Finite Group Databasebeta

← #31893551

Subgroups — #31893551

Order 18176.

456 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22597307C211yesmaximal, characteristic
2#22588360C211yesmaximal, characteristic
3#22586839C211yesmaximal, characteristic, complemented
4#22586795C211yesmaximal, characteristic, complemented
5#22586237C211yesmaximal, characteristic, complemented
6#22586179C211yesmaximal, characteristic, complemented
7C71×#3076C211yesmaximal, characteristic
8#17986130—21no—
9#17986077—21no—
10#17986071—22no—
11#17986052—21no—
12#17986052C2211yescharacteristic
13#17986023C2211yescharacteristic
14C2×#14556055—21no—
15C4×#11765849—21no—
16C2×C4×C71⋊C8C2211yescharacteristic
17C4×C4⋊Dic71C2211yescharacteristic
18#17985758—21no—
19#17985736—22no—
20Q8⋊(C71⋊C8)—21no—
21Q8⋊(C71⋊C8)—21no—
22#533×C71C2211yescharacteristic
23C4×#41×C71C2211yescharacteristic
24#479×C71—21no—
25#478×C71—21no—
26#477×C71C411yescharacteristic, complemented
27#476×C71C411yescharacteristic, complemented
28#475×C71C411yescharacteristic, complemented
29#474×C71C411yescharacteristic, complemented
30#427×C71—21no—
31C2×C4⋊C8×C71C2211yescharacteristic
32#416×C71—21no—
33C2×#11765849—22no—
34C2×#11765849—21no—
35C22×C71⋊C8—21no—
36C2×C4⋊Dic71—21no—
37C2×C4×Dic71—21no—
38C4⋊(C71⋊C8)—22no—
39C4⋊(C71⋊C8)—21no—
40C4⋊(C71⋊C8)—21no—
41C4⋊(C71⋊C8)—21no—
42#14556055—42no—
43C22⋊(C71⋊C8)—22no—
44C22⋊(C71⋊C8)D412yes—
45#14556039—22no—
46C4×C71⋊C8—21no—
47C4×C71⋊C8—21no—
48C4×C71⋊C8D411yescharacteristic
49C4×C71⋊C8D411yescharacteristic
50C71⋊#95—21no—
51C2×#41×C71D411yescharacteristic
52C22×C8×C71—21no—
53C4⋊Q8×C71C2×C411yescharacteristic
54C4⋊Q8×C71C2×C411yescharacteristic
55#125×C71—21no—
56C22⋊Q8×C71—21no—
57C22⋊Q8×C71—21no—
58C4×Q8×C71—21no—
59C4×Q8×C71—21no—
60C4×Q8×C71C2×C411yescharacteristic
61C4×Q8×C71D411yescharacteristic
62C4×D4×C71—21no—
63C4×D4×C71—21no—
64C4×D4×C71C2×C411yescharacteristic
65C4×D4×C71D411yescharacteristic
66#117×C71—21no—
67#117×C71—21no—
68#117×C71D411yescharacteristic
69C2×C42×C71—21no—
70C2×C42×C71C2311yescharacteristic
71#107×C71—21no—
72#107×C71—21no—
73#106×C71—21no—
74#106×C71—21no—
75C4⋊C8×C71—21no—
76C4⋊C8×C71C2×C411yescharacteristic
77C4⋊C8×C71C2×C411yescharacteristic
78#103×C71—21no—
79#103×C71—21no—
80#102×C71—21no—
81#102×C71—21no—
82C22×Dic71—41no—
83#11765849—44no—
84#11765849—41no—
85#11765849—41no—
86C2×C71⋊C8—21no—
87C2×C71⋊C8—21no—
88C2×C71⋊C8—22no—
89C2×C71⋊C8—42no—
90C2×C71⋊C8—41no—
91C4⋊Dic71—22no—
92C4⋊Dic71—41no—
93C4×Dic71—41no—
94C4×Dic71—41no—
95#41×C71—21no—
96#41×C71—21no—
97#41×C71—21no—
98#41×C71—21no—
99C2×Q8×C71—21no—
100C2×Q8×C71—21no—
101C2×Q8×C71C22⋊C411yescharacteristic
102C2×D4×C71—21no—
103C2×D4×C71C22⋊C411yescharacteristic
104C22×C4×C71—21no—
105C22×C4×C71—21no—
106C22×C4×C71—21no—
107C22×C4×C71C2×D411yescharacteristic
108C22×C4×C71C2×D411yescharacteristic
109C22×C4×C71#4111yescharacteristic
110C2×C8×C71—21no—
111C2×C8×C71—21no—
112C2×C8×C71—41no—
113C2×C8×C71—41no—
114C4⋊C4×C71—21no—
115C4⋊C4×C71—21no—
116C4⋊C4×C71—21no—
117C4⋊C4×C71—21no—
118C4⋊C4×C71—21no—
119C4⋊C4×C71—21no—
120C4⋊C4×C71—41no—
121C4⋊C4×C71—41no—
122C4⋊C4×C71C22⋊C411yescharacteristic
123C4⋊C4×C71C22⋊C411yescharacteristic
124C22⋊C4×C71—21no—
125C22⋊C4×C71—21no—
126C22⋊C4×C71—21no—
127C22⋊C4×C71—41no—
128C42×C71—21no—
129C42×C71—21no—
130C42×C71—21no—
131C42×C71—41no—
132C42×C71C22×C411yescharacteristic
133C42×C71C2×D411yescharacteristic
134C42×C71C2×D411yescharacteristic
135C42×C71#4111yescharacteristic
136C23×C71—21no—
137C23×C71#12111yescharacteristic
138Q8×C71—21no—
139Q8×C71—21no—
140Q8×C71—41no—
141Q8×C71—41no—
142Q8×C71—81no—
143D4×C71—21no—
144D4×C71—21no—
145D4×C71—41no—
146D4×C71—41no—
147C2×C4×C71—21no—
148C2×C4×C71—21no—
149C2×C4×C71—21no—
150C2×C4×C71—21no—
151C2×C4×C71—21no—
152C2×C4×C71—41no—
153C2×C4×C71—42no—
154C2×C4×C71—41no—
155C2×C4×C71—41no—
156C2×C4×C71—41no—
157C2×C4×C71—41no—
158C2×C4×C71—41no—
159C2×C4×C71—41no—
160C2×C4×C71C2×C22⋊C411yescharacteristic
161C2×C4×C71C4×D411yescharacteristic
162C2×C4×C71C4×D411yescharacteristic
163C2×C4×C71#12011yescharacteristic
164C2×C4×C71#12111yescharacteristic
165C2×C4×C71#12311yescharacteristic
166C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C71#13711yescharacteristic
168C2×C4×C71#13711yescharacteristic
169C2×C4×C71#13711yescharacteristic
170C2×Dic71—42no—
171C2×Dic71—81no—
172C2×Dic71—81no—
173C8×C71—41no—
174C8×C71—41no—
175C71⋊C8—22no—
176C71⋊C8—41no—
177C71⋊C8—42no—
178C71⋊C8—81no—
179C22×C71—21no—
180C22×C71—21no—
181C22×C71—41no—
182C22×C71—41no—
183C22×C71—41no—
184C22×C71#38611yescharacteristic
185C22×C71#46811yescharacteristic
186C22×C71#47011yescharacteristic
187C4×C71—21no—
188C4×C71—21no—
189C4×C71—21no—
190C4×C71—41no—
191C4×C71—41no—
192C4×C71—41no—
193C4×C71—81no—
194C4×C71—81no—
195C4×C71#41711yescharacteristic
196C4×C71#41911yescharacteristic
197C4×C71#44311yescharacteristic
198C4×C71#44411yescharacteristic
199C4×C71#46811yescharacteristic
200C4×C71#47011yescharacteristic
201Dic71—82no—
202#9209—711nomaximal, Sylow
203C2×C71—21no—
204C2×C71—21no—
205C2×C71—81no—
206C2×C71#187611yescharacteristic
207C2×C71#189411yescharacteristic
208C2×C71#190611yescharacteristic
209#3076D7111yescharacteristic
210#2965—711no—
211#1806—711no—
212#1503—711no—
213#1493—711no—
214#1301—711no—
215#1279—711no—
216C71#920911yescharacteristic, complemented, Sylow
217#533C2×D7111yescharacteristic
218C4×#41C2×D7111yescharacteristic
219#479—21no—
220#478—21no—
221#477Dic7111yescharacteristic, complemented
222#476Dic7111yescharacteristic, complemented
223#475Dic7111yescharacteristic, complemented
224#474Dic7111yescharacteristic, complemented
225#446—1421no—
226#436—1421no—
227#435—1422no—
228#431—711no—
229#431—1421no—
230#427—21no—
231#423—711no—
232C2×C4⋊C8C2×D7111yescharacteristic
233C2×#104—1421no—
234#416—21no—
235C4×M16—1421no—
236C2×C4×C8—711no—
237C4×C4⋊C4—711no—
238#339—1421no—
239#328—1422no—
240Q8⋊C8—1421no—
241Q8⋊C8—1421no—
242C2×#41C22⋊D7111yescharacteristic
243C2×M16—1421no—
244C2×M16—1422no—
245C22×C8—21no—
246C22×C8—1421no—
247C4⋊Q8C2×Dic7111yescharacteristic
248C4⋊Q8C2×Dic7111yescharacteristic
249#125—21no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8C2×Dic7111yescharacteristic
255C4×Q8C22⋊D7111yescharacteristic
256C4×D4—21no—
257C4×D4—21no—
258C4×D4C2×Dic7111yescharacteristic
259C4×D4C22⋊D7111yescharacteristic
260#117—21no—
261#117—21no—
262#117C22⋊D7111yescharacteristic
263C2×C4⋊C4—1421no—
264C2×C42—21no—
265C2×C42—1421no—
266C2×C42C22×D7111yescharacteristic
267#107—21no—
268#107—21no—
269#106—21no—
270#106—21no—
271C4⋊C8—21no—
272C4⋊C8—1422no—
273C4⋊C8—1421no—
274C4⋊C8—1421no—
275C4⋊C8—1421no—
276C4⋊C8C2×Dic7111yescharacteristic
277C4⋊C8C2×Dic7111yescharacteristic
278#104—2842no—
279#103—21no—
280#103—21no—
281#102—21no—
282#102—21no—
283C22⋊C8—712no—
284C22⋊C8—1422no—
285#97—1422no—
286C4×C8—711no—
287C4×C8—711no—
288C4×C8—1421no—
289C4×C8—1421no—
290#95—1421no—
291#41—21no—
292#41—21no—
293#41—21no—
294#41—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8C22⋊Dic7111yescharacteristic
298C2×D4—21no—
299C2×D4C22⋊Dic7111yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—2841no—
304C22×C4D4×D7111yescharacteristic
305C22×C4C2×C22⋊D7111yescharacteristic
306C22×C4#1176586411yescharacteristic
307M16—2844no—
308M16—2841no—
309M16—2841no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—2842no—
315C2×C8—2841no—
316C2×C8—1421no—
317C2×C8—1422no—
318C2×C8—1421no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—2841no—
328C4⋊C4—1422no—
329C4⋊C4C22⋊Dic7111yescharacteristic
330C4⋊C4C22⋊Dic7111yescharacteristic
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—41no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—41no—
339C42—2841no—
340C42—2841no—
341C42C22×Dic7111yescharacteristic
342C42D4×D7111yescharacteristic
343C42C2×C22⋊D7111yescharacteristic
344C42#1176586411yescharacteristic
345C23—21no—
346C23#1455610011yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—41no—
350Q8—41no—
351Q8—81no—
352D4—21no—
353D4—21no—
354D4—41no—
355D4—41no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—2842no—
370C2×C4—5681no—
371C2×C4—5681no—
372C2×C4C2×C22⋊Dic7111yescharacteristic
373C2×C4D4×Dic7111yescharacteristic
374C2×C4D4×Dic7111yescharacteristic
375C2×C4#1455609611yescharacteristic
376C2×C4#1455610011yescharacteristic
377C2×C4#1455610911yescharacteristic
378C2×C4#1455615311yescharacteristic
379C2×C4#1455615811yescharacteristic
380C2×C4#1455615911yescharacteristic
381C2×C4#1455615911yescharacteristic
382C8—41no—
383C8—41no—
384C8—2841no—
385C8—2842no—
386C8—5681no—
387C8—1422no—
388C22—21no—
389C22—21no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22#1798587911yescharacteristic
394C22#1798624711yescharacteristic
395C22#1798625511yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—81no—
403C4—81no—
404C4—5682no—
405C4#1798600011yescharacteristic
406C4#1798600811yescharacteristic
407C4#1798611611yescharacteristic
408C4#1798612111yescharacteristic
409C4#1798624711yescharacteristic
410C4#1798625511yescharacteristic
411C2—21no—
412C2—21no—
413C2—81no—
414C2#2258866911yescharacteristic
415C2#2258877411yescharacteristic
416C2#2258884311yescharacteristic
subgroupG/H
centerC2×C4#14556100
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#3076C2
socleC22×C71#386
prime radicalC2×#41×C71D4
cosocleC2×C42×C71C23
hypercenter#3076D71
hypocenterC71#9209
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
3. #22586839C2semidirect
4. #22586795C2semidirect
5. #22586237C2semidirect
6. #22586179C2semidirect
26. #477×C71C4semidirect
27. #476×C71C4semidirect
28. #475×C71C4semidirect
29. #474×C71C4semidirect
216. C71#9209semidirect
221. #477Dic71semidirect
222. #476Dic71semidirect
223. #475Dic71semidirect
224. #474Dic71semidirect

Maximal subgroups

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

subgroupindex
#225973072
#225883602
#225868392
#225867952
#225862372
#225861792
C71×#30762
#920971