Finite Group Databasebeta

← C109×#12696 #42130690

Subgroups — C109×#12696 #42130690

Order 27904.

1214 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1214 conjugacy classes of subgroups — too many to draw. Only its 150 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×#535×C109C211yesmaximal, characteristic
2C2×#354×C109C212yesmaximal, complemented
3C2×#353×C109C212yesmaximal, complemented
4C109×#1781C212yesmaximal, complemented
5C22×#41×C109C411yescharacteristic, complemented
6C23×D4×C109C2211yescharacteristic
7#535×C109—24no—
8#535×C109—24no—
9C2×#127×C109C411yescharacteristic, complemented
10C2×#127×C109C2211yescharacteristic
11C2×#124×C109C411yescharacteristic, complemented
12C2×#124×C109C2211yescharacteristic
13C2×#121×C109—22no—
14C2×#120×C109—22no—
15C2×#117×C109C411yescharacteristic, complemented
16C2×#99×C109—22no—
17C2×#99×C109C2214yes—
18#379×C109—22no—
19#354×C109—24no—
20#353×C109—24no—
21C25×C109D412yes—
22C2×#41×C109—22no—
23C2×#41×C109—22no—
24C2×#41×C109—42no—
25C2×#41×C109—42no—
26C22×Q8×C109C2×C411yescharacteristic
27C22×D4×C109—21no—
28C22×D4×C109—22no—
29C22×D4×C109—22no—
30C22×D4×C109—22no—
31C22×D4×C109—22no—
32C22×D4×C109—42no—
33C22×D4×C109—42no—
34C22×D4×C109C2×C411yescharacteristic
35C22×D4×C109C2×C411yescharacteristic
36C22×D4×C109D412yes—
37C22×D4×C109C2311yescharacteristic
38C23×C4×C109—21no—
39C23×C4×C109C2×C411yescharacteristic
40#127×C109—22no—
41#127×C109—22no—
42#127×C109—22no—
43#127×C109D414yes—
44#124×C109—22no—
45#124×C109—22no—
46#124×C109—22no—
47#124×C109D414yes—
48#121×C109—216no—
49#120×C109—216no—
50#117×C109—22no—
51#117×C109—22no—
52C2×C4⋊C4×C109—21no—
53C2×C22⋊C4×C109—22no—
54C2×C22⋊C4×C109—21no—
55C2×C22⋊C4×C109—22no—
56C2×C22⋊C4×C109—24no—
57C2×C22⋊C4×C109—44no—
58C2×C42×C109C2×C411yescharacteristic
59C2×C42×C109C2×C411yescharacteristic
60#99×C109—24no—
61#99×C109—24no—
62#99×C109—44no—
63C24×C109—21no—
64C24×C109—24no—
65C24×C109—24no—
66C24×C109—42no—
67C24×C109—84no—
68C24×C109C22⋊C412yes—
69C24×C109C22⋊C411yescharacteristic
70C24×C109C2×D412yes—
71#41×C109—48no—
72#41×C109—48no—
73C2×Q8×C109—21no—
74C2×Q8×C109—41no—
75C2×Q8×C109—41no—
76C2×Q8×C109C22⋊C412yescomplemented
77C2×D4×C109—24no—
78C2×D4×C109—24no—
79C2×D4×C109—28no—
80C2×D4×C109—28no—
81C2×D4×C109—24no—
82C2×D4×C109—21no—
83C2×D4×C109—21no—
84C2×D4×C109—21no—
85C2×D4×C109—41no—
86C2×D4×C109—41no—
87C2×D4×C109—42no—
88C2×D4×C109—41no—
89C2×D4×C109—41no—
90C2×D4×C109—41no—
91C2×D4×C109—41no—
92C2×D4×C109—48no—
93C2×D4×C109—48no—
94C2×D4×C109—42no—
95C2×D4×C109—42no—
96C2×D4×C109—84no—
97C2×D4×C109—84no—
98C2×D4×C109C22⋊C412yescomplemented
99C2×D4×C109C2×D412yes—
100C2×D4×C109C2×D412yes—
101C22×C4×C109—22no—
102C22×C4×C109—22no—
103C22×C4×C109—21no—
104C22×C4×C109—21no—
105C22×C4×C109—24no—
106C22×C4×C109—21no—
107C22×C4×C109—21no—
108C22×C4×C109—21no—
109C22×C4×C109—42no—
110C22×C4×C109—42no—
111C22×C4×C109—44no—
112C22×C4×C109C22⋊C411yescharacteristic
113C22×C4×C109C22×C411yescharacteristic
114C4⋊C4×C109—24no—
115C22⋊C4×C109—24no—
116C22⋊C4×C109—28no—
117C22⋊C4×C109—28no—
118C22⋊C4×C109—44no—
119C22⋊C4×C109—44no—
120C22⋊C4×C109—88no—
121C42×C109—21no—
122C42×C109—21no—
123C42×C109C22⋊C412yescomplemented
124C42×C109C22⋊C412yescomplemented
125C23×C109—21no—
126C23×C109—21no—
127C23×C109—22no—
128C23×C109—22no—
129C23×C109—22no—
130C23×C109—22no—
131C23×C109—22no—
132C23×C109—22no—
133C23×C109—42no—
134C23×C109—48no—
135C23×C109—44no—
136C23×C109—44no—
137C23×C109—48no—
138C23×C109—44no—
139C23×C109—41no—
140C23×C109—41no—
141C23×C109—84no—
142C23×C109—88no—
143C23×C109—88no—
144C23×C109#9912yes—
145C23×C109C2×C22⋊C411yescharacteristic
146C23×C109#12014yes—
147Q8×C109—42no—
148Q8×C109—42no—
149D4×C109—42no—
150D4×C109—42no—
151D4×C109—48no—
152D4×C109—42no—
153D4×C109—42no—
154D4×C109—42no—
155D4×C109—42no—
156D4×C109—48no—
157D4×C109—88no—
158D4×C109—88no—
159C2×C4×C109—24no—
160C2×C4×C109—24no—
161C2×C4×C109—21no—
162C2×C4×C109—21no—
163C2×C4×C109—21no—
164C2×C4×C109—21no—
165C2×C4×C109—22no—
166C2×C4×C109—22no—
167C2×C4×C109—22no—
168C2×C4×C109—48no—
169C2×C4×C109—42no—
170C2×C4×C109—42no—
171C2×C4×C109—41no—
172C2×C4×C109—41no—
173C2×C4×C109—84no—
174C2×C4×C109—88no—
175C2×C4×C109#9912yes—
176C2×C4×C109C2×C22⋊C411yescharacteristic
177C2×C4×C109C2×C22⋊C411yescharacteristic
178C22×C109—22no—
179C22×C109—24no—
180C22×C109—22no—
181C22×C109—24no—
182C22×C109—44no—
183C22×C109—42no—
184C22×C109—42no—
185C22×C109—44no—
186C22×C109—44no—
187C22×C109—44no—
188C22×C109—44no—
189C22×C109—42no—
190C22×C109—42no—
191C22×C109—82no—
192C22×C109—82no—
193C22×C109—82no—
194C22×C109—816no—
195C22×C109—82no—
196C22×C109#37911yescharacteristic
197C22×C109C2×#9911yescharacteristic
198C22×C109C2×#9911yescharacteristic
199C4×C109—22no—
200C4×C109—21no—
201C4×C109—21no—
202C4×C109—42no—
203C4×C109—42no—
204C4×C109—42no—
205C4×C109—164no—
206#12696C10911yesmaximal, characteristic, direct product factor, Sylow
207C2×C109—21no—
208C2×C109—21no—
209C2×C109—44no—
210C2×C109—42no—
211C2×C109—82no—
212C2×C109—82no—
213C2×C109#178111yescharacteristic
214C2×C109#212712yes—
215C2×#535C2×C10911yescharacteristic
216C2×#354C2×C10912yescomplemented
217C2×#353C2×C10912yescomplemented
218#1781C2×C10912yescomplemented
219C109#1269611yescharacteristic, direct product factor, Sylow
220C22×#41C4×C10911yescharacteristic, complemented
221C23×D4C22×C10911yescharacteristic
222#535—24no—
223#535—24no—
224C2×#127C4×C10911yescharacteristic, complemented
225C2×#127C22×C10911yescharacteristic
226C2×#124C4×C10911yescharacteristic, complemented
227C2×#124C22×C10911yescharacteristic
228C2×#121—22no—
229C2×#120—22no—
230C2×#117C4×C10911yescharacteristic, complemented
231C2×#99—22no—
232C2×#99C22×C10914yes—
233#379—22no—
234#354—24no—
235#353—24no—
236C25D4×C10912yes—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—42no—
240C2×#41—42no—
241C22×Q8C2×C4×C10911yescharacteristic
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—42no—
248C22×D4—42no—
249C22×D4C2×C4×C10911yescharacteristic
250C22×D4C2×C4×C10911yescharacteristic
251C22×D4D4×C10912yes—
252C22×D4C23×C10911yescharacteristic
253C23×C4—21no—
254C23×C4C2×C4×C10911yescharacteristic
255#127—22no—
256#127—22no—
257#127—22no—
258#127D4×C10914yes—
259#124—22no—
260#124—22no—
261#124—22no—
262#124D4×C10914yes—
263#121—216no—
264#120—216no—
265#117—22no—
266#117—22no—
267C2×C4⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—44no—
273C2×C42C2×C4×C10911yescharacteristic
274C2×C42C2×C4×C10911yescharacteristic
275#99—24no—
276#99—24no—
277#99—44no—
278C24—21no—
279C24—24no—
280C24—24no—
281C24—42no—
282C24—84no—
283C24C22⋊C4×C10912yes—
284C24C22⋊C4×C10911yescharacteristic
285C24C2×D4×C10912yes—
286#41—48no—
287#41—48no—
288C2×Q8—21no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8C22⋊C4×C10912yescomplemented
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—28no—
295C2×D4—28no—
296C2×D4—24no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—48no—
308C2×D4—48no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—84no—
312C2×D4—84no—
313C2×D4C22⋊C4×C10912yescomplemented
314C2×D4C2×D4×C10912yes—
315C2×D4C2×D4×C10912yes—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—24no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4C22⋊C4×C10911yescharacteristic
328C22×C4C22×C4×C10911yescharacteristic
329C4⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—28no—
332C22⋊C4—28no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—88no—
336C42—21no—
337C42—21no—
338C42C22⋊C4×C10912yescomplemented
339C42C22⋊C4×C10912yescomplemented
340C23—21no—
341C23—21no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—42no—
349C23—48no—
350C23—44no—
351C23—44no—
352C23—48no—
353C23—44no—
354C23—41no—
355C23—41no—
356C23—84no—
357C23—88no—
358C23—88no—
359C23#99×C10912yes—
360C23C2×C22⋊C4×C10911yescharacteristic
361C23#120×C10914yes—
362Q8—42no—
363Q8—42no—
364D4—42no—
365D4—42no—
366D4—48no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—48no—
372D4—88no—
373D4—88no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—48no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—84no—
389C2×C4—88no—
390C2×C4#99×C10912yes—
391C2×C4C2×C22⋊C4×C10911yescharacteristic
392C2×C4C2×C22⋊C4×C10911yescharacteristic
393C22—22no—
394C22—22no—
395C22—24no—
396C22—24no—
397C22—44no—
398C22—42no—
399C22—42no—
400C22—44no—
401C22—44no—
402C22—44no—
403C22—42no—
404C22—42no—
405C22—44no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—816no—
410C22—82no—
411C22#379×C10911yescharacteristic
412C22C2×#99×C10911yescharacteristic
413C22C2×#99×C10911yescharacteristic
414C4—22no—
415C4—21no—
416C4—21no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—164no—
421C2—21no—
422C2—21no—
423C2—44no—
424C2—42no—
425C2—82no—
426C2—82no—
427C2C109×#178111yescharacteristic
428C2C109×#212712yes—
subgroupG/H
center, socleC22×C109C2×#99
commutatorC22×C4C22×C4×C109
Frattini, cosocleC22×D4C23×C109
prime radicalC2×#535×C109C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C24×C109 ◁ C23×D4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#354×C109C2semidirect
3. C2×#353×C109C2semidirect
4. C109×#1781C2semidirect
5. C22×#41×C109C4semidirect
9. C2×#127×C109C4semidirect
11. C2×#124×C109C4semidirect
15. C2×#117×C109C4semidirect
76. C2×Q8×C109C22⋊C4semidirect
98. C2×D4×C109C22⋊C4semidirect
123. C42×C109C22⋊C4semidirect
124. C42×C109C22⋊C4semidirect
206. #12696C109direct (N × H)
216. C2×#354C2×C109semidirect
217. C2×#353C2×C109semidirect
218. #1781C2×C109semidirect
219. C109#12696direct (N × H)
220. C22×#41C4×C109semidirect
224. C2×#127C4×C109semidirect
226. C2×#124C4×C109semidirect
230. C2×#117C4×C109semidirect
291. C2×Q8C22⋊C4×C109semidirect
313. C2×D4C22⋊C4×C109semidirect
338. C42C22⋊C4×C109semidirect
339. C42C22⋊C4×C109semidirect

Maximal subgroups

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

subgroupindex
C2×#535×C1092
C2×#354×C1092
C2×#353×C1092
C109×#17812
#12696109