Finite Group Databasebeta

← #34012197

Subgroups — #34012197

Order 18688.

630 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×#17989643C211yesmaximal, characteristic
2C22×#14556659C211yesmaximal, characteristic
3#22625002C214yesmaximal, complemented
4#427×D73C211yesmaximal, characteristic
5C22×C4⋊#64408—21no—
6C2×#14556666C2212yes—
7#17989643—22no—
8#17989643C414yescomplemented
9C2×#14556665—21no—
10C2×#14556664—21no—
11C2×#14556659—21no—
12C2×#14556659—22no—
13C2×#14556659C414yescomplemented
14C2×#14556659C2212yes—
15C22×C8×D73C2211yescharacteristic
16#117×D73—21no—
17#17988625C411yescharacteristic, complemented
18#17988624C2212yes—
19#17988623—21no—
20#17988622—21no—
21#107×D73C412yescomplemented
22#106×D73—21no—
23#427×C73C411yescharacteristic, complemented
24C23×#64408—41no—
25C2×C4⋊#64408—24no—
26C2×C4⋊#64408—22no—
27C2×C4⋊#64408—41no—
28C2×C4⋊#64408—42no—
29#14556666—22no—
30#14556666C2×C414yescomplemented
31#14556665—24no—
32#14556664—24no—
33C2×#11924031—22no—
34#14556659—24no—
35#14556659—22no—
36#14556659C2×C414yescomplemented
37C22×C4×D73D411yescharacteristic
38C22×#10596831D411yescharacteristic
39C2×C8×D73C2×C412yes—
40C2×C8×D73C2×C412yes—
41C2×C8×D73C2×C411yescharacteristic
42C2×C8×D73D411yescharacteristic
43C2×C8×D73D411yescharacteristic
44C2×C8×D73D412yes—
45C2×C8×D73Q812yes—
46C2×C8×D73C2311yescharacteristic
47#14556550—21no—
48#14556549—22no—
49#14556548—41no—
50#14556547—41no—
51C4⋊C4×D73—22no—
52C22⋊C4×D73—41no—
53C42×D73—41no—
54#14556525C2×C412yescomplemented
55#14556524—22no—
56#14556523—21no—
57#14556522—22no—
58C22×C8×C73C2×C411yescharacteristic
59#117×C73—21no—
60#107×C73C2×C412yescomplemented
61#106×C73—21no—
62C22×#64408—44no—
63C22×#64408—42no—
64C22×#64408—84no—
65C4⋊#64408—24no—
66C4⋊#64408—44no—
67C4⋊#64408—42no—
68#11924031—24no—
69C23×D73D811yescharacteristic
70C22×Dic73SD1611yescharacteristic
71C2×C4×D73—42no—
72C2×C4×D73C22⋊C411yescharacteristic
73C2×C4×D73C22⋊C412yes—
74C2×C4×D73C4⋊C412yes—
75C2×C4×D73D811yescharacteristic
76C2×C4×D73SD1612yes—
77C2×C4×D73SD1611yescharacteristic
78C2×C4×D73Q1612yes—
79C2×C4×D73C2×D411yescharacteristic
80C2×#10596831C22⋊C411yescharacteristic
81C2×#10596831C22⋊C412yes—
82C2×#10596831C4⋊C412yes—
83C2×#10596831C2×D411yescharacteristic
84C8×D73C4212yes—
85C8×D73C22⋊C412yes—
86C8×D73C22⋊C412yes—
87C8×D73C4⋊C412yes—
88C8×D73C4⋊C412yes—
89C8×D73C4⋊C412yes—
90C8×D73C22×C411yescharacteristic
91C8×D73C22×C411yescharacteristic
92C8×D73C2×D411yescharacteristic
93C8×D73C2×Q811yescharacteristic
94C4⋊Dic73—22no—
95#11924007—42no—
96C22⋊Dic73—41no—
97#11924005—42no—
98C4×Dic73—41no—
99C4×Dic73—42no—
100C22×C4×C73C22⋊C411yescharacteristic
101C2×C8×C73C4212yes—
102C2×C8×C73C22⋊C411yescharacteristic
103C2×C8×C73C4⋊C412yes—
104C2×C8×C73C22×C411yescharacteristic
105C4⋊C4×C73—22no—
106C22⋊C4×C73—41no—
107C42×C73—41no—
108C23×C73#10211yescharacteristic
109C22×D73—24no—
110C22×D73#10212yes—
111C22×D73#10211yescharacteristic
112C22×D73#10712yes—
113C22×D73C2×D811yescharacteristic
114C2×#64408—44no—
115C2×#64408—84no—
116C2×#64408—88no—
117C2×C4×C73—42no—
118C2×C4×C73#9512yes—
119C2×C4×C73#10211yescharacteristic
120C2×C4×C73#10312yes—
121C2×C4×C73C2×C22⋊C411yescharacteristic
122C2×Dic73—42no—
123C2×Dic73#10211yescharacteristic
124C2×Dic73#10312yes—
125C2×Dic73#10612yes—
126C2×Dic73C2×SD1611yescharacteristic
127C4×D73—84no—
128C4×D73#9512yes—
129C4×D73#10212yes—
130C4×D73#10312yes—
131C4×D73#10312yes—
132C4×D73#10612yes—
133C4×D73#10712yes—
134C4×D73C2×C22⋊C411yescharacteristic
135C4×D73C2×C4⋊C411yescharacteristic
136C4×D73C2×SD1611yescharacteristic
137C4×D73C2×Q1611yescharacteristic
138C8×C73#9512yes—
139C8×C73C2×C4211yescharacteristic
140C8×C73C2×C4⋊C411yescharacteristic
141#10596831#9512yes—
142#10596831C2×C22⋊C411yescharacteristic
143#10596831C2×C4⋊C411yescharacteristic
144C22×C73—22no—
145C22×C73#34012yes—
146C22×C73C2×#10211yescharacteristic
147C2×D73—22no—
148C2×D73—24no—
149C2×D73—24no—
150C2×D73#34012yes—
151C2×D73#36014yes—
152C2×D73C2×#10211yescharacteristic
153C2×D73C2×#10711yescharacteristic
154#64408—84no—
155C4×C73—82no—
156C4×C73#34012yes—
157C4×C73C2×#9511yescharacteristic
158C4×C73C2×#10311yescharacteristic
159Dic73—82no—
160Dic73#34012yes—
161Dic73C2×#10311yescharacteristic
162Dic73C2×#10611yescharacteristic
163C2×#1386—731nomaximal, Sylow
164C2×C73—22no—
165C2×C73C2×#34011yescharacteristic
166C2×C73C2×#36012yes—
167D73—21no—
168D73—21no—
169D73#138614yescomplemented
170C22×M32—731no—
171C2×#427—731no—
172C22×#106—731no—
173#1386—734no—
174C73C2×#138611yescharacteristic, complemented, Sylow
175C23×C8—731no—
176C2×#117—1461no—
177C22×C4⋊C4—1461no—
178C2×M32—734no—
179C2×M32—732no—
180C2×M32—1461no—
181C2×M32—1462no—
182C22×C16—1461no—
183#427—731no—
184#427—732no—
185#427—1461no—
186#427—1461no—
187#427#6440811yescharacteristic, complemented
188C2×#107—732no—
189C2×#106—732no—
190C2×#106—734no—
191C2×#106—1461no—
192C2×#106—1461no—
193C2×#106—1462no—
194C23×C4—2921no—
195C23×C4—731no—
196C22×C8—731no—
197C22×C8—732no—
198C22×C8—732no—
199C22×C8—731no—
200C22×C8—731no—
201C22×C8—732no—
202C22×C8—732no—
203C22×C8—731no—
204C22×C8—731no—
205C22×C8C2×#6440811yescharacteristic
206#117—21no—
207#117—2921no—
208#117—2921no—
209#117—1462no—
210#117—1461no—
211C2×C4⋊C4—2921no—
212C2×C4⋊C4—2922no—
213C2×C4⋊C4—1462no—
214C2×C4⋊C4—1464no—
215C2×C4⋊C4—1462no—
216C2×C22⋊C4—2921no—
217C2×C42—2921no—
218M32—734no—
219M32—1464no—
220M32—1462no—
221C2×C16—1462no—
222C2×C16—1464no—
223#107—732no—
224#107—1462no—
225#107C2×#6440812yescomplemented
226#106—21no—
227#106—734no—
228#106—1462no—
229#106—1464no—
230#106—1462no—
231#106—1461no—
232C24—731no—
233C22×C4—2922no—
234C22×C4—2924no—
235C22×C4—2922no—
236C22×C4—5844no—
237C22×C4—731no—
238C22×C4—732no—
239C22×C4—732no—
240C22×C4—731no—
241C22×C4—731no—
242C22×C4—732no—
243C22×C4—732no—
244C22×C4—731no—
245C22×C4—731no—
246C22×C4C22⋊#6440811yescharacteristic
247C2×C8—732no—
248C2×C8—732no—
249C2×C8—731no—
250C2×C8—731no—
251C2×C8—732no—
252C2×C8—732no—
253C2×C8—732no—
254C2×C8—732no—
255C2×C8—732no—
256C2×C8—731no—
257C2×C8—731no—
258C2×C8—732no—
259C2×C8—731no—
260C2×C8—731no—
261C2×C8C4×#6440812yes—
262C2×C8C22⋊#6440811yescharacteristic
263C2×C8C4⋊#6440812yes—
264C2×C8C22×#6440811yescharacteristic
265C4⋊C4—22no—
266C4⋊C4—2922no—
267C4⋊C4—2924no—
268C4⋊C4—2922no—
269C4⋊C4—1464no—
270C4⋊C4—1462no—
271C22⋊C4—41no—
272C22⋊C4—2922no—
273C22⋊C4—2921no—
274C42—41no—
275C42—2922no—
276C42—2921no—
277C16—1464no—
278C23—731no—
279C23—732no—
280C23—732no—
281C23—731no—
282C23—1464no—
283C23#1455665311yescharacteristic
284C2×C4—42no—
285C2×C4—2924no—
286C2×C4—2922no—
287C2×C4—5844no—
288C2×C4—5848no—
289C2×C4—5844no—
290C2×C4—732no—
291C2×C4—732no—
292C2×C4—731no—
293C2×C4—731no—
294C2×C4—732no—
295C2×C4—732no—
296C2×C4—732no—
297C2×C4—732no—
298C2×C4—731no—
299C2×C4—732no—
300C2×C4—731no—
301C2×C4—731no—
302C2×C4—732no—
303C2×C4—731no—
304C2×C4(C2×#64408)⋊C412yes—
305C2×C4#1455665311yescharacteristic
306C2×C4#1455665412yes—
307C2×C4C2×C22⋊#6440811yescharacteristic
308C8—731no—
309C8—732no—
310C8—731no—
311C8(C2×#64408)⋊C412yes—
312C8C2×C4×#6440811yescharacteristic
313C8C2×C4⋊#6440811yescharacteristic
314C22—22no—
315C22—732no—
316C22—734no—
317C22—731no—
318C22—731no—
319C22—1464no—
320C22—1464no—
321C22—1462no—
322C22#1798946012yes—
323C22C2×#1455665311yescharacteristic
324C4—82no—
325C4—5844no—
326C4—5842no—
327C4—732no—
328C4—731no—
329C4—731no—
330C4#1798946012yes—
331C4C2×(C2×#64408)⋊C411yescharacteristic
332C4C2×#1455665411yescharacteristic
333C2—22no—
334C2—734no—
335C2—1461no—
336C2—1461no—
337C2C2×#1798946011yescharacteristic
338C2D73⋊#36012yes—
subgroupG/H
centerC22C2×#14556653
commutatorC8×C73C2×C42
FrattiniC2×C8C22×#64408
Fitting#427×C73C4
socleC22×C73C2×#102
prime radicalC23×D73D8
cosocleC2×C8×D73C23
hypercenter#427#64408
hypocenterC73C2×#1386
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #427

Complements

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

NH ≅ G/Ntype
3. #22625002C2semidirect
8. #17989643C4semidirect
13. C2×#14556659C4semidirect
17. #17988625C4semidirect
21. #107×D73C4semidirect
23. #427×C73C4semidirect
30. #14556666C2×C4semidirect
36. #14556659C2×C4semidirect
54. #14556525C2×C4semidirect
60. #107×C73C2×C4semidirect
169. D73#1386semidirect
174. C73C2×#1386semidirect
187. #427#64408semidirect
225. #107C2×#64408semidirect

Maximal subgroups

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

subgroupindex
C2×#179896432
C22×#145566592
#226250022
#427×D732
C2×#138673