Finite Group Databasebeta

← #31914539

Subgroups — #31914539

Order 18176.

462 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22597257C211yesmaximal, characteristic, complemented
2#22590076C211yesmaximal, characteristic, complemented
3#22590068C211yesmaximal, characteristic, complemented
4#22589239C211yesmaximal, characteristic, complemented
5#22586436C211yesmaximal, characteristic, complemented
6#22586428C211yesmaximal, characteristic, complemented
7C2×#508×C71C211yesmaximal, characteristic
8#17986766—21no—
9C2×#14556068C2211yescharacteristic
10#17986121—22no—
11#17986116—22no—
12C2×#14556064C2211yescharacteristic
13#17986037C2211yescharacteristic
14#17986016—21no—
15#17986016—21no—
16#17985973—21no—
17#17985954C2211yescharacteristic
18#17985809—22no—
19#17985806—22no—
20C2×#135×C71C2211yescharacteristic
21C2×#135×C71C2211yescharacteristic
22#508×C71—22no—
23#508×C71—22no—
24C2×Q32×C71C411yescharacteristic, complemented
25C2×SD32×C71C411yescharacteristic, complemented
26C2×SD32×C71C411yescharacteristic, complemented
27C2×D16×C71C411yescharacteristic, complemented
28C22×C16×C71C2211yescharacteristic
29#14556135—24no—
30#14556135—42no—
31C2×#11765849—21no—
32C2×#11765849—42no—
33C2×#11765849D411yescharacteristic
34#14556087—21no—
35#14556087D411yescharacteristic
36#14556068—21no—
37#14556068—21no—
38C2×C71⋊C16—21no—
39#14556064—21no—
40#14556064D412yes—
41#14556062—21no—
42#14556060—21no—
43#14556055—24no—
44#14556055—42no—
45#14556043—42no—
46#14556038—21no—
47C8×Dic71—21no—
48C2×#41×C71—21no—
49C2×#41×C71—21no—
50#135×C71—21no—
51#135×C71—21no—
52#135×C71—21no—
53#135×C71—21no—
54#135×C71—21no—
55#135×C71—21no—
56#135×C71D412yes—
57#135×C71D412yes—
58C2×Q16×C71C2×C411yescharacteristic
59C2×Q16×C71C2×C411yescharacteristic
60C2×SD16×C71—21no—
61C2×SD16×C71—21no—
62C2×D8×C71C2×C411yescharacteristic
63C2×D8×C71C2×C411yescharacteristic
64C22×C8×C71C2311yescharacteristic
65Q32×C71—22no—
66SD32×C71—22no—
67SD32×C71—22no—
68D16×C71—22no—
69C2×C16×C71—21no—
70C2×C16×C71—21no—
71C2×C16×C71C2×C411yescharacteristic
72C2×C16×C71C2×C411yescharacteristic
73#11765849—21no—
74#11765849—22no—
75#11765849—44no—
76#11765849—41no—
77#11765849D812yes—
78C2×C71⋊C8—21no—
79C2×C71⋊C8—44no—
80C2×C71⋊C8—41no—
81C4⋊Dic71—21no—
82C4⋊Dic71—41no—
83C22⋊Dic71—21no—
84C22⋊Dic71—41no—
85C4×Dic71—21no—
86C4×Dic71—41no—
87C71⋊C16—41no—
88#41×C71—22no—
89#41×C71—22no—
90#41×C71—41no—
91#41×C71—41no—
92#41×C71—41no—
93#41×C71—41no—
94#41×C71—41no—
95#41×C71—41no—
96C2×Q8×C71—21no—
97C2×Q8×C71—21no—
98C2×D4×C71—21no—
99C2×D4×C71—21no—
100C2×D4×C71—41no—
101C2×D4×C71—41no—
102C22×C4×C71—41no—
103C22×C4×C71—41no—
104C22×C4×C71C2×D411yescharacteristic
105Q16×C71—21no—
106Q16×C71—21no—
107Q16×C71—21no—
108Q16×C71C22⋊C412yes—
109SD16×C71—22no—
110SD16×C71—22no—
111SD16×C71—41no—
112SD16×C71—41no—
113D8×C71—21no—
114D8×C71—21no—
115D8×C71—21no—
116D8×C71C22⋊C412yes—
117C2×C8×C71C22×C411yescharacteristic
118C2×C8×C71C2×D411yescharacteristic
119C2×C8×C71C2×D411yescharacteristic
120C2×C8×C71C2×D411yescharacteristic
121C2×C8×C71#4111yescharacteristic
122C2×C8×C71#4111yescharacteristic
123C16×C71—21no—
124C16×C71—21no—
125C23×C71—41no—
126C23×C71—41no—
127C23×C71AGL(1,ℤ/8)11yescharacteristic
128Q8×C71—22no—
129Q8×C71—41no—
130Q8×C71—41no—
131Q8×C71—41no—
132D4×C71—22no—
133D4×C71—42no—
134D4×C71—42no—
135D4×C71—41no—
136D4×C71—41no—
137D4×C71—41no—
138D4×C71—81no—
139D4×C71—81no—
140C2×C4×C71—41no—
141C2×C4×C71—42no—
142C2×C4×C71—42no—
143C2×C4×C71—41no—
144C2×C4×C71—81no—
145C2×C4×C71—81no—
146C2×C4×C71C4×D411yescharacteristic
147C2×C4×C71#12011yescharacteristic
148C2×C4×C71#12111yescharacteristic
149C2×C4×C71C2×D811yescharacteristic
150C2×C4×C71#13511yescharacteristic
151C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
152C2×Dic71—41no—
153C2×Dic71—81no—
154C8×C71C2×C22⋊C411yescharacteristic
155C8×C71C4×D411yescharacteristic
156C8×C71#12111yescharacteristic
157C8×C71#12311yescharacteristic
158C71⋊C8—22no—
159C71⋊C8—42no—
160C22×C71—41no—
161C22×C71—42no—
162C22×C71—81no—
163C22×C71—161no—
164C22×C71—161no—
165C22×C71#44211yescharacteristic
166C22×C71#44911yescharacteristic
167C22×C71#45911yescharacteristic
168C4×C71—42no—
169C4×C71—81no—
170C4×C71#38611yescharacteristic
171C4×C71C4×D811yescharacteristic
172C4×C71#44711yescharacteristic
173C4×C71#46311yescharacteristic
174Dic71—81no—
175Dic71—161no—
176#13016—711nomaximal, Sylow
177C2×C71—21no—
178C2×C71—21no—
179C2×C71—21no—
180C2×C71—82no—
181C2×C71—161no—
182C2×C71#187911yescharacteristic
183C2×#508D7111yescharacteristic
184#2957—711no—
185#2145—711no—
186#2143—711no—
187#1976—711no—
188#1382—711no—
189#1379—711no—
190C71#1301611yescharacteristic, complemented, Sylow
191C2×#135C2×D7111yescharacteristic
192C2×#135C2×D7111yescharacteristic
193#568—1421no—
194#508—22no—
195#508—22no—
196C2×Q32Dic7111yescharacteristic, complemented
197C2×SD32Dic7111yescharacteristic, complemented
198C2×SD32Dic7111yescharacteristic, complemented
199C2×D16Dic7111yescharacteristic, complemented
200C2×M32—711no—
201C22×C16C2×D7111yescharacteristic
202#444—1422no—
203#443—1422no—
204C2×#108—711no—
205#427—711no—
206#421—1421no—
207#421—1421no—
208#410—1421no—
209#405—711no—
210#360—1422no—
211#359—1422no—
212C2×#41—21no—
213C2×#41—21no—
214#135—21no—
215#135—21no—
216#135—21no—
217#135—21no—
218#135—21no—
219#135—21no—
220#135C22⋊D7112yes—
221#135C22⋊D7112yes—
222C2×Q16C2×Dic7111yescharacteristic
223C2×Q16C2×Dic7111yescharacteristic
224C2×SD16—21no—
225C2×SD16—21no—
226C2×D8C2×Dic7111yescharacteristic
227C2×D8C2×Dic7111yescharacteristic
228#131—2842no—
229#131—1424no—
230C2×M16—2842no—
231C2×M16—711no—
232C2×M16—1421no—
233C22×C8C22×D7111yescharacteristic
234#117—711no—
235#117—1421no—
236Q32—22no—
237SD32—22no—
238SD32—22no—
239D16—22no—
240M32—1421no—
241M32—1421no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—1421no—
245C2×C16C2×Dic7111yescharacteristic
246C2×C16C2×Dic7111yescharacteristic
247#108—712no—
248#108—1421no—
249#107—1421no—
250#106—1421no—
251#104—2842no—
252#104—1424no—
253#99—2842no—
254#97—1421no—
255C4×C8—1421no—
256#41—22no—
257#41—22no—
258#41—41no—
259#41—41no—
260#41—41no—
261#41—41no—
262#41—41no—
263#41—41no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—41no—
269C2×D4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4D4×D7111yescharacteristic
273Q16—21no—
274Q16—21no—
275Q16—21no—
276Q16C22⋊Dic7112yes—
277SD16—22no—
278SD16—22no—
279SD16—41no—
280SD16—41no—
281D8—21no—
282D8—21no—
283D8—21no—
284D8C22⋊Dic7112yes—
285M16—2841no—
286M16—2844no—
287M16—712no—
288M16—1422no—
289M16—1421no—
290C2×C8—2841no—
291C2×C8—2844no—
292C2×C8—1421no—
293C2×C8C22×Dic7111yescharacteristic
294C2×C8D4×D7111yescharacteristic
295C2×C8C2×C22⋊D7111yescharacteristic
296C2×C8C2×C22⋊D7111yescharacteristic
297C2×C8#1176586411yescharacteristic
298C2×C8#1176586411yescharacteristic
299C4⋊C4—2841no—
300C4⋊C4—1421no—
301C22⋊C4—2841no—
302C22⋊C4—1421no—
303C42—2841no—
304C42—1421no—
305C16—21no—
306C16—21no—
307C16—2841no—
308C23—41no—
309C23—41no—
310C23#1455615211yescharacteristic
311Q8—22no—
312Q8—41no—
313Q8—41no—
314Q8—41no—
315D4—22no—
316D4—42no—
317D4—42no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—81no—
322D4—81no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—2841no—
330C2×C4—5681no—
331C2×C4D4×Dic7111yescharacteristic
332C2×C4#1455609611yescharacteristic
333C2×C4#1455610011yescharacteristic
334C2×C4D8×D7111yescharacteristic
335C2×C4#1455614711yescharacteristic
336C2×C4#1455615211yescharacteristic
337C8—2842no—
338C8—1422no—
339C8C2×C22⋊Dic7111yescharacteristic
340C8D4×Dic7111yescharacteristic
341C8#1455610011yescharacteristic
342C8#1455610911yescharacteristic
343C22—41no—
344C22—42no—
345C22—81no—
346C22—161no—
347C22—161no—
348C22#1798611111yescharacteristic
349C22#1798614111yescharacteristic
350C22#1798619211yescharacteristic
351C4—42no—
352C4—81no—
353C4—5681no—
354C4—11361no—
355C4#1798587911yescharacteristic
356C4D8×Dic7111yescharacteristic
357C4#1798613111yescharacteristic
358C4#1798621211yescharacteristic
359C2—21no—
360C2—21no—
361C2—21no—
362C2—82no—
363C2—161no—
364C2#2258869011yescharacteristic
subgroupG/H
centerC4#17986131
commutatorC2×C8×C71C22×C4
FrattiniC22×C8C22×D71
Fitting, prime radicalC2×#508×C71C2
socleC2×C71#1879
cosocleC22×C8×C71C23
hypercenterC2×#508D71
hypocenterC71#13016
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ C2×#508

Complements

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

NH ≅ G/Ntype
1. #22597257C2semidirect
2. #22590076C2semidirect
3. #22590068C2semidirect
4. #22589239C2semidirect
5. #22586436C2semidirect
6. #22586428C2semidirect
24. C2×Q32×C71C4semidirect
25. C2×SD32×C71C4semidirect
26. C2×SD32×C71C4semidirect
27. C2×D16×C71C4semidirect
190. C71#13016semidirect
196. C2×Q32Dic71semidirect
197. C2×SD32Dic71semidirect
198. C2×SD32Dic71semidirect
199. C2×D16Dic71semidirect

Maximal subgroups

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

subgroupindex
#225972572
#225900762
#225900682
#225892392
#225864362
#225864282
C2×#508×C712
#1301671