Finite Group Databasebeta

← C109×#13013 #42131007

Subgroups — C109×#13013 #42131007

Order 27904.

526 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×SD32×C109C211yesmaximal, characteristic
2C2×C4×Q16×C109C211yesmaximal, characteristic, complemented
3C2×#358×C109C211yesmaximal, characteristic, complemented
4C2×#357×C109C211yesmaximal, characteristic
5C109×#1970C211yesmaximal, characteristic
6C109×#1380C211yesmaximal, characteristic, complemented
7C109×#1374C211yesmaximal, characteristic, complemented
8C22×Q16×C109C2211yescharacteristic
9C22×D8×C109C2211yescharacteristic
10C2×C4×Q8×C109—21no—
11C2×SD32×C109—22no—
12C2×SD32×C109—22no—
13C2×SD32×C109C414yescomplemented
14C22×C16×C109C2211yescharacteristic
15C22×C16×C109C2211yescharacteristic
16C4×Q16×C109—22no—
17C4×Q16×C109—22no—
18C2×#107×C109C2211yescharacteristic
19C2×#107×C109C2211yescharacteristic
20C2×#103×C109—21no—
21C2×#102×C109—21no—
22C2×C4×C8×C109C2211yescharacteristic
23#390×C109—21no—
24#358×C109—22no—
25#357×C109—22no—
26C22×Q8×C109—21no—
27C22×D4×C109—21no—
28C2×Q16×C109—21no—
29C2×Q16×C109—21no—
30C2×Q16×C109—21no—
31C2×Q16×C109C2×C412yes—
32C2×Q16×C109D412yes—
33C2×Q16×C109D412yes—
34C2×D8×C109—21no—
35C2×D8×C109—21no—
36C2×D8×C109—21no—
37C2×D8×C109—21no—
38C2×D8×C109—21no—
39C2×D8×C109C2×C412yes—
40C22×C8×C109D411yescharacteristic
41C22×C8×C109C2311yescharacteristic
42C4×Q8×C109—24no—
43C4×Q8×C109—42no—
44C2×C4⋊C4×C109—21no—
45C2×C4⋊C4×C109—21no—
46C2×C4⋊C4×C109—41no—
47C2×C22⋊C4×C109—41no—
48C2×C22⋊C4×C109—41no—
49C2×C42×C109—41no—
50C2×C42×C109D411yescharacteristic
51SD32×C109—28no—
52C2×C16×C109—21no—
53C2×C16×C109—21no—
54C2×C16×C109—21no—
55C2×C16×C109—21no—
56C2×C16×C109—21no—
57C2×C16×C109C2×C412yes—
58#107×C109—21no—
59#107×C109—21no—
60#107×C109—21no—
61#107×C109D412yes—
62#103×C109—24no—
63#102×C109—42no—
64C4×C8×C109—21no—
65C4×C8×C109—21no—
66C24×C109—41no—
67C2×Q8×C109—22no—
68C2×Q8×C109—22no—
69C2×Q8×C109—22no—
70C2×Q8×C109—41no—
71C2×Q8×C109—41no—
72C2×Q8×C109—41no—
73C2×D4×C109—22no—
74C2×D4×C109—41no—
75C2×D4×C109—41no—
76C2×D4×C109—41no—
77C2×D4×C109—41no—
78C2×D4×C109—41no—
79C22×C4×C109—41no—
80C22×C4×C109—41no—
81C22×C4×C109—41no—
82C22×C4×C109D811yescharacteristic
83C22×C4×C109D811yescharacteristic
84C22×C4×C109C2×D411yescharacteristic
85Q16×C109—22no—
86Q16×C109—22no—
87Q16×C109—22no—
88Q16×C109C22⋊C414yescomplemented
89D8×C109—22no—
90D8×C109—22no—
91D8×C109—41no—
92D8×C109—41no—
93C2×C8×C109—21no—
94C2×C8×C109—21no—
95C2×C8×C109—21no—
96C2×C8×C109C22×C411yescharacteristic
97C2×C8×C109C2×D411yescharacteristic
98C2×C8×C109C2×D411yescharacteristic
99C2×C8×C109C2×D411yescharacteristic
100C2×C8×C109#4111yescharacteristic
101C2×C8×C109#4111yescharacteristic
102C4⋊C4×C109—22no—
103C4⋊C4×C109—44no—
104C4⋊C4×C109—41no—
105C4⋊C4×C109—41no—
106C4⋊C4×C109—41no—
107C22⋊C4×C109—82no—
108C22⋊C4×C109—82no—
109C42×C109—21no—
110C42×C109—21no—
111C42×C109—44no—
112C16×C109—22no—
113C16×C109—41no—
114C23×C109—42no—
115C23×C109—81no—
116C23×C109—81no—
117C23×C109—81no—
118C23×C109—81no—
119C23×C109—81no—
120C23×C109—81no—
121C23×C109C2×D811yescharacteristic
122Q8×C109—24no—
123Q8×C109—42no—
124Q8×C109—42no—
125Q8×C109—42no—
126D4×C109—42no—
127D4×C109—42no—
128D4×C109—81no—
129D4×C109—81no—
130C2×C4×C109—21no—
131C2×C4×C109—21no—
132C2×C4×C109—21no—
133C2×C4×C109—21no—
134C2×C4×C109—21no—
135C2×C4×C109—42no—
136C2×C4×C109—42no—
137C2×C4×C109—42no—
138C2×C4×C109—42no—
139C2×C4×C109—81no—
140C2×C4×C109—81no—
141C2×C4×C109—81no—
142C2×C4×C109—81no—
143C2×C4×C109—81no—
144C2×C4×C109SD3212yescomplemented
145C2×C4×C109C4×D411yescharacteristic
146C2×C4×C109#12011yescharacteristic
147C2×C4×C109#12111yescharacteristic
148C2×C4×C109#13511yescharacteristic
149C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
151C8×C109—41no—
152C8×C109C2×C22⋊C411yescharacteristic
153C8×C109C4×D411yescharacteristic
154C8×C109#12111yescharacteristic
155C8×C109#12311yescharacteristic
156C22×C109—82no—
157C22×C109—82no—
158C22×C109—82no—
159C22×C109—161no—
160C22×C109—161no—
161C22×C109—161no—
162C22×C109—161no—
163C22×C109C4×D811yescharacteristic
164C22×C109#44711yescharacteristic
165C22×C109#45911yescharacteristic
166C22×C109C2×SD3211yescharacteristic
167C22×C109#50811yescharacteristic
168C22×C109#50911yescharacteristic
169C22×C109#51011yescharacteristic
170C4×C109—22no—
171C4×C109—44no—
172C4×C109—41no—
173C4×C109—82no—
174C4×C109—161no—
175C4×C109#38611yescharacteristic
176C4×C109#44211yescharacteristic
177C4×C109#44911yescharacteristic
178C4×C109#46311yescharacteristic
179#13013C10911yesmaximal, characteristic, direct product factor, Sylow
180C2×C109—162no—
181C2×C109#187911yescharacteristic
182C2×C109C4×SD3211yescharacteristic
183C2×C109#217511yescharacteristic
184C2×C109#218511yescharacteristic
185C2×C109#218611yescharacteristic
186C2×C109#220711yescharacteristic
187C2×C109#221111yescharacteristic
188C22×SD32C2×C10911yescharacteristic
189C2×C4×Q16C2×C10911yescharacteristic, complemented
190C2×#358C2×C10911yescharacteristic, complemented
191C2×#357C2×C10911yescharacteristic
192#1970C2×C10911yescharacteristic
193#1380C2×C10911yescharacteristic, complemented
194#1374C2×C10911yescharacteristic, complemented
195C109#1301311yescharacteristic, direct product factor, Sylow
196C22×Q16C22×C10911yescharacteristic
197C22×D8C22×C10911yescharacteristic
198C2×C4×Q8—21no—
199C2×SD32—22no—
200C2×SD32—22no—
201C2×SD32C4×C10914yescomplemented
202C22×C16C22×C10911yescharacteristic
203C22×C16C22×C10911yescharacteristic
204C4×Q16—22no—
205C4×Q16—22no—
206C2×#107C22×C10911yescharacteristic
207C2×#107C22×C10911yescharacteristic
208C2×#103—21no—
209C2×#102—21no—
210C2×C4×C8C22×C10911yescharacteristic
211#390—21no—
212#358—22no—
213#357—22no—
214C22×Q8—21no—
215C22×D4—21no—
216C2×Q16—21no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16C2×C4×C10912yes—
220C2×Q16D4×C10912yes—
221C2×Q16D4×C10912yes—
222C2×D8—21no—
223C2×D8—21no—
224C2×D8—21no—
225C2×D8—21no—
226C2×D8—21no—
227C2×D8C2×C4×C10912yes—
228C22×C8D4×C10911yescharacteristic
229C22×C8C23×C10911yescharacteristic
230C4×Q8—24no—
231C4×Q8—42no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C42—41no—
238C2×C42D4×C10911yescharacteristic
239SD32—28no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16C2×C4×C10912yes—
246#107—21no—
247#107—21no—
248#107—21no—
249#107D4×C10912yes—
250#103—24no—
251#102—42no—
252C4×C8—21no—
253C4×C8—21no—
254C24—41no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×D4—22no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4D8×C10911yescharacteristic
271C22×C4D8×C10911yescharacteristic
272C22×C4C2×D4×C10911yescharacteristic
273Q16—22no—
274Q16—22no—
275Q16—22no—
276Q16C22⋊C4×C10914yescomplemented
277D8—22no—
278D8—22no—
279D8—41no—
280D8—41no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8C22×C4×C10911yescharacteristic
285C2×C8C2×D4×C10911yescharacteristic
286C2×C8C2×D4×C10911yescharacteristic
287C2×C8C2×D4×C10911yescharacteristic
288C2×C8#41×C10911yescharacteristic
289C2×C8#41×C10911yescharacteristic
290C4⋊C4—22no—
291C4⋊C4—44no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C22⋊C4—82no—
296C22⋊C4—82no—
297C42—21no—
298C42—21no—
299C42—44no—
300C16—22no—
301C16—41no—
302C23—42no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23—81no—
308C23—81no—
309C23C2×D8×C10911yescharacteristic
310Q8—24no—
311Q8—42no—
312Q8—42no—
313Q8—42no—
314D4—42no—
315D4—42no—
316D4—81no—
317D4—81no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4SD32×C10912yescomplemented
333C2×C4C4×D4×C10911yescharacteristic
334C2×C4#120×C10911yescharacteristic
335C2×C4#121×C10911yescharacteristic
336C2×C4#135×C10911yescharacteristic
337C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
338C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
339C8—41no—
340C8C2×C22⋊C4×C10911yescharacteristic
341C8C4×D4×C10911yescharacteristic
342C8#121×C10911yescharacteristic
343C8#123×C10911yescharacteristic
344C22—82no—
345C22—82no—
346C22—82no—
347C22—161no—
348C22—161no—
349C22—161no—
350C22—161no—
351C22C4×D8×C10911yescharacteristic
352C22#447×C10911yescharacteristic
353C22#459×C10911yescharacteristic
354C22C2×SD32×C10911yescharacteristic
355C22#508×C10911yescharacteristic
356C22#509×C10911yescharacteristic
357C22#510×C10911yescharacteristic
358C4—22no—
359C4—41no—
360C4—44no—
361C4—82no—
362C4—161no—
363C4#386×C10911yescharacteristic
364C4#442×C10911yescharacteristic
365C4#449×C10911yescharacteristic
366C4#463×C10911yescharacteristic
367C2—162no—
368C2C109×#187911yescharacteristic
369C2C4×SD32×C10911yescharacteristic
370C2C109×#217511yescharacteristic
371C2C109×#218511yescharacteristic
372C2C109×#218611yescharacteristic
373C2C109×#220711yescharacteristic
374C2C109×#221111yescharacteristic
subgroupG/H
center, socleC23×C109C2×D8
commutatorC2×C8C22×C4×C109
Frattini, cosocleC22×C8C23×C109
prime radicalC22×D8×C109C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C109 ◁ C2×C42×C109 ◁ C2×C4×C8×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C4×Q16×C109C2semidirect
3. C2×#358×C109C2semidirect
6. C109×#1380C2semidirect
7. C109×#1374C2semidirect
13. C2×SD32×C109C4semidirect
88. Q16×C109C22⋊C4semidirect
144. C2×C4×C109SD32semidirect
179. #13013C109direct (N × H)
189. C2×C4×Q16C2×C109semidirect
190. C2×#358C2×C109semidirect
193. #1380C2×C109semidirect
194. #1374C2×C109semidirect
195. C109#13013direct (N × H)
201. C2×SD32C4×C109semidirect
276. Q16C22⋊C4×C109semidirect
332. C2×C4SD32×C109semidirect

Maximal subgroups

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

subgroupindex
C22×SD32×C1092
C2×C4×Q16×C1092
C2×#358×C1092
C2×#357×C1092
C109×#19702
C109×#13802
C109×#13742
#13013109