Finite Group Databasebeta

← C109×#12764 #42130758

Subgroups — C109×#12764 #42130758

Order 27904.

768 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 142 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
1C2×#563×C109C211yesmaximal, characteristic
2C2×#354×C109C211yesmaximal, characteristic
3C2×#352×C109C212yesmaximal
4C109×#1890C212yesmaximal
5C109×#1885C211yesmaximal, characteristic
6#563×C109—22no—
7#563×C109—24no—
8#563×C109C414yescomplemented
9C2×C4⋊Q8×C109—21no—
10C2×#126×C109—22no—
11C2×#125×C109—21no—
12C2×#124×C109C2211yescharacteristic
13C2×#123×C109C2212yes—
14C2×C22⋊Q8×C109—22no—
15#518×C109C2211yescharacteristic
16#419×C109—21no—
17C2×#100×C109C2211yescharacteristic
18C2×#99×C109C2212yes—
19#381×C109—22no—
20#354×C109—22no—
21#352×C109—24no—
22#342×C109—22no—
23#339×C109—21no—
24C22×Q8×C109—21no—
25C22×Q8×C109D411yescharacteristic
26C22×D4×C109C2311yescharacteristic
27C23×C4×C109D412yes—
28C2×M16×C109—21no—
29C4⋊Q8×C109—24no—
30C4⋊Q8×C109—24no—
31#126×C109—28no—
32#126×C109—28no—
33#125×C109—24no—
34#125×C109—24no—
35#124×C109—22no—
36#124×C109C2×C412yes—
37#124×C109D412yes—
38#123×C109—22no—
39#123×C109—22no—
40#123×C109C2×C414yes—
41#123×C109D414yes—
42C22⋊Q8×C109—28no—
43C22⋊Q8×C109—28no—
44C4×D4×C109—42no—
45#117×C109—21no—
46C2×C4⋊C4×C109—21no—
47C2×C4⋊C4×C109—22no—
48C2×C4⋊C4×C109—22no—
49C2×C4⋊C4×C109—21no—
50C2×C4⋊C4×C109—22no—
51C2×C22⋊C4×C109—22no—
52C2×C22⋊C4×C109—22no—
53C2×C22⋊C4×C109—22no—
54C2×C22⋊C4×C109—22no—
55C2×C22⋊C4×C109D412yes—
56C2×C42×C109—21no—
57C2×C42×C109D411yescharacteristic
58#103×C109—42no—
59#100×C109—22no—
60#99×C109—22no—
61#99×C109—22no—
62#95×C109—44no—
63C24×C109C2×D412yes—
64C2×Q8×C109—24no—
65C2×Q8×C109—21no—
66C2×Q8×C109—41no—
67C2×Q8×C109—41no—
68C2×Q8×C109—81no—
69C2×Q8×C109—81no—
70C2×Q8×C109C22⋊C412yes—
71C2×D4×C109—41no—
72C2×D4×C109—41no—
73C2×D4×C109C22×C411yescharacteristic
74C2×D4×C109C2×D411yescharacteristic
75C2×D4×C109C2×D412yes—
76C22×C4×C109—21no—
77C22×C4×C109—22no—
78C22×C4×C109—21no—
79C22×C4×C109—22no—
80C22×C4×C109—22no—
81C22×C4×C109—24no—
82C22×C4×C109—42no—
83C22×C4×C109—42no—
84C22×C4×C109—42no—
85C22×C4×C109C22⋊C414yes—
86C22×C4×C109C2×D411yescharacteristic
87M16×C109—41no—
88M16×C109—41no—
89C2×C8×C109—41no—
90C4⋊C4×C109—24no—
91C4⋊C4×C109—28no—
92C4⋊C4×C109—28no—
93C4⋊C4×C109—24no—
94C4⋊C4×C109—24no—
95C4⋊C4×C109—24no—
96C4⋊C4×C109—22no—
97C22⋊C4×C109—24no—
98C22⋊C4×C109—24no—
99C22⋊C4×C109—24no—
100C22⋊C4×C109—24no—
101C22⋊C4×C109—22no—
102C22⋊C4×C109—42no—
103C22⋊C4×C109—41no—
104C22⋊C4×C109—42no—
105C22⋊C4×C109—42no—
106C22⋊C4×C109C22⋊C414yes—
107C42×C109—24no—
108C42×C109—21no—
109C42×C109—41no—
110C42×C109C22⋊C412yes—
111C23×C109—22no—
112C23×C109—22no—
113C23×C109—44no—
114C23×C109C2×C22⋊C412yes—
115C23×C109#12012yes—
116C23×C109#12011yescharacteristic
117Q8×C109—42no—
118Q8×C109—81no—
119Q8×C109—84no—
120D4×C109—41no—
121D4×C109—42no—
122D4×C109—41no—
123C2×C4×C109—21no—
124C2×C4×C109—21no—
125C2×C4×C109—22no—
126C2×C4×C109—24no—
127C2×C4×C109—22no—
128C2×C4×C109—24no—
129C2×C4×C109—44no—
130C2×C4×C109—42no—
131C2×C4×C109—42no—
132C2×C4×C109—41no—
133C2×C4×C109—42no—
134C2×C4×C109—41no—
135C2×C4×C109—44no—
136C2×C4×C109—42no—
137C2×C4×C109—82no—
138C2×C4×C109—84no—
139C2×C4×C109—82no—
140C2×C4×C109—84no—
141C2×C4×C109C2×C22⋊C411yescharacteristic
142C2×C4×C109#12011yescharacteristic
143C8×C109—81no—
144C22×C109—22no—
145C22×C109—22no—
146C22×C109—22no—
147C22×C109—42no—
148C22×C109—42no—
149C22×C109—44no—
150C22×C109—82no—
151C22×C109#37911yescharacteristic
152C22×C109B(4,𝔽2)11yescharacteristic
153C22×C109#45811yescharacteristic
154C4×C109—21no—
155C4×C109—21no—
156C4×C109—44no—
157C4×C109—42no—
158C4×C109—44no—
159C4×C109—42no—
160C4×C109—82no—
161C4×C109—162no—
162#12764C10911yesmaximal, characteristic, direct product factor, Sylow
163C2×C109—21no—
164C2×C109—21no—
165C2×C109—42no—
166C2×C109—42no—
167C2×C109#179011yescharacteristic
168C2×C109#220112yes—
169C2×#563C2×C10911yescharacteristic
170C2×#354C2×C10911yescharacteristic
171C2×#352C2×C10912yes—
172#1890C2×C10912yes—
173#1885C2×C10911yescharacteristic
174C109#1276411yescharacteristic, direct product factor, Sylow
175#563—22no—
176#563—24no—
177#563C4×C10914yescomplemented
178C2×C4⋊Q8—21no—
179C2×#126—22no—
180C2×#125—21no—
181C2×#124C22×C10911yescharacteristic
182C2×#123C22×C10912yes—
183C2×C22⋊Q8—22no—
184#518C22×C10911yescharacteristic
185#419—21no—
186C2×#100C22×C10911yescharacteristic
187C2×#99C22×C10912yes—
188#381—22no—
189#354—22no—
190#352—24no—
191#342—22no—
192#339—21no—
193C22×Q8—21no—
194C22×Q8D4×C10911yescharacteristic
195C22×D4C23×C10911yescharacteristic
196C23×C4D4×C10912yes—
197C2×M16—21no—
198C4⋊Q8—24no—
199C4⋊Q8—24no—
200#126—28no—
201#126—28no—
202#125—24no—
203#125—24no—
204#124—22no—
205#124C2×C4×C10912yes—
206#124D4×C10912yes—
207#123—22no—
208#123—22no—
209#123C2×C4×C10914yes—
210#123D4×C10914yes—
211C22⋊Q8—28no—
212C22⋊Q8—28no—
213C4×D4—42no—
214#117—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4D4×C10912yes—
225C2×C42—21no—
226C2×C42D4×C10911yescharacteristic
227#103—42no—
228#100—22no—
229#99—22no—
230#99—22no—
231#95—44no—
232C24C2×D4×C10912yes—
233C2×Q8—24no—
234C2×Q8—21no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—81no—
238C2×Q8—81no—
239C2×Q8C22⋊C4×C10912yes—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4C22×C4×C10911yescharacteristic
243C2×D4C2×D4×C10911yescharacteristic
244C2×D4C2×D4×C10912yes—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4C22⋊C4×C10914yes—
255C22×C4C2×D4×C10911yescharacteristic
256M16—41no—
257M16—41no—
258C2×C8—41no—
259C4⋊C4—24no—
260C4⋊C4—28no—
261C4⋊C4—28no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—22no—
271C22⋊C4—42no—
272C22⋊C4—41no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4C22⋊C4×C10914yes—
276C42—24no—
277C42—21no—
278C42—41no—
279C42C22⋊C4×C10912yes—
280C23—22no—
281C23—22no—
282C23—44no—
283C23C2×C22⋊C4×C10912yes—
284C23#120×C10911yescharacteristic
285C23#120×C10912yes—
286Q8—42no—
287Q8—81no—
288Q8—84no—
289D4—41no—
290D4—42no—
291D4—41no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—82no—
307C2×C4—84no—
308C2×C4—82no—
309C2×C4—84no—
310C2×C4C2×C22⋊C4×C10911yescharacteristic
311C2×C4#120×C10911yescharacteristic
312C8—81no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—42no—
317C22—42no—
318C22—44no—
319C22—82no—
320C22#379×C10911yescharacteristic
321C22B(4,𝔽2)×C10911yescharacteristic
322C22#458×C10911yescharacteristic
323C4—21no—
324C4—21no—
325C4—44no—
326C4—42no—
327C4—44no—
328C4—42no—
329C4—82no—
330C4—162no—
331C2—21no—
332C2—21no—
333C2—42no—
334C2—42no—
335C2C109×#179011yescharacteristic
336C2C109×#220112yes—
subgroupG/H
center, socleC22×C109B(4,𝔽2)
commutatorC2×D4C22×C4×C109
Frattini, cosocleC22×D4C23×C109
prime radicalC22×D4×C109C23
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C23×C109 ◁ C22×D4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
8. #563×C109C4semidirect
162. #12764C109direct (N × H)
174. C109#12764direct (N × H)
177. #563C4×C109semidirect

Maximal subgroups

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

subgroupindex
C2×#563×C1092
C2×#354×C1092
C2×#352×C1092
C109×#18902
C109×#18852
#12764109