Finite Group Databasebeta

← C113×#13356 #43281376

Subgroups — C113×#13356 #43281376

Order 28928.

676 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C113×#3202C211yesmaximal, characteristic, complemented
2C113×#3028C211yesmaximal, characteristic
3C113×#3022C211yesmaximal, characteristic
4C113×#2135C211yesmaximal, characteristic, complemented
5C113×#2123C211yesmaximal, characteristic, complemented
6C113×#1791C211yesmaximal, characteristic
7C113×#1395C211yesmaximal, characteristic, complemented
8#585×C113C2211yescharacteristic
9C2×#137×C113—21no—
10#536×C113—21no—
11#536×C113C2211yescharacteristic
12#536×C113C2211yescharacteristic
13#485×C113—21no—
14#485×C113—21no—
15#470×C113—21no—
16#467×C113—21no—
17#458×C113—22no—
18#458×C113—22no—
19#456×C113—22no—
20#455×C113—22no—
21#451×C113—21no—
22#451×C113—21no—
23#421×C113—21no—
24#413×C113—21no—
25#413×C113C2211yescharacteristic
26#413×C113C2211yescharacteristic
27#410×C113—21no—
28#410×C113C2211yescharacteristic
29#408×C113—21no—
30#408×C113C2211yescharacteristic
31#356×C113—22no—
32#352×C113—22no—
332−1+4×C113—21no—
342−1+4×C113—41no—
35H(5,𝔽2)×C113—21no—
36H(5,𝔽2)×C113—21no—
37H(5,𝔽2)×C113—41no—
38H(5,𝔽2)×C113D412yes—
39C2×#41×C113—21no—
40C2×#41×C113—21no—
41C2×#41×C113—21no—
42C2×#41×C113—42no—
43C2×#41×C113C2311yescharacteristic
44C22×Q8×C113D411yescharacteristic
45C22×Q8×C113D411yescharacteristic
46#137×C113—24no—
47#137×C113—42no—
48C2×Q16×C113—21no—
49C2×Q16×C113—21no—
50C2×Q16×C113—41no—
51C2×SD16×C113—41no—
52C2×M16×C113—21no—
53C2×M16×C113—21no—
54C2×M16×C113D411yescharacteristic
55C22×C8×C113D411yescharacteristic
56C4⋊Q8×C113—21no—
57C4⋊Q8×C113—21no—
58C4⋊Q8×C113—41no—
59#124×C113—21no—
60#124×C113—21no—
61#124×C113—41no—
62#123×C113—24no—
63#123×C113—21no—
64#123×C113—21no—
65#123×C113—21no—
66#123×C113—21no—
67C22⋊Q8×C113—21no—
68C22⋊Q8×C113—21no—
69C22⋊Q8×C113—21no—
70C22⋊Q8×C113—21no—
71C22⋊Q8×C113—42no—
72#117×C113—21no—
73#117×C113D411yescharacteristic
74#117×C113D411yescharacteristic
75#107×C113—21no—
76#106×C113—21no—
77#104×C113—24no—
78#103×C113—21no—
79#103×C113—21no—
80#103×C113—21no—
81#103×C113—21no—
82#101×C113—21no—
83#101×C113—41no—
84#101×C113D412yes—
85#100×C113—21no—
86#100×C113—21no—
87#100×C113—41no—
88#99×C113—24no—
89#99×C113—21no—
90#99×C113D412yes—
91C22⋊C8×C113—21no—
92C22⋊C8×C113—21no—
93C22⋊C8×C113—42no—
94#41×C113—22no—
95#41×C113—22no—
96#41×C113—41no—
97#41×C113—41no—
98#41×C113—42no—
99#41×C113—41no—
100#41×C113—41no—
101#41×C113—41no—
102#41×C113—41no—
103#41×C113—41no—
104#41×C113—42no—
105#41×C113—42no—
106#41×C113—42no—
107#41×C113—41no—
108#41×C113—41no—
109#41×C113—41no—
110C2×Q8×C113—21no—
111C2×Q8×C113—21no—
112C2×Q8×C113—21no—
113C2×Q8×C113—21no—
114C2×Q8×C113—21no—
115C2×Q8×C113—21no—
116C2×Q8×C113—21no—
117C2×Q8×C113—41no—
118C2×Q8×C113—41no—
119C2×Q8×C113—41no—
120C2×Q8×C113—41no—
121C2×Q8×C113—42no—
122C2×Q8×C113C2×D411yescharacteristic
123C2×D4×C113—21no—
124C2×D4×C113—21no—
125C2×D4×C113—21no—
126C2×D4×C113—22no—
127C2×D4×C113—22no—
128C2×D4×C113—42no—
129C2×D4×C113—42no—
130C2×D4×C113—42no—
131C2×D4×C113—41no—
132C2×D4×C113C2×D411yescharacteristic
133C2×D4×C113C2×D411yescharacteristic
134C2×D4×C113#4111yescharacteristic
135C22×C4×C113—22no—
136C22×C4×C113—22no—
137C22×C4×C113—21no—
138C22×C4×C113—21no—
139C22×C4×C113—41no—
140C22×C4×C113C2×D411yescharacteristic
141C22×C4×C113C2×D411yescharacteristic
142C22×C4×C113C2×D411yescharacteristic
143Q16×C113—44no—
144Q16×C113—81no—
145Q16×C113—81no—
146SD16×C113—44no—
147M16×C113—22no—
148M16×C113—41no—
149M16×C113—41no—
150M16×C113—41no—
151M16×C113—41no—
152C2×C8×C113—21no—
153C2×C8×C113—21no—
154C2×C8×C113—41no—
155C2×C8×C113—41no—
156C2×C8×C113—41no—
157C4⋊C4×C113—21no—
158C4⋊C4×C113—21no—
159C4⋊C4×C113—41no—
160C4⋊C4×C113—44no—
161C4⋊C4×C113—41no—
162C4⋊C4×C113—41no—
163C4⋊C4×C113—41no—
164C4⋊C4×C113—41no—
165C22⋊C4×C113—21no—
166C22⋊C4×C113—22no—
167C22⋊C4×C113—21no—
168C22⋊C4×C113—44no—
169C22⋊C4×C113—41no—
170C22⋊C4×C113—41no—
171C22⋊C4×C113—41no—
172C22⋊C4×C113—41no—
173C42×C113—21no—
174C42×C113—21no—
175C42×C113—41no—
176C23×C113—22no—
177C23×C113—22no—
178C23×C113—41no—
179C23×C113#12011yescharacteristic
180C23×C113#12111yescharacteristic
181C23×C113#12111yescharacteristic
182Q8×C113—22no—
183Q8×C113—41no—
184Q8×C113—41no—
185Q8×C113—41no—
186Q8×C113—42no—
187Q8×C113—41no—
188Q8×C113—41no—
189Q8×C113—41no—
190Q8×C113—41no—
191Q8×C113—41no—
192Q8×C113—81no—
193D4×C113—22no—
194D4×C113—41no—
195D4×C113—41no—
196D4×C113—41no—
197D4×C113—41no—
198D4×C113—41no—
199D4×C113—42no—
200D4×C113—42no—
201D4×C113—42no—
202D4×C113—42no—
203D4×C113—41no—
204C2×C4×C113—21no—
205C2×C4×C113—21no—
206C2×C4×C113—21no—
207C2×C4×C113—21no—
208C2×C4×C113—21no—
209C2×C4×C113—21no—
210C2×C4×C113—22no—
211C2×C4×C113—22no—
212C2×C4×C113—42no—
213C2×C4×C113—42no—
214C2×C4×C113—42no—
215C2×C4×C113—41no—
216C2×C4×C113—42no—
217C2×C4×C113—42no—
218C2×C4×C113—42no—
219C2×C4×C113—41no—
220C2×C4×C113—41no—
221C2×C4×C113—41no—
222C2×C4×C113—41no—
223C2×C4×C113—81no—
224C2×C4×C113#12011yescharacteristic
225C2×C4×C113#12011yescharacteristic
226C2×C4×C113#12111yescharacteristic
227C2×C4×C113#12711yescharacteristic
228C8×C113—42no—
229C8×C113—41no—
230C8×C113—41no—
231C8×C113—81no—
232C22×C113—21no—
233C22×C113—21no—
234C22×C113—21no—
235C22×C113—41no—
236C22×C113—41no—
237C22×C113—41no—
238C22×C113—42no—
239C22×C113—82no—
240C22×C113—82no—
241C22×C113—161no—
242C22×C113#39211yescharacteristic
243C4×C113—21no—
244C4×C113—21no—
245C4×C113—21no—
246C4×C113—42no—
247C4×C113—81no—
248C4×C113—81no—
249C4×C113—82no—
250C4×C113—81no—
251C4×C113—81no—
252C4×C113B(4,𝔽2)11yescharacteristic
253C4×C113B(4,𝔽2)11yescharacteristic
254#13356C11311yesmaximal, characteristic, direct product factor, Sylow
255C2×C113—21no—
256C2×C113—41no—
257C2×C113—41no—
258C2×C113—41no—
259C2×C113—82no—
260C2×C113#202111yescharacteristic
261#3202C2×C11311yescharacteristic, complemented
262#3028C2×C11311yescharacteristic
263#3022C2×C11311yescharacteristic
264#2135C2×C11311yescharacteristic, complemented
265#2123C2×C11311yescharacteristic, complemented
266#1791C2×C11311yescharacteristic
267#1395C2×C11311yescharacteristic, complemented
268C113#1335611yescharacteristic, direct product factor, Sylow
269#585C22×C11311yescharacteristic
270C2×#137—21no—
271#536—21no—
272#536C22×C11311yescharacteristic
273#536C22×C11311yescharacteristic
274#485—21no—
275#485—21no—
276#470—21no—
277#467—21no—
278#458—22no—
279#458—22no—
280#456—22no—
281#455—22no—
282#451—21no—
283#451—21no—
284#421—21no—
285#413—21no—
286#413C22×C11311yescharacteristic
287#413C22×C11311yescharacteristic
288#410—21no—
289#410C22×C11311yescharacteristic
290#408—21no—
291#408C22×C11311yescharacteristic
292#356—22no—
293#352—22no—
2942−1+4—21no—
2952−1+4—41no—
296H(5,𝔽2)—21no—
297H(5,𝔽2)—21no—
298H(5,𝔽2)—41no—
299H(5,𝔽2)D4×C11312yes—
300C2×#41—21no—
301C2×#41—21no—
302C2×#41—21no—
303C2×#41—42no—
304C2×#41C23×C11311yescharacteristic
305C22×Q8D4×C11311yescharacteristic
306C22×Q8D4×C11311yescharacteristic
307#137—24no—
308#137—42no—
309C2×Q16—21no—
310C2×Q16—21no—
311C2×Q16—41no—
312C2×SD16—41no—
313C2×M16—21no—
314C2×M16—21no—
315C2×M16D4×C11311yescharacteristic
316C22×C8D4×C11311yescharacteristic
317C4⋊Q8—21no—
318C4⋊Q8—21no—
319C4⋊Q8—41no—
320#124—21no—
321#124—21no—
322#124—41no—
323#123—24no—
324#123—21no—
325#123—21no—
326#123—21no—
327#123—21no—
328C22⋊Q8—21no—
329C22⋊Q8—21no—
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332C22⋊Q8—42no—
333#117—21no—
334#117D4×C11311yescharacteristic
335#117D4×C11311yescharacteristic
336#107—21no—
337#106—21no—
338#104—24no—
339#103—21no—
340#103—21no—
341#103—21no—
342#103—21no—
343#101—21no—
344#101—41no—
345#101D4×C11312yes—
346#100—21no—
347#100—21no—
348#100—41no—
349#99—24no—
350#99—21no—
351#99D4×C11312yes—
352C22⋊C8—21no—
353C22⋊C8—21no—
354C22⋊C8—42no—
355#41—22no—
356#41—22no—
357#41—41no—
358#41—41no—
359#41—42no—
360#41—41no—
361#41—41no—
362#41—41no—
363#41—41no—
364#41—41no—
365#41—42no—
366#41—42no—
367#41—42no—
368#41—41no—
369#41—41no—
370#41—41no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—41no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8—42no—
383C2×Q8C2×D4×C11311yescharacteristic
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—41no—
393C2×D4C2×D4×C11311yescharacteristic
394C2×D4C2×D4×C11311yescharacteristic
395C2×D4#41×C11311yescharacteristic
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—41no—
401C22×C4C2×D4×C11311yescharacteristic
402C22×C4C2×D4×C11311yescharacteristic
403C22×C4C2×D4×C11311yescharacteristic
404Q16—44no—
405Q16—81no—
406Q16—81no—
407SD16—44no—
408M16—22no—
409M16—41no—
410M16—41no—
411M16—41no—
412M16—41no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—41no—
421C4⋊C4—44no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C22⋊C4—21no—
427C22⋊C4—22no—
428C22⋊C4—21no—
429C22⋊C4—44no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C42—21no—
435C42—21no—
436C42—41no—
437C23—22no—
438C23—22no—
439C23—41no—
440C23#120×C11311yescharacteristic
441C23#121×C11311yescharacteristic
442C23#121×C11311yescharacteristic
443Q8—22no—
444Q8—41no—
445Q8—41no—
446Q8—41no—
447Q8—42no—
448Q8—41no—
449Q8—41no—
450Q8—41no—
451Q8—41no—
452Q8—41no—
453Q8—81no—
454D4—22no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—41no—
459D4—41no—
460D4—42no—
461D4—42no—
462D4—42no—
463D4—42no—
464D4—41no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—81no—
485C2×C4#120×C11311yescharacteristic
486C2×C4#120×C11311yescharacteristic
487C2×C4#121×C11311yescharacteristic
488C2×C4#127×C11311yescharacteristic
489C8—42no—
490C8—41no—
491C8—41no—
492C8—81no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—42no—
500C22—82no—
501C22—82no—
502C22—161no—
503C22#392×C11311yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—42no—
508C4—81no—
509C4—81no—
510C4—82no—
511C4—81no—
512C4—81no—
513C4B(4,𝔽2)×C11311yescharacteristic
514C4B(4,𝔽2)×C11311yescharacteristic
515C2—21no—
516C2—41no—
517C2—41no—
518C2—41no—
519C2—82no—
520C2C113×#202111yescharacteristic
subgroupG/H
center, socleC2×C113#2021
commutator, Frattini, cosocleC2×#41C23×C113
prime radical#585×C113C22
LCSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C113 ◁ C2×C4×C113 ◁ C2×#41×C113 ◁ G

Complements

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

NH ≅ G/Ntype
1. C113×#3202C2semidirect
4. C113×#2135C2semidirect
5. C113×#2123C2semidirect
7. C113×#1395C2semidirect
254. #13356C113direct (N × H)
261. #3202C2×C113semidirect
264. #2135C2×C113semidirect
265. #2123C2×C113semidirect
267. #1395C2×C113semidirect
268. C113#13356direct (N × H)

Maximal subgroups

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

subgroupindex
C113×#32022
C113×#30282
C113×#30222
C113×#21352
C113×#21232
C113×#17912
C113×#13952
#13356113