Finite Group Databasebeta

← C109×#10180 #42128174

Subgroups — C109×#10180 #42128174

Order 27904.

562 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3206C211yesmaximal, characteristic, complemented
2C2×#476×C109C211yesmaximal, characteristic, complemented
3C109×#3036C211yesmaximal, characteristic
4C109×#1549C211yesmaximal, characteristic, complemented
5C109×#1480C211yesmaximal, characteristic, complemented
6C109×#1460C211yesmaximal, characteristic, complemented
7C109×#1301C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C109—21no—
9#532×C109C2211yescharacteristic
10C2×C4⋊Q8×C109C2211yescharacteristic
11C2×C4×D4×C109C2211yescharacteristic
12#492×C109—21no—
13#487×C109—21no—
14#476×C109—22no—
15#476×C109—22no—
16#463×C109—22no—
17#461×C109—22no—
18#461×C109—21no—
19#459×C109—21no—
20#448×C109—22no—
21#431×C109C2211yescharacteristic
22#431×C109C2211yescharacteristic
23C2×#106×C109—21no—
24#423×C109C2211yescharacteristic
25C2×C4⋊C8×C109C2211yescharacteristic
26#417×C109—21no—
27C2×#102×C109—21no—
28#332×C109—22no—
29D4⋊C8×C109—22no—
30#323×C109—22no—
31C2×#41×C109—21no—
32C22×Q8×C109—21no—
33C22×D4×C109D411yescharacteristic
34C23×C4×C109—21no—
35AGL(1,ℤ/8)×C109—42no—
36AGL(1,ℤ/8)×C109—42no—
37C2×SD16×C109—22no—
38C2×SD16×C109—24no—
39C2×D8×C109—22no—
40C2×M16×C109—21no—
41C22×C8×C109—21no—
42C4⋊Q8×C109—21no—
43C4⋊Q8×C109—21no—
44C4⋊Q8×C109—21no—
45C4⋊Q8×C109D412yes—
46C4⋊Q8×C109D411yescharacteristic
47#127×C109D411yescharacteristic
48#124×C109—21no—
49#121×C109—21no—
50#121×C109—21no—
51C4×D4×C109—21no—
52C4×D4×C109—21no—
53C4×D4×C109—21no—
54C4×D4×C109—21no—
55C4×D4×C109—21no—
56C4×D4×C109—21no—
57C2×C4⋊C4×C109—21no—
58C2×C4⋊C4×C109—21no—
59C2×C4⋊C4×C109D411yescharacteristic
60C2×C22⋊C4×C109—21no—
61C2×C42×C109C2311yescharacteristic
62#106×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—22no—
70#103×C109—24no—
71#102×C109—24no—
72#102×C109—22no—
73C22⋊C8×C109—21no—
74C22⋊C8×C109D412yes—
75C4×C8×C109—21no—
76C4×C8×C109—21no—
77C24×C109—21no—
78#41×C109—41no—
79#41×C109—41no—
80#41×C109—41no—
81#41×C109—41no—
82C2×Q8×C109—21no—
83C2×Q8×C109—22no—
84C2×Q8×C109—41no—
85C2×Q8×C109—82no—
86C2×D4×C109—21no—
87C2×D4×C109—21no—
88C2×D4×C109—21no—
89C2×D4×C109—21no—
90C2×D4×C109—21no—
91C2×D4×C109—21no—
92C2×D4×C109—21no—
93C2×D4×C109—21no—
94C2×D4×C109—41no—
95C2×D4×C109SD1612yescomplemented
96C22×C4×C109—21no—
97C22×C4×C109—41no—
98C22×C4×C109—41no—
99C22×C4×C109—41no—
100C22×C4×C109—41no—
101C22×C4×C109—82no—
102C22×C4×C109C2×D411yescharacteristic
103C22×C4×C109C2×D411yescharacteristic
104C22×C4×C109#4111yescharacteristic
105SD16×C109—44no—
106SD16×C109—84no—
107D8×C109—44no—
108M16×C109—41no—
109M16×C109—41no—
110C2×C8×C109—22no—
111C2×C8×C109—22no—
112C2×C8×C109—22no—
113C2×C8×C109—41no—
114C2×C8×C109—41no—
115C2×C8×C109—41no—
116C4⋊C4×C109—21no—
117C4⋊C4×C109—22no—
118C4⋊C4×C109—21no—
119C4⋊C4×C109—21no—
120C4⋊C4×C109—22no—
121C4⋊C4×C109—21no—
122C4⋊C4×C109—41no—
123C4⋊C4×C109—41no—
124C22⋊C4×C109—41no—
125C22⋊C4×C109—41no—
126C22⋊C4×C109—41no—
127C22⋊C4×C109—41no—
128C42×C109C2×D411yescharacteristic
129C42×C109C2×D411yescharacteristic
130C42×C109C2×D411yescharacteristic
131C42×C109C2×D411yescharacteristic
132C23×C109—22no—
133C23×C109—41no—
134C23×C109—41no—
135C23×C109—41no—
136C23×C109—81no—
137C23×C109—81no—
138C23×C109#12111yescharacteristic
139Q8×C109—44no—
140Q8×C109—41no—
141Q8×C109—41no—
142Q8×C109—82no—
143D4×C109—22no—
144D4×C109—22no—
145D4×C109—42no—
146D4×C109—41no—
147D4×C109—41no—
148D4×C109—41no—
149D4×C109—41no—
150D4×C109—42no—
151D4×C109—81no—
152D4×C109—81no—
153C2×C4×C109—22no—
154C2×C4×C109—41no—
155C2×C4×C109—41no—
156C2×C4×C109—41no—
157C2×C4×C109—42no—
158C2×C4×C109—82no—
159C2×C4×C109—82no—
160C2×C4×C109—81no—
161C2×C4×C109—81no—
162C2×C4×C109—82no—
163C2×C4×C109—81no—
164C2×C4×C109#12011yescharacteristic
165C2×C4×C109#12011yescharacteristic
166C2×C4×C109#12011yescharacteristic
167C2×C4×C109#12111yescharacteristic
168C2×C4×C109#12111yescharacteristic
169C2×C4×C109#12711yescharacteristic
170C2×C4×C109C2×SD1611yescharacteristic
171C2×C4×C109#13511yescharacteristic
172C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
173C2×C4×C109#13711yescharacteristic
174C8×C109—42no—
175C8×C109—42no—
176C8×C109—42no—
177C8×C109—81no—
178C22×C109—21no—
179C22×C109—21no—
180C22×C109—42no—
181C22×C109—82no—
182C22×C109—82no—
183C22×C109—81no—
184C22×C109—81no—
185C22×C109—81no—
186C22×C109—81no—
187C22×C109—81no—
188C22×C109#39211yescharacteristic
189C22×C109#46511yescharacteristic
190C22×C109#46811yescharacteristic
191C4×C109—21no—
192C4×C109—21no—
193C4×C109—21no—
194C4×C109—81no—
195C4×C109—82no—
196C4×C109—81no—
197C4×C109#44811yescharacteristic
198C4×C109#44911yescharacteristic
199C4×C109#45011yescharacteristic
200C4×C109#45211yescharacteristic
201C4×C109#47112yes—
202#10180C10911yesmaximal, characteristic, direct product factor, Sylow
203C2×C109—21no—
204C2×C109—21no—
205C2×C109—82no—
206C2×C109—161no—
207C2×C109#200011yescharacteristic
208C2×C109#200811yescharacteristic
209C2×C109#201911yescharacteristic
210#3206C2×C10911yescharacteristic, complemented
211C2×#476C2×C10911yescharacteristic, complemented
212#3036C2×C10911yescharacteristic
213#1549C2×C10911yescharacteristic, complemented
214#1480C2×C10911yescharacteristic, complemented
215#1460C2×C10911yescharacteristic, complemented
216#1301C2×C10911yescharacteristic, complemented
217C109#1018011yescharacteristic, direct product factor, Sylow
218C2×AGL(1,ℤ/8)—21no—
219#532C22×C10911yescharacteristic
220C2×C4⋊Q8C22×C10911yescharacteristic
221C2×C4×D4C22×C10911yescharacteristic
222#492—21no—
223#487—21no—
224#476—22no—
225#476—22no—
226#463—22no—
227#461—22no—
228#461—21no—
229#459—21no—
230#448—22no—
231#431C22×C10911yescharacteristic
232#431C22×C10911yescharacteristic
233C2×#106—21no—
234#423C22×C10911yescharacteristic
235C2×C4⋊C8C22×C10911yescharacteristic
236#417—21no—
237C2×#102—21no—
238#332—22no—
239D4⋊C8—22no—
240#323—22no—
241C2×#41—21no—
242C22×Q8—21no—
243C22×D4D4×C10911yescharacteristic
244C23×C4—21no—
245AGL(1,ℤ/8)—42no—
246AGL(1,ℤ/8)—42no—
247C2×SD16—22no—
248C2×SD16—24no—
249C2×D8—22no—
250C2×M16—21no—
251C22×C8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8D4×C10912yes—
256C4⋊Q8D4×C10911yescharacteristic
257#127D4×C10911yescharacteristic
258#124—21no—
259#121—21no—
260#121—21no—
261C4×D4—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4D4×C10911yescharacteristic
270C2×C22⋊C4—21no—
271C2×C42C23×C10911yescharacteristic
272#106—24no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8D4×C10912yes—
278C4⋊C8D4×C10912yes—
279#103—22no—
280#103—24no—
281#102—24no—
282#102—22no—
283C22⋊C8—21no—
284C22⋊C8D4×C10912yes—
285C4×C8—21no—
286C4×C8—21no—
287C24—21no—
288#41—41no—
289#41—41no—
290#41—41no—
291#41—41no—
292C2×Q8—21no—
293C2×Q8—22no—
294C2×Q8—41no—
295C2×Q8—82no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4SD16×C10912yescomplemented
306C22×C4—21no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—82no—
312C22×C4C2×D4×C10911yescharacteristic
313C22×C4C2×D4×C10911yescharacteristic
314C22×C4#41×C10911yescharacteristic
315SD16—44no—
316SD16—84no—
317D8—44no—
318M16—41no—
319M16—41no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C42C2×D4×C10911yescharacteristic
339C42C2×D4×C10911yescharacteristic
340C42C2×D4×C10911yescharacteristic
341C42C2×D4×C10911yescharacteristic
342C23—22no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—81no—
347C23—81no—
348C23#121×C10911yescharacteristic
349Q8—44no—
350Q8—41no—
351Q8—41no—
352Q8—82no—
353D4—22no—
354D4—22no—
355D4—42no—
356D4—41no—
357D4—41no—
358D4—41no—
359D4—41no—
360D4—42no—
361D4—81no—
362D4—81no—
363C2×C4—22no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4#120×C10911yescharacteristic
375C2×C4#120×C10911yescharacteristic
376C2×C4#120×C10911yescharacteristic
377C2×C4#121×C10911yescharacteristic
378C2×C4#121×C10911yescharacteristic
379C2×C4#127×C10911yescharacteristic
380C2×C4C2×SD16×C10911yescharacteristic
381C2×C4#135×C10911yescharacteristic
382C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
383C2×C4#137×C10911yescharacteristic
384C8—42no—
385C8—42no—
386C8—42no—
387C8—81no—
388C22—21no—
389C22—21no—
390C22—42no—
391C22—82no—
392C22—82no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22#392×C10911yescharacteristic
399C22#465×C10911yescharacteristic
400C22#468×C10911yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—81no—
405C4—82no—
406C4—81no—
407C4#448×C10911yescharacteristic
408C4#449×C10911yescharacteristic
409C4#450×C10911yescharacteristic
410C4#452×C10911yescharacteristic
411C4#471×C10912yes—
412C2—21no—
413C2—21no—
414C2—82no—
415C2—161no—
416C2C109×#200011yescharacteristic
417C2C109×#200811yescharacteristic
418C2C109×#201911yescharacteristic
subgroupG/H
center, socleC22×C109#392
commutator, Frattini, cosocleC2×C42C23×C109
prime radicalC109×#3036C2
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×#3206C2semidirect
2. C2×#476×C109C2semidirect
4. C109×#1549C2semidirect
5. C109×#1480C2semidirect
6. C109×#1460C2semidirect
7. C109×#1301C2semidirect
95. C2×D4×C109SD16semidirect
202. #10180C109direct (N × H)
210. #3206C2×C109semidirect
211. C2×#476C2×C109semidirect
213. #1549C2×C109semidirect
214. #1480C2×C109semidirect
215. #1460C2×C109semidirect
216. #1301C2×C109semidirect
217. C109#10180direct (N × H)
305. C2×D4SD16×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#32062
C2×#476×C1092
C109×#30362
C109×#15492
C109×#14802
C109×#14602
C109×#13012
#10180109