Finite Group Databasebeta

← #31901787

Subgroups — #31901787

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#22587021C211yesmaximal, characteristic
5#22586742C211yesmaximal, characteristic
6#22586190C211yesmaximal, characteristic
7C2×#332×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#17985748—22no—
20#17985747—21no—
21#17985746—21no—
22#17985729—22no—
23C2×C4⋊Q8×C71C2211yescharacteristic
24C2×C4⋊C8×C71C2211yescharacteristic
25C2×C4⋊C8×C71C2211yescharacteristic
26#332×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—81no—
87C2×Q8×C71—81no—
88C22×C4×C71—41no—
89C22×C4×C71C2×D411yescharacteristic
90C22×C4×C71#4111yescharacteristic
91C22×C4×C71#4111yescharacteristic
92C2×C8×C71—22no—
93C2×C8×C71—41no—
94C2×C8×C71—41no—
95C2×C8×C71—41no—
96C2×C8×C71—81no—
97C4⋊C4×C71—22no—
98C4⋊C4×C71—22no—
99C4⋊C4×C71—41no—
100C4⋊C4×C71—41no—
101C42×C71C2×D411yescharacteristic
102C42×C71C2×D411yescharacteristic
103C42×C71C2×D411yescharacteristic
104C42×C71#4111yescharacteristic
105C23×C71#12411yescharacteristic
106Q8×C71—82no—
107Q8×C71—81no—
108Q8×C71—81no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—41no—
112C2×C4×C71—42no—
113C2×C4×C71—82no—
114C2×C4×C71#12011yescharacteristic
115C2×C4×C71#12111yescharacteristic
116C2×C4×C71#12111yescharacteristic
117C2×C4×C71#12111yescharacteristic
118C2×C4×C71#12311yescharacteristic
119C2×C4×C71#12311yescharacteristic
120C2×C4×C71C2×SD1611yescharacteristic
121C2×C4×C71C2×Q1611yescharacteristic
122C2×C4×C71#13711yescharacteristic
123C2×C4×C71#13711yescharacteristic
124C2×Dic71—41no—
125C2×Dic71—41no—
126Dic142—41no—
127Dic142—41no—
128Dic142—81no—
129C8×C71—42no—
130C8×C71—81no—
131C71⋊C8—22no—
132C71⋊C8—22no—
133C71⋊C8—81no—
134C71⋊C8—81no—
135C22×C71—21no—
136C22×C71—21no—
137C22×C71#39411yescharacteristic
138C22×C71#48711yescharacteristic
139C22×C71#49111yescharacteristic
140C4×C71—21no—
141C4×C71—21no—
142C4×C71—21no—
143C4×C71—82no—
144C4×C71#44811yescharacteristic
145C4×C71#45111yescharacteristic
146C4×C71#46111yescharacteristic
147C4×C71#46211yescharacteristic
148C4×C71#47111yescharacteristic
149C4×C71#47211yescharacteristic
150Dic71—81no—
151Dic71—81no—
152#10556—711nomaximal, Sylow
153C2×C71—21no—
154C2×C71—21no—
155C2×C71#204311yescharacteristic
156C2×C71#204911yescharacteristic
157C2×C71#205111yescharacteristic
158#3165—711no—
159#1950—711no—
160#1550—711no—
161#1548—711no—
162C2×#332D7111yescharacteristic
163#1481—711no—
164#1283—711no—
165C71#1055611yescharacteristic, complemented, Sylow
166#533—711no—
167C2×C4⋊Q8C2×D7111yescharacteristic
168#501—1422no—
169#500—1421no—
170#498—1421no—
171#431—711no—
172C2×#108—1421no—
173#428—1421no—
174#428—1421no—
175#423—711no—
176C2×C4⋊C8C2×D7111yescharacteristic
177C2×C4⋊C8C2×D7111yescharacteristic
178C4×M16—711no—
179#341—1421no—
180#333—1421no—
181#333—1422no—
182#332—22no—
183#332—1421no—
184Q8⋊C8—1422no—
185C22×Q8—21no—
186C2×M16—711no—
187C2×M16—711no—
188C2×M16—1421no—
189C22×C8—21no—
190C22×C8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—711no—
195C4⋊Q8—711no—
196C4⋊Q8D28412yes—
197#125—1421no—
198C22⋊Q8—1421no—
199C22⋊Q8—1421no—
200C4×Q8—1421no—
201C4×Q8—1421no—
202#117—1421no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C42C22×D7111yescharacteristic
206#108—2842no—
207#107—1422no—
208#107—1422no—
209#106—1422no—
210#106—1422no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—1421no—
215C4⋊C8—1421no—
216C4⋊C8—1421no—
217C4⋊C8C22⋊D7112yes—
218C22⋊C8—1421no—
219#97—1421no—
220C4×C8—1421no—
221C4×C8—1421no—
222C2×Q8—22no—
223C2×Q8—41no—
224C2×Q8—81no—
225C2×Q8—81no—
226C2×Q8—1421no—
227C2×Q8—1421no—
228C22×C4—41no—
229C22×C4D4×D7111yescharacteristic
230C22×C4#1176586411yescharacteristic
231C22×C4#1176586511yescharacteristic
232M16—2841no—
233M16—2841no—
234M16—712no—
235M16—712no—
236M16—1421no—
237M16—1421no—
238C2×C8—22no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—81no—
243C2×C8—2841no—
244C2×C8—2841no—
245C2×C8—1421no—
246C2×C8—1421no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—2841no—
252C4⋊C4—2841no—
253C4⋊C4—1421no—
254C4⋊C4—1421no—
255C4⋊C4—1421no—
256C4⋊C4—1421no—
257C22⋊C4—2841no—
258C42—2841no—
259C42D4×D7111yescharacteristic
260C42C2×C22⋊D7111yescharacteristic
261C42C2×D28411yescharacteristic
262C42#1176586411yescharacteristic
263C23#1455611411yescharacteristic
264Q8—82no—
265Q8—81no—
266Q8—81no—
267Q8—2841no—
268Q8—2841no—
269Q8—5681no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—82no—
275C2×C4—2841no—
276C2×C4—2841no—
277C2×C4#1455609811yescharacteristic
278C2×C4#1455609911yescharacteristic
279C2×C4#1455610011yescharacteristic
280C2×C4#1455610211yescharacteristic
281C2×C4#1455611011yescharacteristic
282C2×C4#1455611211yescharacteristic
283C2×C4SD16×D7111yescharacteristic
284C2×C4Q16×D7111yescharacteristic
285C2×C4#1455615711yescharacteristic
286C2×C4#1455616011yescharacteristic
287C8—42no—
288C8—81no—
289C8—5681no—
290C8—5681no—
291C8—1422no—
292C8—1422no—
293C22—21no—
294C22—21no—
295C22#1798591511yescharacteristic
296C22#1798634811yescharacteristic
297C22#1798636611yescharacteristic
298C4—21no—
299C4—21no—
300C4—21no—
301C4—82no—
302C4—5681no—
303C4—5681no—
304C4#1798613911yescharacteristic
305C4#1798615611yescharacteristic
306C4#1798620211yescharacteristic
307C4#1798620711yescharacteristic
308C4#1798625911yescharacteristic
309C4#1798626611yescharacteristic
310C2—21no—
311C2—21no—
312C2#2258958511yescharacteristic
313C2#2258961911yescharacteristic
314C2#2258963311yescharacteristic
subgroupG/H
centerC22#17985915
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC2×#332×C71C2
socleC22×C71#394
prime radicalC23×C71#124
hypercenterC2×#332D71
hypocenterC71#10556
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
165. C71#10556semidirect

Maximal subgroups

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

subgroupindex
#225988822
#225891162
#225870282
#225870212
#225867422
#225861902
C2×#332×C712
#1055671