Finite Group Databasebeta

← C109×#12815 #42130809

Subgroups — C109×#12815 #42130809

Order 27904.

422 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#135×C109C211yesmaximal, characteristic
2C2×C4⋊C16×C109C211yesmaximal, characteristic, complemented
3C2×#359×C109C212yesmaximal
4C109×#1848C211yesmaximal, characteristic, complemented
5C109×#1382C212yesmaximal, complemented
6C2×#135×C109C2211yescharacteristic
7C4×#41×C109—21no—
8C22×C16×C109C2212yes—
9C4×Q16×C109C411yescharacteristic
10C4×Q16×C109C411yescharacteristic
11C4×SD16×C109—21no—
12C4×SD16×C109—21no—
13C4×D8×C109C411yescharacteristic
14C4×D8×C109C411yescharacteristic
15C2×#108×C109—21no—
16C2×#108×C109C2212yes—
17#427×C109C2211yescharacteristic
18#423×C109—21no—
19#416×C109—21no—
20C2×C4×C8×C109C2211yescharacteristic
21C4⋊C16×C109—21no—
22C4⋊C16×C109—21no—
23#359×C109—24no—
24C2×#41×C109—21no—
25#135×C109—21no—
26#135×C109—21no—
27#135×C109D412yes—
28#135×C109Q812yes—
29C2×Q16×C109C2×C411yescharacteristic
30C2×SD16×C109—21no—
31C2×D8×C109C2×C411yescharacteristic
32C2×M16×C109—22no—
33C22×C8×C109D411yescharacteristic
34C22×C8×C109C2311yescharacteristic
35C4×Q8×C109—21no—
36C4×Q8×C109—21no—
37C4×D4×C109—21no—
38C4×D4×C109—21no—
39C4×D4×C109—41no—
40C4×D4×C109—41no—
41#117×C109—21no—
42#117×C109—41no—
43C2×C42×C109—41no—
44C2×C42×C109D411yescharacteristic
45C2×C16×C109—22no—
46C2×C16×C109—22no—
47C2×C16×C109—22no—
48#108×C109—22no—
49#108×C109—22no—
50#108×C109—42no—
51#107×C109C2×C411yescharacteristic
52#107×C109C2×C411yescharacteristic
53#106×C109—21no—
54C4⋊C8×C109—41no—
55C4⋊C8×C109—41no—
56#103×C109—21no—
57#103×C109—21no—
58#102×C109—21no—
59#102×C109—21no—
60C4×C8×C109C2×C411yescharacteristic
61C4×C8×C109C2×C411yescharacteristic
62C4×C8×C109D411yescharacteristic
63C4×C8×C109D411yescharacteristic
64#41×C109—22no—
65#41×C109—22no—
66#41×C109—41no—
67#41×C109—41no—
68C2×Q8×C109—21no—
69C2×D4×C109—21no—
70C2×D4×C109—41no—
71C22×C4×C109—41no—
72C22×C4×C109—41no—
73C22×C4×C109—41no—
74C22×C4×C109SD1611yescharacteristic
75C22×C4×C109SD1611yescharacteristic
76C22×C4×C109C2×D411yescharacteristic
77Q16×C109—21no—
78Q16×C109C4⋊C412yes—
79SD16×C109—22no—
80SD16×C109—22no—
81D8×C109—21no—
82D8×C109C4⋊C412yes—
83M16×C109—42no—
84M16×C109—42no—
85C2×C8×C109—21no—
86C2×C8×C109—21no—
87C2×C8×C109—42no—
88C2×C8×C109C22⋊C411yescharacteristic
89C2×C8×C109C22⋊C411yescharacteristic
90C2×C8×C109C22×C411yescharacteristic
91C2×C8×C109C2×D411yescharacteristic
92C2×C8×C109C2×D411yescharacteristic
93C2×C8×C109C2×Q811yescharacteristic
94C2×C8×C109#4111yescharacteristic
95C2×C8×C109#4111yescharacteristic
96C4⋊C4×C109—21no—
97C4⋊C4×C109—21no—
98C4⋊C4×C109—41no—
99C4⋊C4×C109—41no—
100C22⋊C4×C109—41no—
101C22⋊C4×C109—41no—
102C22⋊C4×C109—41no—
103C42×C109—41no—
104C42×C109—41no—
105C42×C109—41no—
106C42×C109C22⋊C411yescharacteristic
107C42×C109C22⋊C411yescharacteristic
108C42×C109D811yescharacteristic
109C42×C109D811yescharacteristic
110C16×C109—42no—
111C23×C109—41no—
112C23×C109C2×SD1611yescharacteristic
113Q8×C109—22no—
114Q8×C109—41no—
115D4×C109—22no—
116D4×C109—42no—
117D4×C109—42no—
118D4×C109—41no—
119C2×C4×C109—22no—
120C2×C4×C109—41no—
121C2×C4×C109—41no—
122C2×C4×C109—42no—
123C2×C4×C109—42no—
124C2×C4×C109—41no—
125C2×C4×C109—41no—
126C2×C4×C109—82no—
127C2×C4×C109—82no—
128C2×C4×C109#10211yescharacteristic
129C2×C4×C109#10211yescharacteristic
130C2×C4×C109#10211yescharacteristic
131C2×C4×C109#10211yescharacteristic
132C2×C4×C109C2×C22⋊C411yescharacteristic
133C2×C4×C109#12111yescharacteristic
134C2×C4×C109C22⋊Q811yescharacteristic
135C2×C4×C109C2×D811yescharacteristic
136C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C109#13711yescharacteristic
138C8×C109—21no—
139C8×C109—21no—
140C8×C109—82no—
141C8×C109C2×C4⋊C411yescharacteristic
142C8×C109#11711yescharacteristic
143C8×C109#12111yescharacteristic
144C8×C109C22⋊Q811yescharacteristic
145C22×C109—21no—
146C22×C109—21no—
147C22×C109—42no—
148C22×C109—81no—
149C22×C109—81no—
150C22×C109C2×#10211yescharacteristic
151C22×C109#46111yescharacteristic
152C22×C109#47611yescharacteristic
153C4×C109—21no—
154C4×C109—21no—
155C4×C109—21no—
156C4×C109—42no—
157C4×C109—81no—
158C4×C109—81no—
159C4×C109#35912yescomplemented
160C4×C109#38011yescharacteristic
161C4×C109#41711yescharacteristic
162C4×C109#45911yescharacteristic
163C4×C109#47411yescharacteristic
164#12815C10911yesmaximal, characteristic, direct product factor, Sylow
165C2×C109—21no—
166C2×C109—21no—
167C2×C109—82no—
168C2×C109#180211yescharacteristic
169C2×C109C2×#35911yescharacteristic
170C2×C109#214311yescharacteristic
171C4×#135C2×C10911yescharacteristic
172C2×C4⋊C16C2×C10911yescharacteristic, complemented
173C2×#359C2×C10912yes—
174#1848C2×C10911yescharacteristic, complemented
175#1382C2×C10912yescomplemented
176C109#1281511yescharacteristic, direct product factor, Sylow
177C2×#135C22×C10911yescharacteristic
178C4×#41—21no—
179C22×C16C22×C10912yes—
180C4×Q16C4×C10911yescharacteristic
181C4×Q16C4×C10911yescharacteristic
182C4×SD16—21no—
183C4×SD16—21no—
184C4×D8C4×C10911yescharacteristic
185C4×D8C4×C10911yescharacteristic
186C2×#108—21no—
187C2×#108C22×C10912yes—
188#427C22×C10911yescharacteristic
189#423—21no—
190#416—21no—
191C2×C4×C8C22×C10911yescharacteristic
192C4⋊C16—21no—
193C4⋊C16—21no—
194#359—24no—
195C2×#41—21no—
196#135—21no—
197#135—21no—
198#135D4×C10912yes—
199#135Q8×C10912yes—
200C2×Q16C2×C4×C10911yescharacteristic
201C2×SD16—21no—
202C2×D8C2×C4×C10911yescharacteristic
203C2×M16—22no—
204C22×C8D4×C10911yescharacteristic
205C22×C8C23×C10911yescharacteristic
206C4×Q8—21no—
207C4×Q8—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—41no—
211C4×D4—41no—
212#117—21no—
213#117—41no—
214C2×C42—41no—
215C2×C42D4×C10911yescharacteristic
216C2×C16—22no—
217C2×C16—22no—
218C2×C16—22no—
219#108—22no—
220#108—22no—
221#108—42no—
222#107C2×C4×C10911yescharacteristic
223#107C2×C4×C10911yescharacteristic
224#106—21no—
225C4⋊C8—41no—
226C4⋊C8—41no—
227#103—21no—
228#103—21no—
229#102—21no—
230#102—21no—
231C4×C8C2×C4×C10911yescharacteristic
232C4×C8C2×C4×C10911yescharacteristic
233C4×C8D4×C10911yescharacteristic
234C4×C8D4×C10911yescharacteristic
235#41—22no—
236#41—22no—
237#41—41no—
238#41—41no—
239C2×Q8—21no—
240C2×D4—21no—
241C2×D4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4SD16×C10911yescharacteristic
246C22×C4SD16×C10911yescharacteristic
247C22×C4C2×D4×C10911yescharacteristic
248Q16—21no—
249Q16C4⋊C4×C10912yes—
250SD16—22no—
251SD16—22no—
252D8—21no—
253D8C4⋊C4×C10912yes—
254M16—42no—
255M16—42no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—42no—
259C2×C8C22⋊C4×C10911yescharacteristic
260C2×C8C22⋊C4×C10911yescharacteristic
261C2×C8C22×C4×C10911yescharacteristic
262C2×C8C2×D4×C10911yescharacteristic
263C2×C8C2×D4×C10911yescharacteristic
264C2×C8C2×Q8×C10911yescharacteristic
265C2×C8#41×C10911yescharacteristic
266C2×C8#41×C10911yescharacteristic
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C42—41no—
275C42—41no—
276C42—41no—
277C42C22⋊C4×C10911yescharacteristic
278C42C22⋊C4×C10911yescharacteristic
279C42D8×C10911yescharacteristic
280C42D8×C10911yescharacteristic
281C16—42no—
282C23—41no—
283C23C2×SD16×C10911yescharacteristic
284Q8—22no—
285Q8—41no—
286D4—22no—
287D4—42no—
288D4—42no—
289D4—41no—
290C2×C4—22no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4#102×C10911yescharacteristic
300C2×C4#102×C10911yescharacteristic
301C2×C4#102×C10911yescharacteristic
302C2×C4#102×C10911yescharacteristic
303C2×C4C2×C22⋊C4×C10911yescharacteristic
304C2×C4#121×C10911yescharacteristic
305C2×C4C22⋊Q8×C10911yescharacteristic
306C2×C4C2×D8×C10911yescharacteristic
307C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
308C2×C4#137×C10911yescharacteristic
309C8—21no—
310C8—21no—
311C8—82no—
312C8C2×C4⋊C4×C10911yescharacteristic
313C8#117×C10911yescharacteristic
314C8#121×C10911yescharacteristic
315C8C22⋊Q8×C10911yescharacteristic
316C22—21no—
317C22—21no—
318C22—42no—
319C22—81no—
320C22—81no—
321C22C2×#102×C10911yescharacteristic
322C22#461×C10911yescharacteristic
323C22#476×C10911yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—42no—
328C4—81no—
329C4—81no—
330C4#359×C10912yescomplemented
331C4#380×C10911yescharacteristic
332C4#417×C10911yescharacteristic
333C4#459×C10911yescharacteristic
334C4#474×C10911yescharacteristic
335C2—21no—
336C2—21no—
337C2—82no—
338C2C109×#180211yescharacteristic
339C2C2×#359×C10911yescharacteristic
340C2C109×#214311yescharacteristic
subgroupG/H
centerC2×C4×C109C2×D8
commutatorC2×C8C22×C4×C109
Frattini, cosocleC22×C8C23×C109
socleC22×C109C2×#102
prime radicalC2×#135×C109C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×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⋊C16×C109C2semidirect
4. C109×#1848C2semidirect
5. C109×#1382C2semidirect
159. C4×C109#359semidirect
164. #12815C109direct (N × H)
172. C2×C4⋊C16C2×C109semidirect
174. #1848C2×C109semidirect
175. #1382C2×C109semidirect
176. C109#12815direct (N × H)
330. C4#359×C109semidirect

Maximal subgroups

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

subgroupindex
C4×#135×C1092
C2×C4⋊C16×C1092
C2×#359×C1092
C109×#18482
C109×#13822
#12815109