Finite Group Databasebeta

← #31892696

Subgroups — #31892696

Order 18176.

432 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22597289C211yesmaximal, characteristic
2#22589285C211yesmaximal, characteristic
3#22586839C211yesmaximal, characteristic
4#22586743C211yesmaximal, characteristic
5#22586180C211yesmaximal, characteristic
6#22586177C211yesmaximal, characteristic
7C71×#3170C211yesmaximal, characteristic
8#17986130—21no—
9#17986130—22no—
10Q8×C71⋊C8—21no—
11#17986052C2211yescharacteristic
12#17986026—21no—
13C2×C4⋊(C71⋊C8)C2211yescharacteristic
14#17985955—21no—
15C4×#11765849C2211yescharacteristic
16C4×C4⋊Dic71C2211yescharacteristic
17#17985758—21no—
18#17985758—21no—
19#17985736—22no—
20Q8⋊(C71⋊C8)—21no—
21Q8⋊(C71⋊C8)—21no—
22#533×C71C2211yescharacteristic
23#533×C71C2211yescharacteristic
24#500×C71C411yescharacteristic, complemented
25#494×C71C411yescharacteristic, complemented
26#487×C71C411yescharacteristic, complemented
27#487×C71C411yescharacteristic, complemented
28#467×C71—22no—
29C4×Q16×C71—22no—
30C2×C4×C8×C71C2211yescharacteristic
31C2×#11765849D411yescharacteristic
32C2×#11765849D411yescharacteristic
33C22×C71⋊C8—21no—
34C2×C4⋊Dic71—21no—
35C2×C4×Dic71—21no—
36C4⋊(C71⋊C8)—22no—
37C4⋊(C71⋊C8)—22no—
38C4⋊(C71⋊C8)—21no—
39C4⋊(C71⋊C8)—21no—
40C4⋊(C71⋊C8)D411yescharacteristic
41C4⋊(C71⋊C8)D411yescharacteristic
42C22⋊(C71⋊C8)—21no—
43#14556039—22no—
44#14556039—21no—
45C4×C71⋊C8—22no—
46C4×C71⋊C8—21no—
47C4×C71⋊C8—21no—
48C71⋊#95—21no—
49C2×Q16×C71—22no—
50C22×C8×C71—21no—
51C4⋊Q8×C71C2×C411yescharacteristic
52C4⋊Q8×C71C2×C411yescharacteristic
53C4⋊Q8×C71C2×C411yescharacteristic
54C4⋊Q8×C71C2×C411yescharacteristic
55#125×C71—21no—
56#125×C71—21no—
57C22⋊Q8×C71—21no—
58C22⋊Q8×C71—21no—
59C22⋊Q8×C71—21no—
60C22⋊Q8×C71D412yes—
61C4×Q8×C71—21no—
62C4×Q8×C71—21no—
63C4×Q8×C71—21no—
64C4×Q8×C71D412yes—
65#117×C71—21no—
66#117×C71—21no—
67C2×C42×C71C2311yescharacteristic
68#107×C71—22no—
69#103×C71—22no—
70#103×C71—22no—
71C4×C8×C71—21no—
72C4×C8×C71C2×C411yescharacteristic
73C4×C8×C71C2×C411yescharacteristic
74C22×Dic71—41no—
75#11765849—21no—
76#11765849—21no—
77#11765849—21no—
78#11765849Q1612yescomplemented
79C2×C71⋊C8—21no—
80C2×C71⋊C8—21no—
81C2×C71⋊C8—21no—
82C2×C71⋊C8—21no—
83C2×C71⋊C8—41no—
84C2×C71⋊C8—41no—
85C2×C71⋊C8—41no—
86C4⋊Dic71—22no—
87C4⋊Dic71—41no—
88C4×Dic71—41no—
89C4×Dic71—41no—
90C2×Q8×C71—21no—
91C2×Q8×C71—21no—
92C2×Q8×C71—21no—
93C2×Q8×C71C22⋊C412yes—
94C22×C4×C71C2×D411yescharacteristic
95C22×C4×C71C2×D411yescharacteristic
96C22×C4×C71#4111yescharacteristic
97Q16×C71—44no—
98C2×C8×C71—21no—
99C2×C8×C71—21no—
100C2×C8×C71—42no—
101C4⋊C4×C71—21no—
102C4⋊C4×C71—21no—
103C4⋊C4×C71—21no—
104C4⋊C4×C71—22no—
105C4⋊C4×C71—22no—
106C4⋊C4×C71—21no—
107C4⋊C4×C71—21no—
108C4⋊C4×C71—21no—
109C4⋊C4×C71—21no—
110C4⋊C4×C71—41no—
111C4⋊C4×C71—41no—
112C4⋊C4×C71C22⋊C412yes—
113C22⋊C4×C71—22no—
114C22⋊C4×C71—41no—
115C42×C71—22no—
116C42×C71—41no—
117C42×C71C22×C411yescharacteristic
118C42×C71C2×D411yescharacteristic
119C42×C71C2×D411yescharacteristic
120C42×C71#4111yescharacteristic
121C23×C71#12111yescharacteristic
122Q8×C71—42no—
123Q8×C71—42no—
124Q8×C71—41no—
125Q8×C71—41no—
126Q8×C71—41no—
127Q8×C71—41no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—22no—
131C2×C4×C71—22no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71—41no—
135C2×C4×C71C2×C22⋊C411yescharacteristic
136C2×C4×C71C4×D411yescharacteristic
137C2×C4×C71C4×D411yescharacteristic
138C2×C4×C71#12011yescharacteristic
139C2×C4×C71#12111yescharacteristic
140C2×C4×C71#12311yescharacteristic
141C2×C4×C71C2×Q1611yescharacteristic
142C2×C4×C71#13511yescharacteristic
143C2×C4×C71#13711yescharacteristic
144C2×C4×C71#13711yescharacteristic
145C2×Dic71—42no—
146C2×Dic71—81no—
147C2×Dic71—81no—
148C8×C71—22no—
149C8×C71—41no—
150C71⋊C8—22no—
151C71⋊C8—41no—
152C71⋊C8—41no—
153C71⋊C8—41no—
154C71⋊C8—81no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71#38611yescharacteristic
158C22×C71#46211yescharacteristic
159C22×C71#46411yescharacteristic
160C4×C71—21no—
161C4×C71—21no—
162C4×C71—21no—
163C4×C71—42no—
164C4×C71—42no—
165C4×C71—81no—
166C4×C71—81no—
167C4×C71C4×Q1611yescharacteristic
168C4×C71#44111yescharacteristic
169C4×C71#44311yescharacteristic
170C4×C71#44411yescharacteristic
171C4×C71#45111yescharacteristic
172C4×C71#45211yescharacteristic
173Dic71—82no—
174#9056—711nomaximal, Sylow
175C2×C71—21no—
176C2×C71—21no—
177C2×C71#187611yescharacteristic
178C2×C71#187811yescharacteristic
179C2×C71#188411yescharacteristic
180#3170D7111yescharacteristic
181#2962—711no—
182#1986—711no—
183#1503—711no—
184#1481—711no—
185#1279—711no—
186#1278—711no—
187C71#905611yescharacteristic, complemented, Sylow
188#533C2×D7111yescharacteristic
189#533C2×D7111yescharacteristic
190#500Dic7111yescharacteristic, complemented
191#494Dic7111yescharacteristic, complemented
192#487Dic7111yescharacteristic, complemented
193#487Dic7111yescharacteristic, complemented
194#467—22no—
195#446—1421no—
196#446—1422no—
197C8×Q8—1421no—
198C4×Q16—22no—
199#431—711no—
200#424—1421no—
201C2×C4⋊C8—711no—
202#405—1421no—
203C4×M16—711no—
204C2×C4×C8C2×D7111yescharacteristic
205C4×C4⋊C4—711no—
206#339—1421no—
207#339—1421no—
208#328—1422no—
209Q8⋊C8—1421no—
210Q8⋊C8—1421no—
211C2×Q16—22no—
212C2×M16—711no—
213C2×M16—711no—
214C22×C8—21no—
215C22×C8—1421no—
216C4⋊Q8C2×Dic7111yescharacteristic
217C4⋊Q8C2×Dic7111yescharacteristic
218C4⋊Q8C2×Dic7111yescharacteristic
219C4⋊Q8C2×Dic7111yescharacteristic
220#125—21no—
221#125—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C22⋊Q8C22⋊D7112yes—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8C22⋊D7112yes—
230#117—21no—
231#117—21no—
232C2×C4⋊C4—1421no—
233C2×C42—1421no—
234C2×C42C22×D7111yescharacteristic
235#107—22no—
236C4⋊C8—711no—
237C4⋊C8—711no—
238C4⋊C8—1422no—
239C4⋊C8—1422no—
240C4⋊C8—1421no—
241C4⋊C8—1421no—
242#103—22no—
243#103—22no—
244C22⋊C8—1421no—
245#97—1422no—
246#97—1421no—
247C4×C8—21no—
248C4×C8—1422no—
249C4×C8—1421no—
250C4×C8—1421no—
251C4×C8C2×Dic7111yescharacteristic
252C4×C8C2×Dic7111yescharacteristic
253#95—1421no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8C22⋊Dic7112yes—
258C22×C4—2841no—
259C22×C4D4×D7111yescharacteristic
260C22×C4C2×C22⋊D7111yescharacteristic
261C22×C4#1176586411yescharacteristic
262Q16—44no—
263M16—712no—
264M16—1421no—
265M16—1421no—
266M16—1421no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—42no—
270C2×C8—2841no—
271C2×C8—2841no—
272C2×C8—2841no—
273C2×C8—1421no—
274C2×C8—1421no—
275C2×C8—1421no—
276C2×C8—1421no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—2841no—
289C4⋊C4—1422no—
290C4⋊C4C22⋊Dic7112yes—
291C22⋊C4—22no—
292C22⋊C4—41no—
293C42—22no—
294C42—41no—
295C42—2841no—
296C42—2841no—
297C42C22×Dic7111yescharacteristic
298C42D4×D7111yescharacteristic
299C42C2×C22⋊D7111yescharacteristic
300C42#1176586411yescharacteristic
301C23#1455610011yescharacteristic
302Q8—42no—
303Q8—42no—
304Q8—41no—
305Q8—41no—
306Q8—41no—
307Q8—41no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—2842no—
316C2×C4—5681no—
317C2×C4—5681no—
318C2×C4C2×C22⋊Dic7111yescharacteristic
319C2×C4D4×Dic7111yescharacteristic
320C2×C4D4×Dic7111yescharacteristic
321C2×C4#1455609611yescharacteristic
322C2×C4#1455610011yescharacteristic
323C2×C4#1455610911yescharacteristic
324C2×C4Q16×D7111yescharacteristic
325C2×C4#1455615111yescharacteristic
326C2×C4#1455616011yescharacteristic
327C2×C4#1455616011yescharacteristic
328C8—22no—
329C8—41no—
330C8—2841no—
331C8—2841no—
332C8—2841no—
333C8—5681no—
334C8—1422no—
335C22—21no—
336C22—21no—
337C22#1798587911yescharacteristic
338C22#1798620711yescharacteristic
339C22#1798622311yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—42no—
344C4—42no—
345C4—81no—
346C4—81no—
347C4—5682no—
348C4Q16×Dic7111yescharacteristic
349C4#1798610611yescharacteristic
350C4#1798611611yescharacteristic
351C4#1798612111yescharacteristic
352C4#1798615311yescharacteristic
353C4#1798616011yescharacteristic
354C2—21no—
355C2—21no—
356C2#2258866911yescharacteristic
357C2#2258868311yescharacteristic
358C2#2258872411yescharacteristic
subgroupG/H
centerC2×C4#14556096
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#3170C2
socleC22×C71#386
prime radicalC23×C71#121
cosocleC2×C42×C71C23
hypercenter#3170D71
hypocenterC71#9056
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3170

Complements

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

NH ≅ G/Ntype
24. #500×C71C4semidirect
25. #494×C71C4semidirect
26. #487×C71C4semidirect
27. #487×C71C4semidirect
78. #11765849Q16semidirect
187. C71#9056semidirect
190. #500Dic71semidirect
191. #494Dic71semidirect
192. #487Dic71semidirect
193. #487Dic71semidirect

Maximal subgroups

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

subgroupindex
#225972892
#225892852
#225868392
#225867432
#225861802
#225861772
C71×#31702
#905671