Finite Group Databasebeta

← C109×#10292 #42128286

Subgroups — C109×#10292 #42128286

Order 27904.

640 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3230C211yesmaximal, characteristic, complemented
2C109×#3208C211yesmaximal, characteristic
3C109×#2858C211yesmaximal, characteristic
4C109×#1994C211yesmaximal, characteristic, complemented
5C109×#1549C211yesmaximal, characteristic, complemented
6C109×#1545C211yesmaximal, characteristic
7C109×#1276C211yesmaximal, characteristic, complemented
8#533×C109C2211yescharacteristic
9C2×C4⋊Q8×C109C2211yescharacteristic
10C2×#127×C109C2211yescharacteristic
11C2×#124×C109—22no—
12#500×C109—21no—
13#492×C109—21no—
14#487×C109—21no—
15#486×C109—21no—
16#484×C109—22no—
17#477×C109—22no—
18#460×C109—22no—
19#450×C109—22no—
20#431×C109C2211yescharacteristic
21#431×C109C2211yescharacteristic
22#423×C109C2211yescharacteristic
23#390×C109—22no—
24#390×C109—22no—
25C43×C109C2211yescharacteristic
26#339×C109—22no—
27#332×C109—22no—
28#331×C109—22no—
29C22×Q8×C109—21no—
30C22×D4×C109—21no—
31C22×D4×C109—21no—
32C22×D4×C109D412yes—
33C2×SD16×C109—24no—
34C2×M16×C109—21no—
35C4⋊Q8×C109—21no—
36C4⋊Q8×C109—21no—
37C4⋊Q8×C109—21no—
38C4⋊Q8×C109—21no—
39C4⋊Q8×C109D411yescharacteristic
40C4⋊Q8×C109D411yescharacteristic
41#127×C109—21no—
42#127×C109—21no—
43#127×C109—21no—
44#127×C109—21no—
45#125×C109—21no—
46#124×C109—48no—
47C22⋊Q8×C109—21no—
48C22⋊Q8×C109—21no—
49C4×Q8×C109—21no—
50C4×Q8×C109—21no—
51#117×C109—21no—
52C2×C4⋊C4×C109—21no—
53C2×C4⋊C4×C109D412yes—
54C2×C22⋊C4×C109—28no—
55C2×C42×C109—21no—
56C2×C42×C109—21no—
57C2×C42×C109D412yes—
58C2×C42×C109C2311yescharacteristic
59#107×C109—24no—
60C4⋊C8×C109—21no—
61C4⋊C8×C109—21no—
62C4⋊C8×C109D412yes—
63C4⋊C8×C109D412yes—
64#103×C109—24no—
65#102×C109—24no—
66C22⋊C8×C109—21no—
67C22⋊C8×C109—21no—
68C4×C8×C109—21no—
69C4×C8×C109—21no—
70C24×C109—22no—
71C2×Q8×C109—21no—
72C2×Q8×C109—21no—
73C2×Q8×C109—44no—
74C2×Q8×C109—41no—
75C2×Q8×C109—41no—
76C2×D4×C109—22no—
77C2×D4×C109—22no—
78C2×D4×C109—44no—
79C2×D4×C109—44no—
80C2×D4×C109—41no—
81C2×D4×C109—41no—
82C2×D4×C109—41no—
83C2×D4×C109—41no—
84C2×D4×C109—42no—
85C2×D4×C109—42no—
86C22×C4×C109—22no—
87C22×C4×C109—22no—
88C22×C4×C109C2×D411yescharacteristic
89C22×C4×C109C2×D411yescharacteristic
90C22×C4×C109C2×D411yescharacteristic
91SD16×C109—84no—
92M16×C109—42no—
93C2×C8×C109—22no—
94C2×C8×C109—22no—
95C2×C8×C109—41no—
96C4⋊C4×C109—21no—
97C4⋊C4×C109—22no—
98C4⋊C4×C109—21no—
99C4⋊C4×C109—22no—
100C4⋊C4×C109—21no—
101C4⋊C4×C109—21no—
102C4⋊C4×C109—41no—
103C4⋊C4×C109—41no—
104C4⋊C4×C109—41no—
105C4⋊C4×C109—41no—
106C22⋊C4×C109—416no—
107C22⋊C4×C109—41no—
108C42×C109—24no—
109C42×C109—22no—
110C42×C109—22no—
111C42×C109—42no—
112C42×C109—41no—
113C42×C109C2×D411yescharacteristic
114C42×C109C2×D411yescharacteristic
115C42×C109C2×D411yescharacteristic
116C42×C109#4111yescharacteristic
117C23×C109—44no—
118C23×C109—44no—
119C23×C109—42no—
120C23×C109—44no—
121C23×C109#12011yescharacteristic
122Q8×C109—42no—
123Q8×C109—42no—
124Q8×C109—82no—
125Q8×C109—82no—
126D4×C109—44no—
127D4×C109—82no—
128D4×C109—82no—
129D4×C109—82no—
130D4×C109—82no—
131D4×C109—81no—
132D4×C109—81no—
133C2×C4×C109—22no—
134C2×C4×C109—22no—
135C2×C4×C109—24no—
136C2×C4×C109—42no—
137C2×C4×C109—42no—
138C2×C4×C109—44no—
139C2×C4×C109—41no—
140C2×C4×C109—41no—
141C2×C4×C109—42no—
142C2×C4×C109#12011yescharacteristic
143C2×C4×C109#12011yescharacteristic
144C2×C4×C109#12111yescharacteristic
145C2×C4×C109#12111yescharacteristic
146C2×C4×C109#12111yescharacteristic
147C2×C4×C109#12711yescharacteristic
148C2×C4×C109#13512yes—
149C2×C4×C109#13712yes—
150C8×C109—42no—
151C8×C109—42no—
152C8×C109—81no—
153C22×C109—21no—
154C22×C109—21no—
155C22×C109—88no—
156C22×C109—82no—
157C22×C109—82no—
158C22×C109—82no—
159C22×C109#39211yescharacteristic
160C22×C109#45212yes—
161C4×C109—21no—
162C4×C109—21no—
163C4×C109—44no—
164C4×C109—82no—
165C4×C109—81no—
166C4×C109—81no—
167C4×C109#45312yes—
168C4×C109#45412yes—
169C4×C109#46412yes—
170C4×C109#49511yescharacteristic
171C4×C109#49711yescharacteristic
172#10292C10911yesmaximal, characteristic, direct product factor, Sylow
173C2×C109—22no—
174C2×C109—162no—
175C2×C109#200211yescharacteristic
176C2×C109#200311yescharacteristic
177C2×C109#201511yescharacteristic
178#3230C2×C10911yescharacteristic, complemented
179#3208C2×C10911yescharacteristic
180#2858C2×C10911yescharacteristic
181#1994C2×C10911yescharacteristic, complemented
182#1549C2×C10911yescharacteristic, complemented
183#1545C2×C10911yescharacteristic
184#1276C2×C10911yescharacteristic, complemented
185C109#1029211yescharacteristic, direct product factor, Sylow
186#533C22×C10911yescharacteristic
187C2×C4⋊Q8C22×C10911yescharacteristic
188C2×#127C22×C10911yescharacteristic
189C2×#124—22no—
190#500—21no—
191#492—21no—
192#487—21no—
193#486—21no—
194#484—22no—
195#477—22no—
196#460—22no—
197#450—22no—
198#431C22×C10911yescharacteristic
199#431C22×C10911yescharacteristic
200#423C22×C10911yescharacteristic
201#390—22no—
202#390—22no—
203C43C22×C10911yescharacteristic
204#339—22no—
205#332—22no—
206#331—22no—
207C22×Q8—21no—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4D4×C10912yes—
211C2×SD16—24no—
212C2×M16—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8D4×C10911yescharacteristic
218C4⋊Q8D4×C10911yescharacteristic
219#127—21no—
220#127—21no—
221#127—21no—
222#127—21no—
223#125—21no—
224#124—48no—
225C22⋊Q8—21no—
226C22⋊Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229#117—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4D4×C10912yes—
232C2×C22⋊C4—28no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42D4×C10912yes—
236C2×C42C23×C10911yescharacteristic
237#107—24no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8D4×C10912yes—
241C4⋊C8D4×C10912yes—
242#103—24no—
243#102—24no—
244C22⋊C8—21no—
245C22⋊C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C24—22no—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—44no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—42no—
263C2×D4—42no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4C2×D4×C10911yescharacteristic
267C22×C4C2×D4×C10911yescharacteristic
268C22×C4C2×D4×C10911yescharacteristic
269SD16—84no—
270M16—42no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—41no—
274C4⋊C4—21no—
275C4⋊C4—22no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C22⋊C4—416no—
285C22⋊C4—41no—
286C42—24no—
287C42—22no—
288C42—22no—
289C42—42no—
290C42—41no—
291C42C2×D4×C10911yescharacteristic
292C42C2×D4×C10911yescharacteristic
293C42C2×D4×C10911yescharacteristic
294C42#41×C10911yescharacteristic
295C23—44no—
296C23—44no—
297C23—42no—
298C23—44no—
299C23#120×C10911yescharacteristic
300Q8—42no—
301Q8—42no—
302Q8—82no—
303Q8—82no—
304D4—44no—
305D4—82no—
306D4—82no—
307D4—82no—
308D4—82no—
309D4—81no—
310D4—81no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4#120×C10911yescharacteristic
321C2×C4#120×C10911yescharacteristic
322C2×C4#121×C10911yescharacteristic
323C2×C4#121×C10911yescharacteristic
324C2×C4#121×C10911yescharacteristic
325C2×C4#127×C10911yescharacteristic
326C2×C4#135×C10912yes—
327C2×C4#137×C10912yes—
328C8—42no—
329C8—42no—
330C8—81no—
331C22—21no—
332C22—21no—
333C22—88no—
334C22—82no—
335C22—82no—
336C22—82no—
337C22#392×C10911yescharacteristic
338C22#452×C10912yes—
339C4—21no—
340C4—21no—
341C4—44no—
342C4—82no—
343C4—81no—
344C4—81no—
345C4#453×C10912yes—
346C4#454×C10912yes—
347C4#464×C10912yes—
348C4#495×C10911yescharacteristic
349C4#497×C10911yescharacteristic
350C2—22no—
351C2—162no—
352C2C109×#200211yescharacteristic
353C2C109×#200311yescharacteristic
354C2C109×#201511yescharacteristic
subgroupG/H
center, socleC22×C109#392
commutator, Frattini, cosocleC2×C42C23×C109
prime radicalC2×#127×C109C22
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×#3230C2semidirect
4. C109×#1994C2semidirect
5. C109×#1549C2semidirect
7. C109×#1276C2semidirect
172. #10292C109direct (N × H)
178. #3230C2×C109semidirect
181. #1994C2×C109semidirect
182. #1549C2×C109semidirect
184. #1276C2×C109semidirect
185. C109#10292direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#32302
C109×#32082
C109×#28582
C109×#19942
C109×#15492
C109×#15452
C109×#12762
#10292109