Finite Group Databasebeta

← C109×#12995 #42130989

Subgroups — C109×#12995 #42130989

Order 27904.

476 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3067C211yesmaximal, characteristic
2C109×#2916C211yesmaximal, characteristic, complemented
3C109×#2143C211yesmaximal, characteristic, complemented
4C2×#359×C109C211yesmaximal, characteristic, complemented
5C2×#350×C109C211yesmaximal, characteristic, complemented
6C109×#1388C211yesmaximal, characteristic, complemented
7C109×#1381C211yesmaximal, characteristic, complemented
8#575×C109—21no—
9C2×#135×C109C2211yescharacteristic
10#568×C109—21no—
11C2×#131×C109C2211yescharacteristic
12C2×M32×C109C2211yescharacteristic
13C22×C16×C109C2211yescharacteristic
14#473×C109C412yes—
15#472×C109C412yes—
16#471×C109—22no—
17#430×C109—21no—
18#430×C109—22no—
19C2×#108×C109—22no—
20C2×#108×C109C2211yescharacteristic
21C2×#108×C109C2211yescharacteristic
22C2×#108×C109C2211yescharacteristic
23#413×C109—21no—
24#359×C109—22no—
25#359×C109—22no—
26#350×C109—22no—
27C2×#41×C109—21no—
28C2×#41×C109D411yescharacteristic
29#137×C109—22no—
30#137×C109—41no—
31AGL(1,ℤ/8)×C109—22no—
32AGL(1,ℤ/8)×C109—41no—
33#135×C109—21no—
34#135×C109—21no—
35#135×C109—21no—
36#135×C109—21no—
37#135×C109—42no—
38C2×Q16×C109C2×C411yescharacteristic
39C2×SD16×C109—21no—
40C2×D8×C109C2×C411yescharacteristic
41#131×C109—21no—
42#131×C109—21no—
43#131×C109—42no—
44#131×C109—42no—
45#131×C109C2×C412yes—
46#131×C109D412yes—
47C2×M16×C109—21no—
48C2×M16×C109—24no—
49C2×M16×C109—21no—
50C2×M16×C109—21no—
51C2×M16×C109—21no—
52C2×M16×C109D411yescharacteristic
53C22×C8×C109—21no—
54C22×C8×C109C2311yescharacteristic
55M32×C109—21no—
56M32×C109—21no—
57C2×C16×C109—21no—
58C2×C16×C109—21no—
59C2×C16×C109—21no—
60C2×C16×C109—21no—
61#108×C109—24no—
62#108×C109—21no—
63#108×C109—21no—
64#108×C109—21no—
65#108×C109C2×C412yes—
66#108×C109D412yes—
67#108×C109D412yes—
68#101×C109—22no—
69#100×C109—22no—
70#41×C109—21no—
71#41×C109—21no—
72#41×C109—41no—
73#41×C109—41no—
74#41×C109—41no—
75#41×C109—41no—
76#41×C109C22⋊C412yes—
77#41×C109D812yes—
78C2×Q8×C109—21no—
79C2×Q8×C109SD1611yescharacteristic
80C2×D4×C109—21no—
81C2×D4×C109—21no—
82C2×D4×C109—41no—
83C2×D4×C109SD1611yescharacteristic
84C22×C4×C109—21no—
85C22×C4×C109—41no—
86C22×C4×C109C2×D411yescharacteristic
87Q16×C109—21no—
88Q16×C109—21no—
89Q16×C109—42no—
90SD16×C109—42no—
91SD16×C109—42no—
92SD16×C109—41no—
93SD16×C109—41no—
94D8×C109—21no—
95D8×C109—21no—
96D8×C109—42no—
97M16×C109—21no—
98M16×C109—22no—
99M16×C109—22no—
100M16×C109—22no—
101M16×C109—22no—
102M16×C109—22no—
103M16×C109—44no—
104M16×C109—41no—
105M16×C109—41no—
106M16×C109—41no—
107M16×C109C22⋊C412yes—
108C2×C8×C109—21no—
109C2×C8×C109—22no—
110C2×C8×C109—22no—
111C2×C8×C109—44no—
112C2×C8×C109—41no—
113C2×C8×C109—41no—
114C2×C8×C109—41no—
115C2×C8×C109C22×C411yescharacteristic
116C2×C8×C109C2×D411yescharacteristic
117C2×C8×C109C2×D411yescharacteristic
118C2×C8×C109C2×D411yescharacteristic
119C2×C8×C109#4111yescharacteristic
120C2×C8×C109#4111yescharacteristic
121C16×C109—41no—
122C16×C109—41no—
123C23×C109—21no—
124C23×C109—41no—
125C23×C109#13711yescharacteristic
126Q8×C109—21no—
127Q8×C109—41no—
128Q8×C109—41no—
129Q8×C109#10212yes—
130D4×C109—22no—
131D4×C109—22no—
132D4×C109—21no—
133D4×C109—41no—
134D4×C109—41no—
135D4×C109—81no—
136D4×C109—81no—
137D4×C109#10212yes—
138C2×C4×C109—21no—
139C2×C4×C109—22no—
140C2×C4×C109—22no—
141C2×C4×C109—41no—
142C2×C4×C109—81no—
143C2×C4×C109—81no—
144C2×C4×C109C2×C22⋊C411yescharacteristic
145C2×C4×C109#12011yescharacteristic
146C2×C4×C109#12311yescharacteristic
147C2×C4×C109C2×D811yescharacteristic
148C2×C4×C109C2×SD1611yescharacteristic
149C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
150C8×C109—22no—
151C8×C109—42no—
152C8×C109—42no—
153C8×C109—42no—
154C8×C109C4×D411yescharacteristic
155C8×C109C4×D411yescharacteristic
156C8×C109#12111yescharacteristic
157C8×C109#12111yescharacteristic
158C22×C109—22no—
159C22×C109—41no—
160C22×C109—81no—
161C22×C109—81no—
162C22×C109—161no—
163C22×C109#41711yescharacteristic
164C22×C109#44811yescharacteristic
165C22×C109#48011yescharacteristic
166C4×C109—22no—
167C4×C109—81no—
168C4×C109#38611yescharacteristic
169C4×C109C2×#10211yescharacteristic
170C4×C109#44711yescharacteristic
171C4×C109#48111yescharacteristic
172#12995C10911yesmaximal, characteristic, direct product factor, Sylow
173C2×C109—21no—
174C2×C109—21no—
175C2×C109—21no—
176C2×C109—42no—
177C2×C109—161no—
178C2×C109#187411yescharacteristic
179#3067C2×C10911yescharacteristic
180#2916C2×C10911yescharacteristic, complemented
181#2143C2×C10911yescharacteristic, complemented
182C2×#359C2×C10911yescharacteristic, complemented
183C2×#350C2×C10911yescharacteristic, complemented
184#1388C2×C10911yescharacteristic, complemented
185#1381C2×C10911yescharacteristic, complemented
186C109#1299511yescharacteristic, direct product factor, Sylow
187#575—21no—
188C2×#135C22×C10911yescharacteristic
189#568—21no—
190C2×#131C22×C10911yescharacteristic
191C2×M32C22×C10911yescharacteristic
192C22×C16C22×C10911yescharacteristic
193#473C4×C10912yes—
194#472C4×C10912yes—
195#471—22no—
196#430—21no—
197#430—22no—
198C2×#108—22no—
199C2×#108C22×C10911yescharacteristic
200C2×#108C22×C10911yescharacteristic
201C2×#108C22×C10911yescharacteristic
202#413—21no—
203#359—22no—
204#359—22no—
205#350—22no—
206C2×#41—21no—
207C2×#41D4×C10911yescharacteristic
208#137—22no—
209#137—41no—
210AGL(1,ℤ/8)—22no—
211AGL(1,ℤ/8)—41no—
212#135—21no—
213#135—21no—
214#135—21no—
215#135—21no—
216#135—42no—
217C2×Q16C2×C4×C10911yescharacteristic
218C2×SD16—21no—
219C2×D8C2×C4×C10911yescharacteristic
220#131—21no—
221#131—21no—
222#131—42no—
223#131—42no—
224#131C2×C4×C10912yes—
225#131D4×C10912yes—
226C2×M16—21no—
227C2×M16—24no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16D4×C10911yescharacteristic
232C22×C8—21no—
233C22×C8C23×C10911yescharacteristic
234M32—21no—
235M32—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240#108—24no—
241#108—21no—
242#108—21no—
243#108—21no—
244#108C2×C4×C10912yes—
245#108D4×C10912yes—
246#108D4×C10912yes—
247#101—22no—
248#100—22no—
249#41—21no—
250#41—21no—
251#41—41no—
252#41—41no—
253#41—41no—
254#41—41no—
255#41C22⋊C4×C10912yes—
256#41D8×C10912yes—
257C2×Q8—21no—
258C2×Q8SD16×C10911yescharacteristic
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—41no—
262C2×D4SD16×C10911yescharacteristic
263C22×C4—21no—
264C22×C4—41no—
265C22×C4C2×D4×C10911yescharacteristic
266Q16—21no—
267Q16—21no—
268Q16—42no—
269SD16—42no—
270SD16—42no—
271SD16—41no—
272SD16—41no—
273D8—21no—
274D8—21no—
275D8—42no—
276M16—21no—
277M16—22no—
278M16—22no—
279M16—22no—
280M16—22no—
281M16—22no—
282M16—44no—
283M16—41no—
284M16—41no—
285M16—41no—
286M16C22⋊C4×C10912yes—
287C2×C8—21no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—44no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8C22×C4×C10911yescharacteristic
295C2×C8C2×D4×C10911yescharacteristic
296C2×C8C2×D4×C10911yescharacteristic
297C2×C8C2×D4×C10911yescharacteristic
298C2×C8#41×C10911yescharacteristic
299C2×C8#41×C10911yescharacteristic
300C16—41no—
301C16—41no—
302C23—21no—
303C23—41no—
304C23#137×C10911yescharacteristic
305Q8—21no—
306Q8—41no—
307Q8—41no—
308Q8#102×C10912yes—
309D4—22no—
310D4—22no—
311D4—21no—
312D4—41no—
313D4—41no—
314D4—81no—
315D4—81no—
316D4#102×C10912yes—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4C2×C22⋊C4×C10911yescharacteristic
324C2×C4#120×C10911yescharacteristic
325C2×C4#123×C10911yescharacteristic
326C2×C4C2×D8×C10911yescharacteristic
327C2×C4C2×SD16×C10911yescharacteristic
328C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
329C8—22no—
330C8—42no—
331C8—42no—
332C8—42no—
333C8C4×D4×C10911yescharacteristic
334C8C4×D4×C10911yescharacteristic
335C8#121×C10911yescharacteristic
336C8#121×C10911yescharacteristic
337C22—22no—
338C22—41no—
339C22—81no—
340C22—81no—
341C22—161no—
342C22#417×C10911yescharacteristic
343C22#448×C10911yescharacteristic
344C22#480×C10911yescharacteristic
345C4—22no—
346C4—81no—
347C4#386×C10911yescharacteristic
348C4C2×#102×C10911yescharacteristic
349C4#447×C10911yescharacteristic
350C4#481×C10911yescharacteristic
351C2—21no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—161no—
356C2C109×#187411yescharacteristic
subgroupG/H
centerC4×C109#447
commutatorC2×C8C22×C4×C109
Frattini, cosocleC22×C8C23×C109
socleC2×C109#1874
prime radicalC109×#3067C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C109 ◁ C22×C4×C109 ◁ C2×#131×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#2916C2semidirect
3. C109×#2143C2semidirect
4. C2×#359×C109C2semidirect
5. C2×#350×C109C2semidirect
6. C109×#1388C2semidirect
7. C109×#1381C2semidirect
172. #12995C109direct (N × H)
180. #2916C2×C109semidirect
181. #2143C2×C109semidirect
182. C2×#359C2×C109semidirect
183. C2×#350C2×C109semidirect
184. #1388C2×C109semidirect
185. #1381C2×C109semidirect
186. C109#12995direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#30672
C109×#29162
C109×#21432
C2×#359×C1092
C2×#350×C1092
C109×#13882
C109×#13812
#12995109