Finite Group Databasebeta

← #31901780

Subgroups — #31901780

Order 18176.

364 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 99 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#22598882C211yesmaximal, characteristic
2#22589116C211yesmaximal, characteristic
3#22587028C211yesmaximal, characteristic
4#22587025C211yesmaximal, characteristic
5#22586742C211yesmaximal, characteristic
6#22586190C211yesmaximal, characteristic
7C2×#333×C71C211yesmaximal, characteristic
8#17986556C2211yescharacteristic
9#17986405—22no—
10#17986402—21no—
11#17986395—21no—
12#17986052C2211yescharacteristic
13C2×#14556064—21no—
14#17986039—21no—
15#17986039—21no—
16#17986023C2211yescharacteristic
17C4×#11765849C2211yescharacteristic
18#17985764—21no—
19#17985747—21no—
20#17985746—21no—
21#17985745—22no—
22#17985729—22no—
23C2×C4⋊Q8×C71C2211yescharacteristic
24C2×C4⋊C8×C71C2211yescharacteristic
25C2×C4⋊C8×C71C2211yescharacteristic
26#333×C71—22no—
27C2×#11765849—21no—
28C2×#11765849D411yescharacteristic
29C2×#11765849D411yescharacteristic
30#14556128D411yescharacteristic
31#14556128D411yescharacteristic
32#14556120—21no—
33#14556107—21no—
34#14556107—21no—
35C4×Dic142—21no—
36C4×Dic142—21no—
37#14556087—21no—
38#14556064—42no—
39#14556061—22no—
40#14556061—22no—
41#14556059—22no—
42#14556059—22no—
43C4⋊(C71⋊C8)—21no—
44C4⋊(C71⋊C8)—21no—
45C4⋊(C71⋊C8)—21no—
46C22⋊(C71⋊C8)—21no—
47#14556039—21no—
48C4×C71⋊C8—21no—
49C4×C71⋊C8—21no—
50C22×Q8×C71—21no—
51C22×C8×C71—21no—
52C22×C8×C71—21no—
53C4⋊Q8×C71—21no—
54C4⋊Q8×C71—21no—
55C4⋊Q8×C71—21no—
56C4⋊Q8×C71D412yes—
57C2×C4⋊C4×C71—21no—
58C2×C4⋊C4×C71—21no—
59C2×C42×C71C2311yescharacteristic
60C4⋊C8×C71—21no—
61C4⋊C8×C71—21no—
62C4⋊C8×C71—21no—
63C4⋊C8×C71D412yes—
64C2×Dic142—21no—
65C2×Dic142—21no—
66#11765849—21no—
67#11765849—21no—
68#11765849—41no—
69#11765849—41no—
70#11765849SD1612yes—
71#11765849Q1612yes—
72C2×C71⋊C8—21no—
73C2×C71⋊C8—21no—
74C2×C71⋊C8—41no—
75C2×C71⋊C8—41no—
76C4⋊Dic71—21no—
77C4⋊Dic71—21no—
78C4⋊Dic71—21no—
79C4⋊Dic71—21no—
80Dic71⋊C4—41no—
81Dic71⋊C4—41no—
82C22⋊Dic71—41no—
83C4×Dic71—41no—
84C2×Q8×C71—22no—
85C2×Q8×C71—41no—
86C2×Q8×C71—82no—
87C22×C4×C71—41no—
88C22×C4×C71C2×D411yescharacteristic
89C22×C4×C71#4111yescharacteristic
90C22×C4×C71#4111yescharacteristic
91C2×C8×C71—22no—
92C2×C8×C71—41no—
93C2×C8×C71—41no—
94C2×C8×C71—41no—
95C2×C8×C71—81no—
96C4⋊C4×C71—22no—
97C4⋊C4×C71—22no—
98C4⋊C4×C71—41no—
99C4⋊C4×C71—41no—
100C42×C71C2×D411yescharacteristic
101C42×C71C2×D411yescharacteristic
102C42×C71C2×D411yescharacteristic
103C42×C71#4111yescharacteristic
104C23×C71#12411yescharacteristic
105Q8×C71—82no—
106Q8×C71—82no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—41no—
110C2×C4×C71—42no—
111C2×C4×C71—82no—
112C2×C4×C71#12011yescharacteristic
113C2×C4×C71#12111yescharacteristic
114C2×C4×C71#12111yescharacteristic
115C2×C4×C71#12111yescharacteristic
116C2×C4×C71#12311yescharacteristic
117C2×C4×C71#12311yescharacteristic
118C2×C4×C71C2×SD1611yescharacteristic
119C2×C4×C71C2×Q1611yescharacteristic
120C2×C4×C71#13711yescharacteristic
121C2×C4×C71#13711yescharacteristic
122C2×Dic71—41no—
123C2×Dic71—41no—
124Dic142—41no—
125Dic142—41no—
126Dic142—81no—
127C8×C71—42no—
128C8×C71—81no—
129C71⋊C8—22no—
130C71⋊C8—22no—
131C71⋊C8—81no—
132C71⋊C8—81no—
133C22×C71—21no—
134C22×C71—21no—
135C22×C71#39411yescharacteristic
136C22×C71#48711yescharacteristic
137C22×C71#49111yescharacteristic
138C4×C71—21no—
139C4×C71—21no—
140C4×C71—21no—
141C4×C71—82no—
142C4×C71#44811yescharacteristic
143C4×C71#45111yescharacteristic
144C4×C71#46111yescharacteristic
145C4×C71#46211yescharacteristic
146C4×C71#47111yescharacteristic
147C4×C71#47311yescharacteristic
148Dic71—81no—
149Dic71—81no—
150#10555—711nomaximal, Sylow
151C2×C71—21no—
152C2×C71—21no—
153C2×C71#204311yescharacteristic
154C2×C71#204811yescharacteristic
155C2×C71#205211yescharacteristic
156#3165—711no—
157#1950—711no—
158#1550—711no—
159#1549—711no—
160C2×#333D7111yescharacteristic
161#1481—711no—
162#1283—711no—
163C71#1055511yescharacteristic, complemented, Sylow
164#533—711no—
165C2×C4⋊Q8C2×D7111yescharacteristic
166#501—1422no—
167#500—1421no—
168#498—1421no—
169#431—711no—
170C2×#108—1421no—
171#428—1421no—
172#428—1421no—
173#423—711no—
174C2×C4⋊C8C2×D7111yescharacteristic
175C2×C4⋊C8C2×D7111yescharacteristic
176C4×M16—711no—
177#341—1421no—
178#333—22no—
179#333—1421no—
180#332—1421no—
181#332—1422no—
182Q8⋊C8—1422no—
183C22×Q8—21no—
184C2×M16—711no—
185C2×M16—711no—
186C2×M16—1421no—
187C22×C8—21no—
188C22×C8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—711no—
193C4⋊Q8—711no—
194C4⋊Q8D28412yes—
195#125—1421no—
196C22⋊Q8—1421no—
197C22⋊Q8—1421no—
198C4×Q8—1421no—
199C4×Q8—1421no—
200#117—1421no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C42C22×D7111yescharacteristic
204#108—2842no—
205#107—1422no—
206#107—1422no—
207#106—1422no—
208#106—1422no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—1421no—
213C4⋊C8—1421no—
214C4⋊C8—1421no—
215C4⋊C8C22⋊D7112yes—
216C22⋊C8—1421no—
217#97—1421no—
218C4×C8—1421no—
219C4×C8—1421no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×Q8—82no—
223C2×Q8—1421no—
224C2×Q8—1421no—
225C22×C4—41no—
226C22×C4D4×D7111yescharacteristic
227C22×C4#1176586411yescharacteristic
228C22×C4#1176586511yescharacteristic
229M16—2841no—
230M16—2841no—
231M16—712no—
232M16—712no—
233M16—1421no—
234M16—1421no—
235C2×C8—22no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—81no—
240C2×C8—2841no—
241C2×C8—2841no—
242C2×C8—1421no—
243C2×C8—1421no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—2841no—
249C4⋊C4—2841no—
250C4⋊C4—1421no—
251C4⋊C4—1421no—
252C4⋊C4—1421no—
253C4⋊C4—1421no—
254C22⋊C4—2841no—
255C42—2841no—
256C42D4×D7111yescharacteristic
257C42C2×C22⋊D7111yescharacteristic
258C42C2×D28411yescharacteristic
259C42#1176586411yescharacteristic
260C23#1455611411yescharacteristic
261Q8—82no—
262Q8—82no—
263Q8—2841no—
264Q8—2841no—
265Q8—5681no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—82no—
271C2×C4—2841no—
272C2×C4—2841no—
273C2×C4#1455609811yescharacteristic
274C2×C4#1455609911yescharacteristic
275C2×C4#1455610011yescharacteristic
276C2×C4#1455610211yescharacteristic
277C2×C4#1455611011yescharacteristic
278C2×C4#1455611211yescharacteristic
279C2×C4SD16×D7111yescharacteristic
280C2×C4Q16×D7111yescharacteristic
281C2×C4#1455615711yescharacteristic
282C2×C4#1455616011yescharacteristic
283C8—42no—
284C8—81no—
285C8—5681no—
286C8—5681no—
287C8—1422no—
288C8—1422no—
289C22—21no—
290C22—21no—
291C22#1798591511yescharacteristic
292C22#1798634811yescharacteristic
293C22#1798636611yescharacteristic
294C4—21no—
295C4—21no—
296C4—21no—
297C4—82no—
298C4—5681no—
299C4—5681no—
300C4#1798613911yescharacteristic
301C4#1798615611yescharacteristic
302C4#1798620211yescharacteristic
303C4#1798620711yescharacteristic
304C4#1798625911yescharacteristic
305C4#1798627111yescharacteristic
306C2—21no—
307C2—21no—
308C2#2258958511yescharacteristic
309C2#2258961211yescharacteristic
310C2#2258964011yescharacteristic
subgroupG/H
centerC22#17985915
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC2×#333×C71C2
socleC22×C71#394
prime radicalC23×C71#124
hypercenterC2×#333D71
hypocenterC71#10555
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
163. C71#10555semidirect

Maximal subgroups

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

subgroupindex
#225988822
#225891162
#225870282
#225870252
#225867422
#225861902
C2×#333×C712
#1055571