Finite Group Databasebeta

← C109×#9254 #42127248

Subgroups — C109×#9254 #42127248

Order 27904.

534 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#460×C109C211yesmaximal, characteristic
2Q8×M16×C109C211yesmaximal, characteristic, complemented
3C109×#1965C211yesmaximal, characteristic
4C109×#1545C211yesmaximal, characteristic
5C109×#1487C211yesmaximal, characteristic, complemented
6C109×#1292C211yesmaximal, characteristic, complemented
7C109×#1283C211yesmaximal, characteristic, complemented
8C22×SD16×C109C412yes—
9C2×#127×C109C2211yescharacteristic
10C2×C4×Q8×C109C2211yescharacteristic
11#460×C109—21no—
12#460×C109—21no—
13#460×C109—21no—
14#460×C109—21no—
15#446×C109—21no—
16#446×C109—22no—
17C8×Q8×C109—21no—
18#431×C109C2211yescharacteristic
19#423×C109—21no—
20#423×C109C2211yescharacteristic
21#423×C109C2211yescharacteristic
22C2×C4⋊C8×C109C2211yescharacteristic
23C2×#102×C109C412yes—
24C2×#100×C109—21no—
25C2×#100×C109—21no—
26C4×M16×C109—21no—
27C4×M16×C109C2211yescharacteristic
28#331×C109—22no—
29Q8⋊C8×C109—21no—
30Q8⋊C8×C109—21no—
31C22×Q8×C109C2×C411yescharacteristic
32C22×D4×C109—21no—
33C22×D4×C109—21no—
34C22×D4×C109C2×C412yes—
35C2×SD16×C109—22no—
36C2×SD16×C109—22no—
37C2×SD16×C109—44no—
38C2×M16×C109—21no—
39C2×M16×C109—21no—
40C2×M16×C109D411yescharacteristic
41C2×M16×C109D411yescharacteristic
42C22×C8×C109C2×C412yes—
43#127×C109—21no—
44#127×C109—21no—
45#127×C109—21no—
46#127×C109—21no—
47C4×Q8×C109—21no—
48C4×Q8×C109—21no—
49C4×Q8×C109D411yescharacteristic
50C4×Q8×C109D411yescharacteristic
51C4×Q8×C109D411yescharacteristic
52C4×Q8×C109D411yescharacteristic
53C2×C4⋊C4×C109—21no—
54C2×C4⋊C4×C109C2×C411yescharacteristic
55C2×C42×C109—21no—
56C2×C42×C109C2311yescharacteristic
57C4⋊C8×C109—22no—
58C4⋊C8×C109—22no—
59C4⋊C8×C109—21no—
60C4⋊C8×C109—21no—
61C4⋊C8×C109—21no—
62C4⋊C8×C109—21no—
63C4⋊C8×C109—21no—
64C4⋊C8×C109—21no—
65C4⋊C8×C109D412yes—
66#102×C109—22no—
67#102×C109—22no—
68#100×C109—42no—
69#100×C109—42no—
70C22⋊C8×C109—21no—
71#97×C109—22no—
72#97×C109—21no—
73C4×C8×C109—22no—
74C4×C8×C109—21no—
75C4×C8×C109—21no—
76C24×C109—22no—
77C2×Q8×C109—21no—
78C2×Q8×C109—22no—
79C2×Q8×C109—41no—
80C2×Q8×C109C22⋊C411yescharacteristic
81C2×Q8×C109C22⋊C411yescharacteristic
82C2×D4×C109—22no—
83C2×D4×C109—22no—
84C2×D4×C109—42no—
85C2×D4×C109—42no—
86C2×D4×C109—41no—
87C2×D4×C109—41no—
88C2×D4×C109—41no—
89C2×D4×C109—41no—
90C2×D4×C109—41no—
91C2×D4×C109—41no—
92C2×D4×C109—41no—
93C2×D4×C109—41no—
94C2×D4×C109—82no—
95C22×C4×C109—21no—
96C22×C4×C109—21no—
97C22×C4×C109C22×C411yescharacteristic
98C22×C4×C109C2×D411yescharacteristic
99C22×C4×C109#4111yescharacteristic
100SD16×C109—44no—
101SD16×C109—44no—
102M16×C109—21no—
103M16×C109—21no—
104M16×C109—21no—
105M16×C109—41no—
106M16×C109—41no—
107M16×C109—41no—
108M16×C109—41no—
109M16×C109SD1612yescomplemented
110C2×C8×C109—21no—
111C2×C8×C109—22no—
112C2×C8×C109—21no—
113C2×C8×C109—22no—
114C2×C8×C109—42no—
115C2×C8×C109—41no—
116C2×C8×C109—41no—
117C4⋊C4×C109—21no—
118C4⋊C4×C109—21no—
119C4⋊C4×C109—21no—
120C4⋊C4×C109—21no—
121C4⋊C4×C109—21no—
122C4⋊C4×C109C22⋊C411yescharacteristic
123C4⋊C4×C109C22⋊C411yescharacteristic
124C42×C109—21no—
125C42×C109—21no—
126C42×C109—21no—
127C42×C109—21no—
128C42×C109C2×D411yescharacteristic
129C42×C109C2×D411yescharacteristic
130C42×C109C2×D411yescharacteristic
131C42×C109#4111yescharacteristic
132C23×C109—42no—
133C23×C109—42no—
134C23×C109—42no—
135C23×C109—82no—
136C23×C109—82no—
137C23×C109C4×D411yescharacteristic
138Q8×C109—21no—
139Q8×C109—21no—
140Q8×C109—21no—
141Q8×C109—42no—
142Q8×C109#10012yescomplemented
143D4×C109—42no—
144D4×C109—42no—
145D4×C109—82no—
146D4×C109—82no—
147D4×C109—82no—
148D4×C109—81no—
149D4×C109—81no—
150D4×C109—81no—
151D4×C109—81no—
152C2×C4×C109—21no—
153C2×C4×C109—21no—
154C2×C4×C109—21no—
155C2×C4×C109—21no—
156C2×C4×C109—21no—
157C2×C4×C109—21no—
158C2×C4×C109—41no—
159C2×C4×C109—42no—
160C2×C4×C109—81no—
161C2×C4×C109C2×C22⋊C411yescharacteristic
162C2×C4×C109C4×D411yescharacteristic
163C2×C4×C109#12011yescharacteristic
164C2×C4×C109#12111yescharacteristic
165C2×C4×C109#12111yescharacteristic
166C2×C4×C109#12311yescharacteristic
167C2×C4×C109C2×SD1611yescharacteristic
168C2×C4×C109#13511yescharacteristic
169C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C109#13711yescharacteristic
171C8×C109—22no—
172C8×C109—42no—
173C8×C109—41no—
174C8×C109—42no—
175C8×C109—81no—
176C8×C109—81no—
177C22×C109—21no—
178C22×C109—21no—
179C22×C109—82no—
180C22×C109—82no—
181C22×C109—82no—
182C22×C109—162no—
183C22×C109—162no—
184C22×C109#38611yescharacteristic
185C22×C109C4×SD1611yescharacteristic
186C22×C109#44011yescharacteristic
187C4×C109—21no—
188C4×C109—22no—
189C4×C109—21no—
190C4×C109—21no—
191C4×C109—41no—
192C4×C109—41no—
193C4×C109—41no—
194C4×C109C2×#10011yescharacteristic
195C4×C109#41311yescharacteristic
196C4×C109#44811yescharacteristic
197C4×C109#44911yescharacteristic
198C4×C109#46011yescharacteristic
199C4×C109#46411yescharacteristic
200#9254C10911yesmaximal, characteristic, direct product factor, Sylow
201C2×C109—21no—
202C2×C109—21no—
203C2×C109—162no—
204C2×C109#188011yescharacteristic
205C2×C109#190011yescharacteristic
206C2×C109#190311yescharacteristic
207C2×#460C2×C10911yescharacteristic
208Q8×M16C2×C10911yescharacteristic, complemented
209#1965C2×C10911yescharacteristic
210#1545C2×C10911yescharacteristic
211#1487C2×C10911yescharacteristic, complemented
212#1292C2×C10911yescharacteristic, complemented
213#1283C2×C10911yescharacteristic, complemented
214C109#925411yescharacteristic, direct product factor, Sylow
215C22×SD16C4×C10912yes—
216C2×#127C22×C10911yescharacteristic
217C2×C4×Q8C22×C10911yescharacteristic
218#460—21no—
219#460—21no—
220#460—21no—
221#460—21no—
222#446—21no—
223#446—22no—
224C8×Q8—21no—
225#431C22×C10911yescharacteristic
226#423—21no—
227#423C22×C10911yescharacteristic
228#423C22×C10911yescharacteristic
229C2×C4⋊C8C22×C10911yescharacteristic
230C2×#102C4×C10912yes—
231C2×#100—21no—
232C2×#100—21no—
233C4×M16—21no—
234C4×M16C22×C10911yescharacteristic
235#331—22no—
236Q8⋊C8—21no—
237Q8⋊C8—21no—
238C22×Q8C2×C4×C10911yescharacteristic
239C22×D4—21no—
240C22×D4—21no—
241C22×D4C2×C4×C10912yes—
242C2×SD16—22no—
243C2×SD16—22no—
244C2×SD16—44no—
245C2×M16—21no—
246C2×M16—21no—
247C2×M16D4×C10911yescharacteristic
248C2×M16D4×C10911yescharacteristic
249C22×C8C2×C4×C10912yes—
250#127—21no—
251#127—21no—
252#127—21no—
253#127—21no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8D4×C10911yescharacteristic
257C4×Q8D4×C10911yescharacteristic
258C4×Q8D4×C10911yescharacteristic
259C4×Q8D4×C10911yescharacteristic
260C2×C4⋊C4—21no—
261C2×C4⋊C4C2×C4×C10911yescharacteristic
262C2×C42—21no—
263C2×C42C23×C10911yescharacteristic
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8D4×C10912yes—
273#102—22no—
274#102—22no—
275#100—42no—
276#100—42no—
277C22⋊C8—21no—
278#97—22no—
279#97—21no—
280C4×C8—22no—
281C4×C8—21no—
282C4×C8—21no—
283C24—22no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22⋊C4×C10911yescharacteristic
288C2×Q8C22⋊C4×C10911yescharacteristic
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—82no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4C22×C4×C10911yescharacteristic
305C22×C4C2×D4×C10911yescharacteristic
306C22×C4#41×C10911yescharacteristic
307SD16—44no—
308SD16—44no—
309M16—21no—
310M16—21no—
311M16—21no—
312M16—41no—
313M16—41no—
314M16—41no—
315M16—41no—
316M16SD16×C10912yescomplemented
317C2×C8—21no—
318C2×C8—22no—
319C2×C8—21no—
320C2×C8—22no—
321C2×C8—42no—
322C2×C8—41no—
323C2×C8—41no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4C22⋊C4×C10911yescharacteristic
330C4⋊C4C22⋊C4×C10911yescharacteristic
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42C2×D4×C10911yescharacteristic
336C42C2×D4×C10911yescharacteristic
337C42C2×D4×C10911yescharacteristic
338C42#41×C10911yescharacteristic
339C23—42no—
340C23—42no—
341C23—42no—
342C23—82no—
343C23—82no—
344C23C4×D4×C10911yescharacteristic
345Q8—21no—
346Q8—21no—
347Q8—21no—
348Q8—42no—
349Q8#100×C10912yescomplemented
350D4—42no—
351D4—42no—
352D4—82no—
353D4—82no—
354D4—82no—
355D4—81no—
356D4—81no—
357D4—81no—
358D4—81no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—81no—
368C2×C4C2×C22⋊C4×C10911yescharacteristic
369C2×C4C4×D4×C10911yescharacteristic
370C2×C4#120×C10911yescharacteristic
371C2×C4#121×C10911yescharacteristic
372C2×C4#121×C10911yescharacteristic
373C2×C4#123×C10911yescharacteristic
374C2×C4C2×SD16×C10911yescharacteristic
375C2×C4#135×C10911yescharacteristic
376C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
377C2×C4#137×C10911yescharacteristic
378C8—22no—
379C8—42no—
380C8—41no—
381C8—42no—
382C8—81no—
383C8—81no—
384C22—21no—
385C22—21no—
386C22—82no—
387C22—82no—
388C22—82no—
389C22—162no—
390C22—162no—
391C22#386×C10911yescharacteristic
392C22C4×SD16×C10911yescharacteristic
393C22#440×C10911yescharacteristic
394C4—21no—
395C4—22no—
396C4—21no—
397C4—21no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4C2×#100×C10911yescharacteristic
402C4#413×C10911yescharacteristic
403C4#448×C10911yescharacteristic
404C4#449×C10911yescharacteristic
405C4#460×C10911yescharacteristic
406C4#464×C10911yescharacteristic
407C2—21no—
408C2—21no—
409C2—162no—
410C2C109×#188011yescharacteristic
411C2C109×#190011yescharacteristic
412C2C109×#190311yescharacteristic
subgroupG/H
center, socleC22×C109#386
commutatorC22×C4C22×C4×C109
Frattini, cosocleC2×C42C23×C109
prime radicalC2×#127×C109C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C109 ◁ C2×C42×C109 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. Q8×M16×C109C2semidirect
5. C109×#1487C2semidirect
6. C109×#1292C2semidirect
7. C109×#1283C2semidirect
109. M16×C109SD16semidirect
142. Q8×C109#100semidirect
200. #9254C109direct (N × H)
208. Q8×M16C2×C109semidirect
211. #1487C2×C109semidirect
212. #1292C2×C109semidirect
213. #1283C2×C109semidirect
214. C109#9254direct (N × H)
316. M16SD16×C109semidirect
349. Q8#100×C109semidirect

Maximal subgroups

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

subgroupindex
C2×#460×C1092
Q8×M16×C1092
C109×#19652
C109×#15452
C109×#14872
C109×#12922
C109×#12832
#9254109