Finite Group Databasebeta

← C109×#9501 #42127495

Subgroups — C109×#9501 #42127495

Order 27904.

928 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2405C211yesmaximal, characteristic
2C109×#2309C211yesmaximal, characteristic
3C109×#1819C212yesmaximal, complemented
4C109×#1781C211yesmaximal, characteristic
5C109×#1304C212yesmaximal, complemented
6C23×D4×C109C2211yescharacteristic
7C2×#123×C109C412yescomplemented
8C2×#117×C109—21no—
9C22×C22⋊C4×C109C2211yescharacteristic
10C2×#99×C109—22no—
11#394×C109—22no—
12#390×C109—24no—
13#386×C109C412yescomplemented
14#385×C109—21no—
15#379×C109—21no—
16#379×C109—21no—
17#379×C109C2212yes—
18#379×C109C2211yescharacteristic
19#379×C109C2212yes—
20#342×C109—24no—
21C25×C109D411yescharacteristic
22C25×C109C2311yescharacteristic
23C22×D4×C109—22no—
24C22×D4×C109—22no—
25C22×D4×C109—44no—
26C22×D4×C109C2×C412yes—
27C22×D4×C109D412yes—
28C23×C4×C109C2×C411yescharacteristic
29C23×C4×C109C2×C411yescharacteristic
30C23×C4×C109D411yescharacteristic
31#123×C109—44no—
32#117×C109—44no—
33C2×C4⋊C4×C109—22no—
34C2×C4⋊C4×C109—22no—
35C2×C22⋊C4×C109—22no—
36C2×C22⋊C4×C109—22no—
37C2×C22⋊C4×C109—21no—
38C2×C22⋊C4×C109—21no—
39C2×C22⋊C4×C109—22no—
40C2×C22⋊C4×C109—21no—
41C2×C22⋊C4×C109—22no—
42C2×C22⋊C4×C109—22no—
43C2×C22⋊C4×C109—22no—
44C2×C22⋊C4×C109—22no—
45C2×C22⋊C4×C109—22no—
46C2×C22⋊C4×C109—22no—
47C2×C22⋊C4×C109—22no—
48C2×C22⋊C4×C109—21no—
49C2×C22⋊C4×C109—22no—
50C2×C22⋊C4×C109C2×C412yes—
51C2×C22⋊C4×C109D414yes—
52C2×C42×C109—22no—
53#99×C109—44no—
54#95×C109—22no—
55C24×C109—21no—
56C24×C109—21no—
57C24×C109—42no—
58C24×C109—42no—
59C24×C109—42no—
60C24×C109—42no—
61C24×C109—81no—
62C24×C109—81no—
63C24×C109C22⋊C412yes—
64C24×C109C22×C411yescharacteristic
65C24×C109C2×D411yescharacteristic
66C24×C109C2×D411yescharacteristic
67C24×C109C2×D412yes—
68C24×C109#4112yes—
69C2×D4×C109—24no—
70C2×D4×C109—24no—
71C2×D4×C109—24no—
72C2×D4×C109—24no—
73C2×D4×C109—42no—
74C2×D4×C109—42no—
75C2×D4×C109—44no—
76C2×D4×C109—44no—
77C2×D4×C109—48no—
78C22×C4×C109—21no—
79C22×C4×C109—21no—
80C22×C4×C109—24no—
81C22×C4×C109—21no—
82C22×C4×C109—22no—
83C22×C4×C109—21no—
84C22×C4×C109—24no—
85C22×C4×C109—22no—
86C22×C4×C109—22no—
87C22×C4×C109—42no—
88C22×C4×C109—42no—
89C22×C4×C109—81no—
90C22×C4×C109C22⋊C412yes—
91C4⋊C4×C109—44no—
92C4⋊C4×C109—44no—
93C22⋊C4×C109—42no—
94C22⋊C4×C109—44no—
95C22⋊C4×C109—44no—
96C22⋊C4×C109—41no—
97C22⋊C4×C109—42no—
98C22⋊C4×C109—41no—
99C22⋊C4×C109—42no—
100C22⋊C4×C109—44no—
101C22⋊C4×C109—44no—
102C22⋊C4×C109—44no—
103C22⋊C4×C109—82no—
104C22⋊C4×C109—81no—
105C22⋊C4×C109—82no—
106C22⋊C4×C109—82no—
107C22⋊C4×C109—82no—
108C22⋊C4×C109—82no—
109C42×C109—44no—
110C23×C109—22no—
111C23×C109—22no—
112C23×C109—22no—
113C23×C109—22no—
114C23×C109—24no—
115C23×C109—22no—
116C23×C109—22no—
117C23×C109—22no—
118C23×C109—41no—
119C23×C109—41no—
120C23×C109—41no—
121C23×C109—41no—
122C23×C109—42no—
123C23×C109—42no—
124C23×C109—42no—
125C23×C109—42no—
126C23×C109—41no—
127C23×C109—42no—
128C23×C109—42no—
129C23×C109—42no—
130C23×C109—42no—
131C23×C109—42no—
132C23×C109—42no—
133C23×C109—42no—
134C23×C109—81no—
135C23×C109—81no—
136C23×C109—88no—
137C23×C109—88no—
138C23×C109C2×C22⋊C411yescharacteristic
139C23×C109C4×D412yes—
140C23×C109#12011yescharacteristic
141C23×C109#12112yes—
142C23×C109#12311yescharacteristic
143D4×C109—48no—
144D4×C109—48no—
145C2×C4×C109—22no—
146C2×C4×C109—22no—
147C2×C4×C109—24no—
148C2×C4×C109—22no—
149C2×C4×C109—22no—
150C2×C4×C109—44no—
151C2×C4×C109—42no—
152C2×C4×C109—42no—
153C2×C4×C109—44no—
154C2×C4×C109—42no—
155C2×C4×C109—41no—
156C2×C4×C109—41no—
157C2×C4×C109—42no—
158C2×C4×C109—42no—
159C2×C4×C109—44no—
160C2×C4×C109—82no—
161C2×C4×C109—82no—
162C2×C4×C109—82no—
163C2×C4×C109—82no—
164C2×C4×C109—82no—
165C2×C4×C109#9914yes—
166C22×C109—21no—
167C22×C109—21no—
168C22×C109—21no—
169C22×C109—21no—
170C22×C109—22no—
171C22×C109—21no—
172C22×C109—22no—
173C22×C109—21no—
174C22×C109—22no—
175C22×C109—22no—
176C22×C109—22no—
177C22×C109—22no—
178C22×C109—21no—
179C22×C109—44no—
180C22×C109—44no—
181C22×C109—44no—
182C22×C109—42no—
183C22×C109—42no—
184C22×C109—42no—
185C22×C109—84no—
186C22×C109—82no—
187C22×C109—82no—
188C22×C109—84no—
189C22×C109—84no—
190C22×C109—84no—
191C22×C109#38611yescharacteristic
192C22×C109C2×#9912yes—
193C22×C109#45812yes—
194C4×C109—24no—
195C4×C109—84no—
196C4×C109—82no—
197C4×C109—81no—
198C4×C109—81no—
199C4×C109—162no—
200#9501C10911yesmaximal, characteristic, direct product factor, Sylow
201C2×C109—22no—
202C2×C109—22no—
203C2×C109—42no—
204C2×C109—42no—
205C2×C109—41no—
206C2×C109—82no—
207C2×C109#189011yescharacteristic
208C2×C109#189811yescharacteristic
209C2×C109#191311yescharacteristic
210#2405C2×C10911yescharacteristic
211#2309C2×C10911yescharacteristic
212#1819C2×C10912yescomplemented
213#1781C2×C10911yescharacteristic
214#1304C2×C10912yescomplemented
215C109#950111yescharacteristic, direct product factor, Sylow
216C23×D4C22×C10911yescharacteristic
217C2×#123C4×C10912yescomplemented
218C2×#117—21no—
219C22×C22⋊C4C22×C10911yescharacteristic
220C2×#99—22no—
221#394—22no—
222#390—24no—
223#386C4×C10912yescomplemented
224#385—21no—
225#379—21no—
226#379—21no—
227#379C22×C10912yes—
228#379C22×C10911yescharacteristic
229#379C22×C10912yes—
230#342—24no—
231C25D4×C10911yescharacteristic
232C25C23×C10911yescharacteristic
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—44no—
236C22×D4C2×C4×C10912yes—
237C22×D4D4×C10912yes—
238C23×C4C2×C4×C10911yescharacteristic
239C23×C4C2×C4×C10911yescharacteristic
240C23×C4D4×C10911yescharacteristic
241#123—44no—
242#117—44no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4C2×C4×C10912yes—
261C2×C22⋊C4D4×C10914yes—
262C2×C42—22no—
263#99—44no—
264#95—22no—
265C24—21no—
266C24—21no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—81no—
272C24—81no—
273C24C22⋊C4×C10912yes—
274C24C22×C4×C10911yescharacteristic
275C24C2×D4×C10911yescharacteristic
276C24C2×D4×C10911yescharacteristic
277C24C2×D4×C10912yes—
278C24#41×C10912yes—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—44no—
286C2×D4—44no—
287C2×D4—48no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—81no—
300C22×C4C22⋊C4×C10912yes—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C22⋊C4—42no—
304C22⋊C4—44no—
305C22⋊C4—44no—
306C22⋊C4—41no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—42no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—82no—
314C22⋊C4—81no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C42—44no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—24no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—41no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—81no—
346C23—88no—
347C23—88no—
348C23C2×C22⋊C4×C10911yescharacteristic
349C23C4×D4×C10912yes—
350C23#120×C10911yescharacteristic
351C23#121×C10912yes—
352C23#123×C10911yescharacteristic
353D4—48no—
354D4—48no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4#99×C10914yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—22no—
381C22—21no—
382C22—22no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—21no—
389C22—44no—
390C22—44no—
391C22—44no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—84no—
396C22—82no—
397C22—82no—
398C22—84no—
399C22—84no—
400C22—84no—
401C22#386×C10911yescharacteristic
402C22C2×#99×C10912yes—
403C22#458×C10912yes—
404C4—24no—
405C4—84no—
406C4—82no—
407C4—81no—
408C4—81no—
409C4—162no—
410C2—22no—
411C2—22no—
412C2—42no—
413C2—41no—
414C2—42no—
415C2—82no—
416C2C109×#189011yescharacteristic
417C2C109×#189811yescharacteristic
418C2C109×#191311yescharacteristic
subgroupG/H
center, socleC22×C109#386
commutatorC24C22×C4×C109
Frattini, cosocleC25C23×C109
prime radicalC23×D4×C109C22
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C109 ◁ C25×C109 ◁ G

Complements

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

NH ≅ G/Ntype
3. C109×#1819C2semidirect
5. C109×#1304C2semidirect
7. C2×#123×C109C4semidirect
13. #386×C109C4semidirect
200. #9501C109direct (N × H)
212. #1819C2×C109semidirect
214. #1304C2×C109semidirect
215. C109#9501direct (N × H)
217. C2×#123C4×C109semidirect
223. #386C4×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#24052
C109×#23092
C109×#18192
C109×#17812
C109×#13042
#9501109