Finite Group Databasebeta

← C109×#3346 #28360600

Subgroups — C109×#3346 #28360600

Order 13952.

356 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#554×C109C211yesmaximal, characteristic, complemented
2#553×C109C211yesmaximal, characteristic
3#501×C109C211yesmaximal, characteristic, complemented
4#485×C109C211yesmaximal, characteristic, complemented
5#484×C109C211yesmaximal, characteristic, complemented
6#481×C109C211yesmaximal, characteristic, complemented
7#480×C109C211yesmaximal, characteristic, complemented
8#479×C109C211yesmaximal, characteristic, complemented
9#474×C109C211yesmaximal, characteristic, complemented
10#468×C109C211yesmaximal, characteristic, complemented
11#465×C109C211yesmaximal, characteristic, complemented
12#440×C109C211yesmaximal, characteristic, complemented
13#435×C109C211yesmaximal, characteristic, complemented
14#428×C109C211yesmaximal, characteristic, complemented
15C2×#107×C109C211yesmaximal, characteristic, complemented
16C2×SD16×C109C2211yescharacteristic
17C2×M16×C109C2211yescharacteristic
18C22×C8×C109C2211yescharacteristic
19C4⋊Q8×C109C2211yescharacteristic
20#126×C109—21no—
21#125×C109—21no—
22#125×C109C2211yescharacteristic
23#123×C109—21no—
24C22⋊Q8×C109—21no—
25C22⋊Q8×C109—21no—
26C22⋊Q8×C109C2211yescharacteristic
27C22⋊Q8×C109C2211yescharacteristic
28#121×C109—21no—
29#121×C109C2211yescharacteristic
30#121×C109C2211yescharacteristic
31C4×Q8×C109C2211yescharacteristic
32C4×D4×C109—21no—
33C4×D4×C109—21no—
34C4×D4×C109C2211yescharacteristic
35C4×D4×C109C2211yescharacteristic
36#117×C109C2211yescharacteristic
37C2×C4⋊C4×C109C2211yescharacteristic
38C2×C4⋊C4×C109C2211yescharacteristic
39C2×C4⋊C4×C109C2211yescharacteristic
40#107×C109C2211yescharacteristic
41#107×C109C2211yescharacteristic
42#107×C109C2211yescharacteristic
43#107×C109C2211yescharacteristic
44#107×C109C2211yescharacteristic
45#107×C109C2211yescharacteristic
46#106×C109C2211yescharacteristic
47#106×C109C2211yescharacteristic
48#106×C109C2211yescharacteristic
49C4⋊C8×C109C2211yescharacteristic
50#103×C109C2211yescharacteristic
51#103×C109C2211yescharacteristic
52#103×C109C2211yescharacteristic
53#102×C109C2211yescharacteristic
54#102×C109C2211yescharacteristic
55#102×C109C2211yescharacteristic
56C22⋊C8×C109C2211yescharacteristic
57C22⋊C8×C109C2211yescharacteristic
58#97×C109C2211yescharacteristic
59C2×Q8×C109—21no—
60C2×Q8×C109C2311yescharacteristic
61C2×D4×C109—21no—
62C2×D4×C109—21no—
63C2×D4×C109D411yescharacteristic
64C2×D4×C109C2311yescharacteristic
65C22×C4×C109—21no—
66C22×C4×C109—21no—
67C22×C4×C109—21no—
68C22×C4×C109—21no—
69C22×C4×C109C2311yescharacteristic
70C22×C4×C109C2311yescharacteristic
71SD16×C109—22no—
72M16×C109—22no—
73C2×C8×C109—22no—
74C2×C8×C109C2311yescharacteristic
75C2×C8×C109C2311yescharacteristic
76C2×C8×C109C2311yescharacteristic
77C2×C8×C109C2311yescharacteristic
78C4⋊C4×C109—21no—
79C4⋊C4×C109—21no—
80C4⋊C4×C109—21no—
81C4⋊C4×C109—21no—
82C4⋊C4×C109—21no—
83C4⋊C4×C109—21no—
84C4⋊C4×C109—21no—
85C4⋊C4×C109—21no—
86C4⋊C4×C109—21no—
87C4⋊C4×C109D411yescharacteristic
88C4⋊C4×C109C2311yescharacteristic
89C4⋊C4×C109C2311yescharacteristic
90C4⋊C4×C109C2311yescharacteristic
91C4⋊C4×C109C2311yescharacteristic
92C4⋊C4×C109C2311yescharacteristic
93C4⋊C4×C109C2311yescharacteristic
94C22⋊C4×C109—21no—
95C22⋊C4×C109—21no—
96C22⋊C4×C109—21no—
97C22⋊C4×C109—21no—
98C22⋊C4×C109—21no—
99C22⋊C4×C109—21no—
100C22⋊C4×C109D411yescharacteristic
101C22⋊C4×C109D411yescharacteristic
102C42×C109—21no—
103C42×C109—21no—
104C42×C109C2311yescharacteristic
105C23×C109—21no—
106C23×C109C2×D411yescharacteristic
107C23×C109C2×D411yescharacteristic
108Q8×C109—21no—
109Q8×C109—21no—
110Q8×C109—81no—
111D4×C109—22no—
112D4×C109—21no—
113D4×C109—21no—
114D4×C109—42no—
115D4×C109—81no—
116C2×C4×C109—21no—
117C2×C4×C109—21no—
118C2×C4×C109—21no—
119C2×C4×C109—21no—
120C2×C4×C109—21no—
121C2×C4×C109—21no—
122C2×C4×C109—21no—
123C2×C4×C109—21no—
124C2×C4×C109—21no—
125C2×C4×C109—21no—
126C2×C4×C109—21no—
127C2×C4×C109—81no—
128C2×C4×C109—81no—
129C2×C4×C109—81no—
130C2×C4×C109—81no—
131C2×C4×C109C2×D411yescharacteristic
132C2×C4×C109C2×D411yescharacteristic
133C2×C4×C109C2×D411yescharacteristic
134C2×C4×C109C2×D411yescharacteristic
135C2×C4×C109C2411yescharacteristic
136C8×C109—21no—
137C8×C109—21no—
138C8×C109—21no—
139C8×C109#4112yes—
140C22×C109—21no—
141C22×C109—21no—
142C22×C109—21no—
143C22×C109—21no—
144C22×C109—21no—
145C22×C109—41no—
146C22×C109—41no—
147C22×C109—41no—
148C22×C109#13712yes—
149C22×C109C22×D411yescharacteristic
150C4×C109—21no—
151C4×C109—21no—
152C4×C109—21no—
153C4×C109—21no—
154C4×C109—41no—
155C4×C109—41no—
156C4×C109—41no—
157C4×C109—41no—
158C4×C109—41no—
159C4×C109—41no—
160C4×C109—41no—
161C4×C109C2×#4111yescharacteristic
162C4×C1092−1+411yescharacteristic
163C2×C109—22no—
164C2×C109—41no—
165C2×C109—81no—
166C2×C109#54711yescharacteristic
167C2×C109C2×#13711yescharacteristic
168C2×C109#57611yescharacteristic
169#3346C10911yesmaximal, characteristic, direct product factor, Sylow
170C109#334611yescharacteristic, direct product factor, Sylow
171#554C2×C10911yescharacteristic, complemented
172#553C2×C10911yescharacteristic
173#501C2×C10911yescharacteristic, complemented
174#485C2×C10911yescharacteristic, complemented
175#484C2×C10911yescharacteristic, complemented
176#481C2×C10911yescharacteristic, complemented
177#480C2×C10911yescharacteristic, complemented
178#479C2×C10911yescharacteristic, complemented
179#474C2×C10911yescharacteristic, complemented
180#468C2×C10911yescharacteristic, complemented
181#465C2×C10911yescharacteristic, complemented
182#440C2×C10911yescharacteristic, complemented
183#435C2×C10911yescharacteristic, complemented
184#428C2×C10911yescharacteristic, complemented
185C2×#107C2×C10911yescharacteristic, complemented
186C2×SD16C22×C10911yescharacteristic
187C2×M16C22×C10911yescharacteristic
188C22×C8C22×C10911yescharacteristic
189C4⋊Q8C22×C10911yescharacteristic
190#126—21no—
191#125—21no—
192#125C22×C10911yescharacteristic
193#123—21no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196C22⋊Q8C22×C10911yescharacteristic
197C22⋊Q8C22×C10911yescharacteristic
198#121—21no—
199#121C22×C10911yescharacteristic
200#121C22×C10911yescharacteristic
201C4×Q8C22×C10911yescharacteristic
202C4×D4—21no—
203C4×D4—21no—
204C4×D4C22×C10911yescharacteristic
205C4×D4C22×C10911yescharacteristic
206#117C22×C10911yescharacteristic
207C2×C4⋊C4C22×C10911yescharacteristic
208C2×C4⋊C4C22×C10911yescharacteristic
209C2×C4⋊C4C22×C10911yescharacteristic
210#107C22×C10911yescharacteristic
211#107C22×C10911yescharacteristic
212#107C22×C10911yescharacteristic
213#107C22×C10911yescharacteristic
214#107C22×C10911yescharacteristic
215#107C22×C10911yescharacteristic
216#106C22×C10911yescharacteristic
217#106C22×C10911yescharacteristic
218#106C22×C10911yescharacteristic
219C4⋊C8C22×C10911yescharacteristic
220#103C22×C10911yescharacteristic
221#103C22×C10911yescharacteristic
222#103C22×C10911yescharacteristic
223#102C22×C10911yescharacteristic
224#102C22×C10911yescharacteristic
225#102C22×C10911yescharacteristic
226C22⋊C8C22×C10911yescharacteristic
227C22⋊C8C22×C10911yescharacteristic
228#97C22×C10911yescharacteristic
229C2×Q8—21no—
230C2×Q8C23×C10911yescharacteristic
231C2×D4—21no—
232C2×D4—21no—
233C2×D4D4×C10911yescharacteristic
234C2×D4C23×C10911yescharacteristic
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4C23×C10911yescharacteristic
240C22×C4C23×C10911yescharacteristic
241SD16—22no—
242M16—22no—
243C2×C8—22no—
244C2×C8C23×C10911yescharacteristic
245C2×C8C23×C10911yescharacteristic
246C2×C8C23×C10911yescharacteristic
247C2×C8C23×C10911yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4D4×C10911yescharacteristic
258C4⋊C4C23×C10911yescharacteristic
259C4⋊C4C23×C10911yescharacteristic
260C4⋊C4C23×C10911yescharacteristic
261C4⋊C4C23×C10911yescharacteristic
262C4⋊C4C23×C10911yescharacteristic
263C4⋊C4C23×C10911yescharacteristic
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4D4×C10911yescharacteristic
271C22⋊C4D4×C10911yescharacteristic
272C42—21no—
273C42—21no—
274C42C23×C10911yescharacteristic
275C23—21no—
276C23C2×D4×C10911yescharacteristic
277C23C2×D4×C10911yescharacteristic
278Q8—21no—
279Q8—21no—
280Q8—81no—
281D4—22no—
282D4—21no—
283D4—21no—
284D4—42no—
285D4—81no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4C2×D4×C10911yescharacteristic
302C2×C4C2×D4×C10911yescharacteristic
303C2×C4C2×D4×C10911yescharacteristic
304C2×C4C2×D4×C10911yescharacteristic
305C2×C4C24×C10911yescharacteristic
306C8—21no—
307C8—21no—
308C8—21no—
309C8#41×C10912yes—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22#137×C10912yes—
319C22C22×D4×C10911yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4C2×#41×C10911yescharacteristic
332C42−1+4×C10911yescharacteristic
333C2—22no—
334C2—41no—
335C2—81no—
336C2#547×C10911yescharacteristic
337C2C2×#137×C10911yescharacteristic
338C2#576×C10911yescharacteristic
subgroupG/H
center, socleC22×C109C22×D4
commutator, Frattini, cosocleC2×C4C24×C109
prime radical#553×C109C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C4×D4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
1. #554×C109C2semidirect
3. #501×C109C2semidirect
4. #485×C109C2semidirect
5. #484×C109C2semidirect
6. #481×C109C2semidirect
7. #480×C109C2semidirect
8. #479×C109C2semidirect
9. #474×C109C2semidirect
10. #468×C109C2semidirect
11. #465×C109C2semidirect
12. #440×C109C2semidirect
13. #435×C109C2semidirect
14. #428×C109C2semidirect
15. C2×#107×C109C2semidirect
169. #3346C109direct (N × H)
170. C109#3346direct (N × H)
171. #554C2×C109semidirect
173. #501C2×C109semidirect
174. #485C2×C109semidirect
175. #484C2×C109semidirect
176. #481C2×C109semidirect
177. #480C2×C109semidirect
178. #479C2×C109semidirect
179. #474C2×C109semidirect
180. #468C2×C109semidirect
181. #465C2×C109semidirect
182. #440C2×C109semidirect
183. #435C2×C109semidirect
184. #428C2×C109semidirect
185. C2×#107C2×C109semidirect

Maximal subgroups

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

subgroupindex
#554×C1092
#553×C1092
#501×C1092
#485×C1092
#484×C1092
#481×C1092
#480×C1092
#479×C1092
#474×C1092
#468×C1092
#465×C1092
#440×C1092
#435×C1092
#428×C1092
C2×#107×C1092
#3346109