Finite Group Databasebeta

← #31895097

Subgroups — #31895097

Order 18176.

474 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 103 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#22596978C211yesmaximal, characteristic
2#22588417C211yesmaximal, characteristic
3#22586873C212yesmaximal
4#22586193C212yesmaximal
5C71×#2499C211yesmaximal, characteristic
6#17986060—24no—
7#17986023—21no—
8#17985963C2212yes—
9C4×#11765849—21no—
10#17985862C2212yes—
11#17985766—21no—
12#17985766—21no—
13#17985736—24no—
14C2×C22⋊Q8×C71C2211yescharacteristic
15C2×#117×C71C2211yescharacteristic
16#397×C71—21no—
17#397×C71—21no—
18#391×C71C411yescharacteristic, complemented
19#391×C71C411yescharacteristic, complemented
20#387×C71—21no—
21#387×C71—21no—
22#384×C71C411yescharacteristic, complemented
23#382×C71C411yescharacteristic, complemented
24C4×C22⋊C4×C71C2211yescharacteristic
25C2×#11765849—22no—
26C2×C22⋊Dic71—22no—
27C4⋊(C71⋊C8)—24no—
28C22⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)D414yes—
30#14556039—22no—
31C4×C71⋊C8—22no—
32C71⋊#95—22no—
33C71⋊#95—22no—
34C22×Q8×C71C2×C411yescharacteristic
35C23×C4×C71C2311yescharacteristic
36C22⋊Q8×C71—22no—
37C22⋊Q8×C71—22no—
38#117×C71—22no—
39#117×C71—22no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71C2×C411yescharacteristic
44C2×C4⋊C4×C71C2×C411yescharacteristic
45C2×C4⋊C4×C71C2×C411yescharacteristic
46C2×C22⋊C4×C71—21no—
47C2×C22⋊C4×C71—21no—
48C2×C22⋊C4×C71D411yescharacteristic
49C2×C22⋊C4×C71D411yescharacteristic
50C2×C42×C71C2×C411yescharacteristic
51C2×C42×C71C2×C411yescharacteristic
52C2×C42×C71D411yescharacteristic
53C2×C42×C71D411yescharacteristic
54#95×C71—21no—
55#95×C71—21no—
56#95×C71—21no—
57#95×C71—21no—
58#95×C71—21no—
59#95×C71—21no—
60#95×C71—21no—
61C22×Dic71—42no—
62#11765849—42no—
63#11765849—42no—
64C2×C71⋊C8—24no—
65C22⋊Dic71—42no—
66C22⋊Dic71—42no—
67C24×C71C2×D411yescharacteristic
68C2×Q8×C71—22no—
69C2×Q8×C71—41no—
70C2×Q8×C71—41no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—41no—
76C22×C4×C71—41no—
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C22×C411yescharacteristic
82C22×C4×C71C2×D412yes—
83C22×C4×C71C2×D411yescharacteristic
84C22×C4×C71#4112yes—
85C4⋊C4×C71—22no—
86C4⋊C4×C71—22no—
87C4⋊C4×C71—22no—
88C4⋊C4×C71—42no—
89C4⋊C4×C71—42no—
90C4⋊C4×C71—41no—
91C4⋊C4×C71—41no—
92C22⋊C4×C71—22no—
93C22⋊C4×C71—22no—
94C22⋊C4×C71—42no—
95C22⋊C4×C71—42no—
96C42×C71—21no—
97C42×C71—21no—
98C42×C71—21no—
99C42×C71—21no—
100C42×C71—41no—
101C42×C71—41no—
102C23×C71—21no—
103C23×C71—21no—
104C23×C71—41no—
105C23×C71—41no—
106C23×C71C2×C22⋊C411yescharacteristic
107C23×C71#12112yes—
108Q8×C71—82no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—22no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—42no—
119C2×C4×C71—42no—
120C2×C4×C71—41no—
121C2×C4×C71—41no—
122C2×C4×C71—41no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71—41no—
129C2×C4×C71—41no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71—41no—
135C2×C4×C71C4×D412yes—
136C2×C4×C71#12011yescharacteristic
137C2×C4×C71#12311yescharacteristic
138C2×Dic71—82no—
139C2×Dic71—82no—
140C2×Dic71—82no—
141C71⋊C8—44no—
142C22×C71—21no—
143C22×C71—21no—
144C22×C71—22no—
145C22×C71—22no—
146C22×C71—22no—
147C22×C71—81no—
148C22×C71—81no—
149C22×C71#38611yescharacteristic
150C22×C71#41011yescharacteristic
151C22×C71#41011yescharacteristic
152C4×C71—21no—
153C4×C71—22no—
154C4×C71—21no—
155C4×C71—42no—
156C4×C71—41no—
157C4×C71—41no—
158C4×C71—41no—
159C4×C71—41no—
160C4×C71—81no—
161C4×C71—81no—
162C4×C71—81no—
163Dic71—162no—
164#9444—711nomaximal, Sylow
165C2×C71—21no—
166C2×C71—21no—
167C2×C71—42no—
168C2×C71#188411yescharacteristic
169C2×C71#189911yescharacteristic
170C2×C71#190911yescharacteristic
171#2921—711no—
172#2499D7111yescharacteristic
173#1820—711no—
174#1511—712no—
175#1284—712no—
176C71#944411yescharacteristic, complemented, Sylow
177C2×C22⋊Q8C2×D7111yescharacteristic
178C2×#117C2×D7111yescharacteristic
179#433—1424no—
180#423—1421no—
181#407—712no—
182C4×M16—1421no—
183#397—21no—
184#397—21no—
185#391Dic7111yescharacteristic, complemented
186#391Dic7111yescharacteristic, complemented
187#387—21no—
188#387—21no—
189#384Dic7111yescharacteristic, complemented
190#382Dic7111yescharacteristic, complemented
191#381—712no—
192C4×C22⋊C4C2×D7111yescharacteristic
193#342—1421no—
194#342—1421no—
195#328—1424no—
196C22×Q8C2×Dic7111yescharacteristic
197C23×C4C22×D7111yescharacteristic
198C2×M16—1422no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201#117—22no—
202#117—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4C2×Dic7111yescharacteristic
207C2×C4⋊C4C2×Dic7111yescharacteristic
208C2×C4⋊C4C2×Dic7111yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—1422no—
212C2×C22⋊C4C22⋊D7111yescharacteristic
213C2×C22⋊C4C22⋊D7111yescharacteristic
214C2×C42C2×Dic7111yescharacteristic
215C2×C42C2×Dic7111yescharacteristic
216C2×C42C22⋊D7111yescharacteristic
217C2×C42C22⋊D7111yescharacteristic
218C4⋊C8—1424no—
219C22⋊C8—714no—
220C22⋊C8—1422no—
221#97—1422no—
222C4×C8—1422no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—1422no—
231#95—1422no—
232C24C2×C22⋊D7111yescharacteristic
233C2×Q8—22no—
234C2×Q8—41no—
235C2×Q8—41no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—2842no—
243C22×C4C22⋊Dic7111yescharacteristic
244C22×C4C22⋊Dic7111yescharacteristic
245C22×C4C22⋊Dic7111yescharacteristic
246C22×C4C22⋊Dic7111yescharacteristic
247C22×C4C22×Dic7111yescharacteristic
248C22×C4D4×D7112yes—
249C22×C4C2×C22⋊D7111yescharacteristic
250C22×C4#1176586412yes—
251M16—2842no—
252M16—2842no—
253C2×C8—1424no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—2842no—
266C22⋊C4—2842no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—41no—
272C42—41no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23C2×C22⋊Dic7111yescharacteristic
278C23#1455610012yes—
279Q8—82no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—5682no—
307C2×C4—5682no—
308C2×C4—5682no—
309C2×C4D4×Dic7112yes—
310C2×C4#1455609611yescharacteristic
311C2×C4#1455610911yescharacteristic
312C8—2844no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—81no—
319C22—81no—
320C22#1798587911yescharacteristic
321C22#1798597311yescharacteristic
322C22#1798597311yescharacteristic
323C4—21no—
324C4—22no—
325C4—21no—
326C4—42no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4—11362no—
335C2—21no—
336C2—21no—
337C2—42no—
338C2#2258872411yescharacteristic
339C2#2258879911yescharacteristic
340C2#2258886311yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2499C2
socleC22×C71#386
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenter#2499D71
hypocenterC71#9444
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2499

Complements

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

NH ≅ G/Ntype
18. #391×C71C4semidirect
19. #391×C71C4semidirect
22. #384×C71C4semidirect
23. #382×C71C4semidirect
176. C71#9444semidirect
185. #391Dic71semidirect
186. #391Dic71semidirect
189. #384Dic71semidirect
190. #382Dic71semidirect

Maximal subgroups

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

subgroupindex
#225969782
#225884172
#225868732
#225861932
C71×#24992
#944471