Finite Group Databasebeta

← #31902215

Subgroups — #31902215

Order 18176.

350 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22589276C211yesmaximal, characteristic
2#22588468C211yesmaximal, characteristic
3#22588346C211yesmaximal, characteristic
4#22588346C211yesmaximal, characteristic
5#22586224C211yesmaximal, characteristic
6#22586224C211yesmaximal, characteristic
7C71×#1277C211yesmaximal, characteristic
8C2×C4×Dic142C2211yescharacteristic
9C2×C4⋊(C71⋊C8)C2211yescharacteristic
10C2×#14556053—21no—
11C2×#14556053—21no—
12#17985869C2211yescharacteristic
13#17985869C2211yescharacteristic
14#17985762—21no—
15#17985762—21no—
16#17985762—21no—
17#17985762—21no—
18#17985753—21no—
19#17985753—21no—
20C2×C4⋊C8×C71C2211yescharacteristic
21C2×C4⋊C8×C71C2211yescharacteristic
22C2×C4×C8×C71C2211yescharacteristic
23C22×Dic142D411yescharacteristic
24C22×C71⋊C8—21no—
25C4×Dic142—21no—
26C4×Dic142—21no—
27C4×Dic142—21no—
28C4×Dic142—21no—
29C2×C4⋊Dic71—21no—
30C2×C4⋊Dic71—21no—
31C2×C4⋊Dic71—21no—
32C2×C4⋊Dic71—21no—
33C2×C4⋊Dic71—21no—
34C2×C4⋊Dic71—21no—
35C2×C4⋊Dic71D411yescharacteristic
36C2×Dic71⋊C4—21no—
37C2×C4×Dic71—21no—
38C4⋊(C71⋊C8)—21no—
39C4⋊(C71⋊C8)—21no—
40#14556053—42no—
41#14556053—42no—
42C22×C8×C71—21no—
43C22×C8×C71—21no—
44C22×C8×C71D411yescharacteristic
45C22×C8×C71D411yescharacteristic
46C2×C42×C71C2311yescharacteristic
47C4⋊C8×C71—21no—
48C4⋊C8×C71—21no—
49C4⋊C8×C71—21no—
50C4⋊C8×C71—21no—
51C4×C8×C71—21no—
52C4×C8×C71—21no—
53C2×Dic142—21no—
54C2×Dic142—21no—
55C2×Dic142—21no—
56C2×Dic142—21no—
57C2×Dic142—21no—
58C2×Dic142SD1612yes—
59C22×Dic71—21no—
60C22×Dic71—21no—
61C22×Dic71—41no—
62C22×Dic71—41no—
63C2×C71⋊C8—41no—
64C2×C71⋊C8—41no—
65C2×C71⋊C8—41no—
66C4⋊Dic71—21no—
67C4⋊Dic71—21no—
68C4⋊Dic71—41no—
69C4⋊Dic71—41no—
70C4⋊Dic71—41no—
71C4⋊Dic71—41no—
72C4⋊Dic71—41no—
73C4⋊Dic71—41no—
74C4⋊Dic71—41no—
75C4⋊Dic71—41no—
76C4⋊Dic71—81no—
77C4⋊Dic71—81no—
78Dic71⋊C4—41no—
79Dic71⋊C4—41no—
80C4×Dic71—41no—
81C4×Dic71—41no—
82C22×C4×C71C2×D411yescharacteristic
83C22×C4×C71C2×D411yescharacteristic
84C22×C4×C71#4111yescharacteristic
85C2×C8×C71—21no—
86C2×C8×C71—21no—
87C2×C8×C71—21no—
88C2×C8×C71—21no—
89C2×C8×C71—21no—
90C2×C8×C71—21no—
91C2×C8×C71—41no—
92C2×C8×C71—41no—
93C2×C8×C71—41no—
94C2×C8×C71—41no—
95C2×C8×C71—41no—
96C2×C8×C71—41no—
97C42×C71#4111yescharacteristic
98C42×C71#4111yescharacteristic
99C42×C71#4111yescharacteristic
100C42×C71#4111yescharacteristic
101C23×C71#12111yescharacteristic
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71#12311yescharacteristic
105C2×C4×C71#12311yescharacteristic
106C2×C4×C71#12311yescharacteristic
107C2×C4×C71#12411yescharacteristic
108C2×C4×C71#12611yescharacteristic
109C2×C4×C71#12611yescharacteristic
110C2×C4×C71C2×SD1611yescharacteristic
111C2×C4×C71#13511yescharacteristic
112C2×C4×C71#13511yescharacteristic
113C2×C4×C71#13511yescharacteristic
114C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
115C2×C4×C71#13711yescharacteristic
116C2×C4×C71#13711yescharacteristic
117C2×C4×C71#13711yescharacteristic
118C2×Dic71—22no—
119C2×Dic71—41no—
120C2×Dic71—41no—
121C2×Dic71—41no—
122C2×Dic71—41no—
123C2×Dic71—41no—
124C2×Dic71—81no—
125C2×Dic71—81no—
126C2×Dic71—81no—
127C2×Dic71—81no—
128C2×Dic71—81no—
129C2×Dic71—81no—
130Dic142—22no—
131Dic142—22no—
132Dic142—41no—
133Dic142—41no—
134C8×C71—41no—
135C8×C71—41no—
136C8×C71—81no—
137C8×C71—81no—
138C71⋊C8—81no—
139C22×C71#39711yescharacteristic
140C22×C71#46411yescharacteristic
141C22×C71#46411yescharacteristic
142C22×C71#46511yescharacteristic
143C22×C71#46811yescharacteristic
144C22×C71#47211yescharacteristic
145C22×C71#47311yescharacteristic
146C4×C71—21no—
147C4×C71—21no—
148C4×C71#48111yescharacteristic
149C4×C71#48211yescharacteristic
150C4×C71#48311yescharacteristic
151C4×C71#48511yescharacteristic
152C4×C71#48511yescharacteristic
153C4×C71#48511yescharacteristic
154C4×C71#48811yescharacteristic
155C4×C71#49111yescharacteristic
156Dic71—42no—
157Dic71—81no—
158Dic71—161no—
159Dic71—161no—
160#10626—711nomaximal, Sylow
161C2×C71#167811yescharacteristic
162C2×C71#168011yescharacteristic
163C2×C71#169811yescharacteristic
164C2×C71#170011yescharacteristic
165C2×C71#207411yescharacteristic
166C2×C71#207711yescharacteristic
167C2×C71#207811yescharacteristic
168#1984—711no—
169#1834—711no—
170#1803—711no—
171#1803—711no—
172#1296—711no—
173#1296—711no—
174#1277D7111yescharacteristic
175C71#1062611yescharacteristic, complemented, Sylow
176C2×C4×Q8—711no—
177C2×C4⋊C8—711no—
178C2×C4⋊C8C2×D7111yescharacteristic
179C2×C4⋊C8C2×D7111yescharacteristic
180C2×#103—1421no—
181C2×#103—1421no—
182C2×C4×C8C2×D7111yescharacteristic
183#384—711no—
184#384—711no—
185#340—1421no—
186#340—1421no—
187#340—1421no—
188#340—1421no—
189#336—1421no—
190#336—1421no—
191C22×Q8—711no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—1421no—
195C22×C8D28411yescharacteristic
196C22×C8D28411yescharacteristic
197C4×Q8—1421no—
198C4×Q8—1421no—
199C4×Q8—1421no—
200C4×Q8—1421no—
201C2×C4⋊C4—711no—
202C2×C4⋊C4—1421no—
203C2×C4⋊C4—1421no—
204C2×C4⋊C4—1421no—
205C2×C4⋊C4—1421no—
206C2×C4⋊C4—1421no—
207C2×C4⋊C4—1421no—
208C2×C4⋊C4—1421no—
209C2×C42—1421no—
210C2×C42C22×D7111yescharacteristic
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—1421no—
216C4⋊C8—1421no—
217#103—2842no—
218#103—2842no—
219C4×C8—21no—
220C4×C8—21no—
221C2×Q8—712no—
222C2×Q8—1421no—
223C2×Q8—1421no—
224C2×Q8—1421no—
225C2×Q8—1421no—
226C2×Q8—1421no—
227C22×C4—2841no—
228C22×C4—2841no—
229C22×C4—1421no—
230C22×C4—1421no—
231C22×C4D4×D7111yescharacteristic
232C22×C4C2×D28411yescharacteristic
233C22×C4#1176586611yescharacteristic
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—2841no—
247C2×C8—2841no—
248C2×C8—2841no—
249C4⋊C4—2841no—
250C4⋊C4—2841no—
251C4⋊C4—2841no—
252C4⋊C4—2841no—
253C4⋊C4—2841no—
254C4⋊C4—2841no—
255C4⋊C4—2841no—
256C4⋊C4—2841no—
257C4⋊C4—2841no—
258C4⋊C4—2841no—
259C4⋊C4—5681no—
260C4⋊C4—5681no—
261C4⋊C4—1421no—
262C4⋊C4—1421no—
263C42—2841no—
264C42—2841no—
265C42#1176586411yescharacteristic
266C42#1176586411yescharacteristic
267C42#1176586511yescharacteristic
268C42#1176586511yescharacteristic
269C23#1455610311yescharacteristic
270Q8—2841no—
271Q8—2841no—
272Q8—1422no—
273Q8—1422no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—2841no—
277C2×C4—2841no—
278C2×C4—2841no—
279C2×C4—2841no—
280C2×C4—2841no—
281C2×C4—5681no—
282C2×C4—5681no—
283C2×C4—5681no—
284C2×C4—5681no—
285C2×C4—5681no—
286C2×C4—5681no—
287C2×C4—1422no—
288C2×C4#1455611011yescharacteristic
289C2×C4#1455611011yescharacteristic
290C2×C4#1455611211yescharacteristic
291C2×C4#1455611611yescharacteristic
292C2×C4#1455612311yescharacteristic
293C2×C4#1455612311yescharacteristic
294C2×C4SD16×D7111yescharacteristic
295C2×C4#1455614911yescharacteristic
296C2×C4#1455615011yescharacteristic
297C2×C4#1455615011yescharacteristic
298C2×C4#1455615211yescharacteristic
299C2×C4#1455616011yescharacteristic
300C2×C4#1455616111yescharacteristic
301C2×C4#1455616111yescharacteristic
302C8—41no—
303C8—41no—
304C8—81no—
305C8—81no—
306C8—5681no—
307C22#1798592611yescharacteristic
308C22#1798622211yescharacteristic
309C22#1798622211yescharacteristic
310C22#1798622911yescharacteristic
311C22#1798624611yescharacteristic
312C22#1798627011yescharacteristic
313C22#1798627511yescharacteristic
314C4—21no—
315C4—21no—
316C4—2842no—
317C4—5681no—
318C4—11361no—
319C4—11361no—
320C4#1798631611yescharacteristic
321C4#1798632211yescharacteristic
322C4#1798632811yescharacteristic
323C4#1798633811yescharacteristic
324C4#1798634111yescharacteristic
325C4#1798634111yescharacteristic
326C4#1798635411yescharacteristic
327C4#1798636811yescharacteristic
328C2#2258775711yescharacteristic
329C2#2258777111yescharacteristic
330C2#2258788111yescharacteristic
331C2#2258789511yescharacteristic
332C2#2258976011yescharacteristic
333C2#2258977511yescharacteristic
334C2#2258978211yescharacteristic
subgroupG/H
centerC23#14556103
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1277C2
socle, prime radicalC23×C71#121
hypercenter#1277D71
hypocenterC71#10626
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
175. C71#10626semidirect

Maximal subgroups

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

subgroupindex
#225892762
#225884682
#225883462
#225862242
C71×#12772
#1062671