Finite Group Databasebeta

← C109×#12259 #42130253

Subgroups — C109×#12259 #42130253

Order 27904.

332 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2843C211yesmaximal, characteristic
2C109×#2060C211yesmaximal, characteristic
3C109×#1617C211yesmaximal, characteristic
4C109×#1605C211yesmaximal, characteristic
5C109×#1354C211yesmaximal, characteristic
6C109×#1353C211yesmaximal, characteristic
7C109×#1325C211yesmaximal, characteristic
8#544×C109C2211yescharacteristic
9#484×C109—21no—
10#451×C109—21no—
11#428×C109—21no—
12#400×C109—21no—
13#400×C109C2211yescharacteristic
14#398×C109—21no—
15#398×C109—21no—
16#398×C109—21no—
17#395×C109—21no—
18#395×C109C2211yescharacteristic
19#384×C109C2211yescharacteristic
20#383×C109—21no—
21#343×C109C2211yescharacteristic
22#339×C109—21no—
23#328×C109—21no—
24#328×C109—21no—
25#324×C109C2211yescharacteristic
26#324×C109C2211yescharacteristic
27C22×Q8×C109D411yescharacteristic
28C2×Q16×C109—41no—
29C2×M16×C109—21no—
30C4⋊Q8×C109—41no—
31#125×C109—41no—
32C22⋊Q8×C109—21no—
33C22⋊Q8×C109—21no—
34#117×C109—21no—
35C2×C4⋊C4×C109—21no—
36C2×C4⋊C4×C109—21no—
37C2×C4⋊C4×C109—21no—
38C2×C4⋊C4×C109—21no—
39C2×C4⋊C4×C109—21no—
40C2×C4⋊C4×C109—21no—
41C2×C4⋊C4×C109D411yescharacteristic
42C2×C4⋊C4×C109D411yescharacteristic
43C2×C4⋊C4×C109C2311yescharacteristic
44C2×C42×C109D411yescharacteristic
45#107×C109—41no—
46#107×C109—41no—
47#106×C109—41no—
48#103×C109—41no—
49#103×C109—41no—
50C22⋊C8×C109—21no—
51C22⋊C8×C109—21no—
52#95×C109—21no—
53#95×C109—21no—
54#95×C109—21no—
55#95×C109—21no—
56#95×C109—21no—
57#95×C109—21no—
58#95×C109—21no—
59#95×C109—21no—
60#95×C109D411yescharacteristic
61#95×C109D411yescharacteristic
62C2×Q8×C109—21no—
63C2×Q8×C109—21no—
64C2×Q8×C109—41no—
65C2×Q8×C109—81no—
66C22×C4×C109—21no—
67C22×C4×C109—21no—
68C22×C4×C109—21no—
69C22×C4×C109—21no—
70C22×C4×C109—21no—
71C22×C4×C109—21no—
72C22×C4×C109C2×D411yescharacteristic
73C22×C4×C109C2×D411yescharacteristic
74C22×C4×C109C2×D411yescharacteristic
75Q16×C109—82no—
76M16×C109—22no—
77M16×C109—41no—
78C2×C8×C109—41no—
79C2×C8×C109—41no—
80C2×C8×C109—41no—
81C4⋊C4×C109—21no—
82C4⋊C4×C109—21no—
83C4⋊C4×C109—21no—
84C4⋊C4×C109—21no—
85C4⋊C4×C109—41no—
86C4⋊C4×C109—41no—
87C4⋊C4×C109—41no—
88C4⋊C4×C109—41no—
89C4⋊C4×C109—41no—
90C4⋊C4×C109—81no—
91C4⋊C4×C109—81no—
92C4⋊C4×C109—81no—
93C4⋊C4×C109—81no—
94C4⋊C4×C109—81no—
95C4⋊C4×C109—81no—
96C22⋊C4×C109—41no—
97C42×C109—21no—
98C42×C109—41no—
99C42×C109Q1612yescomplemented
100C23×C109#12011yescharacteristic
101Q8×C109—42no—
102Q8×C109—81no—
103Q8×C109—81no—
104Q8×C109—81no—
105C2×C4×C109—21no—
106C2×C4×C109—21no—
107C2×C4×C109—22no—
108C2×C4×C109—21no—
109C2×C4×C109—21no—
110C2×C4×C109—41no—
111C2×C4×C109—41no—
112C2×C4×C109—41no—
113C2×C4×C109—41no—
114C2×C4×C109—41no—
115C2×C4×C109—41no—
116C2×C4×C109—41no—
117C2×C4×C109—41no—
118C2×C4×C109—41no—
119C2×C4×C109—41no—
120C2×C4×C109—41no—
121C2×C4×C109—41no—
122C2×C4×C109—42no—
123C2×C4×C109—41no—
124C2×C4×C109—41no—
125C2×C4×C109—41no—
126C2×C4×C109—41no—
127C2×C4×C109—41no—
128C2×C4×C109—41no—
129C2×C4×C109—41no—
130C2×C4×C109C2×Q1611yescharacteristic
131C2×C4×C109#13711yescharacteristic
132C8×C109—42no—
133C8×C109—81no—
134C8×C109—81no—
135C22×C109—21no—
136C22×C109—21no—
137C22×C109#45111yescharacteristic
138C22×C109#45611yescharacteristic
139C22×C109B(4,𝔽2)11yescharacteristic
140C4×C109—21no—
141C4×C109—21no—
142C4×C109—42no—
143C4×C109—41no—
144C4×C109—41no—
145C4×C109—81no—
146C4×C109—81no—
147C4×C109—81no—
148C4×C109—81no—
149C4×C109—81no—
150C4×C109—81no—
151C4×C109—81no—
152#12259C10911yesmaximal, characteristic, direct product factor, Sylow
153C2×C109—21no—
154C2×C109—21no—
155C2×C109#160511yescharacteristic
156C2×C109#220011yescharacteristic
157C2×C109#220311yescharacteristic
158#2843C2×C10911yescharacteristic
159#2060C2×C10911yescharacteristic
160#1617C2×C10911yescharacteristic
161#1605C2×C10911yescharacteristic
162#1354C2×C10911yescharacteristic
163#1353C2×C10911yescharacteristic
164#1325C2×C10911yescharacteristic
165C109#1225911yescharacteristic, direct product factor, Sylow
166#544C22×C10911yescharacteristic
167#484—21no—
168#451—21no—
169#428—21no—
170#400—21no—
171#400C22×C10911yescharacteristic
172#398—21no—
173#398—21no—
174#398—21no—
175#395—21no—
176#395C22×C10911yescharacteristic
177#384C22×C10911yescharacteristic
178#383—21no—
179#343C22×C10911yescharacteristic
180#339—21no—
181#328—21no—
182#328—21no—
183#324C22×C10911yescharacteristic
184#324C22×C10911yescharacteristic
185C22×Q8D4×C10911yescharacteristic
186C2×Q16—41no—
187C2×M16—21no—
188C4⋊Q8—41no—
189#125—41no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192#117—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4D4×C10911yescharacteristic
200C2×C4⋊C4D4×C10911yescharacteristic
201C2×C4⋊C4C23×C10911yescharacteristic
202C2×C42D4×C10911yescharacteristic
203#107—41no—
204#107—41no—
205#106—41no—
206#103—41no—
207#103—41no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95D4×C10911yescharacteristic
219#95D4×C10911yescharacteristic
220C2×Q8—21no—
221C2×Q8—21no—
222C2×Q8—41no—
223C2×Q8—81no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4C2×D4×C10911yescharacteristic
231C22×C4C2×D4×C10911yescharacteristic
232C22×C4C2×D4×C10911yescharacteristic
233Q16—82no—
234M16—22no—
235M16—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—81no—
249C4⋊C4—81no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C4⋊C4—81no—
253C4⋊C4—81no—
254C22⋊C4—41no—
255C42—21no—
256C42—41no—
257C42Q16×C10912yescomplemented
258C23#120×C10911yescharacteristic
259Q8—42no—
260Q8—81no—
261Q8—81no—
262Q8—81no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4C2×Q16×C10911yescharacteristic
289C2×C4#137×C10911yescharacteristic
290C8—42no—
291C8—81no—
292C8—81no—
293C22—21no—
294C22—21no—
295C22#451×C10911yescharacteristic
296C22#456×C10911yescharacteristic
297C22B(4,𝔽2)×C10911yescharacteristic
298C4—21no—
299C4—21no—
300C4—42no—
301C4—41no—
302C4—41no—
303C4—81no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4—81no—
308C4—81no—
309C4—81no—
310C2—21no—
311C2—21no—
312C2C109×#160511yescharacteristic
313C2C109×#220011yescharacteristic
314C2C109×#220311yescharacteristic
subgroupG/H
center, socleC22×C109B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C109
prime radicalC23×C109#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C23×C109 ◁ C2×C4⋊C4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
99. C42×C109Q16semidirect
152. #12259C109direct (N × H)
165. C109#12259direct (N × H)
257. C42Q16×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#28432
C109×#20602
C109×#16172
C109×#16052
C109×#13542
C109×#13532
C109×#13252
#12259109