Finite Group Databasebeta

← #31916123

Subgroups — #31916123

Order 18176.

556 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588328C211yesmaximal, characteristic
2C4×#14556043C211yesmaximal, characteristic
3#22586465C214yesmaximal
4C71×#2542C211yesmaximal, characteristic
5C2×#14556043C2212yes—
6C2×#14556043C2212yes—
7#17985859—21no—
8C4×C22⋊Dic71—21no—
9#17985766—21no—
10#17985766—21no—
11C2×C4⋊Q8×C71C411yescharacteristic, complemented
12C2×C22⋊Q8×C71—21no—
13C2×C22⋊Q8×C71—21no—
14C2×C4×D4×C71C2211yescharacteristic
15#399×C71—21no—
16#399×C71—21no—
17#390×C71C2212yes—
18#384×C71C412yescomplemented
19C4×C4⋊C4×C71C411yescharacteristic, complemented
20C2×C4⋊Dic71—41no—
21C2×C22⋊Dic71—22no—
22C2×C22⋊Dic71—22no—
23C2×C4×Dic71—41no—
24#14556043—22no—
25#14556043—24no—
26#14556043Q814yescomplemented
27C71⋊#95—41no—
28C71⋊#95—41no—
29C22×Q8×C71—21no—
30C22×D4×C71C2311yescharacteristic
31C23×C4×C71D411yescharacteristic
32C23×C4×C71D411yescharacteristic
33C4⋊Q8×C71—41no—
34C4⋊Q8×C71—41no—
35C22⋊Q8×C71—44no—
36C22⋊Q8×C71—44no—
37C4×D4×C71—41no—
38C4×D4×C71—41no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—22no—
41C2×C4⋊C4×C71—22no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71C2×C412yes—
44C2×C4⋊C4×C71C2×C411yescharacteristic
45C2×C22⋊C4×C71—22no—
46C2×C22⋊C4×C71—22no—
47C2×C22⋊C4×C71D411yescharacteristic
48C2×C22⋊C4×C71D411yescharacteristic
49C2×C42×C71C2×C411yescharacteristic
50C2×C42×C71C2×C412yes—
51#95×C71—21no—
52C22×Dic71—42no—
53C22×Dic71—42no—
54C4⋊Dic71—82no—
55C22⋊Dic71—24no—
56C22⋊Dic71—42no—
57C22⋊Dic71—42no—
58C22⋊Dic71—42no—
59C4×Dic71—82no—
60C24×C71C2×D411yescharacteristic
61C24×C71C2×D411yescharacteristic
62C2×Q8×C71—42no—
63C2×Q8×C71—42no—
64C2×Q8×C71—41no—
65C2×Q8×C71—41no—
66C2×D4×C71—41no—
67C2×D4×C71—41no—
68C2×D4×C71C2×Q812yes—
69C2×D4×C71#4112yes—
70C22×C4×C71—22no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—22no—
76C22×C4×C71—42no—
77C22×C4×C71—42no—
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C22⋊C411yescharacteristic
82C22×C4×C71C22×C411yescharacteristic
83C4⋊C4×C71—24no—
84C4⋊C4×C71—41no—
85C4⋊C4×C71—42no—
86C4⋊C4×C71—44no—
87C4⋊C4×C71—44no—
88C4⋊C4×C71—42no—
89C22⋊C4×C71—41no—
90C22⋊C4×C71—41no—
91C22⋊C4×C71—44no—
92C22⋊C4×C71—44no—
93C42×C71—21no—
94C42×C71—21no—
95C42×C71—42no—
96C23×C71—21no—
97C23×C71—21no—
98C23×C71—21no—
99C23×C71—21no—
100C23×C71—41no—
101C23×C71—41no—
102C23×C71—41no—
103C23×C71—41no—
104C23×C71C2×C22⋊C411yescharacteristic
105C23×C71C22⋊Q812yes—
106C23×C71#12411yescharacteristic
107C23×C71C4⋊Q811yescharacteristic
108Q8×C71—82no—
109Q8×C71—82no—
110D4×C71—42no—
111D4×C71—42no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—22no—
115C2×C4×C71—24no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—42no—
120C2×C4×C71—42no—
121C2×C4×C71—42no—
122C2×C4×C71—42no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71—42no—
127C2×C4×C71—42no—
128C2×C4×C71—42no—
129C2×C4×C71—81no—
130C2×C4×C71—81no—
131C2×C4×C71#9911yescharacteristic
132C2×C4×C71#9911yescharacteristic
133C2×C4×C71C4×Q812yes—
134C2×Dic71—44no—
135C2×Dic71—82no—
136C2×Dic71—82no—
137C2×Dic71—82no—
138C2×Dic71—82no—
139C2×Dic71—82no—
140C22×C71—21no—
141C22×C71—21no—
142C22×C71—22no—
143C22×C71—21no—
144C22×C71—21no—
145C22×C71—42no—
146C22×C71—41no—
147C22×C71—41no—
148C22×C71—41no—
149C22×C71—41no—
150C22×C71—41no—
151C22×C71—41no—
152C22×C71—81no—
153C22×C71—81no—
154C22×C71#39111yescharacteristic
155C22×C71C2×#9911yescharacteristic
156C22×C71#41011yescharacteristic
157C4×C71—21no—
158C4×C71—22no—
159C4×C71—44no—
160C4×C71—41no—
161C4×C71—41no—
162C4×C71—41no—
163C4×C71—82no—
164C4×C71#35312yes—
165Dic71—84no—
166Dic71—162no—
167#13298—711nomaximal, Sylow
168C2×C71—21no—
169C2×C71—21no—
170C2×C71—42no—
171C2×C71—41no—
172C2×C71—41no—
173C2×C71#199611yescharacteristic
174C2×C71C2×#35311yescharacteristic
175C2×C71#212911yescharacteristic
176#2542D7111yescharacteristic
177#1799—711no—
178C4×#99—711no—
179#1393—714no—
180C71#1329811yescharacteristic, complemented, Sylow
181C2×C4⋊Q8Dic7111yescharacteristic, complemented
182C2×C22⋊Q8—21no—
183C2×C22⋊Q8—21no—
184C2×C4×D4C2×D7111yescharacteristic
185C2×#99—712no—
186C2×#99—712no—
187#399—21no—
188#399—21no—
189#390C2×D7112yes—
190#384Dic7112yescomplemented
191#380—1421no—
192C4×C4⋊C4Dic7111yescharacteristic, complemented
193C4×C22⋊C4—1421no—
194#342—1421no—
195#342—1421no—
196C22×Q8—21no—
197C22×D4C22×D7111yescharacteristic
198C23×C4C22⋊D7111yescharacteristic
199C23×C4C22⋊D7111yescharacteristic
200C4⋊Q8—41no—
201C4⋊Q8—41no—
202C22⋊Q8—44no—
203C22⋊Q8—44no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—2841no—
211C2×C4⋊C4C2×Dic7112yes—
212C2×C4⋊C4C2×Dic7111yescharacteristic
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—1422no—
216C2×C22⋊C4—1422no—
217C2×C22⋊C4C22⋊D7111yescharacteristic
218C2×C22⋊C4C22⋊D7111yescharacteristic
219C2×C42—2841no—
220C2×C42C2×Dic7111yescharacteristic
221C2×C42C2×Dic7112yes—
222#99—714no—
223#99—1424no—
224#99—1422no—
225#95—21no—
226#95—2841no—
227#95—2841no—
228C24C2×C22⋊D7111yescharacteristic
229C24C2×C22⋊D7111yescharacteristic
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4Q8×D7112yes—
237C2×D4#1176586612yes—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—2842no—
247C22×C4—2842no—
248C22×C4C22⋊Dic7111yescharacteristic
249C22×C4C22⋊Dic7111yescharacteristic
250C22×C4C22⋊Dic7111yescharacteristic
251C22×C4C22⋊Dic7111yescharacteristic
252C22×C4C22×Dic7111yescharacteristic
253C4⋊C4—24no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—42no—
259C4⋊C4—5682no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—44no—
263C22⋊C4—44no—
264C22⋊C4—2842no—
265C22⋊C4—2842no—
266C22⋊C4—2842no—
267C22⋊C4—1424no—
268C42—21no—
269C42—21no—
270C42—42no—
271C42—5682no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×C22⋊Dic7111yescharacteristic
281C23#1455610412yes—
282C23#1455611711yescharacteristic
283C23#1455612611yescharacteristic
284Q8—82no—
285Q8—82no—
286D4—42no—
287D4—42no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—2844no—
308C2×C4—5682no—
309C2×C4—5682no—
310C2×C4—5682no—
311C2×C4—5682no—
312C2×C4—5682no—
313C2×C4#1455604311yescharacteristic
314C2×C4#1455604311yescharacteristic
315C2×C4Q8×Dic7112yes—
316C22—21no—
317C22—21no—
318C22—22no—
319C22—21no—
320C22—21no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22—81no—
330C22#1798590611yescharacteristic
331C22C2×#1455604311yescharacteristic
332C22#1798597311yescharacteristic
333C4—21no—
334C4—22no—
335C4—44no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—82no—
340C4—5684no—
341C4—11362no—
342C4#1798579212yes—
343C2—21no—
344C2—21no—
345C2—42no—
346C2—41no—
347C2—41no—
348C2#2258932611yescharacteristic
349C2C2×#1798579211yescharacteristic
350C2#2259001511yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC22×D4C22×D71
FittingC71×#2542C2
socleC22×C71C2×#99
prime radical, cosocleC22×D4×C71C23
hypercenter#2542D71
hypocenterC71#13298
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2542

Complements

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

NH ≅ G/Ntype
11. C2×C4⋊Q8×C71C4semidirect
18. #384×C71C4semidirect
19. C4×C4⋊C4×C71C4semidirect
26. #14556043Q8semidirect
180. C71#13298semidirect
181. C2×C4⋊Q8Dic71semidirect
190. #384Dic71semidirect
192. C4×C4⋊C4Dic71semidirect

Maximal subgroups

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

subgroupindex
#225883282
C4×#145560432
#225864652
C71×#25422
#1329871