Finite Group Databasebeta

← C109×#10181 #42128175

Subgroups — C109×#10181 #42128175

Order 27904.

530 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3206C211yesmaximal, characteristic
2C2×#477×C109C211yesmaximal, characteristic, complemented
3C109×#3037C211yesmaximal, characteristic
4C109×#1549C211yesmaximal, characteristic, complemented
5C109×#1480C211yesmaximal, characteristic
6C109×#1461C211yesmaximal, characteristic, complemented
7C109×#1301C211yesmaximal, characteristic, complemented
8C2×#137×C109—21no—
9#532×C109C2211yescharacteristic
10C2×C4⋊Q8×C109C2211yescharacteristic
11C2×C4×Q8×C109C2211yescharacteristic
12#492×C109—21no—
13#487×C109—21no—
14#477×C109—22no—
15#477×C109—22no—
16#464×C109—22no—
17#462×C109—21no—
18#461×C109—22no—
19#460×C109—21no—
20#448×C109—22no—
21#431×C109C2211yescharacteristic
22#431×C109C2211yescharacteristic
23C2×#107×C109—21no—
24#423×C109C2211yescharacteristic
25C2×C4⋊C8×C109C2211yescharacteristic
26#417×C109—21no—
27C2×#103×C109—21no—
28#332×C109—22no—
29D4⋊C8×C109—22no—
30#324×C109—22no—
31C2×#41×C109—21no—
32C22×Q8×C109—21no—
33C22×Q8×C109D411yescharacteristic
34#137×C109—42no—
35#137×C109—42no—
36C2×Q16×C109—22no—
37C2×SD16×C109—22no—
38C2×SD16×C109—24no—
39C2×M16×C109—21no—
40C22×C8×C109—21no—
41C4⋊Q8×C109—21no—
42C4⋊Q8×C109—21no—
43C4⋊Q8×C109—21no—
44C4⋊Q8×C109D412yes—
45C4⋊Q8×C109D411yescharacteristic
46#127×C109D411yescharacteristic
47#124×C109—21no—
48#121×C109—21no—
49#121×C109—21no—
50C4×Q8×C109—21no—
51C4×Q8×C109—21no—
52C4×Q8×C109—21no—
53C4×Q8×C109—21no—
54C4×D4×C109—21no—
55C4×D4×C109—21no—
56C2×C4⋊C4×C109—21no—
57C2×C4⋊C4×C109—21no—
58C2×C4⋊C4×C109—21no—
59C2×C4⋊C4×C109D411yescharacteristic
60C2×C42×C109—21no—
61C2×C42×C109C2311yescharacteristic
62#107×C109—24no—
63C4⋊C8×C109—21no—
64C4⋊C8×C109—21no—
65C4⋊C8×C109—21no—
66C4⋊C8×C109—21no—
67C4⋊C8×C109D412yes—
68C4⋊C8×C109D412yes—
69#103×C109—24no—
70#103×C109—22no—
71#103×C109—24no—
72#102×C109—22no—
73C22⋊C8×C109—21no—
74C22⋊C8×C109D412yes—
75C4×C8×C109—21no—
76C4×C8×C109—21no—
77#41×C109—41no—
78#41×C109—41no—
79#41×C109—41no—
80#41×C109—41no—
81C2×Q8×C109—21no—
82C2×Q8×C109—21no—
83C2×Q8×C109—21no—
84C2×Q8×C109—21no—
85C2×Q8×C109—22no—
86C2×Q8×C109—21no—
87C2×Q8×C109—21no—
88C2×Q8×C109—41no—
89C2×Q8×C109—82no—
90C2×Q8×C109SD1612yescomplemented
91C2×D4×C109—21no—
92C2×D4×C109—21no—
93C2×D4×C109—21no—
94C2×D4×C109—41no—
95C22×C4×C109—21no—
96C22×C4×C109—21no—
97C22×C4×C109—41no—
98C22×C4×C109—41no—
99C22×C4×C109C2×D411yescharacteristic
100C22×C4×C109C2×D411yescharacteristic
101C22×C4×C109#4111yescharacteristic
102Q16×C109—44no—
103SD16×C109—44no—
104SD16×C109—84no—
105M16×C109—41no—
106M16×C109—41no—
107C2×C8×C109—22no—
108C2×C8×C109—22no—
109C2×C8×C109—22no—
110C2×C8×C109—41no—
111C2×C8×C109—41no—
112C2×C8×C109—41no—
113C4⋊C4×C109—21no—
114C4⋊C4×C109—21no—
115C4⋊C4×C109—22no—
116C4⋊C4×C109—21no—
117C4⋊C4×C109—22no—
118C4⋊C4×C109—21no—
119C4⋊C4×C109—41no—
120C4⋊C4×C109—41no—
121C4⋊C4×C109—41no—
122C4⋊C4×C109—41no—
123C22⋊C4×C109—41no—
124C22⋊C4×C109—41no—
125C42×C109—41no—
126C42×C109—41no—
127C42×C109C2×D411yescharacteristic
128C42×C109C2×D411yescharacteristic
129C42×C109C2×D411yescharacteristic
130C42×C109C2×D411yescharacteristic
131C23×C109—41no—
132C23×C109#12111yescharacteristic
133Q8×C109—22no—
134Q8×C109—22no—
135Q8×C109—44no—
136Q8×C109—41no—
137Q8×C109—41no—
138Q8×C109—41no—
139Q8×C109—41no—
140Q8×C109—82no—
141D4×C109—42no—
142D4×C109—41no—
143D4×C109—41no—
144D4×C109—42no—
145D4×C109—81no—
146D4×C109—81no—
147C2×C4×C109—22no—
148C2×C4×C109—22no—
149C2×C4×C109—41no—
150C2×C4×C109—41no—
151C2×C4×C109—41no—
152C2×C4×C109—41no—
153C2×C4×C109—41no—
154C2×C4×C109—42no—
155C2×C4×C109—82no—
156C2×C4×C109—81no—
157C2×C4×C109—81no—
158C2×C4×C109—81no—
159C2×C4×C109#12011yescharacteristic
160C2×C4×C109#12011yescharacteristic
161C2×C4×C109#12011yescharacteristic
162C2×C4×C109#12111yescharacteristic
163C2×C4×C109#12111yescharacteristic
164C2×C4×C109#12711yescharacteristic
165C2×C4×C109C2×SD1611yescharacteristic
166C2×C4×C109#13511yescharacteristic
167C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C109#13711yescharacteristic
169C8×C109—42no—
170C8×C109—42no—
171C8×C109—42no—
172C8×C109—81no—
173C22×C109—21no—
174C22×C109—21no—
175C22×C109—81no—
176C22×C109—81no—
177C22×C109—81no—
178C22×C109#39211yescharacteristic
179C22×C109#46511yescharacteristic
180C22×C109#46811yescharacteristic
181C4×C109—21no—
182C4×C109—21no—
183C4×C109—21no—
184C4×C109—42no—
185C4×C109—81no—
186C4×C109—81no—
187C4×C109—82no—
188C4×C109#44811yescharacteristic
189C4×C109#44911yescharacteristic
190C4×C109#45011yescharacteristic
191C4×C109#45211yescharacteristic
192C4×C109#47312yes—
193#10181C10911yesmaximal, characteristic, direct product factor, Sylow
194C2×C109—21no—
195C2×C109—21no—
196C2×C109—161no—
197C2×C109#200011yescharacteristic
198C2×C109#200811yescharacteristic
199C2×C109#202011yescharacteristic
200#3206C2×C10911yescharacteristic
201C2×#477C2×C10911yescharacteristic, complemented
202#3037C2×C10911yescharacteristic
203#1549C2×C10911yescharacteristic, complemented
204#1480C2×C10911yescharacteristic
205#1461C2×C10911yescharacteristic, complemented
206#1301C2×C10911yescharacteristic, complemented
207C109#1018111yescharacteristic, direct product factor, Sylow
208C2×#137—21no—
209#532C22×C10911yescharacteristic
210C2×C4⋊Q8C22×C10911yescharacteristic
211C2×C4×Q8C22×C10911yescharacteristic
212#492—21no—
213#487—21no—
214#477—22no—
215#477—22no—
216#464—22no—
217#462—21no—
218#461—22no—
219#460—21no—
220#448—22no—
221#431C22×C10911yescharacteristic
222#431C22×C10911yescharacteristic
223C2×#107—21no—
224#423C22×C10911yescharacteristic
225C2×C4⋊C8C22×C10911yescharacteristic
226#417—21no—
227C2×#103—21no—
228#332—22no—
229D4⋊C8—22no—
230#324—22no—
231C2×#41—21no—
232C22×Q8—21no—
233C22×Q8D4×C10911yescharacteristic
234#137—42no—
235#137—42no—
236C2×Q16—22no—
237C2×SD16—22no—
238C2×SD16—24no—
239C2×M16—21no—
240C22×C8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8D4×C10912yes—
245C4⋊Q8D4×C10911yescharacteristic
246#127D4×C10911yescharacteristic
247#124—21no—
248#121—21no—
249#121—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×D4—21no—
255C4×D4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4D4×C10911yescharacteristic
260C2×C42—21no—
261C2×C42C23×C10911yescharacteristic
262#107—24no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8D4×C10912yes—
268C4⋊C8D4×C10912yes—
269#103—24no—
270#103—22no—
271#103—24no—
272#102—22no—
273C22⋊C8—21no—
274C22⋊C8D4×C10912yes—
275C4×C8—21no—
276C4×C8—21no—
277#41—41no—
278#41—41no—
279#41—41no—
280#41—41no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—41no—
289C2×Q8—82no—
290C2×Q8SD16×C10912yescomplemented
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—41no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4C2×D4×C10911yescharacteristic
300C22×C4C2×D4×C10911yescharacteristic
301C22×C4#41×C10911yescharacteristic
302Q16—44no—
303SD16—44no—
304SD16—84no—
305M16—41no—
306M16—41no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—22no—
316C4⋊C4—21no—
317C4⋊C4—22no—
318C4⋊C4—21no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C42—41no—
326C42—41no—
327C42C2×D4×C10911yescharacteristic
328C42C2×D4×C10911yescharacteristic
329C42C2×D4×C10911yescharacteristic
330C42C2×D4×C10911yescharacteristic
331C23—41no—
332C23#121×C10911yescharacteristic
333Q8—22no—
334Q8—22no—
335Q8—44no—
336Q8—41no—
337Q8—41no—
338Q8—41no—
339Q8—41no—
340Q8—82no—
341D4—42no—
342D4—41no—
343D4—41no—
344D4—42no—
345D4—81no—
346D4—81no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—82no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4#120×C10911yescharacteristic
360C2×C4#120×C10911yescharacteristic
361C2×C4#120×C10911yescharacteristic
362C2×C4#121×C10911yescharacteristic
363C2×C4#121×C10911yescharacteristic
364C2×C4#127×C10911yescharacteristic
365C2×C4C2×SD16×C10911yescharacteristic
366C2×C4#135×C10911yescharacteristic
367C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
368C2×C4#137×C10911yescharacteristic
369C8—42no—
370C8—42no—
371C8—42no—
372C8—81no—
373C22—21no—
374C22—21no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22#392×C10911yescharacteristic
379C22#465×C10911yescharacteristic
380C22#468×C10911yescharacteristic
381C4—21no—
382C4—21no—
383C4—21no—
384C4—42no—
385C4—81no—
386C4—82no—
387C4—81no—
388C4#448×C10911yescharacteristic
389C4#449×C10911yescharacteristic
390C4#450×C10911yescharacteristic
391C4#452×C10911yescharacteristic
392C4#473×C10912yes—
393C2—21no—
394C2—21no—
395C2—161no—
396C2C109×#200011yescharacteristic
397C2C109×#200811yescharacteristic
398C2C109×#202011yescharacteristic
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
2. C2×#477×C109C2semidirect
4. C109×#1549C2semidirect
6. C109×#1461C2semidirect
7. C109×#1301C2semidirect
90. C2×Q8×C109SD16semidirect
193. #10181C109direct (N × H)
201. C2×#477C2×C109semidirect
203. #1549C2×C109semidirect
205. #1461C2×C109semidirect
206. #1301C2×C109semidirect
207. C109#10181direct (N × H)
290. C2×Q8SD16×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#32062
C2×#477×C1092
C109×#30372
C109×#15492
C109×#14802
C109×#14612
C109×#13012
#10181109