Finite Group Databasebeta

← C109×#12706 #42130700

Subgroups — C109×#12706 #42130700

Order 27904.

1000 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 146 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×#3485C211yesmaximal, characteristic
2C109×#2121C214yesmaximal, complemented
3C109×#1785C211yesmaximal, characteristic, complemented
4C109×#1782C211yesmaximal, characteristic, complemented
5C2×H(5,𝔽2)×C109C411yescharacteristic, complemented
6C22×#41×C109C2211yescharacteristic
7#547×C109—24no—
8#546×C109—24no—
9#536×C109—21no—
10#534×C109—21no—
11#534×C109C2212yes—
12#525×C109—21no—
13C2×#123×C109—21no—
14C2×#123×C109C412yescomplemented
15#518×C109C411yescharacteristic, complemented
16C2×#101×C109—21no—
17C2×#100×C109C2212yescomplemented
18C2×#99×C109—21no—
19C2×#99×C109C2212yescomplemented
20#407×C109—21no—
21#381×C109—21no—
22#352×C109—24no—
23AΣL(1,𝔽16)×C109—24no—
24H(5,𝔽2)×C109—42no—
25H(5,𝔽2)×C109—42no—
26C2×#41×C109—24no—
27C2×#41×C109—21no—
28C2×#41×C109—22no—
29C2×#41×C109—42no—
30C2×#41×C109—42no—
31C22×Q8×C109D411yescharacteristic
32C22×D4×C109—21no—
33C22×D4×C109—22no—
34C22×D4×C109C2×C411yescharacteristic
35C22×D4×C109C2×C412yes—
36C22×D4×C109D411yescharacteristic
37C22×D4×C109C2311yescharacteristic
38C23×C4×C109C2×C411yescharacteristic
39C23×C4×C109D411yescharacteristic
40C23×C4×C109D411yescharacteristic
41C2×M16×C109—22no—
42C4⋊Q8×C109—22no—
43#124×C109—22no—
44#123×C109—24no—
45#123×C109—22no—
46#123×C109—22no—
47#123×C109—22no—
48#123×C109—22no—
49#123×C109—24no—
50#123×C109D414yes—
51C22⋊Q8×C109—22no—
52C22⋊Q8×C109—24no—
53#121×C109—22no—
54#121×C109—24no—
55#121×C109—24no—
56#120×C109—22no—
57#120×C109—22no—
58#120×C109D414yes—
59C4×D4×C109—24no—
60C4×D4×C109—24no—
61#117×C109—21no—
62C2×C4⋊C4×C109—22no—
63C2×C4⋊C4×C109—22no—
64C2×C22⋊C4×C109—22no—
65C2×C22⋊C4×C109—21no—
66C2×C22⋊C4×C109—22no—
67C2×C22⋊C4×C109—22no—
68C2×C22⋊C4×C109C2×C412yes—
69#101×C109—42no—
70#100×C109—24no—
71#99×C109—24no—
72#99×C109—42no—
73C22⋊C8×C109—42no—
74#95×C109—42no—
75C24×C109—21no—
76C24×C109C22⋊C411yescharacteristic
77C24×C109C22⋊C412yes—
78C24×C109C22×C411yescharacteristic
79C24×C109C2×D411yescharacteristic
80#41×C109—42no—
81#41×C109—416no—
82#41×C109—48no—
83#41×C109—42no—
84C2×Q8×C109—21no—
85C2×Q8×C109—22no—
86C2×Q8×C109—22no—
87C2×Q8×C109—41no—
88C2×D4×C109—22no—
89C2×D4×C109—22no—
90C2×D4×C109—22no—
91C2×D4×C109—21no—
92C2×D4×C109—22no—
93C2×D4×C109—22no—
94C2×D4×C109—22no—
95C2×D4×C109—22no—
96C2×D4×C109—24no—
97C2×D4×C109—22no—
98C2×D4×C109—41no—
99C2×D4×C109—41no—
100C2×D4×C109—41no—
101C2×D4×C109—48no—
102C2×D4×C109—42no—
103C2×D4×C109—42no—
104C2×D4×C109—42no—
105C2×D4×C109—42no—
106C2×D4×C109C2×D414yes—
107C22×C4×C109—21no—
108C22×C4×C109—22no—
109C22×C4×C109—22no—
110C22×C4×C109—22no—
111C22×C4×C109—22no—
112C22×C4×C109—22no—
113C22×C4×C109—22no—
114C22×C4×C109—22no—
115C22×C4×C109—22no—
116C22×C4×C109—41no—
117C22×C4×C109—41no—
118C22×C4×C109—41no—
119C22×C4×C109—41no—
120C22×C4×C109—42no—
121C22×C4×C109C22⋊C411yescharacteristic
122C22×C4×C109C22⋊C411yescharacteristic
123C22×C4×C109C22⋊C411yescharacteristic
124C22×C4×C109C22⋊C412yes—
125C22×C4×C109C2×D411yescharacteristic
126M16×C109—42no—
127M16×C109—42no—
128C2×C8×C109—42no—
129C4⋊C4×C109—24no—
130C4⋊C4×C109—24no—
131C4⋊C4×C109—22no—
132C4⋊C4×C109—24no—
133C22⋊C4×C109—24no—
134C22⋊C4×C109—22no—
135C22⋊C4×C109—24no—
136C22⋊C4×C109—22no—
137C22⋊C4×C109—24no—
138C22⋊C4×C109—24no—
139C22⋊C4×C109—42no—
140C22⋊C4×C109—42no—
141C22⋊C4×C109C22⋊C414yes—
142C42×C109—22no—
143C23×C109—21no—
144C23×C109—21no—
145C23×C109—21no—
146C23×C109—21no—
147C23×C109—21no—
148C23×C109—21no—
149C23×C109—22no—
150C23×C109—42no—
151C23×C109—42no—
152C23×C109—81no—
153C23×C109—81no—
154C23×C109—81no—
155C23×C109—82no—
156C23×C109—82no—
157C23×C109#9912yes—
158C23×C109C2×C22⋊C412yes—
159C23×C109C2×C22⋊C411yescharacteristic
160C23×C109#12012yes—
161Q8×C109—44no—
162Q8×C109—42no—
163D4×C109—28no—
164D4×C109—44no—
165D4×C109—42no—
166D4×C109—44no—
167D4×C109—44no—
168D4×C109—44no—
169D4×C109—42no—
170D4×C109—42no—
171D4×C109—84no—
172C2×C4×C109—21no—
173C2×C4×C109—21no—
174C2×C4×C109—21no—
175C2×C4×C109—21no—
176C2×C4×C109—24no—
177C2×C4×C109—21no—
178C2×C4×C109—21no—
179C2×C4×C109—22no—
180C2×C4×C109—21no—
181C2×C4×C109—21no—
182C2×C4×C109—21no—
183C2×C4×C109—21no—
184C2×C4×C109—44no—
185C2×C4×C109—44no—
186C2×C4×C109—44no—
187C2×C4×C109—44no—
188C2×C4×C109—42no—
189C2×C4×C109—42no—
190C2×C4×C109—82no—
191C2×C4×C109—82no—
192C2×C4×C109—82no—
193C2×C4×C109—84no—
194C2×C4×C109#9912yes—
195C2×C4×C109#12012yes—
196C8×C109—82no—
197C22×C109—22no—
198C22×C109—22no—
199C22×C109—22no—
200C22×C109—41no—
201C22×C109—42no—
202C22×C109—42no—
203C22×C109—42no—
204C22×C109—42no—
205C22×C109—42no—
206C22×C109—44no—
207C22×C109—42no—
208C22×C109—81no—
209C22×C109—81no—
210C22×C109—81no—
211C22×C109—81no—
212C22×C109—162no—
213C22×C109—162no—
214C22×C109#37911yescharacteristic
215C22×C109C2×#9911yescharacteristic
216C22×C109C2×#9911yescharacteristic
217C4×C109—22no—
218C4×C109—22no—
219C4×C109—44no—
220C4×C109—41no—
221C4×C109—41no—
222C4×C109—42no—
223C4×C109—162no—
224#12706C10911yesmaximal, characteristic, direct product factor, Sylow
225C2×C109—21no—
226C2×C109—21no—
227C2×C109—21no—
228C2×C109—42no—
229C2×C109—42no—
230C2×C109—42no—
231C2×C109—82no—
232C2×C109#178111yescharacteristic
233#3485C2×C10911yescharacteristic
234#2121C2×C10914yescomplemented
235#1785C2×C10911yescharacteristic, complemented
236#1782C2×C10911yescharacteristic, complemented
237C109#1270611yescharacteristic, direct product factor, Sylow
238C2×H(5,𝔽2)C4×C10911yescharacteristic, complemented
239C22×#41C22×C10911yescharacteristic
240#547—24no—
241#546—24no—
242#536—21no—
243#534—21no—
244#534C22×C10912yes—
245#525—21no—
246C2×#123—21no—
247C2×#123C4×C10912yescomplemented
248#518C4×C10911yescharacteristic, complemented
249C2×#101—21no—
250C2×#100C22×C10912yescomplemented
251C2×#99—21no—
252C2×#99C22×C10912yescomplemented
253#407—21no—
254#381—21no—
255#352—24no—
256AΣL(1,𝔽16)—24no—
257H(5,𝔽2)—42no—
258H(5,𝔽2)—42no—
259C2×#41—24no—
260C2×#41—21no—
261C2×#41—22no—
262C2×#41—42no—
263C2×#41—42no—
264C22×Q8D4×C10911yescharacteristic
265C22×D4—21no—
266C22×D4—22no—
267C22×D4C2×C4×C10911yescharacteristic
268C22×D4C2×C4×C10912yes—
269C22×D4D4×C10911yescharacteristic
270C22×D4C23×C10911yescharacteristic
271C23×C4C2×C4×C10911yescharacteristic
272C23×C4D4×C10911yescharacteristic
273C23×C4D4×C10911yescharacteristic
274C2×M16—22no—
275C4⋊Q8—22no—
276#124—22no—
277#123—24no—
278#123—22no—
279#123—22no—
280#123—22no—
281#123—22no—
282#123—24no—
283#123D4×C10914yes—
284C22⋊Q8—22no—
285C22⋊Q8—24no—
286#121—22no—
287#121—24no—
288#121—24no—
289#120—22no—
290#120—22no—
291#120D4×C10914yes—
292C4×D4—24no—
293C4×D4—24no—
294#117—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4C2×C4×C10912yes—
302#101—42no—
303#100—24no—
304#99—24no—
305#99—42no—
306C22⋊C8—42no—
307#95—42no—
308C24—21no—
309C24C22⋊C4×C10911yescharacteristic
310C24C22⋊C4×C10912yes—
311C24C22×C4×C10911yescharacteristic
312C24C2×D4×C10911yescharacteristic
313#41—42no—
314#41—416no—
315#41—48no—
316#41—42no—
317C2×Q8—21no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—41no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—21no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—24no—
330C2×D4—22no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—48no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4C2×D4×C10914yes—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4C22⋊C4×C10911yescharacteristic
355C22×C4C22⋊C4×C10911yescharacteristic
356C22×C4C22⋊C4×C10911yescharacteristic
357C22×C4C22⋊C4×C10912yes—
358C22×C4C2×D4×C10911yescharacteristic
359M16—42no—
360M16—42no—
361C2×C8—42no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—22no—
368C22⋊C4—24no—
369C22⋊C4—22no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4C22⋊C4×C10914yes—
375C42—22no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—22no—
383C23—42no—
384C23—42no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—82no—
389C23—82no—
390C23#99×C10912yes—
391C23C2×C22⋊C4×C10912yes—
392C23C2×C22⋊C4×C10911yescharacteristic
393C23#120×C10912yes—
394Q8—44no—
395Q8—42no—
396D4—28no—
397D4—44no—
398D4—42no—
399D4—44no—
400D4—44no—
401D4—44no—
402D4—42no—
403D4—42no—
404D4—84no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—84no—
427C2×C4#99×C10912yes—
428C2×C4#120×C10912yes—
429C8—82no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—41no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—44no—
440C22—42no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—162no—
446C22—162no—
447C22#379×C10911yescharacteristic
448C22C2×#99×C10911yescharacteristic
449C22C2×#99×C10911yescharacteristic
450C4—22no—
451C4—22no—
452C4—44no—
453C4—41no—
454C4—41no—
455C4—42no—
456C4—162no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—42no—
461C2—42no—
462C2—42no—
463C2—82no—
464C2C109×#178111yescharacteristic
subgroupG/H
center, socleC2×C109#1781
commutatorC24C22×C4×C109
Frattini, cosocleC22×D4C23×C109
prime radicalC109×#3485C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C109 ◁ C23×C109 ◁ C22×#41×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#2121C2semidirect
3. C109×#1785C2semidirect
4. C109×#1782C2semidirect
5. C2×H(5,𝔽2)×C109C4semidirect
14. C2×#123×C109C4semidirect
15. #518×C109C4semidirect
17. C2×#100×C109C22semidirect
19. C2×#99×C109C22semidirect
224. #12706C109direct (N × H)
234. #2121C2×C109semidirect
235. #1785C2×C109semidirect
236. #1782C2×C109semidirect
237. C109#12706direct (N × H)
238. C2×H(5,𝔽2)C4×C109semidirect
247. C2×#123C4×C109semidirect
248. #518C4×C109semidirect
250. C2×#100C22×C109semidirect
252. C2×#99C22×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#34852
C109×#21212
C109×#17852
C109×#17822
#12706109