Finite Group Databasebeta

← C109×#10286 #42128280

Subgroups — C109×#10286 #42128280

Order 27904.

542 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3227C211yesmaximal, characteristic
2C109×#3226C211yesmaximal, characteristic
3C109×#2862C211yesmaximal, characteristic, complemented
4C109×#1962C211yesmaximal, characteristic
5C109×#1549C211yesmaximal, characteristic, complemented
6C109×#1546C211yesmaximal, characteristic, complemented
7C109×#1276C211yesmaximal, characteristic, complemented
8#532×C109C2211yescharacteristic
9C2×C4⋊Q8×C109C2211yescharacteristic
10C2×C4⋊Q8×C109C2211yescharacteristic
11#500×C109—21no—
12#498×C109—21no—
13#486×C109—21no—
14#486×C109—21no—
15#481×C109—22no—
16#480×C109—22no—
17#476×C109—22no—
18#474×C109—22no—
19#431×C109C2211yescharacteristic
20#431×C109C2211yescharacteristic
21#423×C109C2211yescharacteristic
22C2×#104×C109—22no—
23#391×C109—22no—
24#391×C109—22no—
25#382×C109—22no—
26C43×C109C2211yescharacteristic
27#332×C109—22no—
28#332×C109—22no—
29C2×#41×C109—21no—
30C22×Q8×C109—21no—
31C22×Q8×C109—21no—
32C2×M16×C109—21no—
33C4⋊Q8×C109—21no—
34C4⋊Q8×C109—21no—
35C4⋊Q8×C109—21no—
36C4⋊Q8×C109—21no—
37C4⋊Q8×C109—21no—
38C4⋊Q8×C109—21no—
39C4⋊Q8×C109—21no—
40C4⋊Q8×C109—21no—
41C4⋊Q8×C109D411yescharacteristic
42#127×C109D411yescharacteristic
43#124×C109—21no—
44#121×C109—21no—
45#121×C109—21no—
46C4×D4×C109—21no—
47C4×D4×C109—21no—
48C2×C4⋊C4×C109—21no—
49C2×C4⋊C4×C109—21no—
50C2×C4⋊C4×C109D412yes—
51C2×C4⋊C4×C109D412yes—
52C2×C42×C109—21no—
53C2×C42×C109—21no—
54C2×C42×C109D412yes—
55C2×C42×C109C2311yescharacteristic
56#107×C109—24no—
57#106×C109—24no—
58C4⋊C8×C109—21no—
59C4⋊C8×C109—21no—
60C4⋊C8×C109D412yes—
61C4⋊C8×C109D412yes—
62#104×C109—44no—
63#102×C109—24no—
64#102×C109—24no—
65C22⋊C8×C109—21no—
66C22⋊C8×C109—21no—
67C4×C8×C109—21no—
68C4×C8×C109—21no—
69#95×C109—28no—
70#41×C109—42no—
71#41×C109—42no—
72C2×Q8×C109—21no—
73C2×Q8×C109—44no—
74C2×Q8×C109—41no—
75C2×Q8×C109—41no—
76C2×Q8×C109—41no—
77C2×Q8×C109—41no—
78C2×Q8×C109—42no—
79C2×Q8×C109—42no—
80C2×D4×C109—21no—
81C2×D4×C109—21no—
82C2×D4×C109—21no—
83C2×D4×C109—41no—
84C22×C4×C109—22no—
85C22×C4×C109—22no—
86C22×C4×C109—22no—
87C22×C4×C109—41no—
88C22×C4×C109C2×D411yescharacteristic
89C22×C4×C109C2×D411yescharacteristic
90C22×C4×C109C2×D411yescharacteristic
91M16×C109—42no—
92C2×C8×C109—22no—
93C2×C8×C109—22no—
94C2×C8×C109—41no—
95C4⋊C4×C109—22no—
96C4⋊C4×C109—22no—
97C4⋊C4×C109—21no—
98C4⋊C4×C109—21no—
99C4⋊C4×C109—22no—
100C4⋊C4×C109—22no—
101C4⋊C4×C109—41no—
102C4⋊C4×C109—41no—
103C4⋊C4×C109—41no—
104C4⋊C4×C109—41no—
105C22⋊C4×C109—41no—
106C22⋊C4×C109—41no—
107C42×C109—24no—
108C42×C109—22no—
109C42×C109—22no—
110C42×C109—42no—
111C42×C109C2×D411yescharacteristic
112C42×C109C2×D411yescharacteristic
113C42×C109C2×D411yescharacteristic
114C42×C109#4111yescharacteristic
115C23×C109—41no—
116C23×C109#12011yescharacteristic
117Q8×C109—42no—
118Q8×C109—82no—
119Q8×C109—82no—
120Q8×C109—82no—
121Q8×C109—81no—
122Q8×C109—81no—
123D4×C109—42no—
124D4×C109—42no—
125D4×C109—42no—
126D4×C109—82no—
127C2×C4×C109—22no—
128C2×C4×C109—22no—
129C2×C4×C109—24no—
130C2×C4×C109—42no—
131C2×C4×C109—42no—
132C2×C4×C109—44no—
133C2×C4×C109—41no—
134C2×C4×C109—42no—
135C2×C4×C109—42no—
136C2×C4×C109—44no—
137C2×C4×C109—81no—
138C2×C4×C109—81no—
139C2×C4×C109#12011yescharacteristic
140C2×C4×C109#12011yescharacteristic
141C2×C4×C109#12111yescharacteristic
142C2×C4×C109#12111yescharacteristic
143C2×C4×C109#12111yescharacteristic
144C2×C4×C109#12711yescharacteristic
145C2×C4×C109#13512yes—
146C2×C4×C109AGL(1,ℤ/8)12yes—
147C8×C109—42no—
148C8×C109—42no—
149C8×C109—81no—
150C22×C109—21no—
151C22×C109—21no—
152C22×C109—81no—
153C22×C109—81no—
154C22×C109—81no—
155C22×C109#39211yescharacteristic
156C22×C109#44912yes—
157C4×C109—21no—
158C4×C109—21no—
159C4×C109—44no—
160C4×C109—82no—
161C4×C109—82no—
162C4×C109—81no—
163C4×C109#45412yes—
164C4×C109#45612yes—
165C4×C109#46312yes—
166C4×C109#49511yescharacteristic
167C4×C109#49611yescharacteristic
168#10286C10911yesmaximal, characteristic, direct product factor, Sylow
169C2×C109—22no—
170C2×C109—161no—
171C2×C109#200311yescharacteristic
172C2×C109#200511yescharacteristic
173C2×C109#201211yescharacteristic
174#3227C2×C10911yescharacteristic
175#3226C2×C10911yescharacteristic
176#2862C2×C10911yescharacteristic, complemented
177#1962C2×C10911yescharacteristic
178#1549C2×C10911yescharacteristic, complemented
179#1546C2×C10911yescharacteristic, complemented
180#1276C2×C10911yescharacteristic, complemented
181C109#1028611yescharacteristic, direct product factor, Sylow
182#532C22×C10911yescharacteristic
183C2×C4⋊Q8C22×C10911yescharacteristic
184C2×C4⋊Q8C22×C10911yescharacteristic
185#500—21no—
186#498—21no—
187#486—21no—
188#486—21no—
189#481—22no—
190#480—22no—
191#476—22no—
192#474—22no—
193#431C22×C10911yescharacteristic
194#431C22×C10911yescharacteristic
195#423C22×C10911yescharacteristic
196C2×#104—22no—
197#391—22no—
198#391—22no—
199#382—22no—
200C43C22×C10911yescharacteristic
201#332—22no—
202#332—22no—
203C2×#41—21no—
204C22×Q8—21no—
205C22×Q8—21no—
206C2×M16—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8D4×C10911yescharacteristic
216#127D4×C10911yescharacteristic
217#124—21no—
218#121—21no—
219#121—21no—
220C4×D4—21no—
221C4×D4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4D4×C10912yes—
225C2×C4⋊C4D4×C10912yes—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42D4×C10912yes—
229C2×C42C23×C10911yescharacteristic
230#107—24no—
231#106—24no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8D4×C10912yes—
235C4⋊C8D4×C10912yes—
236#104—44no—
237#102—24no—
238#102—24no—
239C22⋊C8—21no—
240C22⋊C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243#95—28no—
244#41—42no—
245#41—42no—
246C2×Q8—21no—
247C2×Q8—44no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—41no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4C2×D4×C10911yescharacteristic
263C22×C4C2×D4×C10911yescharacteristic
264C22×C4C2×D4×C10911yescharacteristic
265M16—42no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—41no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C42—24no—
282C42—22no—
283C42—22no—
284C42—42no—
285C42C2×D4×C10911yescharacteristic
286C42C2×D4×C10911yescharacteristic
287C42C2×D4×C10911yescharacteristic
288C42#41×C10911yescharacteristic
289C23—41no—
290C23#120×C10911yescharacteristic
291Q8—42no—
292Q8—82no—
293Q8—82no—
294Q8—82no—
295Q8—81no—
296Q8—81no—
297D4—42no—
298D4—42no—
299D4—42no—
300D4—82no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4#120×C10911yescharacteristic
314C2×C4#120×C10911yescharacteristic
315C2×C4#121×C10911yescharacteristic
316C2×C4#121×C10911yescharacteristic
317C2×C4#121×C10911yescharacteristic
318C2×C4#127×C10911yescharacteristic
319C2×C4#135×C10912yes—
320C2×C4AGL(1,ℤ/8)×C10912yes—
321C8—42no—
322C8—42no—
323C8—81no—
324C22—21no—
325C22—21no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22#392×C10911yescharacteristic
330C22#449×C10912yes—
331C4—21no—
332C4—21no—
333C4—44no—
334C4—82no—
335C4—82no—
336C4—81no—
337C4#454×C10912yes—
338C4#456×C10912yes—
339C4#463×C10912yes—
340C4#495×C10911yescharacteristic
341C4#496×C10911yescharacteristic
342C2—22no—
343C2—161no—
344C2C109×#200311yescharacteristic
345C2C109×#200511yescharacteristic
346C2C109×#201211yescharacteristic
subgroupG/H
center, socleC22×C109#392
commutator, Frattini, cosocleC2×C42C23×C109
prime radical#532×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
3. C109×#2862C2semidirect
5. C109×#1549C2semidirect
6. C109×#1546C2semidirect
7. C109×#1276C2semidirect
168. #10286C109direct (N × H)
176. #2862C2×C109semidirect
178. #1549C2×C109semidirect
179. #1546C2×C109semidirect
180. #1276C2×C109semidirect
181. C109#10286direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#32272
C109×#32262
C109×#28622
C109×#19622
C109×#15492
C109×#15462
C109×#12762
#10286109