Finite Group Databasebeta

← #31914058

Subgroups — #31914058

Order 18176.

490 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
1C2×#17985826C211yesmaximal, characteristic, complemented
2C2×#17985800C211yesmaximal, characteristic
3C2×#17985800C211yesmaximal, characteristic
4#22589073C211yesmaximal, characteristic, complemented
5#22586431C211yesmaximal, characteristic, complemented
6#22586418C211yesmaximal, characteristic, complemented
7C2×C4×D8×C71C211yesmaximal, characteristic
8C22×C71⋊C16C2211yescharacteristic
9C22×C71⋊C16C2211yescharacteristic
10C2×#14556062C2211yescharacteristic
11C2×#14556062C2211yescharacteristic
12#17985897—21no—
13#17985826—22no—
14#17985800—22no—
15#17985800C414yescomplemented
16#17985762—21no—
17C22×D8×C71C2211yescharacteristic
18C2×C4×D4×C71—21no—
19C4×D8×C71—22no—
20C4×D8×C71—22no—
21C2×#107×C71C2211yescharacteristic
22C2×#102×C71—21no—
23C2×C4×C8×C71C2211yescharacteristic
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71—21no—
26C2×Dic71⋊C4—41no—
27C2×C71⋊C16—21no—
28C2×C71⋊C16—21no—
29C2×C71⋊C16—21no—
30C2×C71⋊C16—21no—
31C2×C71⋊C16—21no—
32C2×C71⋊C16C2×C412yescomplemented
33#14556062—21no—
34#14556062—21no—
35#14556062—21no—
36#14556062C2×C412yescomplemented
37C71⋊#95—41no—
38C22×D4×C71—21no—
39C23×C4×C71—41no—
40C2×D8×C71—21no—
41C2×D8×C71—21no—
42C2×D8×C71—21no—
43C2×D8×C71C2×C412yes—
44C2×D8×C71D412yes—
45C2×D8×C71Q812yes—
46C22×C8×C71D411yescharacteristic
47C22×C8×C71C2311yescharacteristic
48C4×D4×C71—24no—
49C4×D4×C71—42no—
50C2×C4⋊C4×C71—21no—
51C2×C22⋊C4×C71—41no—
52C2×C42×C71D411yescharacteristic
53#107×C71—21no—
54#107×C71D412yes—
55#102×C71—24no—
56C4×C8×C71—21no—
57C4×C8×C71—21no—
58C22×Dic71—41no—
59C22×Dic71—41no—
60C4⋊Dic71—22no—
61C4⋊Dic71—41no—
62C4⋊Dic71—41no—
63C4⋊Dic71—41no—
64Dic71⋊C4—82no—
65C71⋊C16—22no—
66C71⋊C16—41no—
67C24×C71—41no—
68C2×D4×C71—22no—
69C2×D4×C71—22no—
70C2×D4×C71—22no—
71C2×D4×C71—41no—
72C2×D4×C71—41no—
73C2×D4×C71—41no—
74C22×C4×C71—44no—
75C22×C4×C71—41no—
76C22×C4×C71—82no—
77C22×C4×C71—82no—
78C22×C4×C71D811yescharacteristic
79C22×C4×C71D811yescharacteristic
80C22×C4×C71C2×D411yescharacteristic
81D8×C71—22no—
82D8×C71—22no—
83D8×C71—22no—
84D8×C71C4⋊C414yescomplemented
85C2×C8×C71—21no—
86C2×C8×C71—21no—
87C2×C8×C71—21no—
88C2×C8×C71C22×C411yescharacteristic
89C2×C8×C71C2×D411yescharacteristic
90C2×C8×C71C2×D411yescharacteristic
91C2×C8×C71C2×Q811yescharacteristic
92C2×C8×C71#4111yescharacteristic
93C2×C8×C71#4111yescharacteristic
94C4⋊C4×C71—22no—
95C4⋊C4×C71—41no—
96C22⋊C4×C71—44no—
97C42×C71—21no—
98C42×C71—21no—
99C23×C71—42no—
100C23×C71—42no—
101C23×C71—42no—
102C23×C71—81no—
103C23×C71—81no—
104C23×C71—81no—
105C23×C71—81no—
106C23×C71C2×D811yescharacteristic
107D4×C71—24no—
108D4×C71—42no—
109D4×C71—42no—
110D4×C71—42no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—42no—
117C2×C4×C71—88no—
118C2×C4×C71—81no—
119C2×C4×C71—81no—
120C2×C4×C71SD3212yes—
121C2×C4×C71C4×D411yescharacteristic
122C2×C4×C71#12011yescharacteristic
123C2×C4×C71C22⋊Q811yescharacteristic
124C2×C4×C71#13511yescharacteristic
125C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C71#13711yescharacteristic
127C2×Dic71—42no—
128C2×Dic71—81no—
129C2×Dic71—81no—
130C2×Dic71—81no—
131C2×Dic71—81no—
132C2×Dic71—81no—
133C8×C71—41no—
134C8×C71C2×C4⋊C411yescharacteristic
135C8×C71C4×D411yescharacteristic
136C8×C71C22⋊Q811yescharacteristic
137C8×C71#12311yescharacteristic
138C22×C71—44no—
139C22×C71—82no—
140C22×C71—82no—
141C22×C71—82no—
142C22×C71—82no—
143C22×C71—82no—
144C22×C71—82no—
145C22×C71C4×D811yescharacteristic
146C22×C71#44711yescharacteristic
147C22×C71#47411yescharacteristic
148C22×C71C2×SD3211yescharacteristic
149C22×C71#50811yescharacteristic
150C22×C71#50911yescharacteristic
151C22×C71#50911yescharacteristic
152C4×C71—22no—
153C4×C71—41no—
154C4×C71—82no—
155C4×C71#38511yescharacteristic
156C4×C71#44011yescharacteristic
157C4×C71#45211yescharacteristic
158C4×C71#47811yescharacteristic
159Dic71—82no—
160Dic71—161no—
161#12937—711nomaximal, Sylow
162C2×C71—84no—
163C2×C71#186211yescharacteristic
164C2×C71C4×SD3211yescharacteristic
165C2×C71#217711yescharacteristic
166C2×C71#218611yescharacteristic
167C2×C71#218711yescharacteristic
168C2×C71#222611yescharacteristic
169C2×C71#222811yescharacteristic
170C2×C4×D8D7111yescharacteristic
171C2×#367—711no—
172C2×#357—711no—
173C2×#357—711no—
174#1942—711no—
175#1380—711no—
176#1374—711no—
177C71#1293711yescharacteristic, complemented, Sylow
178C22×D8C2×D7111yescharacteristic
179C2×C4×D4—21no—
180C22×C16—711no—
181C22×C16—711no—
182C4×D8—22no—
183C4×D8—22no—
184C2×#107—711no—
185C2×#107—711no—
186C2×#107C2×D7111yescharacteristic
187C2×#102—21no—
188C2×C4×C8C2×D7111yescharacteristic
189#389—1421no—
190#367—1422no—
191#357—714no—
192#357—1422no—
193#340—1421no—
194C22×D4—21no—
195C23×C4—41no—
196C2×D8—21no—
197C2×D8—21no—
198C2×D8—21no—
199C2×D8Dic14212yes—
200C2×D8C4×D7112yes—
201C2×D8C22⋊D7112yes—
202C22×C8C22⋊D7111yescharacteristic
203C22×C8C22×D7111yescharacteristic
204C4×D4—24no—
205C4×D4—42no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—2841no—
208C2×C4⋊C4—1421no—
209C2×C4⋊C4—1421no—
210C2×C22⋊C4—41no—
211C2×C42C22⋊D7111yescharacteristic
212C2×C16—712no—
213C2×C16—1421no—
214C2×C16—1421no—
215C2×C16—1421no—
216C2×C16—1421no—
217C2×C16—1421no—
218#107—21no—
219#107—712no—
220#107—1421no—
221#107—1421no—
222#107—1421no—
223#107C22⋊D7112yes—
224#102—24no—
225C4×C8—21no—
226C4×C8—21no—
227#95—2841no—
228C24—41no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C22×C4—44no—
236C22×C4—41no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—2841no—
240C22×C4—2841no—
241C22×C4#1176585011yescharacteristic
242C22×C4#1176585011yescharacteristic
243C22×C4C2×C22⋊D7111yescharacteristic
244D8—22no—
245D8—22no—
246D8—22no—
247D8Dic71⋊C414yescomplemented
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8C2×C4×D7111yescharacteristic
252C2×C8C2×C22⋊D7111yescharacteristic
253C2×C8C2×C22⋊D7111yescharacteristic
254C2×C8C2×Dic14211yescharacteristic
255C2×C8#1176586511yescharacteristic
256C2×C8#1176586511yescharacteristic
257C4⋊C4—22no—
258C4⋊C4—41no—
259C4⋊C4—2841no—
260C4⋊C4—2841no—
261C4⋊C4—2841no—
262C4⋊C4—5682no—
263C4⋊C4—1422no—
264C22⋊C4—44no—
265C42—21no—
266C42—21no—
267C16—2841no—
268C16—1422no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—81no—
273C23—81no—
274C23—81no—
275C23—81no—
276C23C2×#1176585011yescharacteristic
277D4—24no—
278D4—42no—
279D4—42no—
280D4—42no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—88no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—2842no—
291C2×C4—5681no—
292C2×C4—5681no—
293C2×C4—5681no—
294C2×C4—5681no—
295C2×C4—5681no—
296C2×C4#1455607212yes—
297C2×C4C4×C22⋊D7111yescharacteristic
298C2×C4#1455609711yescharacteristic
299C2×C4#1455610711yescharacteristic
300C2×C4#1455614811yescharacteristic
301C2×C4#1455615411yescharacteristic
302C2×C4#1455615811yescharacteristic
303C8—41no—
304C8C2×Dic71⋊C411yescharacteristic
305C8C4×C22⋊D7111yescharacteristic
306C8#1455610711yescharacteristic
307C8#1455611111yescharacteristic
308C22—44no—
309C22—82no—
310C22—82no—
311C22—82no—
312C22—82no—
313C22—82no—
314C22—82no—
315C22C4×#1176585011yescharacteristic
316C22#1798613311yescharacteristic
317C22#1798627911yescharacteristic
318C22C2×#1455607211yescharacteristic
319C22#1798642911yescharacteristic
320C22#1798643411yescharacteristic
321C22#1798643511yescharacteristic
322C4—22no—
323C4—41no—
324C4—82no—
325C4—5682no—
326C4—11361no—
327C4#1798587211yescharacteristic
328C4#1798609911yescharacteristic
329C4#1798616111yescharacteristic
330C4#1798629911yescharacteristic
331C2—84no—
332C2#2258858711yescharacteristic
333C2C4×#1455607211yescharacteristic
334C2#2259021411yescharacteristic
335C2#2259026311yescharacteristic
336C2#2259026911yescharacteristic
337C2#2259048711yescharacteristic
338C2#2259049711yescharacteristic
subgroupG/H
centerC23C2×#11765850
commutatorC2×C8×C71C22×C4
FrattiniC22×C8C22×D71
FittingC2×C4×D8×C71C2
socleC23×C71C2×D8
prime radicalC22×D8×C71C22
cosocleC22×C8×C71C23
hypercenterC2×C4×D8D71
hypocenterC71#12937
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#17985826C2semidirect
4. #22589073C2semidirect
5. #22586431C2semidirect
6. #22586418C2semidirect
15. #17985800C4semidirect
32. C2×C71⋊C16C2×C4semidirect
36. #14556062C2×C4semidirect
84. D8×C71C4⋊C4semidirect
177. C71#12937semidirect
247. D8Dic71⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#179858262
C2×#179858002
#225890732
#225864312
#225864182
C2×C4×D8×C712
#1293771