Finite Group Databasebeta

← C109×#10150 #42128144

Subgroups — C109×#10150 #42128144

Order 27904.

630 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
1C109×#3226C211yesmaximal, characteristic, complemented
2C109×#3151C211yesmaximal, characteristic
3C2×#486×C109C211yesmaximal, characteristic, complemented
4C109×#1966C211yesmaximal, characteristic, complemented
5C109×#1545C211yesmaximal, characteristic, complemented
6C109×#1492C211yesmaximal, characteristic, complemented
7C109×#1278C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C109—22no—
9#532×C109C2211yescharacteristic
10C2×C4⋊Q8×C109C2211yescharacteristic
11C2×#127×C109C2211yescharacteristic
12#500×C109—21no—
13#496×C109—22no—
14#493×C109—21no—
15#492×C109—21no—
16#486×C109—21no—
17#486×C109—22no—
18#486×C109—22no—
19#480×C109—22no—
20#474×C109—22no—
21#431×C109C2211yescharacteristic
22#423×C109C2211yescharacteristic
23C2×#102×C109—22no—
24C2×#101×C109—22no—
25C4×M16×C109C2211yescharacteristic
26C2×C4×C8×C109C2211yescharacteristic
27#331×C109—22no—
28D4⋊C8×C109—22no—
29C2×#41×C109—21no—
30C22×Q8×C109—21no—
31C22×D4×C109—21no—
32C22×D4×C109—21no—
33C22×D4×C109—21no—
34AGL(1,ℤ/8)×C109—44no—
35AGL(1,ℤ/8)×C109—44no—
36C2×SD16×C109—24no—
37C2×D8×C109—24no—
38C2×M16×C109—21no—
39C2×M16×C109D412yes—
40C22×C8×C109—21no—
41C4⋊Q8×C109—21no—
42C4⋊Q8×C109—21no—
43C4⋊Q8×C109—21no—
44C4⋊Q8×C109—21no—
45C4⋊Q8×C109D411yescharacteristic
46#127×C109—21no—
47#127×C109—21no—
48#127×C109—21no—
49#127×C109D412yes—
50#127×C109D411yescharacteristic
51#124×C109—21no—
52#121×C109—21no—
53#121×C109—21no—
54C4×D4×C109—21no—
55C4×D4×C109—21no—
56C2×C4⋊C4×C109—21no—
57C2×C4⋊C4×C109D412yes—
58C2×C42×C109C2311yescharacteristic
59#107×C109—24no—
60C4⋊C8×C109—21no—
61C4⋊C8×C109—21no—
62C4⋊C8×C109D412yes—
63#102×C109—28no—
64#102×C109—24no—
65#101×C109—44no—
66C22⋊C8×C109—21no—
67#97×C109—21no—
68C4×C8×C109—21no—
69C4×C8×C109—21no—
70C4×C8×C109—21no—
71C4×C8×C109D412yes—
72C24×C109—41no—
73#41×C109—42no—
74#41×C109—42no—
75C2×Q8×C109—21no—
76C2×Q8×C109—41no—
77C2×Q8×C109—41no—
78C2×Q8×C109—42no—
79C2×Q8×C109—42no—
80C2×D4×C109—21no—
81C2×D4×C109—22no—
82C2×D4×C109—21no—
83C2×D4×C109—22no—
84C2×D4×C109—21no—
85C2×D4×C109—22no—
86C2×D4×C109—44no—
87C2×D4×C109—41no—
88C2×D4×C109—41no—
89C2×D4×C109—41no—
90C2×D4×C109—41no—
91C2×D4×C109—42no—
92C2×D4×C109—42no—
93C2×D4×C109—82no—
94C22×C4×C109—22no—
95C22×C4×C109—41no—
96C22×C4×C109C2×D411yescharacteristic
97C22×C4×C109C2×D411yescharacteristic
98C22×C4×C109#4111yescharacteristic
99SD16×C109—48no—
100D8×C109—48no—
101M16×C109—22no—
102M16×C109—42no—
103M16×C109D814yes—
104C2×C8×C109—22no—
105C2×C8×C109—22no—
106C2×C8×C109—22no—
107C2×C8×C109—42no—
108C2×C8×C109—41no—
109C4⋊C4×C109—22no—
110C4⋊C4×C109—22no—
111C4⋊C4×C109—21no—
112C4⋊C4×C109—21no—
113C4⋊C4×C109—41no—
114C4⋊C4×C109—41no—
115C22⋊C4×C109—41no—
116C22⋊C4×C109—41no—
117C42×C109C2×D411yescharacteristic
118C42×C109C2×D411yescharacteristic
119C42×C109C2×D411yescharacteristic
120C42×C109C2×D411yescharacteristic
121C23×C109—41no—
122C23×C109—42no—
123C23×C109—82no—
124C23×C109—82no—
125C23×C109—82no—
126C23×C109#12111yescharacteristic
127Q8×C109—42no—
128Q8×C109—82no—
129Q8×C109—81no—
130Q8×C109—81no—
131D4×C109—44no—
132D4×C109—42no—
133D4×C109—44no—
134D4×C109—42no—
135D4×C109—42no—
136D4×C109—82no—
137D4×C109—82no—
138D4×C109—82no—
139D4×C109—82no—
140D4×C109—81no—
141D4×C109—81no—
142C2×C4×C109—22no—
143C2×C4×C109—22no—
144C2×C4×C109—41no—
145C2×C4×C109—42no—
146C2×C4×C109—44no—
147C2×C4×C109—81no—
148C2×C4×C109—81no—
149C2×C4×C109#12011yescharacteristic
150C2×C4×C109#12011yescharacteristic
151C2×C4×C109#12011yescharacteristic
152C2×C4×C109#12111yescharacteristic
153C2×C4×C109#12111yescharacteristic
154C2×C4×C109#12711yescharacteristic
155C2×C4×C109C2×D812yes—
156C2×C4×C109AGL(1,ℤ/8)12yes—
157C8×C109—24no—
158C8×C109—42no—
159C8×C109—42no—
160C8×C109—81no—
161C22×C109—21no—
162C22×C109—21no—
163C22×C109—82no—
164C22×C109—81no—
165C22×C109—81no—
166C22×C109—82no—
167C22×C109—81no—
168C22×C109—82no—
169C22×C109—164no—
170C22×C109#39211yescharacteristic
171C22×C109#45912yes—
172C4×C109—21no—
173C4×C109—21no—
174C4×C109—82no—
175C4×C109—81no—
176C4×C109#44712yes—
177C4×C109#45412yes—
178C4×C109#47112yes—
179C4×C109#49311yescharacteristic
180C4×C109#49611yescharacteristic
181#10150C10911yesmaximal, characteristic, direct product factor, Sylow
182C2×C109—22no—
183C2×C109—161no—
184C2×C109—162no—
185C2×C109#200311yescharacteristic
186C2×C109#201111yescharacteristic
187C2×C109#201911yescharacteristic
188#3226C2×C10911yescharacteristic, complemented
189#3151C2×C10911yescharacteristic
190C2×#486C2×C10911yescharacteristic, complemented
191#1966C2×C10911yescharacteristic, complemented
192#1545C2×C10911yescharacteristic, complemented
193#1492C2×C10911yescharacteristic, complemented
194#1278C2×C10911yescharacteristic, complemented
195C109#1015011yescharacteristic, direct product factor, Sylow
196C2×AGL(1,ℤ/8)—22no—
197#532C22×C10911yescharacteristic
198C2×C4⋊Q8C22×C10911yescharacteristic
199C2×#127C22×C10911yescharacteristic
200#500—21no—
201#496—22no—
202#493—21no—
203#492—21no—
204#486—22no—
205#486—21no—
206#486—22no—
207#480—22no—
208#474—22no—
209#431C22×C10911yescharacteristic
210#423C22×C10911yescharacteristic
211C2×#102—22no—
212C2×#101—22no—
213C4×M16C22×C10911yescharacteristic
214C2×C4×C8C22×C10911yescharacteristic
215#331—22no—
216D4⋊C8—22no—
217C2×#41—21no—
218C22×Q8—21no—
219C22×D4—21no—
220C22×D4—21no—
221C22×D4—21no—
222AGL(1,ℤ/8)—44no—
223AGL(1,ℤ/8)—44no—
224C2×SD16—24no—
225C2×D8—24no—
226C2×M16—21no—
227C2×M16D4×C10912yes—
228C22×C8—21no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8D4×C10911yescharacteristic
234#127—21no—
235#127—21no—
236#127—21no—
237#127D4×C10912yes—
238#127D4×C10911yescharacteristic
239#124—21no—
240#121—21no—
241#121—21no—
242C4×D4—21no—
243C4×D4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4D4×C10912yes—
246C2×C42C23×C10911yescharacteristic
247#107—24no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8D4×C10912yes—
251#102—28no—
252#102—24no—
253#101—44no—
254C22⋊C8—21no—
255#97—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8D4×C10912yes—
260C24—41no—
261#41—42no—
262#41—42no—
263C2×Q8—21no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×D4—21no—
269C2×D4—22no—
270C2×D4—21no—
271C2×D4—22no—
272C2×D4—21no—
273C2×D4—22no—
274C2×D4—44no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—82no—
282C22×C4—22no—
283C22×C4—41no—
284C22×C4C2×D4×C10911yescharacteristic
285C22×C4C2×D4×C10911yescharacteristic
286C22×C4#41×C10911yescharacteristic
287SD16—48no—
288D8—48no—
289M16—22no—
290M16—42no—
291M16D8×C10914yes—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—42no—
296C2×C8—41no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C42C2×D4×C10911yescharacteristic
306C42C2×D4×C10911yescharacteristic
307C42C2×D4×C10911yescharacteristic
308C42C2×D4×C10911yescharacteristic
309C23—41no—
310C23—42no—
311C23—82no—
312C23—82no—
313C23—82no—
314C23#121×C10911yescharacteristic
315Q8—42no—
316Q8—82no—
317Q8—81no—
318Q8—81no—
319D4—44no—
320D4—42no—
321D4—44no—
322D4—42no—
323D4—42no—
324D4—82no—
325D4—82no—
326D4—82no—
327D4—82no—
328D4—81no—
329D4—81no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4#120×C10911yescharacteristic
338C2×C4#120×C10911yescharacteristic
339C2×C4#120×C10911yescharacteristic
340C2×C4#121×C10911yescharacteristic
341C2×C4#121×C10911yescharacteristic
342C2×C4#127×C10911yescharacteristic
343C2×C4C2×D8×C10912yes—
344C2×C4AGL(1,ℤ/8)×C10912yes—
345C8—24no—
346C8—42no—
347C8—42no—
348C8—81no—
349C22—21no—
350C22—21no—
351C22—82no—
352C22—81no—
353C22—81no—
354C22—82no—
355C22—81no—
356C22—82no—
357C22—164no—
358C22#392×C10911yescharacteristic
359C22#459×C10912yes—
360C4—21no—
361C4—21no—
362C4—82no—
363C4—81no—
364C4#447×C10912yes—
365C4#454×C10912yes—
366C4#471×C10912yes—
367C4#493×C10911yescharacteristic
368C4#496×C10911yescharacteristic
369C2—22no—
370C2—162no—
371C2—161no—
372C2C109×#200311yescharacteristic
373C2C109×#201111yescharacteristic
374C2C109×#201911yescharacteristic
subgroupG/H
center, socleC22×C109#392
commutator, Frattini, cosocleC2×C42C23×C109
prime radicalC109×#3151C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C109 ◁ C2×C42×C109 ◁ G

Complements

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

NH ≅ G/Ntype
1. C109×#3226C2semidirect
3. C2×#486×C109C2semidirect
4. C109×#1966C2semidirect
5. C109×#1545C2semidirect
6. C109×#1492C2semidirect
7. C109×#1278C2semidirect
181. #10150C109direct (N × H)
188. #3226C2×C109semidirect
190. C2×#486C2×C109semidirect
191. #1966C2×C109semidirect
192. #1545C2×C109semidirect
193. #1492C2×C109semidirect
194. #1278C2×C109semidirect
195. C109#10150direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#32262
C109×#31512
C2×#486×C1092
C109×#19662
C109×#15452
C109×#14922
C109×#12782
#10150109