Finite Group Databasebeta

← C2×C61×#2198 #29536978

Subgroups — C2×C61×#2198 #29536978

Order 15616.

1244 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1244 conjugacy classes of subgroups — too many to draw. Only its 222 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×C61×#559C211yesmaximal, characteristic, complemented
2C2×C61×#458C212yesmaximal, complemented
3C2×C61×#453C211yesmaximal, characteristic
4C61×#2198C218yesmaximal, direct product factor
5C2×C61×#353C211yesmaximal, characteristic, complemented
6C2×C61×#352C212yesmaximal, complemented
7C2×H(5,𝔽2)×C61C2211yescharacteristic
8C2×AGL(1,ℤ/8)×C61—21no—
9C61×#559—22no—
10C61×#559—24no—
11C61×#559C2214yescomplemented
12C2×#127×C61C2211yescharacteristic
13C2×#124×C61—22no—
14C2×#123×C61—21no—
15C2×#123×C61—22no—
16C2×#123×C61C2212yes—
17C2×C4×Q8×C61—21no—
18C2×C4×D4×C61—22no—
19C61×#458—24no—
20C61×#458C2218yescomplemented
21C61×#453—22no—
22C61×#453C2214yes—
23C2×#104×C61—21no—
24C2×#100×C61C2211yescharacteristic
25C2×#99×C61—22no—
26C2×#99×C61C2212yes—
27C61×#353C2214yescomplemented
28C61×#352C2218yescomplemented
29H(5,𝔽2)×C61—22no—
30H(5,𝔽2)×C61—21no—
31H(5,𝔽2)×C61—42no—
32H(5,𝔽2)×C61C2312yes—
33C2×#41×C61—21no—
34C2×#41×C61—41no—
35C22×Q8×C61—21no—
36C22×D4×C61—22no—
37C22×D4×C61—22no—
38C22×D4×C61—41no—
39C22×D4×C61D411yescharacteristic
40C22×D4×C61C2311yescharacteristic
41C23×C4×C61—21no—
42C23×C4×C61D412yes—
43AGL(1,ℤ/8)×C61—24no—
44AGL(1,ℤ/8)×C61—42no—
45C2×SD16×C61—41no—
46C2×D8×C61—41no—
47C2×M16×C61—21no—
48#127×C61—22no—
49#127×C61D412yes—
50#127×C61C2312yes—
51#124×C61—28no—
52#124×C61—28no—
53#123×C61—24no—
54#123×C61—22no—
55#123×C61—22no—
56#123×C61—24no—
57#123×C61—28no—
58#123×C61—44no—
59#123×C61D414yes—
60#123×C61C2314yes—
61C4×Q8×C61—24no—
62C4×Q8×C61—24no—
63C4×D4×C61—28no—
64C4×D4×C61—28no—
65C2×C4⋊C4×C61—21no—
66C2×C4⋊C4×C61—22no—
67C2×C4⋊C4×C61—42no—
68C2×C22⋊C4×C61—22no—
69C2×C22⋊C4×C61—21no—
70C2×C22⋊C4×C61—22no—
71C2×C22⋊C4×C61—22no—
72C2×C22⋊C4×C61—42no—
73C2×C22⋊C4×C61D412yes—
74C2×C42×C61—22no—
75C2×C42×C61D411yescharacteristic
76#104×C61—24no—
77#100×C61—21no—
78#100×C61C2312yes—
79#99×C61—22no—
80#99×C61—28no—
81#99×C61C2314yes—
82C24×C61—22no—
83C24×C61—21no—
84C24×C61C2×D412yes—
85#41×C61—22no—
86#41×C61—22no—
87#41×C61—42no—
88#41×C61—41no—
89#41×C61—44no—
90#41×C61—42no—
91#41×C61—41no—
92C2×Q8×C61—24no—
93C2×Q8×C61—21no—
94C2×Q8×C61—44no—
95C2×D4×C61—21no—
96C2×D4×C61—24no—
97C2×D4×C61—24no—
98C2×D4×C61—24no—
99C2×D4×C61—24no—
100C2×D4×C61—21no—
101C2×D4×C61—41no—
102C2×D4×C61—41no—
103C2×D4×C61—48no—
104C2×D4×C61—41no—
105C2×D4×C61—41no—
106C2×D4×C61—42no—
107C2×D4×C61—42no—
108C2×D4×C61—42no—
109C2×D4×C61—42no—
110C2×D4×C61—44no—
111C2×D4×C61—42no—
112C2×D4×C61—41no—
113C2×D4×C61C2×D412yes—
114C2×D4×C61C2×D412yes—
115C2×D4×C61C2×D411yescharacteristic
116C2×D4×C61C2411yescharacteristic
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—24no—
122C22×C4×C61—22no—
123C22×C4×C61—41no—
124C22×C4×C61—42no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61—81no—
128C22×C4×C61—81no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D414yes—
131SD16×C61—44no—
132D8×C61—44no—
133M16×C61—22no—
134M16×C61—41no—
135C2×C8×C61—41no—
136C4⋊C4×C61—24no—
137C4⋊C4×C61—24no—
138C4⋊C4×C61—24no—
139C4⋊C4×C61—48no—
140C22⋊C4×C61—28no—
141C22⋊C4×C61—24no—
142C22⋊C4×C61—24no—
143C22⋊C4×C61—22no—
144C22⋊C4×C61—24no—
145C22⋊C4×C61—24no—
146C22⋊C4×C61—42no—
147C22⋊C4×C61—48no—
148C22⋊C4×C61C2×D414yes—
149C42×C61—28no—
150C42×C61—21no—
151C42×C61C2×D412yes—
152C23×C61—22no—
153C23×C61—22no—
154C23×C61—24no—
155C23×C61—22no—
156C23×C61—44no—
157C23×C61—44no—
158C23×C61—42no—
159C23×C61—41no—
160C23×C61—41no—
161C23×C61—41no—
162C23×C61—84no—
163C23×C61#12012yes—
164C23×C61#12011yescharacteristic
165C23×C61C22×D412yes—
166Q8×C61—22no—
167Q8×C61—48no—
168Q8×C61—41no—
169D4×C61—22no—
170D4×C61—41no—
171D4×C61—42no—
172D4×C61—41no—
173D4×C61—42no—
174D4×C61—48no—
175D4×C61—48no—
176D4×C61—41no—
177D4×C61—44no—
178D4×C61—44no—
179D4×C61—42no—
180D4×C61—42no—
181D4×C61—81no—
182C2×C4×C61—24no—
183C2×C4×C61—22no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—24no—
187C2×C4×C61—24no—
188C2×C4×C61—24no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—44no—
193C2×C4×C61—42no—
194C2×C4×C61—41no—
195C2×C4×C61—42no—
196C2×C4×C61—44no—
197C2×C4×C61—44no—
198C2×C4×C61—82no—
199C2×C4×C61—84no—
200C2×C4×C61—84no—
201C2×C4×C61#12011yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C8×C61—42no—
204C2×#2198C6111yesmaximal, characteristic, direct product factor, Sylow
205C22×C61—22no—
206C22×C61—22no—
207C22×C61—22no—
208C22×C61—42no—
209C22×C61—42no—
210C22×C61—44no—
211C22×C61—44no—
212C22×C61—42no—
213C22×C61—42no—
214C22×C61—82no—
215C22×C61—82no—
216C22×C61—84no—
217C22×C61—82no—
218C22×C61—82no—
219C22×C61—162no—
220C22×C61B(4,𝔽2)11yescharacteristic
221C22×C61B(4,𝔽2)11yescharacteristic
222C22×C61C2×#12011yescharacteristic
223C4×C61—24no—
224C4×C61—21no—
225C4×C61—21no—
226C4×C61—44no—
227C4×C61—42no—
228C4×C61—44no—
229C4×C61—84no—
230C2×#559C2×C6111yescharacteristic, complemented
231C2×#458C2×C6112yescomplemented
232C2×#453C2×C6111yescharacteristic
233#2198C2×C6118yesdirect product factor
234C2×#353C2×C6111yescharacteristic, complemented
235C2×#352C2×C6112yescomplemented
236C2×C61—21no—
237C2×C61—21no—
238C2×C61—42no—
239C2×C61—42no—
240C2×C61—82no—
241C2×C61—82no—
242C2×C61#219812yesdirect product factor
243C2×C61C2×B(4,𝔽2)11yescharacteristic
244C2×H(5,𝔽2)C22×C6111yescharacteristic
245C2×AGL(1,ℤ/8)—21no—
246#559—22no—
247#559—24no—
248#559C22×C6114yescomplemented
249C2×#127C22×C6111yescharacteristic
250C2×#124—22no—
251C2×#123—21no—
252C2×#123—22no—
253C2×#123C22×C6112yes—
254C2×C4×Q8—21no—
255C2×C4×D4—22no—
256#458—24no—
257#458C22×C6118yescomplemented
258#453—22no—
259#453C22×C6114yes—
260C2×#104—21no—
261C2×#100C22×C6111yescharacteristic
262C2×#99—22no—
263C2×#99C22×C6112yes—
264#353C22×C6114yescomplemented
265#352C22×C6118yescomplemented
266C61C2×#219811yescharacteristic, direct product factor, Sylow
267H(5,𝔽2)—22no—
268H(5,𝔽2)—21no—
269H(5,𝔽2)—42no—
270H(5,𝔽2)C23×C6112yes—
271C2×#41—21no—
272C2×#41—41no—
273C22×Q8—21no—
274C22×D4—22no—
275C22×D4—22no—
276C22×D4—41no—
277C22×D4D4×C6111yescharacteristic
278C22×D4C23×C6111yescharacteristic
279C23×C4—21no—
280C23×C4D4×C6112yes—
281AGL(1,ℤ/8)—24no—
282AGL(1,ℤ/8)—42no—
283C2×SD16—41no—
284C2×D8—41no—
285C2×M16—21no—
286#127—22no—
287#127D4×C6112yes—
288#127C23×C6112yes—
289#124—28no—
290#124—28no—
291#123—24no—
292#123—24no—
293#123—22no—
294#123—22no—
295#123—28no—
296#123—44no—
297#123D4×C6114yes—
298#123C23×C6114yes—
299C4×Q8—24no—
300C4×Q8—24no—
301C4×D4—28no—
302C4×D4—28no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—42no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4D4×C6112yes—
312C2×C42—22no—
313C2×C42D4×C6111yescharacteristic
314#104—24no—
315#100—21no—
316#100C23×C6112yes—
317#99—22no—
318#99—28no—
319#99C23×C6114yes—
320C24—22no—
321C24—21no—
322C24C2×D4×C6112yes—
323#41—22no—
324#41—22no—
325#41—42no—
326#41—41no—
327#41—44no—
328#41—42no—
329#41—41no—
330C2×Q8—24no—
331C2×Q8—21no—
332C2×Q8—44no—
333C2×D4—21no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—21no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—48no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—44no—
349C2×D4—41no—
350C2×D4—42no—
351C2×D4C2×D4×C6112yes—
352C2×D4C2×D4×C6112yes—
353C2×D4C2×D4×C6111yescharacteristic
354C2×D4C24×C6111yescharacteristic
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—81no—
366C22×C4—81no—
367C22×C4C2×D4×C6111yescharacteristic
368C22×C4C2×D4×C6114yes—
369SD16—44no—
370D8—44no—
371M16—22no—
372M16—41no—
373C2×C8—41no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—48no—
378C22⋊C4—28no—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—22no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—42no—
385C22⋊C4—48no—
386C22⋊C4C2×D4×C6114yes—
387C42—28no—
388C42—21no—
389C42C2×D4×C6112yes—
390C23—22no—
391C23—22no—
392C23—24no—
393C23—22no—
394C23—44no—
395C23—44no—
396C23—42no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—84no—
401C23#120×C6111yescharacteristic
402C23#120×C6112yes—
403C23C22×D4×C6112yes—
404Q8—22no—
405Q8—48no—
406Q8—41no—
407D4—22no—
408D4—41no—
409D4—42no—
410D4—41no—
411D4—42no—
412D4—48no—
413D4—48no—
414D4—41no—
415D4—44no—
416D4—44no—
417D4—42no—
418D4—42no—
419D4—81no—
420C2×C4—24no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—82no—
437C2×C4—84no—
438C2×C4—84no—
439C2×C4#120×C6111yescharacteristic
440C2×C4C22×D4×C6111yescharacteristic
441C8—42no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—42no—
446C22—42no—
447C22—44no—
448C22—44no—
449C22—42no—
450C22—42no—
451C22—82no—
452C22—82no—
453C22—84no—
454C22—82no—
455C22—82no—
456C22—162no—
457C22C61×B(4,𝔽2)11yescharacteristic
458C22C61×B(4,𝔽2)11yescharacteristic
459C22C2×#120×C6111yescharacteristic
460C4—24no—
461C4—21no—
462C4—21no—
463C4—44no—
464C4—42no—
465C4—44no—
466C4—84no—
467C2—21no—
468C2—21no—
469C2—42no—
470C2—42no—
471C2—82no—
472C2—82no—
473C2C61×#219812yesdirect product factor
474C2C2×C61×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C61B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C24×C61
prime radicalC2×C61×#453C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C23×C61 ◁ C22×D4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C61×#559C2semidirect
2. C2×C61×#458C2semidirect
4. C61×#2198C2direct (N × H)
5. C2×C61×#353C2semidirect
6. C2×C61×#352C2semidirect
11. C61×#559C22semidirect
20. C61×#458C22semidirect
27. C61×#353C22semidirect
28. C61×#352C22semidirect
204. C2×#2198C61direct (N × H)
230. C2×#559C2×C61semidirect
231. C2×#458C2×C61semidirect
233. #2198C2×C61direct (N × H)
234. C2×#353C2×C61semidirect
235. C2×#352C2×C61semidirect
242. C2×C61#2198direct (N × H)
248. #559C22×C61semidirect
257. #458C22×C61semidirect
264. #353C22×C61semidirect
265. #352C22×C61semidirect
266. C61C2×#2198direct (N × H)
473. C2C61×#2198direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C61×#5592
C2×C61×#4582
C2×C61×#4532
C61×#21982
C2×C61×#3532
C2×C61×#3522
C2×#219861