Dic71—41no—
84C4×Dic71—41no—
85C22×C4×C71C2×D411yescharacteristic
86C22×C4×C71#4111yescharacteristic
87C22×C4×C71#4111yescharacteristic
88M16×C71—41no—
89M16×C71—41no—
90C2×C8×C71—22no—
91C2×C8×C71—22no—
92C2×C8×C71—41no—
93C2×C8×C71—41no—
94C2×C8×C71—41no—
95C42×C71C2×D411yescharacteristic
96C42×C71C2×D411yescharacteristic
97C42×C71C2×D411yescharacteristic
98C42×C71#4111yescharacteristic
99C23×C71#12311yescharacteristic
100C2×C4×C71—22no—
101C2×C4×C71—41no—
102C2×C4×C71#12011yescharacteristic
103C2×C4×C71#12111yescharacteristic
104C2×C4×C71#12111yescharacteristic
105C2×C4×C71#12111yescharacteristic
106C2×C4×C71#12311yescharacteristic
107C2×C4×C71#12411yescharacteristic
108C2×C4×C71C2×Q1611yescharacteristic
109C2×C4×C71#13511yescharacteristic
110C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
111C2×C4×C71#13711yescharacteristic
112C2×Dic71—22no—
113C2×Dic71—41no—
114C2×Dic71—41no—
115C2×Dic71—41no—
116C2×Dic71—42no—
117C2×Dic71—82no—
118C2×Dic71—81no—
119Dic142—22no—
120Dic142—22no—
121Dic142—41no—
122Dic142—41no—
123Dic142—82no—
124Dic142—82no—
125C8×C71—42no—
126C8×C71—42no—
127C8×C71—81no—
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#44911yescharacteristic
139C4×C71#45111yescharacteristic
140C4×C71#46711yescharacteristic
141C4×C71#46811yescharacteristic
142C4×C71#47112yes—
143Dic71—42no—
144Dic71—81no—
145Dic71—82no—
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#1571D7111yescharacteristic
155#1549—711no—
156#1548—711no—
157#1479—711no—
158#1301—711no—
159C71#1045111yescharacteristic, complemented, Sylow
160C2×C4⋊Q8—711no—
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⋊C8C2×D7111yescharacteristic
170C2×#103—1421no—
171C2×#101—1421no—
172#343—1421no—
173#335—22no—
174#333—1422no—
175#332—1422no—
176Q8⋊C8—1422no—
177C22×Q8—711no—
178C22×Q8—1421no—
179C2×M16—21no—
180C2×M16—1421no—
181C22×C8—21no—
182C4⋊Q8—712no—
183C4⋊Q8—1421no—
184C4⋊Q8—1421no—
185C4⋊Q8—1421no—
186C4×Q8—1421no—
187C4×Q8—1421no—
188C4×Q8—1421no—
189C4×Q8—1421no—
190C2×C4⋊C4—711no—
191C2×C4⋊C4—1421no—
192C2×C4⋊C4—1421no—
193C2×C4⋊C4—1421no—
194C2×C42—1421no—
195C2×C42C22×D7111yescharacteristic
196#106—1424no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—1421no—
201C4⋊C8—1421no—
202C4⋊C8—1421no—
203C4⋊C8D28412yes—
204C4⋊C8C22⋊D7112yes—
205#103—1424no—
206#101—2842no—
207C22⋊C8—21no—
208C22⋊C8—1421no—
209C4×C8—21no—
210C4×C8—1421no—
211C2×Q8—2841no—
212C2×Q8—5682no—
213C2×Q8—712no—
214C2×Q8—1421no—
215C2×Q8—1421no—
216C2×Q8—1421no—
217C2×Q8—1422no—
218C2×Q8—1421no—
219C2×Q8—1421no—
220C22×C4—2841no—
221C22×C4—1421no—
222C22×C4—1421no—
223C22×C4D4×D7111yescharacteristic
224C22×C4#1176586511yescharacteristic
225C22×C4#1176586611yescharacteristic
226M16—41no—
227M16—41no—
228M16—2841no—
229M16—2841no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—2841no—
236C2×C8—2841no—
237C4⋊C4—2841no—
238C4⋊C4—2841no—
239C4⋊C4—2841no—
240C4⋊C4—2841no—
241C4⋊C4—1422no—
242C4⋊C4—1421no—
243C4⋊C4—1421no—
244C4⋊C4—1422no—
245C42—2841no—
246C42—2841no—
247C42D4×D7111yescharacteristic
248C42C2×C22⋊D7111yescharacteristic
249C42C2×D28411yescharacteristic
250C42#1176586611yescharacteristic
251C23#1455611311yescharacteristic
252Q8—2841no—
253Q8—2841no—
254Q8—5682no—
255Q8—5682no—
256Q8—1422no—
257Q8—1422no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—2841no—
261C2×C4—2841no—
262C2×C4—2841no—
263C2×C4—2842no—
264C2×C4—5682no—
265C2×C4—5681no—
266C2×C4—1422no—
267C2×C4#1455609611yescharacteristic
268C2×C4#1455610211yescharacteristic
269C2×C4#1455610311yescharacteristic
270C2×C4#1455610311yescharacteristic
271C2×C4#1455611311yescharacteristic
272C2×C4#1455611711yescharacteristic
273C2×C4Q16×D7111yescharacteristic
274C2×C4#1455614711yescharacteristic
275C2×C4#1455615211yescharacteristic
276C2×C4#1455616011yescharacteristic
277C8—42no—
278C8—42no—
279C8—81no—
280C8—5681no—
281C8—5681no—
282C22—21no—
283C22—21no—
284C22#1798591611yescharacteristic
285C22#1798633611yescharacteristic
286C22#1798634211yescharacteristic
287C4—21no—
288C4—21no—
289C4—21no—
290C4—2842no—
291C4—5682no—
292C4—5681no—
293C4#1798614111yescharacteristic
294C4#1798615311yescharacteristic
295C4#1798624111yescharacteristic
296C4#1798624611yescharacteristic
297C4#1798626212yes—
298C2—21no—
299C2—21no—
300C2#2258954511yescharacteristic
301C2#2258960311yescharacteristic
302C2#2258962211yescharacteristic
subgroupG/H
centerC22#17985916
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1571C2
socleC22×C71#394
prime radicalC23×C71#123
hypercenter#1571D71
hypocenterC71#10451
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1571

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×#179862852
#225883332
#225870242
#225870222
#225867342
#225862372
C71×#15712
#1045171