Finite Group Databasebeta

← C109×#11789 #42129783

Subgroups — C109×#11789 #42129783

Order 27904.

462 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2564C211yesmaximal, characteristic
2C4×#99×C109C211yesmaximal, characteristic
3C109×#1466C211yesmaximal, characteristic
4C109×#1352C212yesmaximal, complemented
5C109×#1351C212yesmaximal, complemented
6C2×C22⋊Q8×C109C411yescharacteristic, complemented
7C2×C4×D4×C109C2211yescharacteristic
8#432×C109—21no—
9C2×#99×C109—21no—
10#400×C109C411yescharacteristic, complemented
11#399×C109C2211yescharacteristic
12#395×C109C411yescharacteristic, complemented
13#393×C109C2211yescharacteristic
14#387×C109—21no—
15#386×C109—21no—
16#385×C109—21no—
17#380×C109—21no—
18C4×C22⋊C4×C109—21no—
19C2×#95×C109C411yescharacteristic, complemented
20#342×C109C2212yes—
21#323×C109—21no—
22#323×C109C2212yes—
23C22×Q8×C109C2×C411yescharacteristic
24C22×D4×C109D411yescharacteristic
25C23×C4×C109—21no—
26C23×C4×C109C2×C411yescharacteristic
27C23×C4×C109D411yescharacteristic
28C22⋊Q8×C109—41no—
29C22⋊Q8×C109—41no—
30C4×D4×C109—41no—
31C4×D4×C109—41no—
32C2×C4⋊C4×C109—21no—
33C2×C4⋊C4×C109—21no—
34C2×C4⋊C4×C109—21no—
35C2×C4⋊C4×C109C2×C411yescharacteristic
36C2×C4⋊C4×C109C2×C411yescharacteristic
37C2×C22⋊C4×C109—22no—
38C2×C22⋊C4×C109—21no—
39C2×C22⋊C4×C109—21no—
40C2×C22⋊C4×C109—21no—
41C2×C22⋊C4×C109D411yescharacteristic
42C2×C22⋊C4×C109C2311yescharacteristic
43C2×C42×C109—41no—
44C2×C42×C109D411yescharacteristic
45C4⋊C8×C109—41no—
46#99×C109—24no—
47C22⋊C8×C109—22no—
48#97×C109—41no—
49#95×C109—21no—
50#95×C109—21no—
51#95×C109—21no—
52#95×C109—21no—
53#95×C109—21no—
54#95×C109—41no—
55#95×C109C2×C411yescharacteristic
56#95×C109C2×C411yescharacteristic
57C24×C109C22⋊C411yescharacteristic
58C24×C109C2×D411yescharacteristic
59C2×Q8×C109—41no—
60C2×Q8×C109—81no—
61C2×D4×C109—22no—
62C2×D4×C109—22no—
63C2×D4×C109—41no—
64C22×C4×C109—21no—
65C22×C4×C109—21no—
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×C109—42no—
73C22×C4×C109—42no—
74C22×C4×C109—42no—
75C22×C4×C109—81no—
76C22×C4×C109—81no—
77C22×C4×C109—81no—
78C22×C4×C109C22⋊C411yescharacteristic
79C22×C4×C109C22⋊C411yescharacteristic
80C22×C4×C109C22⋊C411yescharacteristic
81C22×C4×C109C22×C411yescharacteristic
82C22×C4×C109C2×D411yescharacteristic
83C2×C8×C109—42no—
84C4⋊C4×C109—41no—
85C4⋊C4×C109—41no—
86C4⋊C4×C109—42no—
87C4⋊C4×C109—41no—
88C4⋊C4×C109—41no—
89C4⋊C4×C109—81no—
90C22⋊C4×C109—24no—
91C22⋊C4×C109—41no—
92C22⋊C4×C109—41no—
93C22⋊C4×C109—42no—
94C22⋊C4×C109—42no—
95C22⋊C4×C109—42no—
96C22⋊C4×C109—81no—
97C42×C109—21no—
98C42×C109—21no—
99C42×C109—82no—
100C23×C109—21no—
101C23×C109—21no—
102C23×C109—21no—
103C23×C109—21no—
104C23×C109—41no—
105C23×C109—41no—
106C23×C109—41no—
107C23×C109—41no—
108C23×C109#10412yes—
109C23×C109C2×C22⋊C411yescharacteristic
110C23×C109#13712yes—
111Q8×C109—82no—
112D4×C109—44no—
113C2×C4×C109—22no—
114C2×C4×C109—21no—
115C2×C4×C109—21no—
116C2×C4×C109—21no—
117C2×C4×C109—21no—
118C2×C4×C109—21no—
119C2×C4×C109—21no—
120C2×C4×C109—41no—
121C2×C4×C109—42no—
122C2×C4×C109—41no—
123C2×C4×C109—41no—
124C2×C4×C109—41no—
125C2×C4×C109—41no—
126C2×C4×C109—41no—
127C2×C4×C109—44no—
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—81no—
138C2×C4×C109—81no—
139C2×C4×C109—81no—
140C2×C4×C109—84no—
141C2×C4×C109—82no—
142C2×C4×C109—82no—
143C2×C4×C109#9911yescharacteristic
144C2×C4×C109#9911yescharacteristic
145C8×C109—82no—
146C22×C109—21no—
147C22×C109—21no—
148C22×C109—22no—
149C22×C109—22no—
150C22×C109—42no—
151C22×C109—42no—
152C22×C109—42no—
153C22×C109—41no—
154C22×C109—41no—
155C22×C109—41no—
156C22×C109—41no—
157C22×C109—81no—
158C22×C109C2×#9911yescharacteristic
159C22×C109#41911yescharacteristic
160C22×C109C2×#10411yescharacteristic
161C4×C109—21no—
162C4×C109—21no—
163C4×C109—42no—
164C4×C109—41no—
165C4×C109—41no—
166C4×C109—41no—
167C4×C109—41no—
168C4×C109—81no—
169C4×C109—84no—
170C4×C109—81no—
171C4×C109—81no—
172#11789C10911yesmaximal, characteristic, direct product factor, Sylow
173C2×C109—21no—
174C2×C109—21no—
175C2×C109—42no—
176C2×C109—42no—
177C2×C109#150811yescharacteristic
178C2×C109#212111yescharacteristic
179C2×C109#212611yescharacteristic
180#2564C2×C10911yescharacteristic
181C4×#99C2×C10911yescharacteristic
182#1466C2×C10911yescharacteristic
183#1352C2×C10912yescomplemented
184#1351C2×C10912yescomplemented
185C109#1178911yescharacteristic, direct product factor, Sylow
186C2×C22⋊Q8C4×C10911yescharacteristic, complemented
187C2×C4×D4C22×C10911yescharacteristic
188#432—21no—
189C2×#99—21no—
190#400C4×C10911yescharacteristic, complemented
191#399C22×C10911yescharacteristic
192#395C4×C10911yescharacteristic, complemented
193#393C22×C10911yescharacteristic
194#387—21no—
195#386—21no—
196#385—21no—
197#380—21no—
198C4×C22⋊C4—21no—
199C2×#95C4×C10911yescharacteristic, complemented
200#342C22×C10912yes—
201#323—21no—
202#323C22×C10912yes—
203C22×Q8C2×C4×C10911yescharacteristic
204C22×D4D4×C10911yescharacteristic
205C23×C4—21no—
206C23×C4C2×C4×C10911yescharacteristic
207C23×C4D4×C10911yescharacteristic
208C22⋊Q8—41no—
209C22⋊Q8—41no—
210C4×D4—41no—
211C4×D4—41no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4C2×C4×C10911yescharacteristic
216C2×C4⋊C4C2×C4×C10911yescharacteristic
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4D4×C10911yescharacteristic
222C2×C22⋊C4C23×C10911yescharacteristic
223C2×C42—41no—
224C2×C42D4×C10911yescharacteristic
225C4⋊C8—41no—
226#99—24no—
227C22⋊C8—22no—
228#97—41no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—41no—
235#95C2×C4×C10911yescharacteristic
236#95C2×C4×C10911yescharacteristic
237C24C22⋊C4×C10911yescharacteristic
238C24C2×D4×C10911yescharacteristic
239C2×Q8—41no—
240C2×Q8—81no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—41no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4C22⋊C4×C10911yescharacteristic
259C22×C4C22⋊C4×C10911yescharacteristic
260C22×C4C22⋊C4×C10911yescharacteristic
261C22×C4C22×C4×C10911yescharacteristic
262C22×C4C2×D4×C10911yescharacteristic
263C2×C8—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—81no—
270C22⋊C4—24no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—81no—
277C42—21no—
278C42—21no—
279C42—82no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23#104×C10912yes—
289C23C2×C22⋊C4×C10911yescharacteristic
290C23#137×C10912yes—
291Q8—82no—
292D4—44no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—44no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—84no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4#99×C10911yescharacteristic
324C2×C4#99×C10911yescharacteristic
325C8—82no—
326C22—21no—
327C22—21no—
328C22—22no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22C2×#99×C10911yescharacteristic
339C22#419×C10911yescharacteristic
340C22C2×#104×C10911yescharacteristic
341C4—21no—
342C4—21no—
343C4—42no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—81no—
349C4—84no—
350C4—81no—
351C4—81no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—42no—
356C2C109×#150811yescharacteristic
357C2C109×#212111yescharacteristic
358C2C109×#212611yescharacteristic
subgroupG/H
center, socleC22×C109C2×#99
commutatorC22×C4C22×C4×C109
Frattini, cosocleC2×C22⋊C4C23×C109
prime radicalC22×D4×C109D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C22×C4×C109 ◁ C2×C4×D4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
4. C109×#1352C2semidirect
5. C109×#1351C2semidirect
6. C2×C22⋊Q8×C109C4semidirect
10. #400×C109C4semidirect
12. #395×C109C4semidirect
19. C2×#95×C109C4semidirect
172. #11789C109direct (N × H)
183. #1352C2×C109semidirect
184. #1351C2×C109semidirect
185. C109#11789direct (N × H)
186. C2×C22⋊Q8C4×C109semidirect
190. #400C4×C109semidirect
192. #395C4×C109semidirect
199. C2×#95C4×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#25642
C4×#99×C1092
C109×#14662
C109×#13522
C109×#13512
#11789109