Finite Group Databasebeta

← C109×#3201 #28360455

Subgroups — C109×#3201 #28360455

Order 13952.

390 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#537×C109C211yesmaximal, characteristic
2#536×C109C211yesmaximal, characteristic, complemented
3#485×C109C211yesmaximal, characteristic, complemented
4#484×C109C211yesmaximal, characteristic, complemented
5#483×C109C211yesmaximal, characteristic, complemented
6#482×C109C211yesmaximal, characteristic, complemented
7#470×C109C211yesmaximal, characteristic, complemented
8#468×C109C211yesmaximal, characteristic, complemented
9#467×C109C211yesmaximal, characteristic, complemented
10#465×C109C211yesmaximal, characteristic, complemented
11#452×C109C211yesmaximal, characteristic, complemented
12#452×C109C211yesmaximal, characteristic, complemented
13#451×C109C211yesmaximal, characteristic, complemented
14#448×C109C211yesmaximal, characteristic, complemented
15#408×C109C211yesmaximal, characteristic, complemented
16C2×#41×C109C2211yescharacteristic
17C2×#41×C109C2211yescharacteristic
18C22×Q8×C109C2211yescharacteristic
19C2×Q16×C109C2211yescharacteristic
20C2×Q16×C109C2211yescharacteristic
21C2×SD16×C109C2211yescharacteristic
22C2×SD16×C109C2211yescharacteristic
23C2×M16×C109C2211yescharacteristic
24C22×C8×C109C2211yescharacteristic
25C4⋊Q8×C109—21no—
26#124×C109—21no—
27#123×C109—21no—
28#123×C109—21no—
29C22⋊Q8×C109—21no—
30C22⋊Q8×C109C2211yescharacteristic
31C22⋊Q8×C109C2211yescharacteristic
32C22⋊Q8×C109C2211yescharacteristic
33C22⋊Q8×C109C2211yescharacteristic
34C22⋊Q8×C109C2211yescharacteristic
35C22⋊Q8×C109C2211yescharacteristic
36#121×C109—21no—
37#121×C109—21no—
38#121×C109—21no—
39#121×C109C2211yescharacteristic
40#121×C109C2211yescharacteristic
41#117×C109C2211yescharacteristic
42C2×C4⋊C4×C109C2211yescharacteristic
43#107×C109C2211yescharacteristic
44#107×C109C2211yescharacteristic
45#106×C109C2211yescharacteristic
46#106×C109C2211yescharacteristic
47#103×C109C2211yescharacteristic
48#103×C109C2211yescharacteristic
49#103×C109C2211yescharacteristic
50#103×C109C2211yescharacteristic
51#103×C109C2211yescharacteristic
52#103×C109C2211yescharacteristic
53#102×C109C2211yescharacteristic
54#102×C109C2211yescharacteristic
55C22⋊C8×C109C2211yescharacteristic
56C22⋊C8×C109C2211yescharacteristic
57C22⋊C8×C109C2211yescharacteristic
58C22⋊C8×C109C2211yescharacteristic
59#41×C109—41no—
60#41×C109—41no—
61#41×C109—41no—
62#41×C109—41no—
63C2×Q8×C109—21no—
64C2×Q8×C109—41no—
65C2×Q8×C109—41no—
66C2×Q8×C109D411yescharacteristic
67C2×Q8×C109C2311yescharacteristic
68C2×Q8×C109C2311yescharacteristic
69C2×Q8×C109C2311yescharacteristic
70C2×D4×C109—21no—
71C2×D4×C109—21no—
72C2×D4×C109—21no—
73C2×D4×C109D411yescharacteristic
74C2×D4×C109D411yescharacteristic
75C2×D4×C109D411yescharacteristic
76C2×D4×C109C2311yescharacteristic
77C22×C4×C109—21no—
78C22×C4×C109—21no—
79C22×C4×C109—21no—
80C22×C4×C109C2311yescharacteristic
81C22×C4×C109C2311yescharacteristic
82C22×C4×C109C2311yescharacteristic
83Q16×C109—41no—
84Q16×C109—41no—
85SD16×C109—41no—
86SD16×C109—41no—
87M16×C109—41no—
88C2×C8×C109—41no—
89C2×C8×C109C2311yescharacteristic
90C2×C8×C109C2311yescharacteristic
91C2×C8×C109C2311yescharacteristic
92C2×C8×C109C2311yescharacteristic
93C4⋊C4×C109—21no—
94C4⋊C4×C109—21no—
95C4⋊C4×C109—21no—
96C4⋊C4×C109—21no—
97C4⋊C4×C109—21no—
98C4⋊C4×C109—21no—
99C4⋊C4×C109—21no—
100C4⋊C4×C109C2311yescharacteristic
101C4⋊C4×C109C2311yescharacteristic
102C4⋊C4×C109C2311yescharacteristic
103C4⋊C4×C109C2311yescharacteristic
104C22⋊C4×C109—21no—
105C22⋊C4×C109—21no—
106C22⋊C4×C109—21no—
107C22⋊C4×C109—21no—
108C22⋊C4×C109—21no—
109C22⋊C4×C109—21no—
110C22⋊C4×C109—21no—
111C22⋊C4×C109—21no—
112C22⋊C4×C109—21no—
113C42×C109—21no—
114C23×C109—21no—
115C23×C109C2×D411yescharacteristic
116C23×C109C2×D411yescharacteristic
117C23×C109C2×D411yescharacteristic
118Q8×C109—21no—
119Q8×C109—21no—
120Q8×C109—21no—
121Q8×C109—21no—
122Q8×C109—21no—
123Q8×C109—21no—
124Q8×C109—41no—
125Q8×C109—42no—
126D4×C109—21no—
127D4×C109—21no—
128D4×C109—41no—
129D4×C109—41no—
130D4×C109—41no—
131D4×C109—42no—
132D4×C109—81no—
133D4×C109—81no—
134C2×C4×C109—21no—
135C2×C4×C109—21no—
136C2×C4×C109—21no—
137C2×C4×C109—21no—
138C2×C4×C109—21no—
139C2×C4×C109—21no—
140C2×C4×C109—21no—
141C2×C4×C109—21no—
142C2×C4×C109—21no—
143C2×C4×C109—21no—
144C2×C4×C109—21no—
145C2×C4×C109—21no—
146C2×C4×C109—21no—
147C2×C4×C109—42no—
148C2×C4×C109—42no—
149C2×C4×C109—81no—
150C2×C4×C109C2×D411yescharacteristic
151C2×C4×C109C2×D411yescharacteristic
152C2×C4×C109C2×D411yescharacteristic
153C2×C4×C109C2411yescharacteristic
154C8×C109—21no—
155C8×C109—21no—
156C8×C109—21no—
157C8×C109—21no—
158C22×C109—21no—
159C22×C109—21no—
160C22×C109—21no—
161C22×C109—21no—
162C22×C109—21no—
163C22×C109—21no—
164C22×C109—21no—
165C22×C109—21no—
166C22×C109—21no—
167C22×C109—41no—
168C22×C109—41no—
169C22×C109—41no—
170C22×C109C22×D411yescharacteristic
171C4×C109—21no—
172C4×C109—21no—
173C4×C109—21no—
174C4×C109—41no—
175C4×C109—41no—
176C4×C109—41no—
177C4×C109—41no—
178C4×C109—41no—
179C4×C109—41no—
180C4×C109—41no—
181C4×C109H(5,𝔽2)11yescharacteristic
182C4×C109H(5,𝔽2)11yescharacteristic
183C2×C109—41no—
184C2×C109—41no—
185C2×C109—41no—
186C2×C109—81no—
187C2×C109#53411yescharacteristic
188C2×C109#57711yescharacteristic
189C2×C109#57811yescharacteristic
190#3201C10911yesmaximal, characteristic, direct product factor, Sylow
191C109#320111yescharacteristic, direct product factor, Sylow
192#537C2×C10911yescharacteristic
193#536C2×C10911yescharacteristic, complemented
194#485C2×C10911yescharacteristic, complemented
195#484C2×C10911yescharacteristic, complemented
196#483C2×C10911yescharacteristic, complemented
197#482C2×C10911yescharacteristic, complemented
198#470C2×C10911yescharacteristic, complemented
199#468C2×C10911yescharacteristic, complemented
200#467C2×C10911yescharacteristic, complemented
201#465C2×C10911yescharacteristic, complemented
202#452C2×C10911yescharacteristic, complemented
203#452C2×C10911yescharacteristic, complemented
204#451C2×C10911yescharacteristic, complemented
205#448C2×C10911yescharacteristic, complemented
206#408C2×C10911yescharacteristic, complemented
207C2×#41C22×C10911yescharacteristic
208C2×#41C22×C10911yescharacteristic
209C22×Q8C22×C10911yescharacteristic
210C2×Q16C22×C10911yescharacteristic
211C2×Q16C22×C10911yescharacteristic
212C2×SD16C22×C10911yescharacteristic
213C2×SD16C22×C10911yescharacteristic
214C2×M16C22×C10911yescharacteristic
215C22×C8C22×C10911yescharacteristic
216C4⋊Q8—21no—
217#124—21no—
218#123—21no—
219#123—21no—
220C22⋊Q8—21no—
221C22⋊Q8C22×C10911yescharacteristic
222C22⋊Q8C22×C10911yescharacteristic
223C22⋊Q8C22×C10911yescharacteristic
224C22⋊Q8C22×C10911yescharacteristic
225C22⋊Q8C22×C10911yescharacteristic
226C22⋊Q8C22×C10911yescharacteristic
227#121—21no—
228#121—21no—
229#121—21no—
230#121C22×C10911yescharacteristic
231#121C22×C10911yescharacteristic
232#117C22×C10911yescharacteristic
233C2×C4⋊C4C22×C10911yescharacteristic
234#107C22×C10911yescharacteristic
235#107C22×C10911yescharacteristic
236#106C22×C10911yescharacteristic
237#106C22×C10911yescharacteristic
238#103C22×C10911yescharacteristic
239#103C22×C10911yescharacteristic
240#103C22×C10911yescharacteristic
241#103C22×C10911yescharacteristic
242#103C22×C10911yescharacteristic
243#103C22×C10911yescharacteristic
244#102C22×C10911yescharacteristic
245#102C22×C10911yescharacteristic
246C22⋊C8C22×C10911yescharacteristic
247C22⋊C8C22×C10911yescharacteristic
248C22⋊C8C22×C10911yescharacteristic
249C22⋊C8C22×C10911yescharacteristic
250#41—41no—
251#41—41no—
252#41—41no—
253#41—41no—
254C2×Q8—21no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8D4×C10911yescharacteristic
258C2×Q8C23×C10911yescharacteristic
259C2×Q8C23×C10911yescharacteristic
260C2×Q8C23×C10911yescharacteristic
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4D4×C10911yescharacteristic
265C2×D4D4×C10911yescharacteristic
266C2×D4D4×C10911yescharacteristic
267C2×D4C23×C10911yescharacteristic
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4C23×C10911yescharacteristic
272C22×C4C23×C10911yescharacteristic
273C22×C4C23×C10911yescharacteristic
274Q16—41no—
275Q16—41no—
276SD16—41no—
277SD16—41no—
278M16—41no—
279C2×C8—41no—
280C2×C8C23×C10911yescharacteristic
281C2×C8C23×C10911yescharacteristic
282C2×C8C23×C10911yescharacteristic
283C2×C8C23×C10911yescharacteristic
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4C23×C10911yescharacteristic
292C4⋊C4C23×C10911yescharacteristic
293C4⋊C4C23×C10911yescharacteristic
294C4⋊C4C23×C10911yescharacteristic
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C42—21no—
305C23—21no—
306C23C2×D4×C10911yescharacteristic
307C23C2×D4×C10911yescharacteristic
308C23C2×D4×C10911yescharacteristic
309Q8—21no—
310Q8—21no—
311Q8—21no—
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—41no—
316Q8—42no—
317D4—21no—
318D4—21no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—42no—
323D4—81no—
324D4—81no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—81no—
341C2×C4C2×D4×C10911yescharacteristic
342C2×C4C2×D4×C10911yescharacteristic
343C2×C4C2×D4×C10911yescharacteristic
344C2×C4C24×C10911yescharacteristic
345C8—21no—
346C8—21no—
347C8—21no—
348C8—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22C22×D4×C10911yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4H(5,𝔽2)×C10911yescharacteristic
373C4H(5,𝔽2)×C10911yescharacteristic
374C2—41no—
375C2—41no—
376C2—41no—
377C2—81no—
378C2#534×C10911yescharacteristic
379C2#577×C10911yescharacteristic
380C2#578×C10911yescharacteristic
subgroupG/H
center, socleC22×C109C22×D4
commutator, Frattini, cosocleC2×C4C24×C109
prime radical#537×C109C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C2×#41×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. #536×C109C2semidirect
3. #485×C109C2semidirect
4. #484×C109C2semidirect
5. #483×C109C2semidirect
6. #482×C109C2semidirect
7. #470×C109C2semidirect
8. #468×C109C2semidirect
9. #467×C109C2semidirect
10. #465×C109C2semidirect
11. #452×C109C2semidirect
12. #452×C109C2semidirect
13. #451×C109C2semidirect
14. #448×C109C2semidirect
15. #408×C109C2semidirect
190. #3201C109direct (N × H)
191. C109#3201direct (N × H)
193. #536C2×C109semidirect
194. #485C2×C109semidirect
195. #484C2×C109semidirect
196. #483C2×C109semidirect
197. #482C2×C109semidirect
198. #470C2×C109semidirect
199. #468C2×C109semidirect
200. #467C2×C109semidirect
201. #465C2×C109semidirect
202. #452C2×C109semidirect
203. #452C2×C109semidirect
204. #451C2×C109semidirect
205. #448C2×C109semidirect
206. #408C2×C109semidirect

Maximal subgroups

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

subgroupindex
#537×C1092
#536×C1092
#485×C1092
#484×C1092
#483×C1092
#482×C1092
#470×C1092
#468×C1092
#467×C1092
#465×C1092
#452×C1092
#451×C1092
#448×C1092
#408×C1092
#3201109