Finite Group Databasebeta

← C109×#12273 #42130267

Subgroups — C109×#12273 #42130267

Order 27904.

364 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2586C211yesmaximal, characteristic
2C109×#2028C211yesmaximal, characteristic
3C109×#1611C211yesmaximal, characteristic
4C109×#1605C211yesmaximal, characteristic
5C109×#1354C211yesmaximal, characteristic
6C109×#1346C211yesmaximal, characteristic
7C109×#1319C211yesmaximal, characteristic
8#544×C109C2211yescharacteristic
9#483×C109—21no—
10#451×C109—21no—
11#419×C109—21no—
12#398×C109C2211yescharacteristic
13#395×C109—21no—
14#395×C109C2211yescharacteristic
15#391×C109—21no—
16#391×C109C2211yescharacteristic
17#389×C109—21no—
18#387×C109—21no—
19#387×C109—21no—
20C4×C4⋊C4×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
28C22×Q8×C109D411yescharacteristic
29C2×Q16×C109—41no—
30C2×M16×C109—21no—
31C4⋊Q8×C109—41no—
32#125×C109—41no—
33C22⋊Q8×C109—21no—
34C22⋊Q8×C109—21no—
35#117×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×C109C2311yescharacteristic
43C2×C42×C109—21no—
44C2×C42×C109—21no—
45C2×C42×C109D411yescharacteristic
46#106×C109—41no—
47#103×C109—41no—
48#103×C109—41no—
49#103×C109—41no—
50#103×C109—41no—
51C22⋊C8×C109—21no—
52C22⋊C8×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—21no—
65C2×Q8×C109—21no—
66C2×Q8×C109—42no—
67C2×Q8×C109—41no—
68C2×Q8×C109—41no—
69C2×Q8×C109—41no—
70C22×C4×C109—21no—
71C22×C4×C109—21no—
72C22×C4×C109—21no—
73C22×C4×C109—21no—
74C22×C4×C109—21no—
75C22×C4×C109—21no—
76C22×C4×C109C2×D411yescharacteristic
77C22×C4×C109C2×D411yescharacteristic
78C22×C4×C109C2×D411yescharacteristic
79Q16×C109—82no—
80M16×C109—41no—
81M16×C109—41no—
82C2×C8×C109—41no—
83C2×C8×C109—41no—
84C2×C8×C109—41no—
85C4⋊C4×C109—24no—
86C4⋊C4×C109—21no—
87C4⋊C4×C109—21no—
88C4⋊C4×C109—21no—
89C4⋊C4×C109—21no—
90C4⋊C4×C109—41no—
91C4⋊C4×C109—41no—
92C4⋊C4×C109—42no—
93C4⋊C4×C109—41no—
94C4⋊C4×C109—41no—
95C4⋊C4×C109—41no—
96C4⋊C4×C109—41no—
97C4⋊C4×C109—81no—
98C4⋊C4×C109—81no—
99C22⋊C4×C109—41no—
100C42×C109—41no—
101C42×C109—42no—
102C42×C109—41no—
103C42×C109—81no—
104C23×C109#12011yescharacteristic
105Q8×C109—42no—
106Q8×C109—42no—
107Q8×C109—81no—
108Q8×C109—81no—
109Q8×C109—81no—
110Q8×C109—81no—
111C2×C4×C109—21no—
112C2×C4×C109—21no—
113C2×C4×C109—22no—
114C2×C4×C109—22no—
115C2×C4×C109—22no—
116C2×C4×C109—22no—
117C2×C4×C109—21no—
118C2×C4×C109—21no—
119C2×C4×C109—41no—
120C2×C4×C109—41no—
121C2×C4×C109—41no—
122C2×C4×C109—41no—
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×C109—41no—
131C2×C4×C109—41no—
132C2×C4×C109—41no—
133C2×C4×C109—41no—
134C2×C4×C109—41no—
135C2×C4×C109—41no—
136C2×C4×C109—41no—
137C2×C4×C109#13511yescharacteristic
138C2×C4×C109#13711yescharacteristic
139C8×C109—81no—
140C8×C109—81no—
141C8×C109—81no—
142C22×C109—21no—
143C22×C109—21no—
144C22×C109#45211yescharacteristic
145C22×C109#45511yescharacteristic
146C22×C109B(4,𝔽2)11yescharacteristic
147C4×C109—21no—
148C4×C109—21no—
149C4×C109—42no—
150C4×C109—42no—
151C4×C109—42no—
152C4×C109—41no—
153C4×C109—41no—
154C4×C109—42no—
155C4×C109—81no—
156C4×C109—81no—
157C4×C109—81no—
158C4×C109—81no—
159#12273C10911yesmaximal, characteristic, direct product factor, Sylow
160C2×C109—21no—
161C2×C109—21no—
162C2×C109#160611yescharacteristic
163C2×C109#219711yescharacteristic
164C2×C109#220511yescharacteristic
165#2586C2×C10911yescharacteristic
166#2028C2×C10911yescharacteristic
167#1611C2×C10911yescharacteristic
168#1605C2×C10911yescharacteristic
169#1354C2×C10911yescharacteristic
170#1346C2×C10911yescharacteristic
171#1319C2×C10911yescharacteristic
172C109#1227311yescharacteristic, direct product factor, Sylow
173#544C22×C10911yescharacteristic
174#483—21no—
175#451—21no—
176#419—21no—
177#398C22×C10911yescharacteristic
178#395—21no—
179#395C22×C10911yescharacteristic
180#391—21no—
181#391C22×C10911yescharacteristic
182#389—21no—
183#387—21no—
184#387—21no—
185C4×C4⋊C4—21no—
186#343C22×C10911yescharacteristic
187#339—21no—
188#328—21no—
189#328—21no—
190#324C22×C10911yescharacteristic
191#324C22×C10911yescharacteristic
192C22×Q8D4×C10911yescharacteristic
193C22×Q8D4×C10911yescharacteristic
194C2×Q16—41no—
195C2×M16—21no—
196C4⋊Q8—41no—
197#125—41no—
198C22⋊Q8—21no—
199C22⋊Q8—21no—
200#117—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4D4×C10911yescharacteristic
207C2×C4⋊C4C23×C10911yescharacteristic
208C2×C42—21no—
209C2×C42—21no—
210C2×C42D4×C10911yescharacteristic
211#106—41no—
212#103—41no—
213#103—41no—
214#103—41no—
215#103—41no—
216C22⋊C8—21no—
217C22⋊C8—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95D4×C10911yescharacteristic
226#95D4×C10911yescharacteristic
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—42no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4C2×D4×C10911yescharacteristic
242C22×C4C2×D4×C10911yescharacteristic
243C22×C4C2×D4×C10911yescharacteristic
244Q16—82no—
245M16—41no—
246M16—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C4⋊C4—24no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—81no—
263C4⋊C4—81no—
264C22⋊C4—41no—
265C42—41no—
266C42—42no—
267C42—41no—
268C42—81no—
269C23#120×C10911yescharacteristic
270Q8—42no—
271Q8—42no—
272Q8—81no—
273Q8—81no—
274Q8—81no—
275Q8—81no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4#135×C10911yescharacteristic
303C2×C4#137×C10911yescharacteristic
304C8—81no—
305C8—81no—
306C8—81no—
307C22—21no—
308C22—21no—
309C22#452×C10911yescharacteristic
310C22#455×C10911yescharacteristic
311C22B(4,𝔽2)×C10911yescharacteristic
312C4—21no—
313C4—21no—
314C4—42no—
315C4—42no—
316C4—42no—
317C4—41no—
318C4—41no—
319C4—42no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4—81no—
324C2—21no—
325C2—21no—
326C2C109×#160611yescharacteristic
327C2C109×#219711yescharacteristic
328C2C109×#220511yescharacteristic
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
159. #12273C109direct (N × H)
172. C109#12273direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#25862
C109×#20282
C109×#16112
C109×#16052
C109×#13542
C109×#13462
C109×#13192
#12273109