Finite Group Databasebeta

← #31889898

Subgroups — #31889898

Order 18176.

736 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 147 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#22589310C212yesmaximal, complemented
2#22589157C211yesmaximal, characteristic
3C2×#17985741C212yesmaximal, complemented
4C2×#17985739C211yesmaximal, characteristic
5C2×#563×C71C211yesmaximal, characteristic
6C2×#14556052—21no—
7C2×#14556049—21no—
8C2×#14556039C2212yes—
9C2×#14556039C2211yescharacteristic
10#17985869C2211yescharacteristic
11#17985761—22no—
12#17985761—22no—
13#17985741—24no—
14#17985739—22no—
15#563×C71—24no—
16#563×C71—22no—
17#563×C71C414yescomplemented
18C2×C4⋊Q8×C71—21no—
19C2×#126×C71—22no—
20C2×#125×C71C2212yes—
21C2×#124×C71C2211yescharacteristic
22C2×#123×C71—21no—
23C2×C22⋊Q8×C71—22no—
24C22×C71⋊C8—21no—
25C22×C71⋊C8—22no—
26C2×C4⋊Dic71—21no—
27C2×C4⋊Dic71—22no—
28#14556052—42no—
29#14556049—42no—
30#14556039—22no—
31#14556039—22no—
32#14556039—22no—
33C22×Q8×C71—21no—
34C22×Q8×C71D411yescharacteristic
35C22×D4×C71D411yescharacteristic
36C23×C4×C71—21no—
37C4⋊Q8×C71—24no—
38C4⋊Q8×C71—24no—
39#126×C71—28no—
40#126×C71—28no—
41#125×C71—22no—
42#125×C71—22no—
43#125×C71C2×C414yes—
44#125×C71D414yes—
45#124×C71—22no—
46#124×C71C2×C412yes—
47#124×C71D412yes—
48#123×C71—24no—
49#123×C71—24no—
50C22⋊Q8×C71—28no—
51C22⋊Q8×C71—28no—
52C2×C4⋊C4×C71—22no—
53C2×C4⋊C4×C71—22no—
54C2×C4⋊C4×C71—22no—
55C2×C4⋊C4×C71D412yes—
56C2×C4⋊C4×C71D412yes—
57C2×C22⋊C4×C71—22no—
58C2×C22⋊C4×C71—21no—
59C2×C22⋊C4×C71—22no—
60C2×C42×C71—21no—
61C2×C42×C71C2311yescharacteristic
62C22×Dic71—41no—
63C2×C71⋊C8—22no—
64C2×C71⋊C8—24no—
65C2×C71⋊C8—42no—
66C2×C71⋊C8—41no—
67C2×C71⋊C8—42no—
68C2×C71⋊C8—41no—
69C4⋊Dic71—41no—
70C4⋊Dic71—41no—
71C4⋊Dic71—44no—
72C24×C71—21no—
73C2×Q8×C71—24no—
74C2×Q8×C71—21no—
75C2×Q8×C71—41no—
76C2×Q8×C71—41no—
77C2×Q8×C71—81no—
78C2×Q8×C71—81no—
79C2×Q8×C71C22⋊C412yes—
80C2×D4×C71—21no—
81C2×D4×C71—41no—
82C2×D4×C71—41no—
83C2×D4×C71C22⋊C412yes—
84C22×C4×C71—22no—
85C22×C4×C71—21no—
86C22×C4×C71—22no—
87C22×C4×C71—24no—
88C22×C4×C71—81no—
89C22×C4×C71—81no—
90C22×C4×C71C2×D412yes—
91C22×C4×C71C2×D411yescharacteristic
92C4⋊C4×C71—28no—
93C4⋊C4×C71—24no—
94C4⋊C4×C71—24no—
95C4⋊C4×C71—24no—
96C4⋊C4×C71—24no—
97C4⋊C4×C71—22no—
98C4⋊C4×C71—22no—
99C4⋊C4×C71C22⋊C414yes—
100C4⋊C4×C71C22⋊C414yes—
101C22⋊C4×C71—28no—
102C22⋊C4×C71—24no—
103C22⋊C4×C71—24no—
104C22⋊C4×C71—24no—
105C42×C71—24no—
106C42×C71C22×C411yescharacteristic
107C42×C71C2×D412yes—
108C42×C71C2×D411yescharacteristic
109C23×C71—22no—
110C23×C71—41no—
111C23×C71—42no—
112C23×C71—41no—
113C23×C71—42no—
114C23×C71#12011yescharacteristic
115Q8×C71—42no—
116Q8×C71—81no—
117Q8×C71—84no—
118D4×C71—42no—
119D4×C71—81no—
120C2×C4×C71—21no—
121C2×C4×C71—22no—
122C2×C4×C71—21no—
123C2×C4×C71—22no—
124C2×C4×C71—24no—
125C2×C4×C71—22no—
126C2×C4×C71—24no—
127C2×C4×C71—42no—
128C2×C4×C71—41no—
129C2×C4×C71—42no—
130C2×C4×C71—42no—
131C2×C4×C71—41no—
132C2×C4×C71—42no—
133C2×C4×C71—84no—
134C2×C4×C71C2×C22⋊C412yes—
135C2×C4×C71C2×C22⋊C411yescharacteristic
136C2×C4×C71#12012yes—
137C2×C4×C71#12011yescharacteristic
138C2×Dic71—81no—
139C2×Dic71—82no—
140C71⋊C8—42no—
141C71⋊C8—44no—
142C22×C71—42no—
143C22×C71—44no—
144C22×C71—81no—
145C22×C71—81no—
146C22×C71—82no—
147C22×C71#37911yescharacteristic
148C22×C71#45311yescharacteristic
149C22×C71#45412yes—
150C22×C71#45511yescharacteristic
151C22×C71#45612yes—
152C4×C71—21no—
153C4×C71—21no—
154C4×C71—22no—
155C4×C71—22no—
156C4×C71—44no—
157C4×C71—42no—
158C4×C71—44no—
159Dic71—161no—
160#8507—711nomaximal, Sylow
161C2×C71—82no—
162C2×C71#165711yescharacteristic
163C2×C71#166011yescharacteristic
164C2×C71#166412yes—
165C2×C71#179211yescharacteristic
166C2×C71#179312yes—
167C2×#563D7111yescharacteristic
168#1992—712no—
169#1959—711no—
170C2×#330—712no—
171C2×#329—711no—
172C71#850711yescharacteristic, complemented, Sylow
173#563—24no—
174#563—22no—
175#563Dic7114yescomplemented
176C2×C4⋊Q8—21no—
177C2×#126—22no—
178C2×#125C2×D7112yes—
179C2×#124C2×D7111yescharacteristic
180C2×#123—21no—
181C2×C22⋊Q8—22no—
182C2×#103—1421no—
183C2×#102—1421no—
184C2×#97—711no—
185C2×#97—712no—
186#384—711no—
187#340—1422no—
188#340—1422no—
189#330—1424no—
190#329—1422no—
191C22×Q8—21no—
192C22×Q8C22⋊D7111yescharacteristic
193C22×D4C22⋊D7111yescharacteristic
194C23×C4—21no—
195C22×C8—1422no—
196C22×C8—1421no—
197C4⋊Q8—24no—
198C4⋊Q8—24no—
199#126—28no—
200#126—28no—
201#125—22no—
202#125—22no—
203#125C2×Dic7114yes—
204#125C22⋊D7114yes—
205#124—22no—
206#124C2×Dic7112yes—
207#124C22⋊D7112yes—
208#123—24no—
209#123—24no—
210C22⋊Q8—28no—
211C22⋊Q8—28no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—1422no—
216C2×C4⋊C4—1421no—
217C2×C4⋊C4C22⋊D7112yes—
218C2×C4⋊C4C22⋊D7112yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—22no—
222C2×C42—21no—
223C2×C42C22×D7111yescharacteristic
224#103—2842no—
225#102—2842no—
226#97—1422no—
227#97—1422no—
228#97—1422no—
229C24—21no—
230C2×Q8—24no—
231C2×Q8—21no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—81no—
235C2×Q8—81no—
236C2×Q8C22⋊Dic7112yes—
237C2×D4—21no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4C22⋊Dic7112yes—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—2841no—
248C22×C4C2×C22⋊D7112yes—
249C22×C4C2×C22⋊D7111yescharacteristic
250C2×C8—2842no—
251C2×C8—2841no—
252C2×C8—2842no—
253C2×C8—2841no—
254C2×C8—1422no—
255C2×C8—1424no—
256C4⋊C4—28no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—2841no—
264C4⋊C4—2841no—
265C4⋊C4—2844no—
266C4⋊C4C22⋊Dic7114yes—
267C4⋊C4C22⋊Dic7114yes—
268C22⋊C4—28no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C42—24no—
273C42C22×Dic7111yescharacteristic
274C42C2×C22⋊D7112yes—
275C42C2×C22⋊D7111yescharacteristic
276C23—22no—
277C23—41no—
278C23—42no—
279C23—41no—
280C23—42no—
281C23#1455609711yescharacteristic
282Q8—42no—
283Q8—81no—
284Q8—84no—
285D4—42no—
286D4—81no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—84no—
301C2×C4—5681no—
302C2×C4—5682no—
303C2×C4C2×C22⋊Dic7112yes—
304C2×C4C2×C22⋊Dic7111yescharacteristic
305C2×C4#1455609712yes—
306C2×C4#1455609711yescharacteristic
307C8—2842no—
308C8—2844no—
309C22—42no—
310C22—44no—
311C22—81no—
312C22—82no—
313C22—81no—
314C22#1798585511yescharacteristic
315C22#1798616711yescharacteristic
316C22#1798617212yes—
317C22#1798617711yescharacteristic
318C22#1798618212yes—
319C4—21no—
320C4—21no—
321C4—22no—
322C4—22no—
323C4—44no—
324C4—42no—
325C4—44no—
326C4—11361no—
327C2—82no—
328C2#2258763811yescharacteristic
329C2#2258765111yescharacteristic
330C2#2258767412yes—
331C2#2258829411yescharacteristic
332C2#2258829912yes—
subgroupG/H
centerC23#14556097
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC2×#563×C71C2
socleC23×C71#120
prime radicalC22×D4×C71D4
cosocleC2×C42×C71C23
hypercenterC2×#563D71
hypocenterC71#8507
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#563

Complements

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

NH ≅ G/Ntype
1. #22589310C2semidirect
3. C2×#17985741C2semidirect
17. #563×C71C4semidirect
172. C71#8507semidirect
175. #563Dic71semidirect

Maximal subgroups

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

subgroupindex
#225893102
#225891572
C2×#179857412
C2×#179857392
C2×#563×C712
#850771