Finite Group Databasebeta

← C109×#32708 #42150702

Subgroups — C109×#32708 #42150702

Order 27904.

1062 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1062 conjugacy classes of subgroups — too many to draw. Only its 226 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#2848C211yesmaximal, characteristic, complemented
2C109×#2847C211yesmaximal, characteristic
3C109×#2077C213yesmaximal, complemented
4C109×#2035C213yesmaximal, complemented
5C109×#2034C213yesmaximal, complemented
6C109×#2000C213yesmaximal, complemented
7C109×#1779C211yesmaximal, characteristic, complemented
8C24×C4×C109C2211yescharacteristic
9C22×SD16×C109C2211yescharacteristic, complemented
10C2×C22⋊Q8×C109C2213yes—
11C2×#121×C109C2213yes—
12C2×#120×C109—21no—
13C2×#106×C109C2213yescomplemented
14C2×#103×C109C2213yescomplemented
15C2×#102×C109C2213yescomplemented
16C2×C22⋊C8×C109C2213yes—
17C2×C22⋊C8×C109C2213yes—
18#386×C109—23no—
19#385×C109—23no—
20#380×C109C2213yes—
21#380×C109C2213yes—
22#379×C109—21no—
23#340×C109C2213yescomplemented
24#340×C109C2213yescomplemented
25C25×C109D411yescharacteristic
26C22×Q8×C109C2311yescharacteristic
27C22×D4×C109—23no—
28C22×D4×C109C2311yescharacteristic
29C23×C4×C109—43no—
30C23×C4×C109—43no—
31C23×C4×C109D413yes—
32C23×C4×C109C2313yes—
33C2×SD16×C109—43no—
34C22×C8×C109C2311yescharacteristic
35C22×C8×C109C2313yes—
36C22⋊Q8×C109—43no—
37C22⋊Q8×C109—43no—
38#121×C109—43no—
39#121×C109—43no—
40#120×C109—44no—
41C2×C4⋊C4×C109—23no—
42C2×C4⋊C4×C109C2313yes—
43C2×C4⋊C4×C109C2313yes—
44C2×C22⋊C4×C109—23no—
45C2×C22⋊C4×C109—23no—
46C2×C22⋊C4×C109—23no—
47C2×C22⋊C4×C109—23no—
48#106×C109—43no—
49#103×C109—43no—
50#102×C109—43no—
51C22⋊C8×C109—43no—
52C22⋊C8×C109—43no—
53#95×C109—23no—
54#95×C109—23no—
55C24×C109—21no—
56C24×C109—43no—
57C24×C109—43no—
58C24×C109SD1614yescomplemented
59C24×C109C2×D413yes—
60C2×Q8×C109—23no—
61C2×Q8×C109—23no—
62C2×D4×C109—23no—
63C2×D4×C109—23no—
64C2×D4×C109—46no—
65C2×D4×C109—46no—
66C2×D4×C109—46no—
67C22×C4×C109—23no—
68C22×C4×C109—23no—
69C22×C4×C109—23no—
70C22×C4×C109—23no—
71C22×C4×C109—23no—
72C22×C4×C109—23no—
73C22×C4×C109—23no—
74C22×C4×C109—23no—
75C22×C4×C109—21no—
76C22×C4×C109—43no—
77C22×C4×C109—43no—
78C22×C4×C109—43no—
79C22×C4×C109—412no—
80C22×C4×C109—44no—
81C22×C4×C109C2×D413yes—
82C22×C4×C109C2411yescharacteristic
83SD16×C109—44no—
84C2×C8×C109—23no—
85C2×C8×C109—23no—
86C2×C8×C109—23no—
87C2×C8×C109—23no—
88C4⋊C4×C109—23no—
89C4⋊C4×C109—23no—
90C4⋊C4×C109—23no—
91C4⋊C4×C109—23no—
92C4⋊C4×C109—46no—
93C22⋊C4×C109—46no—
94C22⋊C4×C109—46no—
95C22⋊C4×C109—46no—
96C22⋊C4×C109—46no—
97C23×C109—23no—
98C23×C109—23no—
99C23×C109—23no—
100C23×C109—23no—
101C23×C109—23no—
102C23×C109—23no—
103C23×C109—412no—
104C23×C109—44no—
105C23×C109—412no—
106C23×C109—43no—
107C23×C109—41no—
108C23×C109—43no—
109C23×C109C2×SD1616yes—
110C23×C109C22×D411yescharacteristic
111Q8×C109—43no—
112Q8×C109—41no—
113D4×C109—43no—
114D4×C109—41no—
115D4×C109—812no—
116C2×C4×C109—23no—
117C2×C4×C109—23no—
118C2×C4×C109—23no—
119C2×C4×C109—412no—
120C2×C4×C109—43no—
121C2×C4×C109—43no—
122C2×C4×C109—43no—
123C2×C4×C109—43no—
124C2×C4×C109—43no—
125C2×C4×C109—43no—
126C2×C4×C109—43no—
127C2×C4×C109—43no—
128C2×C4×C109—46no—
129C2×C4×C109—43no—
130C2×C4×C109H(5,𝔽2)13yes—
131C2×C4×C109H(5,𝔽2)13yes—
132C8×C109—43no—
133C8×C109—41no—
134C22×C109—26no—
135C22×C109—26no—
136C22×C109—26no—
137C22×C109—416no—
138C22×C109—43no—
139C22×C109—43no—
140C22×C109—43no—
141C22×C109—43no—
142C22×C109—83no—
143C22×C109—83no—
144C22×C109—81no—
145C22×C109#53413yes—
146C22×C109C22×SD1611yescharacteristic
147C22×C109#57513yes—
148C4×C109—43no—
149C4×C109—83no—
150C4×C109—83no—
151C4×C109—81no—
152C4×C109#56013yes—
153C4×C109H(3,𝔽4)11yescharacteristic
154#32708C10911yesmaximal, characteristic, direct product factor, Sylow
155C2×C109—46no—
156C2×C109—161no—
157C2×C109#281211yescharacteristic
158C2×C109#318713yes—
159C2×C109#319713yes—
160#2848C2×C10911yescharacteristic, complemented
161#2847C2×C10911yescharacteristic
162#2077C2×C10913yescomplemented
163#2035C2×C10913yescomplemented
164#2034C2×C10913yescomplemented
165#2000C2×C10913yescomplemented
166#1779C2×C10911yescharacteristic, complemented
167C109#3270811yescharacteristic, direct product factor, Sylow
168C24×C4C22×C10911yescharacteristic
169C22×SD16C22×C10911yescharacteristic, complemented
170C2×C22⋊Q8C22×C10913yes—
171C2×#121C22×C10913yes—
172C2×#120—21no—
173C2×#106C22×C10913yescomplemented
174C2×#103C22×C10913yescomplemented
175C2×#102C22×C10913yescomplemented
176C2×C22⋊C8C22×C10913yes—
177C2×C22⋊C8C22×C10913yes—
178#386—23no—
179#385—23no—
180#380C22×C10913yes—
181#380C22×C10913yes—
182#379—21no—
183#340C22×C10913yescomplemented
184#340C22×C10913yescomplemented
185C25D4×C10911yescharacteristic
186C22×Q8C23×C10911yescharacteristic
187C22×D4—23no—
188C22×D4C23×C10911yescharacteristic
189C23×C4—43no—
190C23×C4—43no—
191C23×C4D4×C10913yes—
192C23×C4C23×C10913yes—
193C2×SD16—43no—
194C22×C8C23×C10911yescharacteristic
195C22×C8C23×C10913yes—
196C22⋊Q8—43no—
197C22⋊Q8—43no—
198#121—43no—
199#121—43no—
200#120—44no—
201C2×C4⋊C4—23no—
202C2×C4⋊C4C23×C10913yes—
203C2×C4⋊C4C23×C10913yes—
204C2×C22⋊C4—23no—
205C2×C22⋊C4—23no—
206C2×C22⋊C4—23no—
207C2×C22⋊C4—23no—
208#106—43no—
209#103—43no—
210#102—43no—
211C22⋊C8—43no—
212C22⋊C8—43no—
213#95—23no—
214#95—23no—
215C24—21no—
216C24—43no—
217C24—43no—
218C24SD16×C10914yescomplemented
219C24C2×D4×C10913yes—
220C2×Q8—23no—
221C2×Q8—23no—
222C2×D4—23no—
223C2×D4—23no—
224C2×D4—46no—
225C2×D4—46no—
226C2×D4—46no—
227C22×C4—23no—
228C22×C4—23no—
229C22×C4—23no—
230C22×C4—23no—
231C22×C4—23no—
232C22×C4—23no—
233C22×C4—23no—
234C22×C4—23no—
235C22×C4—21no—
236C22×C4—43no—
237C22×C4—43no—
238C22×C4—43no—
239C22×C4—412no—
240C22×C4—44no—
241C22×C4C2×D4×C10913yes—
242C22×C4C24×C10911yescharacteristic
243SD16—44no—
244C2×C8—23no—
245C2×C8—23no—
246C2×C8—23no—
247C2×C8—23no—
248C4⋊C4—23no—
249C4⋊C4—23no—
250C4⋊C4—23no—
251C4⋊C4—23no—
252C4⋊C4—46no—
253C22⋊C4—46no—
254C22⋊C4—46no—
255C22⋊C4—46no—
256C22⋊C4—46no—
257C23—23no—
258C23—23no—
259C23—23no—
260C23—23no—
261C23—23no—
262C23—23no—
263C23—412no—
264C23—44no—
265C23—412no—
266C23—43no—
267C23—41no—
268C23—43no—
269C23C2×SD16×C10916yes—
270C23C22×D4×C10911yescharacteristic
271Q8—43no—
272Q8—41no—
273D4—43no—
274D4—41no—
275D4—812no—
276C2×C4—23no—
277C2×C4—23no—
278C2×C4—23no—
279C2×C4—412no—
280C2×C4—43no—
281C2×C4—43no—
282C2×C4—43no—
283C2×C4—43no—
284C2×C4—43no—
285C2×C4—43no—
286C2×C4—43no—
287C2×C4—43no—
288C2×C4—46no—
289C2×C4—43no—
290C2×C4H(5,𝔽2)×C10913yes—
291C2×C4H(5,𝔽2)×C10913yes—
292C8—43no—
293C8—41no—
294C22—26no—
295C22—26no—
296C22—26no—
297C22—416no—
298C22—43no—
299C22—43no—
300C22—43no—
301C22—43no—
302C22—83no—
303C22—81no—
304C22—83no—
305C22#534×C10913yes—
306C22C22×SD16×C10911yescharacteristic
307C22#575×C10913yes—
308C4—43no—
309C4—83no—
310C4—81no—
311C4—83no—
312C4#560×C10913yes—
313C4H(3,𝔽4)×C10911yescharacteristic
314C2—46no—
315C2—161no—
316C2C109×#281211yescharacteristic
317C2C109×#318713yes—
318C2C109×#319713yes—
subgroupG/H
center, socleC23×C109C22×D4
commutator, Frattini, cosocleC22×C4C24×C109
prime radicalC109×#2847C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C109 ◁ C24×C4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
1. C109×#2848C2semidirect
3. C109×#2077C2semidirect
4. C109×#2035C2semidirect
5. C109×#2034C2semidirect
6. C109×#2000C2semidirect
7. C109×#1779C2semidirect
9. C22×SD16×C109C22semidirect
13. C2×#106×C109C22semidirect
14. C2×#103×C109C22semidirect
15. C2×#102×C109C22semidirect
23. #340×C109C22semidirect
24. #340×C109C22semidirect
58. C24×C109SD16semidirect
154. #32708C109direct (N × H)
160. #2848C2×C109semidirect
162. #2077C2×C109semidirect
163. #2035C2×C109semidirect
164. #2034C2×C109semidirect
165. #2000C2×C109semidirect
166. #1779C2×C109semidirect
167. C109#32708direct (N × H)
169. C22×SD16C22×C109semidirect
173. C2×#106C22×C109semidirect
174. C2×#103C22×C109semidirect
175. C2×#102C22×C109semidirect
183. #340C22×C109semidirect
184. #340C22×C109semidirect
218. C24SD16×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#28482
C109×#28472
C109×#20772
C109×#20352
C109×#20342
C109×#20002
C109×#17792
#32708109