Finite Group Databasebeta

← C109×#10288 #42128282

Subgroups — C109×#10288 #42128282

Order 27904.

554 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3226C211yesmaximal, characteristic
2C109×#3206C211yesmaximal, characteristic
3C109×#2862C211yesmaximal, characteristic, complemented
4C109×#1962C211yesmaximal, characteristic
5C109×#1548C211yesmaximal, 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#492×C109—21no—
13#487×C109—21no—
14#486×C109—21no—
15#480×C109—22no—
16#474×C109—22no—
17#461×C109—22no—
18#448×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#333×C109—22no—
28#332×C109—22no—
29C2×#41×C109—21no—
30C22×Q8×C109—21no—
31C22×Q8×C109D412yes—
32C2×SD16×C109—24no—
33C2×M16×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×C109—21no—
42C4⋊Q8×C109D411yescharacteristic
43#127×C109D411yescharacteristic
44#124×C109—21no—
45#121×C109—21no—
46#121×C109—21no—
47C4×D4×C109—21no—
48C4×D4×C109—21no—
49C2×C4⋊C4×C109—21no—
50C2×C4⋊C4×C109—21no—
51C2×C4⋊C4×C109—21no—
52C2×C4⋊C4×C109D412yes—
53C2×C42×C109—21no—
54C2×C42×C109—21no—
55C2×C42×C109D412yes—
56C2×C42×C109C2311yescharacteristic
57#107×C109—24no—
58C4⋊C8×C109—21no—
59C4⋊C8×C109—21no—
60C4⋊C8×C109D412yes—
61C4⋊C8×C109D412yes—
62#104×C109—44no—
63#103×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—22no—
73C2×Q8×C109—21no—
74C2×Q8×C109—22no—
75C2×Q8×C109—44no—
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
91SD16×C109—84no—
92M16×C109—42no—
93C2×C8×C109—22no—
94C2×C8×C109—22no—
95C2×C8×C109—41no—
96C4⋊C4×C109—22no—
97C4⋊C4×C109—21no—
98C4⋊C4×C109—21no—
99C4⋊C4×C109—22no—
100C4⋊C4×C109—41no—
101C4⋊C4×C109—41no—
102C4⋊C4×C109—41no—
103C4⋊C4×C109—41no—
104C4⋊C4×C109—41no—
105C4⋊C4×C109—41no—
106C22⋊C4×C109—41no—
107C22⋊C4×C109—41no—
108C42×C109—24no—
109C42×C109—22no—
110C42×C109—22no—
111C42×C109—42no—
112C42×C109C2×D411yescharacteristic
113C42×C109C2×D411yescharacteristic
114C42×C109C2×D411yescharacteristic
115C42×C109#4111yescharacteristic
116C23×C109—41no—
117C23×C109#12011yescharacteristic
118Q8×C109—44no—
119Q8×C109—42no—
120Q8×C109—82no—
121Q8×C109—82no—
122Q8×C109—81no—
123Q8×C109—81no—
124D4×C109—42no—
125D4×C109—42no—
126D4×C109—42no—
127D4×C109—82no—
128C2×C4×C109—22no—
129C2×C4×C109—22no—
130C2×C4×C109—24no—
131C2×C4×C109—42no—
132C2×C4×C109—42no—
133C2×C4×C109—44no—
134C2×C4×C109—41no—
135C2×C4×C109—42no—
136C2×C4×C109—42no—
137C2×C4×C109—44no—
138C2×C4×C109—81no—
139C2×C4×C109—81no—
140C2×C4×C109#12011yescharacteristic
141C2×C4×C109#12011yescharacteristic
142C2×C4×C109#12111yescharacteristic
143C2×C4×C109#12111yescharacteristic
144C2×C4×C109#12111yescharacteristic
145C2×C4×C109#12711yescharacteristic
146C2×C4×C109#13512yes—
147C2×C4×C109AGL(1,ℤ/8)12yes—
148C8×C109—42no—
149C8×C109—42no—
150C8×C109—81no—
151C22×C109—21no—
152C22×C109—21no—
153C22×C109—81no—
154C22×C109—81no—
155C22×C109—81no—
156C22×C109#39211yescharacteristic
157C22×C109#44912yes—
158C4×C109—21no—
159C4×C109—21no—
160C4×C109—44no—
161C4×C109—82no—
162C4×C109—82no—
163C4×C109—81no—
164C4×C109#45512yes—
165C4×C109#45612yes—
166C4×C109#46312yes—
167C4×C109#49511yescharacteristic
168C4×C109#49611yescharacteristic
169#10288C10911yesmaximal, characteristic, direct product factor, Sylow
170C2×C109—22no—
171C2×C109—161no—
172C2×C109#200411yescharacteristic
173C2×C109#200511yescharacteristic
174C2×C109#201211yescharacteristic
175#3226C2×C10911yescharacteristic
176#3206C2×C10911yescharacteristic
177#2862C2×C10911yescharacteristic, complemented
178#1962C2×C10911yescharacteristic
179#1548C2×C10911yescharacteristic, complemented
180#1546C2×C10911yescharacteristic, complemented
181#1276C2×C10911yescharacteristic, complemented
182C109#1028811yescharacteristic, direct product factor, Sylow
183#532C22×C10911yescharacteristic
184C2×C4⋊Q8C22×C10911yescharacteristic
185C2×C4⋊Q8C22×C10911yescharacteristic
186#500—21no—
187#492—21no—
188#487—21no—
189#486—21no—
190#480—22no—
191#474—22no—
192#461—22no—
193#448—22no—
194#431C22×C10911yescharacteristic
195#431C22×C10911yescharacteristic
196#423C22×C10911yescharacteristic
197C2×#104—22no—
198#391—22no—
199#391—22no—
200#382—22no—
201C43C22×C10911yescharacteristic
202#333—22no—
203#332—22no—
204C2×#41—21no—
205C22×Q8—21no—
206C22×Q8D4×C10912yes—
207C2×SD16—24no—
208C2×M16—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8D4×C10911yescharacteristic
218#127D4×C10911yescharacteristic
219#124—21no—
220#121—21no—
221#121—21no—
222C4×D4—21no—
223C4×D4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4D4×C10912yes—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42D4×C10912yes—
231C2×C42C23×C10911yescharacteristic
232#107—24no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8D4×C10912yes—
236C4⋊C8D4×C10912yes—
237#104—44no—
238#103—24no—
239#102—24no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244#95—28no—
245#41—42no—
246#41—42no—
247C2×Q8—22no—
248C2×Q8—21no—
249C2×Q8—22no—
250C2×Q8—44no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—41no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—41no—
263C22×C4C2×D4×C10911yescharacteristic
264C22×C4C2×D4×C10911yescharacteristic
265C22×C4C2×D4×C10911yescharacteristic
266SD16—84no—
267M16—42no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—41no—
271C4⋊C4—22no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—22no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C42—24no—
284C42—22no—
285C42—22no—
286C42—42no—
287C42C2×D4×C10911yescharacteristic
288C42C2×D4×C10911yescharacteristic
289C42C2×D4×C10911yescharacteristic
290C42#41×C10911yescharacteristic
291C23—41no—
292C23#120×C10911yescharacteristic
293Q8—44no—
294Q8—42no—
295Q8—82no—
296Q8—82no—
297Q8—81no—
298Q8—81no—
299D4—42no—
300D4—42no—
301D4—42no—
302D4—82no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4#120×C10911yescharacteristic
316C2×C4#120×C10911yescharacteristic
317C2×C4#121×C10911yescharacteristic
318C2×C4#121×C10911yescharacteristic
319C2×C4#121×C10911yescharacteristic
320C2×C4#127×C10911yescharacteristic
321C2×C4#135×C10912yes—
322C2×C4AGL(1,ℤ/8)×C10912yes—
323C8—42no—
324C8—42no—
325C8—81no—
326C22—21no—
327C22—21no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22#392×C10911yescharacteristic
332C22#449×C10912yes—
333C4—21no—
334C4—21no—
335C4—44no—
336C4—82no—
337C4—82no—
338C4—81no—
339C4#455×C10912yes—
340C4#456×C10912yes—
341C4#463×C10912yes—
342C4#495×C10911yescharacteristic
343C4#496×C10911yescharacteristic
344C2—22no—
345C2—161no—
346C2C109×#200411yescharacteristic
347C2C109×#200511yescharacteristic
348C2C109×#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×#1548C2semidirect
6. C109×#1546C2semidirect
7. C109×#1276C2semidirect
169. #10288C109direct (N × H)
177. #2862C2×C109semidirect
179. #1548C2×C109semidirect
180. #1546C2×C109semidirect
181. #1276C2×C109semidirect
182. C109#10288direct (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×#32062
C109×#28622
C109×#19622
C109×#15482
C109×#15462
C109×#12762
#10288109