Finite Group Databasebeta

← #31902004

Subgroups — #31902004

Order 18176.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22589143C211yesmaximal, characteristic
2#22589098C211yesmaximal, characteristic
3#22589059C211yesmaximal, characteristic
4C2×#17985751C211yesmaximal, characteristic
5C2×#17985745C211yesmaximal, characteristic
6#22586224C211yesmaximal, characteristic
7C71×#1988C211yesmaximal, characteristic
8C2×#14556061—21no—
9C2×C4⋊(C71⋊C8)—21no—
10C2×C4⋊(C71⋊C8)C2211yescharacteristic
11C2×C4⋊(C71⋊C8)C2211yescharacteristic
12C2×#14556052—21no—
13C2×#14556052—21no—
14C2×C4×C71⋊C8C2211yescharacteristic
15#17985869C2211yescharacteristic
16#17985761—21no—
17#17985754—21no—
18#17985751—22no—
19#17985745—22no—
20C2×C4⋊Q8×C71C2211yescharacteristic
21C2×C4⋊C8×C71C2211yescharacteristic
22C4×C4⋊C4×C71C2211yescharacteristic
23#340×C71—21no—
24#340×C71—21no—
25C22×C71⋊C8—21no—
26C22×C71⋊C8—21no—
27C22×C71⋊C8D411yescharacteristic
28C22×C71⋊C8D411yescharacteristic
29C2×C4⋊Dic71—21no—
30C2×C4⋊Dic71—21no—
31C2×C4⋊Dic71—21no—
32#14556061—42no—
33C4⋊(C71⋊C8)—21no—
34C4⋊(C71⋊C8)—21no—
35C4⋊(C71⋊C8)—21no—
36C4⋊(C71⋊C8)—42no—
37C4⋊(C71⋊C8)D412yes—
38#14556052—42no—
39#14556052—42no—
40C4×C71⋊C8—21no—
41C4×C71⋊C8—21no—
42C22×Q8×C71—21no—
43C22×C8×C71—21no—
44C4⋊Q8×C71—21no—
45C4⋊Q8×C71—21no—
46C4⋊Q8×C71—21no—
47C4⋊Q8×C71Q812yes—
48C2×C4⋊C4×C71—21no—
49C2×C4⋊C4×C71—21no—
50C2×C4⋊C4×C71D411yescharacteristic
51C2×C4⋊C4×C71D411yescharacteristic
52C2×C42×C71—21no—
53C2×C42×C71C2311yescharacteristic
54C4⋊C8×C71—21no—
55C4⋊C8×C71—21no—
56#95×C71—21no—
57C22×Dic71—41no—
58C2×C71⋊C8—21no—
59C2×C71⋊C8—21no—
60C2×C71⋊C8—21no—
61C2×C71⋊C8—22no—
62C2×C71⋊C8—21no—
63C2×C71⋊C8—21no—
64C2×C71⋊C8—41no—
65C2×C71⋊C8—41no—
66C2×C71⋊C8—41no—
67C2×C71⋊C8—41no—
68C2×C71⋊C8—41no—
69C2×C71⋊C8D812yes—
70C4⋊Dic71—41no—
71C4⋊Dic71—41no—
72C4⋊Dic71—41no—
73C4⋊Dic71—41no—
74C4⋊Dic71—81no—
75C2×Q8×C71—22no—
76C2×Q8×C71—41no—
77C2×Q8×C71—41no—
78C2×Q8×C71—41no—
79C2×Q8×C71—41no—
80C2×Q8×C71—41no—
81C22×C4×C71—21no—
82C22×C4×C71—21no—
83C22×C4×C71—41no—
84C22×C4×C71C2×D411yescharacteristic
85C22×C4×C71C2×D411yescharacteristic
86C22×C4×C71#4111yescharacteristic
87C2×C8×C71—41no—
88C2×C8×C71—41no—
89C2×C8×C71—41no—
90C4⋊C4×C71—21no—
91C4⋊C4×C71—22no—
92C4⋊C4×C71—22no—
93C4⋊C4×C71—21no—
94C4⋊C4×C71—21no—
95C4⋊C4×C71—41no—
96C4⋊C4×C71—41no—
97C4⋊C4×C71Q1612yes—
98C42×C71—41no—
99C42×C71—41no—
100C42×C71C2×D411yescharacteristic
101C42×C71C2×Q811yescharacteristic
102C42×C71#4111yescharacteristic
103C42×C71#4111yescharacteristic
104C23×C71#12111yescharacteristic
105Q8×C71—42no—
106Q8×C71—42no—
107Q8×C71—81no—
108Q8×C71—81no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—22no—
112C2×C4×C71—41no—
113C2×C4×C71—41no—
114C2×C4×C71—42no—
115C2×C4×C71—41no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—81no—
119C2×C4×C71—81no—
120C2×C4×C71#12111yescharacteristic
121C2×C4×C71#12111yescharacteristic
122C2×C4×C71C22⋊Q811yescharacteristic
123C2×C4×C71C22⋊Q811yescharacteristic
124C2×C4×C71C22⋊Q811yescharacteristic
125C2×C4×C71#12611yescharacteristic
126C2×C4×C71C2×D811yescharacteristic
127C2×C4×C71C2×Q1611yescharacteristic
128C2×C4×C71#13511yescharacteristic
129C2×C4×C71#13511yescharacteristic
130C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C71#13711yescharacteristic
133C2×C4×C71#13711yescharacteristic
134C2×Dic71—81no—
135C2×Dic71—81no—
136C2×Dic71—81no—
137C8×C71—81no—
138C71⋊C8—22no—
139C71⋊C8—42no—
140C71⋊C8—41no—
141C71⋊C8—81no—
142C22×C71#39611yescharacteristic
143C22×C71#45911yescharacteristic
144C22×C71#46411yescharacteristic
145C22×C71#46711yescharacteristic
146C22×C71#46811yescharacteristic
147C22×C71#47111yescharacteristic
148C22×C71#47311yescharacteristic
149C4×C71—21no—
150C4×C71—21no—
151C4×C71—42no—
152C4×C71—82no—
153C4×C71—81no—
154C4×C71#46211yescharacteristic
155C4×C71#46311yescharacteristic
156C4×C71#46611yescharacteristic
157C4×C71#46811yescharacteristic
158C4×C71#47411yescharacteristic
159C4×C71#47711yescharacteristic
160C4×C71#47811yescharacteristic
161C4×C71#47911yescharacteristic
162Dic71—161no—
163#10592—711nomaximal, Sylow
164C2×C71#166911yescharacteristic
165C2×C71#168411yescharacteristic
166C2×C71#168511yescharacteristic
167C2×C71#170011yescharacteristic
168C2×C71#206411yescharacteristic
169C2×C71#206711yescharacteristic
170C2×C71#206811yescharacteristic
171#1988D7111yescharacteristic
172#1956—711no—
173#1947—711no—
174#1939—711no—
175C2×#335—711no—
176C2×#332—711no—
177#1296—711no—
178C71#1059211yescharacteristic, complemented, Sylow
179C2×C4⋊Q8C2×D7111yescharacteristic
180C2×#107—1421no—
181C2×C4⋊C8—711no—
182C2×C4⋊C8—711no—
183C2×C4⋊C8—1421no—
184C2×C4⋊C8C2×D7111yescharacteristic
185C2×#103—1421no—
186C2×#103—1421no—
187C2×C4×C8—711no—
188#384—711no—
189C4×C4⋊C4C2×D7111yescharacteristic
190#340—21no—
191#340—21no—
192#340—1421no—
193#336—1421no—
194#335—1422no—
195#332—1422no—
196C22×Q8—21no—
197C22×C8—21no—
198C22×C8—711no—
199C22×C8—711no—
200C22×C8—1421no—
201C22×C8—1421no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8Dic14212yes—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—1421no—
209C2×C4⋊C4—1421no—
210C2×C4⋊C4—1421no—
211C2×C4⋊C4C22⋊D7111yescharacteristic
212C2×C4⋊C4C22⋊D7111yescharacteristic
213C2×C42—21no—
214C2×C42C22×D7111yescharacteristic
215#107—2842no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—2842no—
219C4⋊C8—712no—
220C4⋊C8—1421no—
221C4⋊C8—1421no—
222C4⋊C8—1421no—
223#103—2842no—
224#103—2842no—
225C4×C8—1421no—
226C4×C8—1421no—
227#95—21no—
228C2×Q8—22no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—41no—
237C22×C4—2841no—
238C22×C4D4×D7111yescharacteristic
239C22×C4C2×C22⋊D7111yescharacteristic
240C22×C4#1176586411yescharacteristic
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—2841no—
245C2×C8—2841no—
246C2×C8—2841no—
247C2×C8—2841no—
248C2×C8—2841no—
249C2×C8—712no—
250C2×C8—1422no—
251C2×C8—1421no—
252C2×C8—1421no—
253C2×C8—1421no—
254C2×C8—1421no—
255C2×C8—1421no—
256C4⋊C4—21no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—2841no—
264C4⋊C4—2841no—
265C4⋊C4—2841no—
266C4⋊C4—2841no—
267C4⋊C4—5681no—
268C4⋊C4#1176585512yes—
269C42—41no—
270C42—41no—
271C42D4×D7111yescharacteristic
272C42C2×Dic14211yescharacteristic
273C42#1176586411yescharacteristic
274C42#1176586511yescharacteristic
275C23#1455610011yescharacteristic
276Q8—42no—
277Q8—42no—
278Q8—81no—
279Q8—81no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—5681no—
292C2×C4—5681no—
293C2×C4—5681no—
294C2×C4#1455609911yescharacteristic
295C2×C4#1455610111yescharacteristic
296C2×C4#1455610511yescharacteristic
297C2×C4#1455610511yescharacteristic
298C2×C4#1455610711yescharacteristic
299C2×C4#1455612111yescharacteristic
300C2×C4D8×D7111yescharacteristic
301C2×C4C2×#1176585511yescharacteristic
302C2×C4#1455614811yescharacteristic
303C2×C4#1455614911yescharacteristic
304C2×C4#1455615211yescharacteristic
305C2×C4#1455615311yescharacteristic
306C2×C4#1455615711yescharacteristic
307C2×C4#1455615911yescharacteristic
308C8—81no—
309C8—2842no—
310C8—2841no—
311C8—5681no—
312C8—1422no—
313C22#1798592211yescharacteristic
314C22#1798619211yescharacteristic
315C22#1798621911yescharacteristic
316C22#1798623811yescharacteristic
317C22#1798624711yescharacteristic
318C22#1798626411yescharacteristic
319C22#1798627211yescharacteristic
320C4—21no—
321C4—21no—
322C4—42no—
323C4—82no—
324C4—81no—
325C4—11361no—
326C4#1798620611yescharacteristic
327C4#1798621111yescharacteristic
328C4#1798623411yescharacteristic
329C4#1798624411yescharacteristic
330C4#1798627711yescharacteristic
331C4#1798629411yescharacteristic
332C4#1798630111yescharacteristic
333C4#1798630611yescharacteristic
334C2#2258770311yescharacteristic
335C2#2258779611yescharacteristic
336C2#2258780711yescharacteristic
337C2#2258789211yescharacteristic
338C2#2258970711yescharacteristic
339C2#2258972411yescharacteristic
340C2#2258973511yescharacteristic
subgroupG/H
centerC23#14556100
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1988C2
socle, prime radicalC23×C71#121
hypercenter#1988D71
hypocenterC71#10592
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
178. C71#10592semidirect

Maximal subgroups

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

subgroupindex
#225891432
#225890982
#225890592
C2×#179857512
C2×#179857452
#225862242
C71×#19882
#1059271