Finite Group Databasebeta

← #31891127

Subgroups — #31891127

Order 18176.

342 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588491C211yesmaximal, characteristic
2#22588456C211yesmaximal, characteristic
3#22588085C211yesmaximal, characteristic
4#22586175C211yesmaximal, characteristic
5#22586174C211yesmaximal, characteristic
6#22586174C211yesmaximal, characteristic
7C71×#1277C211yesmaximal, characteristic
8C2×C4⋊(C71⋊C8)C2211yescharacteristic
9C2×C4⋊(C71⋊C8)C2211yescharacteristic
10C2×Dic71⋊C8—21no—
11C2×Dic71⋊C8—21no—
12C2×C4×C71⋊C8C2211yescharacteristic
13C42×Dic71C2211yescharacteristic
14#17985753—21no—
15#17985753—21no—
16#17985753C411yescharacteristic
17#17985753C411yescharacteristic
18#17985753C411yescharacteristic
19#17985753C411yescharacteristic
20C2×C4⋊C8×C71C2211yescharacteristic
21C2×C4⋊C8×C71C2211yescharacteristic
22C2×C4×C8×C71C2211yescharacteristic
23C22×C71⋊C8—21no—
24C22×C71⋊C8—21no—
25C22×C71⋊C8C2×C411yescharacteristic
26C22×C71⋊C8C2×C411yescharacteristic
27C2×C4×Dic71C2×C411yescharacteristic
28C2×C4×Dic71C2×C411yescharacteristic
29C2×C4×Dic71D411yescharacteristic
30C2×C4×Dic71D411yescharacteristic
31C2×C4×Dic71Q811yescharacteristic
32C2×C4×Dic71Q811yescharacteristic
33C4⋊(C71⋊C8)—21no—
34C4⋊(C71⋊C8)—21no—
35C4⋊(C71⋊C8)—21no—
36C4⋊(C71⋊C8)—21no—
37Dic71⋊C8—42no—
38Dic71⋊C8—42no—
39C4×C71⋊C8—21no—
40C4×C71⋊C8—21no—
41C22×C8×C71—21no—
42C22×C8×C71—21no—
43C22×C8×C71C2×C411yescharacteristic
44C22×C8×C71C2×C411yescharacteristic
45C2×C42×C71C2311yescharacteristic
46C4⋊C8×C71—21no—
47C4⋊C8×C71—21no—
48C4⋊C8×C71—21no—
49C4⋊C8×C71—21no—
50C4×C8×C71—21no—
51C4×C8×C71—21no—
52C22×Dic71C4⋊C411yescharacteristic
53C22×Dic71C4⋊C411yescharacteristic
54C22×Dic71C4⋊C411yescharacteristic
55C22×Dic71C4⋊C411yescharacteristic
56C2×C71⋊C8—21no—
57C2×C71⋊C8—21no—
58C2×C71⋊C8—21no—
59C2×C71⋊C8—21no—
60C2×C71⋊C8—21no—
61C2×C71⋊C8—21no—
62C2×C71⋊C8—41no—
63C2×C71⋊C8—41no—
64C2×C71⋊C8—41no—
65C2×C71⋊C8—41no—
66C2×C71⋊C8—41no—
67C2×C71⋊C8—41no—
68C4×Dic71—21no—
69C4×Dic71—21no—
70C4×Dic71—21no—
71C4×Dic71—21no—
72C4×Dic71—21no—
73C4×Dic71—21no—
74C4×Dic71—21no—
75C4×Dic71—21no—
76C4×Dic71—22no—
77C4×Dic71—22no—
78C22×C4×C71C22×C411yescharacteristic
79C22×C4×C71C2×D411yescharacteristic
80C22×C4×C71C2×Q811yescharacteristic
81C2×C8×C71—21no—
82C2×C8×C71—21no—
83C2×C8×C71—21no—
84C2×C8×C71—21no—
85C2×C8×C71—21no—
86C2×C8×C71—21no—
87C2×C8×C71—41no—
88C2×C8×C71—41no—
89C2×C8×C71—41no—
90C2×C8×C71—41no—
91C2×C8×C71—41no—
92C2×C8×C71—41no—
93C42×C71#4111yescharacteristic
94C42×C71#4111yescharacteristic
95C42×C71#4111yescharacteristic
96C42×C71#4111yescharacteristic
97C23×C71C2×C4⋊C411yescharacteristic
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71#11711yescharacteristic
101C2×C4×C71#11711yescharacteristic
102C2×C4×C71#12411yescharacteristic
103C2×C4×C71#12411yescharacteristic
104C2×C4×C71#12511yescharacteristic
105C2×C4×C71#12511yescharacteristic
106C2×C4×C71#13111yescharacteristic
107C2×C4×C71#13111yescharacteristic
108C2×C4×C71#13111yescharacteristic
109C2×C4×C71#13111yescharacteristic
110C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
111C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
112C2×C4×C71#13711yescharacteristic
113C2×C4×C71#13711yescharacteristic
114C2×Dic71—21no—
115C2×Dic71—21no—
116C2×Dic71—21no—
117C2×Dic71—21no—
118C2×Dic71—21no—
119C2×Dic71—21no—
120C2×Dic71—21no—
121C2×Dic71—21no—
122C2×Dic71—21no—
123C2×Dic71—21no—
124C2×Dic71#10812yes—
125C2×Dic71#10812yes—
126C8×C71—41no—
127C8×C71—41no—
128C8×C71—81no—
129C8×C71—81no—
130C71⋊C8—41no—
131C71⋊C8—41no—
132C71⋊C8—81no—
133C71⋊C8—81no—
134C22×C71#38211yescharacteristic
135C22×C71#42411yescharacteristic
136C22×C71#42411yescharacteristic
137C22×C71#42811yescharacteristic
138C22×C71#42811yescharacteristic
139C22×C71C2×#10811yescharacteristic
140C22×C71C2×#10811yescharacteristic
141C4×C71—21no—
142C4×C71—21no—
143C4×C71#43311yescharacteristic
144C4×C71#43311yescharacteristic
145C4×C71#43311yescharacteristic
146C4×C71#43311yescharacteristic
147C4×C71#48911yescharacteristic
148C4×C71#48911yescharacteristic
149C4×C71#49011yescharacteristic
150C4×C71#49111yescharacteristic
151Dic71—22no—
152Dic71—22no—
153Dic71—41no—
154Dic71—41no—
155#8789—711nomaximal, Sylow
156C2×C71#157311yescharacteristic
157C2×C71#157311yescharacteristic
158C2×C71#157411yescharacteristic
159C2×C71#157411yescharacteristic
160C2×C71#183211yescharacteristic
161C2×C71#183611yescharacteristic
162C2×C71#183711yescharacteristic
163#1842—711no—
164#1831—711no—
165#1744—711no—
166#1277—711no—
167#1277—711no—
168#1277—711no—
169#1277D7111yescharacteristic
170C71#878911yescharacteristic, complemented, Sylow
171C2×C4⋊C8—711no—
172C2×C4⋊C8—711no—
173C2×C4⋊C8—1421no—
174C2×C4⋊C8—1421no—
175C2×C4⋊C8C2×D7111yescharacteristic
176C2×C4⋊C8C2×D7111yescharacteristic
177C2×C4×C8—711no—
178C2×C4×C8C2×D7111yescharacteristic
179C43—711no—
180#336—711no—
181#336—711no—
182#336—711no—
183#336—711no—
184#336—1421no—
185#336—1421no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—711no—
189C22×C8—711no—
190C22×C8—1421no—
191C22×C8—1421no—
192C22×C8C4×D7111yescharacteristic
193C22×C8C4×D7111yescharacteristic
194C2×C42—711no—
195C2×C42—711no—
196C2×C42—711no—
197C2×C42—711no—
198C2×C42—711no—
199C2×C42—711no—
200C2×C42C22×D7111yescharacteristic
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—2842no—
206C4⋊C8—2842no—
207C4⋊C8—1421no—
208C4⋊C8—1421no—
209C4⋊C8—1421no—
210C4⋊C8—1421no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—1421no—
214C4×C8—1421no—
215C22×C4—711no—
216C22×C4—711no—
217C22×C4—711no—
218C22×C4—711no—
219C22×C4C2×C4×D7111yescharacteristic
220C22×C4D4×D7111yescharacteristic
221C22×C4Q8×D7111yescharacteristic
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—2841no—
235C2×C8—2841no—
236C2×C8—2841no—
237C2×C8—2841no—
238C2×C8—2841no—
239C2×C8—2841no—
240C2×C8—1421no—
241C2×C8—1421no—
242C2×C8—1421no—
243C2×C8—1421no—
244C2×C8—1421no—
245C2×C8—1421no—
246C42—1422no—
247C42—1422no—
248C42—1421no—
249C42—1421no—
250C42—1421no—
251C42—1421no—
252C42—1421no—
253C42—1421no—
254C42—1421no—
255C42—1421no—
256C42#1176586411yescharacteristic
257C42#1176586411yescharacteristic
258C42#1176586511yescharacteristic
259C42#1176586511yescharacteristic
260C23C4⋊C4×D7111yescharacteristic
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—712no—
264C2×C4—712no—
265C2×C4—1421no—
266C2×C4—1421no—
267C2×C4—1421no—
268C2×C4—1421no—
269C2×C4—1421no—
270C2×C4—1421no—
271C2×C4—1421no—
272C2×C4—1421no—
273C2×C4—1421no—
274C2×C4—1421no—
275C2×C4#1455608411yescharacteristic
276C2×C4#1455608511yescharacteristic
277C2×C4#1455611411yescharacteristic
278C2×C4#1455611411yescharacteristic
279C2×C4#1455611811yescharacteristic
280C2×C4#1455611811yescharacteristic
281C2×C4#1455613411yescharacteristic
282C2×C4#1455613411yescharacteristic
283C2×C4#1455613611yescharacteristic
284C2×C4#1455613611yescharacteristic
285C2×C4#1455615211yescharacteristic
286C2×C4#1455615511yescharacteristic
287C2×C4#1455615711yescharacteristic
288C2×C4#1455616011yescharacteristic
289C8—41no—
290C8—41no—
291C8—81no—
292C8—81no—
293C8—2841no—
294C8—2841no—
295C8—5681no—
296C8—5681no—
297C22#1798586311yescharacteristic
298C22#1798602411yescharacteristic
299C22#1798602411yescharacteristic
300C22#1798603811yescharacteristic
301C22#1798604011yescharacteristic
302C22#108×D7111yescharacteristic
303C22#108×D7111yescharacteristic
304C4—21no—
305C4—21no—
306C4—2841no—
307C4—2841no—
308C4—1422no—
309C4—1422no—
310C4#1798605711yescharacteristic
311C4#1798605711yescharacteristic
312C4#1798605811yescharacteristic
313C4#1798606111yescharacteristic
314C4#1798635711yescharacteristic
315C4#1798636011yescharacteristic
316C4#1798636211yescharacteristic
317C4#1798636611yescharacteristic
318C2#2258712211yescharacteristic
319C2#2258712411yescharacteristic
320C2#2258712711yescharacteristic
321C2#2258713011yescharacteristic
322C2#2258845811yescharacteristic
323C2#2258847211yescharacteristic
324C2#2258847611yescharacteristic
subgroupG/H
centerC22×C4D4×D71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1277C2
socle, prime radicalC23×C71C2×C4⋊C4
cosocleC2×C42×C71C23
hypercenter#1277D71
hypocenterC71#8789
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
170. C71#8789semidirect

Maximal subgroups

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

subgroupindex
#225884912
#225884562
#225880852
#225861752
#225861742
C71×#12772
#878971