Finite Group Databasebeta

← C109×#13417 #42131411

Subgroups — C109×#13417 #42131411

Order 27904.

454 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3155C211yesmaximal, characteristic
2C109×#2148C211yesmaximal, characteristic, complemented
3C2×#361×C109C211yesmaximal, characteristic, complemented
4C2×#361×C109C211yesmaximal, characteristic, complemented
5C109×#2143C211yesmaximal, characteristic, complemented
6C109×#1953C211yesmaximal, characteristic, complemented
7C109×#1378C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C109—21no—
9C2×#135×C109C2211yescharacteristic
10C22×D8×C109C2211yescharacteristic
11#535×C109—21no—
12C2×M32×C109C2211yescharacteristic
13C2×M32×C109C2211yescharacteristic
14#496×C109—22no—
15#495×C109—21no—
16#493×C109—21no—
17#430×C109—21no—
18C2×#108×C109C2211yescharacteristic
19C2×#108×C109C2211yescharacteristic
20#405×C109C2211yescharacteristic
21#362×C109—21no—
22#361×C109—22no—
23#361×C109—21no—
24#361×C109—22no—
25#359×C109—22no—
26#344×C109—21no—
27C2×#41×C109—21no—
28C22×D4×C109—21no—
29AGL(1,ℤ/8)×C109—42no—
30AGL(1,ℤ/8)×C109—42no—
31#135×C109—21no—
32#135×C109—21no—
33#135×C109—21no—
34#135×C109—21no—
35C2×Q16×C109D411yescharacteristic
36C2×SD16×C109—22no—
37C2×SD16×C109—21no—
38C2×D8×C109—22no—
39C2×D8×C109—21no—
40C2×D8×C109—21no—
41C2×D8×C109—21no—
42C2×D8×C109—21no—
43C2×D8×C109—21no—
44C2×D8×C109D412yes—
45C2×D8×C109D411yescharacteristic
46C2×M16×C109—21no—
47C2×M16×C109—21no—
48C2×M16×C109D411yescharacteristic
49C22×C8×C109C2311yescharacteristic
50#127×C109—22no—
51#124×C109—22no—
52#121×C109—42no—
53#120×C109—42no—
54#117×C109D411yescharacteristic
55M32×C109—21no—
56M32×C109—21no—
57M32×C109—21no—
58M32×C109—21no—
59C2×C16×C109—21no—
60C2×C16×C109—21no—
61#108×C109—21no—
62#108×C109—21no—
63#108×C109—21no—
64#108×C109—42no—
65#108×C109D412yescomplemented
66#97×C109—21no—
67C4×C8×C109—21no—
68C24×C109—41no—
69#41×C109—41no—
70#41×C109—41no—
71#41×C109—41no—
72#41×C109—41no—
73C2×Q8×C109—21no—
74C2×D4×C109—22no—
75C2×D4×C109—21no—
76C2×D4×C109—44no—
77C2×D4×C109—41no—
78C2×D4×C109—41no—
79C2×D4×C109—41no—
80C2×D4×C109—41no—
81C2×D4×C109—41no—
82C2×D4×C109—41no—
83C22×C4×C109—41no—
84C22×C4×C109C2×D411yescharacteristic
85Q16×C109—21no—
86Q16×C109—21no—
87SD16×C109—44no—
88SD16×C109—41no—
89SD16×C109—41no—
90D8×C109—22no—
91D8×C109—21no—
92D8×C109—22no—
93D8×C109—21no—
94D8×C109—44no—
95D8×C109—41no—
96D8×C109—41no—
97M16×C109—21no—
98M16×C109—22no—
99M16×C109—41no—
100M16×C109—41no—
101M16×C109—41no—
102M16×C109D812yes—
103C2×C8×C109—21no—
104C2×C8×C109—41no—
105C2×C8×C109—41no—
106C2×C8×C109C2×D411yescharacteristic
107C2×C8×C109C2×D411yescharacteristic
108C2×C8×C109C2×D411yescharacteristic
109C2×C8×C109#4111yescharacteristic
110C2×C8×C109#4111yescharacteristic
111C2×C8×C109#4111yescharacteristic
112C4⋊C4×C109—21no—
113C22⋊C4×C109—44no—
114C22⋊C4×C109SD1612yes—
115C42×C109—21no—
116C16×C109—41no—
117C16×C109—41no—
118C23×C109—42no—
119C23×C109—41no—
120C23×C109—81no—
121C23×C109—81no—
122C23×C109—162no—
123C23×C109C2×SD1611yescharacteristic
124Q8×C109—41no—
125Q8×C109—41no—
126D4×C109—42no—
127D4×C109—41no—
128D4×C109—41no—
129D4×C109—42no—
130D4×C109—81no—
131D4×C109—81no—
132D4×C109—81no—
133D4×C109—81no—
134D4×C109—164no—
135C2×C4×C109—22no—
136C2×C4×C109—41no—
137C2×C4×C109—81no—
138C2×C4×C109—81no—
139C2×C4×C109#12011yescharacteristic
140C2×C4×C109#12111yescharacteristic
141C2×C4×C109#12411yescharacteristic
142C2×C4×C109C2×D811yescharacteristic
143C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
145C8×C109—22no—
146C8×C109—42no—
147C8×C109—81no—
148C8×C109#12111yescharacteristic
149C8×C109#12111yescharacteristic
150C8×C109#12311yescharacteristic
151C8×C109#12311yescharacteristic
152C22×C109—41no—
153C22×C109—82no—
154C22×C109—84no—
155C22×C109—81no—
156C22×C109—82no—
157C22×C109—162no—
158C22×C109—162no—
159C22×C109#45011yescharacteristic
160C22×C109#46011yescharacteristic
161C22×C109#48611yescharacteristic
162C4×C109—42no—
163C4×C109—81no—
164C4×C109#39411yescharacteristic
165C4×C109#44711yescharacteristic
166C4×C109#45911yescharacteristic
167C4×C109#49011yescharacteristic
168#13417C10911yesmaximal, characteristic, direct product factor, Sylow
169C2×C109—21no—
170C2×C109—21no—
171C2×C109—21no—
172C2×C109—161no—
173C2×C109—162no—
174C2×C109#204111yescharacteristic
175#3155C2×C10911yescharacteristic
176#2148C2×C10911yescharacteristic, complemented
177C2×#361C2×C10911yescharacteristic, complemented
178C2×#361C2×C10911yescharacteristic, complemented
179#2143C2×C10911yescharacteristic, complemented
180#1953C2×C10911yescharacteristic, complemented
181#1378C2×C10911yescharacteristic, complemented
182C109#1341711yescharacteristic, direct product factor, Sylow
183C2×AGL(1,ℤ/8)—21no—
184C2×#135C22×C10911yescharacteristic
185C22×D8C22×C10911yescharacteristic
186#535—21no—
187C2×M32C22×C10911yescharacteristic
188C2×M32C22×C10911yescharacteristic
189#496—22no—
190#495—21no—
191#493—21no—
192#430—21no—
193C2×#108C22×C10911yescharacteristic
194C2×#108C22×C10911yescharacteristic
195#405C22×C10911yescharacteristic
196#362—21no—
197#361—22no—
198#361—21no—
199#361—22no—
200#359—22no—
201#344—21no—
202C2×#41—21no—
203C22×D4—21no—
204AGL(1,ℤ/8)—42no—
205AGL(1,ℤ/8)—42no—
206#135—21no—
207#135—21no—
208#135—21no—
209#135—21no—
210C2×Q16D4×C10911yescharacteristic
211C2×SD16—22no—
212C2×SD16—21no—
213C2×D8—22no—
214C2×D8—21no—
215C2×D8—21no—
216C2×D8—21no—
217C2×D8—21no—
218C2×D8—21no—
219C2×D8D4×C10912yes—
220C2×D8D4×C10911yescharacteristic
221C2×M16—21no—
222C2×M16—21no—
223C2×M16D4×C10911yescharacteristic
224C22×C8C23×C10911yescharacteristic
225#127—22no—
226#124—22no—
227#121—42no—
228#120—42no—
229#117D4×C10911yescharacteristic
230M32—21no—
231M32—21no—
232M32—21no—
233M32—21no—
234C2×C16—21no—
235C2×C16—21no—
236#108—21no—
237#108—21no—
238#108—21no—
239#108—42no—
240#108D4×C10912yescomplemented
241#97—21no—
242C4×C8—21no—
243C24—41no—
244#41—41no—
245#41—41no—
246#41—41no—
247#41—41no—
248C2×Q8—21no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—44no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C22×C4—41no—
259C22×C4C2×D4×C10911yescharacteristic
260Q16—21no—
261Q16—21no—
262SD16—44no—
263SD16—41no—
264SD16—41no—
265D8—22no—
266D8—21no—
267D8—22no—
268D8—21no—
269D8—44no—
270D8—41no—
271D8—41no—
272M16—21no—
273M16—22no—
274M16—41no—
275M16—41no—
276M16—41no—
277M16D8×C10912yes—
278C2×C8—21no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8C2×D4×C10911yescharacteristic
282C2×C8C2×D4×C10911yescharacteristic
283C2×C8C2×D4×C10911yescharacteristic
284C2×C8#41×C10911yescharacteristic
285C2×C8#41×C10911yescharacteristic
286C2×C8#41×C10911yescharacteristic
287C4⋊C4—21no—
288C22⋊C4—44no—
289C22⋊C4SD16×C10912yes—
290C42—21no—
291C16—41no—
292C16—41no—
293C23—42no—
294C23—41no—
295C23—81no—
296C23—81no—
297C23—162no—
298C23C2×SD16×C10911yescharacteristic
299Q8—41no—
300Q8—41no—
301D4—42no—
302D4—41no—
303D4—41no—
304D4—42no—
305D4—81no—
306D4—81no—
307D4—81no—
308D4—81no—
309D4—164no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4#120×C10911yescharacteristic
315C2×C4#121×C10911yescharacteristic
316C2×C4#124×C10911yescharacteristic
317C2×C4C2×D8×C10911yescharacteristic
318C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
319C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
320C8—22no—
321C8—42no—
322C8—81no—
323C8#121×C10911yescharacteristic
324C8#121×C10911yescharacteristic
325C8#123×C10911yescharacteristic
326C8#123×C10911yescharacteristic
327C22—41no—
328C22—82no—
329C22—84no—
330C22—81no—
331C22—82no—
332C22—162no—
333C22—162no—
334C22#450×C10911yescharacteristic
335C22#460×C10911yescharacteristic
336C22#486×C10911yescharacteristic
337C4—42no—
338C4—81no—
339C4#394×C10911yescharacteristic
340C4#447×C10911yescharacteristic
341C4#459×C10911yescharacteristic
342C4#490×C10911yescharacteristic
343C2—21no—
344C2—21no—
345C2—21no—
346C2—162no—
347C2—161no—
348C2C109×#204111yescharacteristic
subgroupG/H
center, socleC2×C109#2041
commutator, Frattini, cosocleC22×C8C23×C109
prime radicalC109×#3155C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C109 ◁ C22×C4×C109 ◁ #405×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#2148C2semidirect
3. C2×#361×C109C2semidirect
4. C2×#361×C109C2semidirect
5. C109×#2143C2semidirect
6. C109×#1953C2semidirect
7. C109×#1378C2semidirect
65. #108×C109D4semidirect
168. #13417C109direct (N × H)
176. #2148C2×C109semidirect
177. C2×#361C2×C109semidirect
178. C2×#361C2×C109semidirect
179. #2143C2×C109semidirect
180. #1953C2×C109semidirect
181. #1378C2×C109semidirect
182. C109#13417direct (N × H)
240. #108D4×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#31552
C109×#21482
C2×#361×C1092
C109×#21432
C109×#19532
C109×#13782
#13417109