Finite Group Databasebeta

← #31917058

Subgroups — #31917058

Order 18176.

410 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556074C211yesmaximal, characteristic
2C2×C4⋊(C71⋊C16)C211yesmaximal, characteristic
3C2×#17985803C211yesmaximal, characteristic
4C2×#17985802C211yesmaximal, characteristic
5#22589201C211yesmaximal, characteristic
6#22586431C211yesmaximal, characteristic
7C71×#1969C211yesmaximal, characteristic
8C22×Dic284C2211yescharacteristic
9C2×#14556074—22no—
10C2×#14556074—22no—
11C2×#14556074—22no—
12C22×C71⋊C16C2211yescharacteristic
13C22×C71⋊C16C2211yescharacteristic
14C2×#14556062C2211yescharacteristic
15C2×#14556053—21no—
16#17985905—21no—
17C4⋊(C71⋊C16)—22no—
18#17985803—22no—
19#17985802—22no—
20C22×Q16×C71C2211yescharacteristic
21C2×#107×C71C2211yescharacteristic
22C2×#103×C71—21no—
23C2×C4×C8×C71C2211yescharacteristic
24#391×C71—21no—
25C22×Dic142—21no—
26C2×Dic284—21no—
27C2×Dic284—21no—
28C2×Dic284—21no—
29C2×Dic284—21no—
30C2×Dic284—21no—
31C2×Dic284D412yes—
32C2×C4⋊Dic71—21no—
33#14556074—28no—
34C2×C71⋊C16—21no—
35C2×C71⋊C16—21no—
36C2×C71⋊C16—21no—
37C2×C71⋊C16—21no—
38C2×C71⋊C16—21no—
39C2×C71⋊C16D412yes—
40#14556062—21no—
41#14556062—21no—
42#14556053—42no—
43C71⋊#95—41no—
44C71⋊#95—41no—
45C22×Q8×C71—21no—
46C2×Q16×C71—21no—
47C2×Q16×C71—21no—
48C2×Q16×C71—21no—
49C2×Q16×C71—21no—
50C2×Q16×C71—21no—
51C2×Q16×C71D412yes—
52C22×C8×C71D411yescharacteristic
53C22×C8×C71C2311yescharacteristic
54C2×C4⋊C4×C71—21no—
55C2×C42×C71D411yescharacteristic
56#107×C71—21no—
57#107×C71—21no—
58#103×C71—42no—
59C4×C8×C71—21no—
60C4×C8×C71—21no—
61#95×C71—41no—
62#95×C71—41no—
63C2×Dic142—22no—
64C2×Dic142—41no—
65C2×Dic142—41no—
66C2×Dic142—41no—
67C2×Dic142—41no—
68C2×Dic142—41no—
69C22×Dic71—41no—
70C22×Dic71—41no—
71Dic284—22no—
72Dic284—22no—
73Dic284—41no—
74Dic284—41no—
75C4⋊Dic71—41no—
76C4⋊Dic71—41no—
77C71⋊C16—22no—
78C71⋊C16—41no—
79C2×Q8×C71—22no—
80C2×Q8×C71—41no—
81C2×Q8×C71—41no—
82C2×Q8×C71—41no—
83C2×Q8×C71—41no—
84C2×Q8×C71—41no—
85C22×C4×C71—41no—
86C22×C4×C71—41no—
87C22×C4×C71D811yescharacteristic
88C22×C4×C71D811yescharacteristic
89C22×C4×C71C2×D411yescharacteristic
90Q16×C71—22no—
91Q16×C71—22no—
92Q16×C71—41no—
93Q16×C71—41no—
94C2×C8×C71—21no—
95C2×C8×C71—21no—
96C2×C8×C71—21no—
97C2×C8×C71C2×D411yescharacteristic
98C2×C8×C71C2×D411yescharacteristic
99C2×C8×C71C2×D411yescharacteristic
100C2×C8×C71#4111yescharacteristic
101C2×C8×C71#4111yescharacteristic
102C2×C8×C71#4111yescharacteristic
103C4⋊C4×C71—41no—
104C4⋊C4×C71—41no—
105C42×C71—21no—
106C42×C71—21no—
107C23×C71C2×D811yescharacteristic
108Q8×C71—42no—
109Q8×C71—42no—
110Q8×C71—81no—
111Q8×C71—81no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—42no—
118C2×C4×C71—81no—
119C2×C4×C71—81no—
120C2×C4×C71—81no—
121C2×C4×C71—81no—
122C2×C4×C71—81no—
123C2×C4×C71Q3212yescomplemented
124C2×C4×C71#12111yescharacteristic
125C2×C4×C71#12111yescharacteristic
126C2×C4×C71#12111yescharacteristic
127C2×C4×C71#13511yescharacteristic
128C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
130C2×Dic71—42no—
131C2×Dic71—81no—
132C2×Dic71—81no—
133C2×Dic71—81no—
134C2×Dic71—81no—
135C2×Dic71—81no—
136Dic142—42no—
137Dic142—42no—
138Dic142—81no—
139Dic142—81no—
140C8×C71—41no—
141C8×C71#12011yescharacteristic
142C8×C71#12311yescharacteristic
143C8×C71#12311yescharacteristic
144C8×C71#12411yescharacteristic
145C22×C71#45911yescharacteristic
146C22×C71#45911yescharacteristic
147C22×C71#46611yescharacteristic
148C22×C71C2×Q3211yescharacteristic
149C22×C71#50811yescharacteristic
150C22×C71#51011yescharacteristic
151C22×C71#51011yescharacteristic
152C4×C71—22no—
153C4×C71—41no—
154C4×C71—82no—
155C4×C71—161no—
156C4×C71#39411yescharacteristic
157C4×C71#46311yescharacteristic
158C4×C71#46311yescharacteristic
159C4×C71#46911yescharacteristic
160Dic71—82no—
161Dic71—161no—
162#13450—711nomaximal, Sylow
163C2×C71#205411yescharacteristic
164C2×C71#220911yescharacteristic
165C2×C71#220911yescharacteristic
166C2×C71#221111yescharacteristic
167C2×C71#221111yescharacteristic
168C2×C71#221611yescharacteristic
169C2×C71#221911yescharacteristic
170C22×Q32—711no—
171C2×C4⋊C16—711no—
172C2×#358—711no—
173C2×#358—711no—
174#1969—711no—
175#1969D7111yescharacteristic
176#1380—711no—
177C71#1345011yescharacteristic, complemented, Sylow
178C22×Q16—711no—
179C22×Q16C2×D7111yescharacteristic
180C2×Q32—1422no—
181C2×Q32—1422no—
182C2×Q32—1422no—
183C22×C16—711no—
184C22×C16—711no—
185C2×#107—711no—
186C2×#107C2×D7111yescharacteristic
187C2×#103—21no—
188C2×#103—1421no—
189C2×C4×C8C2×D7111yescharacteristic
190#391—21no—
191#391—1421no—
192C4⋊C16—1422no—
193#358—1422no—
194#358—1422no—
195C22×Q8—21no—
196C22×Q8—1421no—
197C2×Q16—21no—
198C2×Q16—21no—
199C2×Q16—21no—
200C2×Q16—21no—
201C2×Q16—21no—
202C2×Q16—712no—
203C2×Q16—1421no—
204C2×Q16—1421no—
205C2×Q16—1421no—
206C2×Q16—1421no—
207C2×Q16—1421no—
208C2×Q16D28412yes—
209C22×C8C22⋊D7111yescharacteristic
210C22×C8C22×D7111yescharacteristic
211C2×C4⋊C4—21no—
212C2×C4⋊C4—1421no—
213C2×C42C22⋊D7111yescharacteristic
214Q32—1428no—
215C2×C16—712no—
216C2×C16—1421no—
217C2×C16—1421no—
218C2×C16—1421no—
219C2×C16—1421no—
220C2×C16—1421no—
221#107—21no—
222#107—21no—
223#107—1421no—
224#107—1421no—
225#103—42no—
226#103—2842no—
227C4×C8—21no—
228C4×C8—21no—
229#95—41no—
230#95—41no—
231#95—2841no—
232#95—2841no—
233C2×Q8—22no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—2841no—
240C2×Q8—2841no—
241C2×Q8—2841no—
242C2×Q8—2841no—
243C2×Q8—2841no—
244C2×Q8—1422no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—2841no—
248C22×C4—2841no—
249C22×C4#1176585011yescharacteristic
250C22×C4#1176585011yescharacteristic
251C22×C4C2×C22⋊D7111yescharacteristic
252Q16—22no—
253Q16—22no—
254Q16—41no—
255Q16—41no—
256Q16—2841no—
257Q16—2841no—
258Q16—1422no—
259Q16—1422no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8D4×D7111yescharacteristic
264C2×C8D4×D7111yescharacteristic
265C2×C8C2×D28411yescharacteristic
266C2×C8#1176586411yescharacteristic
267C2×C8#1176586411yescharacteristic
268C2×C8#1176586511yescharacteristic
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—2841no—
272C4⋊C4—2841no—
273C42—21no—
274C42—21no—
275C16—2841no—
276C16—1422no—
277C23C2×#1176585011yescharacteristic
278Q8—42no—
279Q8—42no—
280Q8—81no—
281Q8—81no—
282Q8—2842no—
283Q8—2842no—
284Q8—5681no—
285Q8—5681no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—2842no—
298C2×C4—5681no—
299C2×C4—5681no—
300C2×C4—5681no—
301C2×C4—5681no—
302C2×C4—5681no—
303C2×C4#1455607412yescomplemented
304C2×C4#1455609911yescharacteristic
305C2×C4#1455610011yescharacteristic
306C2×C4#1455610211yescharacteristic
307C2×C4#1455614811yescharacteristic
308C2×C4#1455615311yescharacteristic
309C2×C4#1455615411yescharacteristic
310C8—41no—
311C8#1455609811yescharacteristic
312C8#1455611011yescharacteristic
313C8#1455611211yescharacteristic
314C8#1455611411yescharacteristic
315C22#1798619111yescharacteristic
316C22#1798619411yescharacteristic
317C22#1798623311yescharacteristic
318C22C2×#1455607411yescharacteristic
319C22#1798642911yescharacteristic
320C22#1798643911yescharacteristic
321C22#1798644011yescharacteristic
322C4—22no—
323C4—41no—
324C4—82no—
325C4—161no—
326C4—5682no—
327C4—11361no—
328C4#1798591511yescharacteristic
329C4#1798621111yescharacteristic
330C4#1798621411yescharacteristic
331C4#1798625011yescharacteristic
332C2#2258965011yescharacteristic
333C2#2259038711yescharacteristic
334C2#2259039011yescharacteristic
335C2#2259039911yescharacteristic
336C2#2259040211yescharacteristic
337C2#2259042911yescharacteristic
338C2#2259044611yescharacteristic
subgroupG/H
centerC23C2×#11765850
commutator, cosocleC22×C8×C71C23
FrattiniC22×C8C22×D71
FittingC71×#1969C2
socle, prime radicalC23×C71C2×D8
hypercenter#1969D71
hypocenterC71#13450
LCSG ▷ C22×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ #1969

Complements

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

NH ≅ G/Ntype
123. C2×C4×C71Q32semidirect
177. C71#13450semidirect
303. C2×C4#14556074semidirect

Maximal subgroups

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

subgroupindex
C22×#145560742
C2×C4⋊(C71⋊C16)2
C2×#179858032
C2×#179858022
#225892012
#225864312
C71×#19692
#1345071