Finite Group Databasebeta

← C109×#10201 #42128195

Subgroups — C109×#10201 #42128195

Order 27904.

528 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3246C211yesmaximal, characteristic
2C109×#3227C211yesmaximal, characteristic
3C109×#3165C211yesmaximal, characteristic, complemented
4C109×#1981C211yesmaximal, characteristic, complemented
5C109×#1980C211yesmaximal, characteristic
6C109×#1503C211yesmaximal, characteristic, complemented
7C109×#1283C211yesmaximal, characteristic, complemented
8#533×C109C2211yescharacteristic
9#532×C109C2211yescharacteristic
10C2×C4⋊Q8×C109C2211yescharacteristic
11#501×C109—22no—
12#500×C109—21no—
13#498×C109—21no—
14#498×C109—21no—
15#486×C109—21no—
16#481×C109—22no—
17#476×C109—22no—
18#476×C109—21no—
19#475×C109—21no—
20#465×C109—22no—
21#461×C109—21no—
22#460×C109—21no—
23#431×C109C2211yescharacteristic
24#428×C109—22no—
25C2×C4⋊C8×C109C2211yescharacteristic
26C2×#104×C109—22no—
27C2×#103×C109—22no—
28C4×M16×C109C2211yescharacteristic
29C4×C4⋊C4×C109C2211yescharacteristic
30#328×C109—22no—
31C2×#41×C109—21no—
32C22×Q8×C109—21no—
33C2×SD16×C109—22no—
34C2×M16×C109D412yes—
35C22×C8×C109D412yes—
36C4⋊Q8×C109—21no—
37C4⋊Q8×C109—21no—
38C4⋊Q8×C109—21no—
39C4⋊Q8×C109D411yescharacteristic
40C4⋊Q8×C109D411yescharacteristic
41C4⋊Q8×C109D411yescharacteristic
42C4⋊Q8×C109D412yes—
43#127×C109D411yescharacteristic
44#125×C109—21no—
45#124×C109—21no—
46C22⋊Q8×C109—21no—
47C22⋊Q8×C109—21no—
48#121×C109—21no—
49#121×C109—21no—
50C4×Q8×C109—21no—
51C4×Q8×C109—21no—
52C4×D4×C109—21no—
53C4×D4×C109—21no—
54#117×C109—21no—
55C2×C4⋊C4×C109—21no—
56C2×C4⋊C4×C109—21no—
57C2×C4⋊C4×C109—21no—
58C2×C42×C109—21no—
59C2×C42×C109C2311yescharacteristic
60#107×C109—24no—
61#106×C109—24no—
62#106×C109—24no—
63#106×C109—22no—
64C4⋊C8×C109—21no—
65C4⋊C8×C109—21no—
66C4⋊C8×C109—21no—
67#104×C109—44no—
68#103×C109—22no—
69#103×C109—44no—
70#102×C109—22no—
71#102×C109—24no—
72C22⋊C8×C109D412yes—
73#97×C109—21no—
74C4×C8×C109—21no—
75C4×C8×C109—21no—
76#95×C109—21no—
77#41×C109—42no—
78#41×C109—42no—
79C2×Q8×C109—21no—
80C2×Q8×C109—21no—
81C2×Q8×C109—22no—
82C2×Q8×C109—21no—
83C2×Q8×C109—42no—
84C2×Q8×C109—42no—
85C2×Q8×C109—41no—
86C2×D4×C109—21no—
87C2×D4×C109—21no—
88C2×D4×C109—21no—
89C2×D4×C109—41no—
90C22×C4×C109—41no—
91C22×C4×C109—41no—
92C22×C4×C109—41no—
93C22×C4×C109C2×D411yescharacteristic
94C22×C4×C109C2×D411yescharacteristic
95C22×C4×C109C2×D411yescharacteristic
96SD16×C109—82no—
97M16×C109—22no—
98M16×C109SD1614yescomplemented
99C2×C8×C109—22no—
100C2×C8×C109—22no—
101C2×C8×C109—22no—
102C2×C8×C109—42no—
103C4⋊C4×C109—22no—
104C4⋊C4×C109—22no—
105C4⋊C4×C109—21no—
106C4⋊C4×C109—21no—
107C4⋊C4×C109—21no—
108C4⋊C4×C109—21no—
109C4⋊C4×C109—22no—
110C4⋊C4×C109—21no—
111C4⋊C4×C109—21no—
112C4⋊C4×C109—41no—
113C4⋊C4×C109—41no—
114C4⋊C4×C109—41no—
115C4⋊C4×C109—41no—
116C4⋊C4×C109—41no—
117C22⋊C4×C109—41no—
118C22⋊C4×C109—41no—
119C22⋊C4×C109—41no—
120C42×C109—41no—
121C42×C109—42no—
122C42×C109C2×D411yescharacteristic
123C42×C109C2×D411yescharacteristic
124C42×C109C2×D411yescharacteristic
125C42×C109#4111yescharacteristic
126C23×C109—41no—
127C23×C109#12711yescharacteristic
128Q8×C109—44no—
129Q8×C109—41no—
130Q8×C109—41no—
131Q8×C109—42no—
132Q8×C109—81no—
133Q8×C109—81no—
134Q8×C109—81no—
135D4×C109—42no—
136D4×C109—41no—
137D4×C109—41no—
138D4×C109—42no—
139D4×C109—81no—
140D4×C109—81no—
141C2×C4×C109—21no—
142C2×C4×C109—21no—
143C2×C4×C109—41no—
144C2×C4×C109—42no—
145C2×C4×C109—41no—
146C2×C4×C109—41no—
147C2×C4×C109—41no—
148C2×C4×C109—42no—
149C2×C4×C109—81no—
150C2×C4×C109—82no—
151C2×C4×C109—81no—
152C2×C4×C109—82no—
153C2×C4×C109#12011yescharacteristic
154C2×C4×C109#12011yescharacteristic
155C2×C4×C109#12011yescharacteristic
156C2×C4×C109#12111yescharacteristic
157C2×C4×C109#12111yescharacteristic
158C2×C4×C109#12111yescharacteristic
159C2×C4×C109C2×SD1612yes—
160C2×C4×C109#13712yes—
161C8×C109—24no—
162C8×C109—42no—
163C8×C109—42no—
164C22×C109—21no—
165C22×C109—21no—
166C22×C109—81no—
167C22×C109—81no—
168C22×C109—81no—
169C22×C109#39211yescharacteristic
170C22×C109#49211yescharacteristic
171C22×C109#49711yescharacteristic
172C4×C109—21no—
173C4×C109—21no—
174C4×C109—21no—
175C4×C109—82no—
176C4×C109—82no—
177C4×C109—81no—
178C4×C109—81no—
179C4×C109—81no—
180C4×C109#44812yes—
181C4×C109#45311yescharacteristic
182C4×C109#45411yescharacteristic
183C4×C109#46112yes—
184#10201C10911yesmaximal, characteristic, direct product factor, Sylow
185C2×C109—21no—
186C2×C109—21no—
187C2×C109—161no—
188C2×C109#200611yescharacteristic
189C2×C109#200911yescharacteristic
190C2×C109#201311yescharacteristic
191#3246C2×C10911yescharacteristic
192#3227C2×C10911yescharacteristic
193#3165C2×C10911yescharacteristic, complemented
194#1981C2×C10911yescharacteristic, complemented
195#1980C2×C10911yescharacteristic
196#1503C2×C10911yescharacteristic, complemented
197#1283C2×C10911yescharacteristic, complemented
198C109#1020111yescharacteristic, direct product factor, Sylow
199#533C22×C10911yescharacteristic
200#532C22×C10911yescharacteristic
201C2×C4⋊Q8C22×C10911yescharacteristic
202#501—22no—
203#500—21no—
204#498—21no—
205#498—21no—
206#486—21no—
207#481—22no—
208#476—22no—
209#476—21no—
210#475—21no—
211#465—22no—
212#461—21no—
213#460—21no—
214#431C22×C10911yescharacteristic
215#428—22no—
216C2×C4⋊C8C22×C10911yescharacteristic
217C2×#104—22no—
218C2×#103—22no—
219C4×M16C22×C10911yescharacteristic
220C4×C4⋊C4C22×C10911yescharacteristic
221#328—22no—
222C2×#41—21no—
223C22×Q8—21no—
224C2×SD16—22no—
225C2×M16D4×C10912yes—
226C22×C8D4×C10912yes—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8D4×C10912yes—
231C4⋊Q8D4×C10911yescharacteristic
232C4⋊Q8D4×C10911yescharacteristic
233C4⋊Q8D4×C10911yescharacteristic
234#127D4×C10911yescharacteristic
235#125—21no—
236#124—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239#121—21no—
240#121—21no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×D4—21no—
244C4×D4—21no—
245#117—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C42—21no—
250C2×C42C23×C10911yescharacteristic
251#107—24no—
252#106—24no—
253#106—22no—
254#106—24no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258#104—44no—
259#103—22no—
260#103—44no—
261#102—22no—
262#102—24no—
263C22⋊C8D4×C10912yes—
264#97—21no—
265C4×C8—21no—
266C4×C8—21no—
267#95—21no—
268#41—42no—
269#41—42no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—22no—
273C2×Q8—21no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—41no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4C2×D4×C10911yescharacteristic
285C22×C4C2×D4×C10911yescharacteristic
286C22×C4C2×D4×C10911yescharacteristic
287SD16—82no—
288M16—22no—
289M16SD16×C10914yescomplemented
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—42no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—22no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C42—41no—
312C42—42no—
313C42C2×D4×C10911yescharacteristic
314C42C2×D4×C10911yescharacteristic
315C42C2×D4×C10911yescharacteristic
316C42#41×C10911yescharacteristic
317C23—41no—
318C23#127×C10911yescharacteristic
319Q8—44no—
320Q8—41no—
321Q8—41no—
322Q8—42no—
323Q8—81no—
324Q8—81no—
325Q8—81no—
326D4—42no—
327D4—41no—
328D4—41no—
329D4—42no—
330D4—81no—
331D4—81no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—82no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—82no—
344C2×C4#120×C10911yescharacteristic
345C2×C4#120×C10911yescharacteristic
346C2×C4#120×C10911yescharacteristic
347C2×C4#121×C10911yescharacteristic
348C2×C4#121×C10911yescharacteristic
349C2×C4#121×C10911yescharacteristic
350C2×C4C2×SD16×C10912yes—
351C2×C4#137×C10912yes—
352C8—24no—
353C8—42no—
354C8—42no—
355C22—21no—
356C22—21no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22#392×C10911yescharacteristic
361C22#492×C10911yescharacteristic
362C22#497×C10911yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—82no—
367C4—82no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4#448×C10912yes—
372C4#453×C10911yescharacteristic
373C4#454×C10911yescharacteristic
374C4#461×C10912yes—
375C2—21no—
376C2—21no—
377C2—161no—
378C2C109×#200611yescharacteristic
379C2C109×#200911yescharacteristic
380C2C109×#201311yescharacteristic
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×#3165C2semidirect
4. C109×#1981C2semidirect
6. C109×#1503C2semidirect
7. C109×#1283C2semidirect
98. M16×C109SD16semidirect
184. #10201C109direct (N × H)
193. #3165C2×C109semidirect
194. #1981C2×C109semidirect
196. #1503C2×C109semidirect
197. #1283C2×C109semidirect
198. C109#10201direct (N × H)
289. M16SD16×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#32462
C109×#32272
C109×#31652
C109×#19812
C109×#19802
C109×#15032
C109×#12832
#10201109