Finite Group Databasebeta

← #137×#10621924 #36370319

Subgroups — #137×#10621924 #36370319

Order 22144.

484 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×#10621924C211yesmaximal, characteristic
2C2×Q8×#10621924C211yesmaximal, characteristic, complemented
3#26012531C212yesmaximal, complemented
4#26012530C212yesmaximal, complemented
5Q16×#10621924C212yesmaximal, complemented
6SD16×#10621924C212yesmaximal, complemented
7#26012502C211yesmaximal, characteristic, complemented
8#26012484C211yesmaximal, characteristic, complemented
9#26012482C211yesmaximal, characteristic
10M16×#10621924C211yesmaximal, characteristic, complemented
11#137×D173C211yesmaximal, characteristic
12D173⋊M16C2211yescharacteristic
13Q8×#10621924—21no—
14Q8×#10621924—21no—
15Q8×#10621924C2212yes—
16Q8×#10621924C2211yescharacteristic
17Q8×#10621924C2212yes—
18Q8×#10621924C2211yescharacteristic
19D4×#10621924—21no—
20D4×#10621924—21no—
21D4×#10621924C2211yescharacteristic
22D4×#10621924C2211yescharacteristic
23#19149758—21no—
24#19149758C2211yescharacteristic
25C2×C4⋊#10621924—21no—
26C2×C4⋊#10621924C2211yescharacteristic
27C2×C4×#10621924—21no—
28C2×C4×#10621924—21no—
29C2×C4×#10621924C2211yescharacteristic
30#19149748C2212yes—
31#19149747C2212yes—
32#19149742C2212yes—
33#19149742C2212yes—
34#19149742C2212yes—
35#19149741C2212yes—
36#19149733C2212yes—
37C8×#10621924C2212yes—
38#41×D173C2211yescharacteristic
39C2×Q8×D173C2211yescharacteristic
40#19149712C411yescharacteristic, complemented
41#19149711C412yescomplemented
42#19149710C411yescharacteristic, complemented
43#19149709C411yescharacteristic, complemented
44#19149708C412yescomplemented
45Q16×D173C2212yes—
46SD16×D173C2212yes—
47M16×D173C2211yescharacteristic
48#137×C173C411yescharacteristic, complemented
49C22×#10621924—21no—
50C22×#10621924—21no—
51C22×#10621924—21no—
52C22×#10621924D411yescharacteristic
53C22×#10621924D411yescharacteristic
54#14583836—21no—
55#14583835—21no—
56D173⋊C8C2312yes—
57C4⋊#10621924—22no—
58C4⋊#10621924—22no—
59C4⋊#10621924—21no—
60C4⋊#10621924—21no—
61C4⋊#10621924—21no—
62C4⋊#10621924C2312yes—
63C4⋊#10621924C2311yescharacteristic
64C4⋊#10621924C2311yescharacteristic
65C22⋊#10621924—21no—
66C22⋊#10621924—21no—
67C22⋊#10621924—21no—
68C4×#10621924—21no—
69C4×#10621924—21no—
70C4×#10621924—22no—
71C4×#10621924—22no—
72C4×#10621924—21no—
73C4×#10621924D411yescharacteristic
74C4×#10621924D411yescharacteristic
75C4×#10621924C2311yescharacteristic
76C4×#10621924C2311yescharacteristic
77#14583827C2×C411yescharacteristic
78#14583826—21no—
79#14583826C2×C411yescharacteristic
80#14583825—21no—
81#14583825C2×C411yescharacteristic
82C2×Dic346—21no—
83C2×Dic346C2×C411yescharacteristic, complemented
84Q8×D173—21no—
85Q8×D173—21no—
86Q8×D173C2×C412yescomplemented
87Q8×D173C2312yes—
88Q8×D173C2311yescharacteristic
89D4×D173—21no—
90D4×D173C2311yescharacteristic
91C2×C4×D173—21no—
92C2×C4×D173—21no—
93C2×C4×D173C2311yescharacteristic
94Dic692C2×C412yescomplemented
95#14583816C2×C412yescomplemented
96#14583816C2×C412yescomplemented
97#14583815C2×C412yescomplemented
98#14583814C2×C412yescomplemented
99#14583813C2×C412yescomplemented
100#14583810C2×C411yescharacteristic, complemented
101#14583809C2×C412yescomplemented
102C8×D173C2312yes—
103#41×C173C2×C411yescharacteristic
104C2×Q8×C173C2×C411yescharacteristic, complemented
105Q16×C173C2×C412yescomplemented
106SD16×C173C2×C412yescomplemented
107M16×C173C2×C411yescharacteristic, complemented
108C2×#10621924—22no—
109C2×#10621924—21no—
110C2×#10621924—21no—
111C2×#10621924—22no—
112C2×#10621924—41no—
113C2×#10621924—42no—
114C2×#10621924—42no—
115C2×#10621924—42no—
116C2×#10621924C2×D411yescharacteristic
117C2×#10621924C2×D411yescharacteristic
118C2×#10621924C2×D411yescharacteristic
119C2×#10621924C2×D411yescharacteristic
120#13063443—22no—
121C22×D173—21no—
122C22×D173C2×D411yescharacteristic
123Dic346—21no—
124Dic346—22no—
125Dic346—22no—
126Dic346—21no—
127Dic346C22×C412yes—
128Dic346C22×C411yescharacteristic
129D692—21no—
130D692C22×C411yescharacteristic
131C22⋊D173—21no—
132C22⋊D173—21no—
133C22⋊D173—21no—
134C2×Dic173—21no—
135C2×Dic173—21no—
136C2×Dic173#4111yescharacteristic
137C4×D173—22no—
138C4×D173—21no—
139C4×D173—22no—
140C4×D173—21no—
141C4×D173—21no—
142C4×D173C22×C411yescharacteristic
143C4×D173C2×D411yescharacteristic
144C4×D173#4111yescharacteristic
145C4×D173C2411yescharacteristic
146#13063436C22×C412yes—
147Q8×C173—21no—
148Q8×C173C22×C412yes—
149Q8×C173C22×C411yescharacteristic
150D4×C173—21no—
151D4×C173C22×C411yescharacteristic
152C2×C4×C173—21no—
153C2×C4×C173—21no—
154C2×C4×C173C22×C411yescharacteristic
155C8×C173C22×C412yes—
156C22×C173—21no—
157C22×C173C4×D411yescharacteristic
158C2×D173—21no—
159C2×D173—21no—
160C2×D173—21no—
161C2×D173—41no—
162C2×D173—41no—
163C2×D173C4×D411yescharacteristic
164C2×D173C22×D411yescharacteristic
165#10621924—21no—
166#10621924—21no—
167#10621924—21no—
168#10621924—41no—
169#10621924—41no—
170#10621924—42no—
171#10621924#13712yesdirect product factor
172C4×C173—21no—
173C4×C173—22no—
174C4×C173C4×D411yescharacteristic
175C4×C173C23×C411yescharacteristic
176Dic173—21no—
177Dic173—22no—
178Dic173C4×D411yescharacteristic
179Dic173C2×#4111yescharacteristic
180C2×C173—21no—
181C2×C173—41no—
182C2×C173C2×C4×D411yescharacteristic
183D173—21no—
184D173—41no—
185D173C2×#13711yescharacteristic
186D173#57511yescharacteristic
187C173C4×#13711yescharacteristic, complemented, Sylow
188C4×#137—1731nomaximal, Sylow
189C2×#137—1731no—
190C4×#41—1731no—
191C2×C4×Q8—1731no—
192#441—1732no—
193#440—1732no—
194C4×Q16—1732no—
195C4×SD16—1732no—
196#428—1731no—
197#419—1731no—
198#417—1731no—
199C4×M16—1731no—
200C2×#41—1731no—
201C22×Q8—1731no—
202#137—1731no—
203#137—1731no—
204#137—1732no—
205#137—1732no—
206#137—1731no—
207#137#1062192411yescharacteristic, direct product factor
208C2×Q16—1732no—
209C2×SD16—1732no—
210C2×M16—1731no—
211C2×M16—1731no—
212C4×Q8—3461no—
213C4×Q8—3461no—
214C4×Q8—1732no—
215C4×Q8—1732no—
216C4×Q8—1731no—
217C4×Q8—1731no—
218C4×D4—3461no—
219C4×D4—3461no—
220C4×D4—1731no—
221C4×D4—1731no—
222#117—3461no—
223#117—1731no—
224C2×C4⋊C4—3461no—
225C2×C4⋊C4—1731no—
226C2×C42—3461no—
227C2×C42—3461no—
228C2×C42—1731no—
229#107—1732no—
230#106—1732no—
231#103—1732no—
232#103—1732no—
233#103—1732no—
234#102—1732no—
235#97—1732no—
236C4×C8—1732no—
237#41—3461no—
238#41—3461no—
239#41—1731no—
240#41—1731no—
241#41—1731no—
242#41C2×#1062192411yescharacteristic
243C2×Q8—3461no—
244C2×Q8—3461no—
245C2×Q8—3461no—
246C2×Q8—1732no—
247C2×Q8—1732no—
248C2×Q8—1731no—
249C2×Q8—1731no—
250C2×Q8C2×#1062192411yescharacteristic, complemented
251C2×D4—3461no—
252C2×D4—1731no—
253C22×C4—3461no—
254C22×C4—3461no—
255C22×C4—3461no—
256C22×C4—3461no—
257C22×C4—3461no—
258C22×C4—1731no—
259C22×C4—1731no—
260C22×C4—1731no—
261Q16—1732no—
262Q16—1732no—
263Q16—1732no—
264Q16C2×#1062192412yescomplemented
265SD16—1732no—
266SD16—1732no—
267SD16—1732no—
268SD16C2×#1062192412yescomplemented
269M16—3461no—
270M16—3461no—
271M16—1731no—
272M16—1732no—
273M16C2×#1062192411yescharacteristic, complemented
274C2×C8—1732no—
275C2×C8—1732no—
276C4⋊C4—3462no—
277C4⋊C4—3462no—
278C4⋊C4—3461no—
279C4⋊C4—3461no—
280C4⋊C4—3461no—
281C4⋊C4—1732no—
282C4⋊C4—1731no—
283C4⋊C4—1731no—
284C22⋊C4—3461no—
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C42—3461no—
288C42—3461no—
289C42—3461no—
290C42—3462no—
291C42—3462no—
292C42—1731no—
293C42—1731no—
294C42—1731no—
295C42—1731no—
296C23—3461no—
297C23—1731no—
298Q8—21no—
299Q8—3462no—
300Q8—3462no—
301Q8—3461no—
302Q8—3461no—
303Q8—1732no—
304Q8—1731no—
305Q8C22×#1062192412yes—
306Q8C22×#1062192411yescharacteristic
307D4—21no—
308D4—3461no—
309D4—3461no—
310D4—3461no—
311D4—3461no—
312D4—1731no—
313D4C22×#1062192411yescharacteristic
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—3461no—
317C2×C4—3462no—
318C2×C4—3462no—
319C2×C4—3461no—
320C2×C4—3461no—
321C2×C4—3461no—
322C2×C4—3462no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—3462no—
327C2×C4—1731no—
328C2×C4—1731no—
329C2×C4—1731no—
330C2×C4—1731no—
331C2×C4—1731no—
332C2×C4—1731no—
333C2×C4—1731no—
334C2×C4—1731no—
335C2×C4—1731no—
336C2×C4—6921no—
337C2×C4—6922no—
338C2×C4—6922no—
339C2×C4—6922no—
340C2×C4C22×#1062192411yescharacteristic
341C8—3462no—
342C8—1732no—
343C8C22×#1062192412yes—
344C22—21no—
345C22—3461no—
346C22—3461no—
347C22—3461no—
348C22—1731no—
349C22—1731no—
350C22—6921no—
351C22—6921no—
352C22D4×#1062192411yescharacteristic
353C4—21no—
354C4—22no—
355C4—3461no—
356C4—3461no—
357C4—3461no—
358C4—3461no—
359C4—3462no—
360C4—1732no—
361C4—1731no—
362C4—1731no—
363C4—6921no—
364C4—6921no—
365C4—6922no—
366C4D4×#1062192411yescharacteristic
367C4C23×#1062192411yescharacteristic
368C2—21no—
369C2—41no—
370C2—3461no—
371C2—1731no—
372C2—1731no—
373C2—6921no—
374C2C2×D4×#1062192411yescharacteristic
subgroupG/H
centerC2C2×D4×#10621924
commutatorC4×C173C23×C4
FrattiniC4C23×#10621924
Fitting#137×C173C4
socleC2×C173C2×C4×D4
prime radical#41×D173C22
cosocleC4×D173C24
hypercenter#137#10621924
hypocenterC173C4×#137
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. C2×Q8×#10621924C2semidirect
3. #26012531C2semidirect
4. #26012530C2semidirect
5. Q16×#10621924C2semidirect
6. SD16×#10621924C2semidirect
7. #26012502C2semidirect
8. #26012484C2semidirect
10. M16×#10621924C2semidirect
40. #19149712C4semidirect
41. #19149711C4semidirect
42. #19149710C4semidirect
43. #19149709C4semidirect
44. #19149708C4semidirect
48. #137×C173C4semidirect
83. C2×Dic346C2×C4semidirect
86. Q8×D173C2×C4semidirect
94. Dic692C2×C4semidirect
95. #14583816C2×C4semidirect
96. #14583816C2×C4semidirect
97. #14583815C2×C4semidirect
98. #14583814C2×C4semidirect
99. #14583813C2×C4semidirect
100. #14583810C2×C4semidirect
101. #14583809C2×C4semidirect
104. C2×Q8×C173C2×C4semidirect
105. Q16×C173C2×C4semidirect
106. SD16×C173C2×C4semidirect
107. M16×C173C2×C4semidirect
171. #10621924#137direct (N × H)
187. C173C4×#137semidirect
207. #137#10621924direct (N × H)
250. C2×Q8C2×#10621924semidirect
264. Q16C2×#10621924semidirect
268. SD16C2×#10621924semidirect
273. M16C2×#10621924semidirect

Maximal subgroups

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

subgroupindex
#41×#106219242
C2×Q8×#106219242
#260125312
#260125302
Q16×#106219242
SD16×#106219242
#260125022
#260124842
#260124822
M16×#106219242
#137×D1732
C4×#137173