Finite Group Databasebeta

← C109×#13412 #42131406

Subgroups — C109×#13412 #42131406

Order 27904.

430 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3156C211yesmaximal, characteristic
2C2×#362×C109C211yesmaximal, characteristic, complemented
3C2×#359×C109C211yesmaximal, characteristic
4C109×#2139C211yesmaximal, characteristic
5C2×#358×C109C211yesmaximal, characteristic, complemented
6C109×#1952C211yesmaximal, characteristic, complemented
7C109×#1379C211yesmaximal, characteristic, complemented
8C2×#137×C109—21no—
9C2×#135×C109C2211yescharacteristic
10C22×Q16×C109C2211yescharacteristic
11#536×C109—21no—
12C2×M32×C109C2211yescharacteristic
13C22×C16×C109C2211yescharacteristic
14#497×C109—22no—
15#495×C109—21no—
16#494×C109—21no—
17C2×#108×C109C2211yescharacteristic
18#428×C109—21no—
19C2×#107×C109C2211yescharacteristic
20#405×C109C2211yescharacteristic
21#362×C109—22no—
22#359×C109—22no—
23#358×C109—22no—
24#358×C109—21no—
25#357×C109—21no—
26#343×C109—21no—
27C2×#41×C109—21no—
28C22×Q8×C109—21no—
29#137×C109—42no—
30#137×C109—42no—
31#135×C109—21no—
32#135×C109—21no—
33#135×C109—21no—
34#135×C109—21no—
35C2×Q16×C109—22no—
36C2×Q16×C109—21no—
37C2×Q16×C109—21no—
38C2×Q16×C109—21no—
39C2×Q16×C109—21no—
40C2×Q16×C109—21no—
41C2×Q16×C109D412yes—
42C2×Q16×C109D411yescharacteristic
43C2×SD16×C109—22no—
44C2×SD16×C109—21no—
45C2×D8×C109D411yescharacteristic
46C2×M16×C109—21no—
47C2×M16×C109D411yescharacteristic
48C22×C8×C109C2311yescharacteristic
49C4⋊Q8×C109—22no—
50#124×C109—22no—
51#123×C109—42no—
52C22⋊Q8×C109—41no—
53C22⋊Q8×C109—41no—
54#117×C109D411yescharacteristic
55C2×C4⋊C4×C109—21no—
56M32×C109—21no—
57M32×C109—21no—
58C2×C16×C109—21no—
59C2×C16×C109—21no—
60C2×C16×C109—21no—
61C2×C16×C109—21no—
62#108×C109—21no—
63#108×C109D412yes—
64#107×C109—21no—
65#107×C109—21no—
66#107×C109—41no—
67#106×C109—41no—
68#97×C109—21no—
69C4×C8×C109—21no—
70#41×C109—41no—
71#41×C109—41no—
72#41×C109—41no—
73#41×C109—41no—
74C2×Q8×C109—22no—
75C2×Q8×C109—21no—
76C2×Q8×C109—41no—
77C2×Q8×C109—41no—
78C2×Q8×C109—41no—
79C2×Q8×C109—41no—
80C2×Q8×C109—41no—
81C2×D4×C109—21no—
82C2×D4×C109—41no—
83C22×C4×C109—41no—
84C22×C4×C109—41no—
85C22×C4×C109—41no—
86C22×C4×C109C2×D411yescharacteristic
87Q16×C109—22no—
88Q16×C109—21no—
89Q16×C109—22no—
90Q16×C109—21no—
91Q16×C109—44no—
92Q16×C109—41no—
93Q16×C109—41no—
94SD16×C109—44no—
95SD16×C109—41no—
96SD16×C109—41no—
97D8×C109—21no—
98D8×C109—21no—
99M16×C109—21no—
100M16×C109—22no—
101M16×C109—41no—
102M16×C109SD1612yes—
103C2×C8×C109—21no—
104C2×C8×C109—41no—
105C2×C8×C109C2×D411yescharacteristic
106C2×C8×C109C2×D411yescharacteristic
107C2×C8×C109C2×D411yescharacteristic
108C2×C8×C109#4111yescharacteristic
109C2×C8×C109#4111yescharacteristic
110C2×C8×C109#4111yescharacteristic
111C4⋊C4×C109—42no—
112C4⋊C4×C109—42no—
113C4⋊C4×C109—41no—
114C4⋊C4×C109—41no—
115C4⋊C4×C109D811yescharacteristic
116C4⋊C4×C109D811yescharacteristic
117C22⋊C4×C109—21no—
118C22⋊C4×C109—42no—
119C22⋊C4×C109—42no—
120C42×C109—21no—
121C16×C109—41no—
122C16×C109—41no—
123C23×C109—41no—
124C23×C109AGL(1,ℤ/8)11yescharacteristic
125Q8×C109—41no—
126Q8×C109—42no—
127Q8×C109—42no—
128Q8×C109—41no—
129Q8×C109—81no—
130Q8×C109—81no—
131D4×C109—41no—
132D4×C109—41no—
133D4×C109—81no—
134D4×C109—81no—
135C2×C4×C109—21no—
136C2×C4×C109—21no—
137C2×C4×C109—42no—
138C2×C4×C109—41no—
139C2×C4×C109—81no—
140C2×C4×C109—81no—
141C2×C4×C109—81no—
142C2×C4×C109—81no—
143C2×C4×C109—81no—
144C2×C4×C109—161no—
145C2×C4×C109#12011yescharacteristic
146C2×C4×C109#12111yescharacteristic
147C2×C4×C109#12411yescharacteristic
148C2×C4×C109C2×D811yescharacteristic
149C2×C4×C109C2×SD1611yescharacteristic
150C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
151C8×C109—22no—
152C8×C109—42no—
153C8×C109#12111yescharacteristic
154C8×C109#12111yescharacteristic
155C8×C109#12311yescharacteristic
156C8×C109#12311yescharacteristic
157C22×C109—21no—
158C22×C109—21no—
159C22×C109—81no—
160C22×C109—81no—
161C22×C109—81no—
162C22×C109#44711yescharacteristic
163C22×C109#46011yescharacteristic
164C22×C109#49011yescharacteristic
165C4×C109—41no—
166C4×C109—41no—
167C4×C109—81no—
168C4×C109—82no—
169C4×C109—161no—
170C4×C109#39411yescharacteristic
171C4×C109#45011yescharacteristic
172C4×C109#45911yescharacteristic
173C4×C109#48611yescharacteristic
174#13412C10911yesmaximal, characteristic, direct product factor, Sylow
175C2×C109—21no—
176C2×C109—21no—
177C2×C109—161no—
178C2×C109#204111yescharacteristic
179C2×C109#219311yescharacteristic
180C2×C109#219511yescharacteristic
181#3156C2×C10911yescharacteristic
182C2×#362C2×C10911yescharacteristic, complemented
183C2×#359C2×C10911yescharacteristic
184#2139C2×C10911yescharacteristic
185C2×#358C2×C10911yescharacteristic, complemented
186#1952C2×C10911yescharacteristic, complemented
187#1379C2×C10911yescharacteristic, complemented
188C109#1341211yescharacteristic, direct product factor, Sylow
189C2×#137—21no—
190C2×#135C22×C10911yescharacteristic
191C22×Q16C22×C10911yescharacteristic
192#536—21no—
193C2×M32C22×C10911yescharacteristic
194C22×C16C22×C10911yescharacteristic
195#497—22no—
196#495—21no—
197#494—21no—
198C2×#108C22×C10911yescharacteristic
199#428—21no—
200C2×#107C22×C10911yescharacteristic
201#405C22×C10911yescharacteristic
202#362—22no—
203#359—22no—
204#358—22no—
205#358—21no—
206#357—21no—
207#343—21no—
208C2×#41—21no—
209C22×Q8—21no—
210#137—42no—
211#137—42no—
212#135—21no—
213#135—21no—
214#135—21no—
215#135—21no—
216C2×Q16—22no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16—21no—
220C2×Q16—21no—
221C2×Q16—21no—
222C2×Q16D4×C10912yes—
223C2×Q16D4×C10911yescharacteristic
224C2×SD16—22no—
225C2×SD16—21no—
226C2×D8D4×C10911yescharacteristic
227C2×M16—21no—
228C2×M16D4×C10911yescharacteristic
229C22×C8C23×C10911yescharacteristic
230C4⋊Q8—22no—
231#124—22no—
232#123—42no—
233C22⋊Q8—41no—
234C22⋊Q8—41no—
235#117D4×C10911yescharacteristic
236C2×C4⋊C4—21no—
237M32—21no—
238M32—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243#108—21no—
244#108D4×C10912yes—
245#107—21no—
246#107—21no—
247#107—41no—
248#106—41no—
249#97—21no—
250C4×C8—21no—
251#41—41no—
252#41—41no—
253#41—41no—
254#41—41no—
255C2×Q8—22no—
256C2×Q8—21no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×D4—21no—
263C2×D4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4C2×D4×C10911yescharacteristic
268Q16—22no—
269Q16—21no—
270Q16—22no—
271Q16—21no—
272Q16—44no—
273Q16—41no—
274Q16—41no—
275SD16—44no—
276SD16—41no—
277SD16—41no—
278D8—21no—
279D8—21no—
280M16—21no—
281M16—22no—
282M16—41no—
283M16SD16×C10912yes—
284C2×C8—21no—
285C2×C8—41no—
286C2×C8C2×D4×C10911yescharacteristic
287C2×C8C2×D4×C10911yescharacteristic
288C2×C8C2×D4×C10911yescharacteristic
289C2×C8#41×C10911yescharacteristic
290C2×C8#41×C10911yescharacteristic
291C2×C8#41×C10911yescharacteristic
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4D8×C10911yescharacteristic
297C4⋊C4D8×C10911yescharacteristic
298C22⋊C4—21no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C42—21no—
302C16—41no—
303C16—41no—
304C23—41no—
305C23AGL(1,ℤ/8)×C10911yescharacteristic
306Q8—41no—
307Q8—42no—
308Q8—42no—
309Q8—41no—
310Q8—81no—
311Q8—81no—
312D4—41no—
313D4—41no—
314D4—81no—
315D4—81no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—161no—
326C2×C4#120×C10911yescharacteristic
327C2×C4#121×C10911yescharacteristic
328C2×C4#124×C10911yescharacteristic
329C2×C4C2×D8×C10911yescharacteristic
330C2×C4C2×SD16×C10911yescharacteristic
331C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
332C8—22no—
333C8—42no—
334C8#121×C10911yescharacteristic
335C8#121×C10911yescharacteristic
336C8#123×C10911yescharacteristic
337C8#123×C10911yescharacteristic
338C22—21no—
339C22—21no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22#447×C10911yescharacteristic
344C22#460×C10911yescharacteristic
345C22#490×C10911yescharacteristic
346C4—41no—
347C4—41no—
348C4—81no—
349C4—82no—
350C4—161no—
351C4#394×C10911yescharacteristic
352C4#450×C10911yescharacteristic
353C4#459×C10911yescharacteristic
354C4#486×C10911yescharacteristic
355C2—21no—
356C2—21no—
357C2—161no—
358C2C109×#204111yescharacteristic
359C2C109×#219311yescharacteristic
360C2C109×#219511yescharacteristic
subgroupG/H
center, socleC22×C109#447
commutator, Frattini, cosocleC22×C8C23×C109
prime radicalC2×#135×C109C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C22×C4×C109 ◁ #405×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#362×C109C2semidirect
5. C2×#358×C109C2semidirect
6. C109×#1952C2semidirect
7. C109×#1379C2semidirect
174. #13412C109direct (N × H)
182. C2×#362C2×C109semidirect
185. C2×#358C2×C109semidirect
186. #1952C2×C109semidirect
187. #1379C2×C109semidirect
188. C109#13412direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#31562
C2×#362×C1092
C2×#359×C1092
C109×#21392
C2×#358×C1092
C109×#19522
C109×#13792
#13412109