Finite Group Databasebeta

← C61×#10324 #29513757

Subgroups — C61×#10324 #29513757

Order 15616.

910 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C61×#3189C211yesmaximal, characteristic, complemented
2C61×#2848C212yesmaximal
3C61×#1779C211yesmaximal, characteristic
4C61×#1520C212yesmaximal, complemented
5C61×#1289C211yesmaximal, characteristic, complemented
6C24×C4×C61C2211yescharacteristic
7C2×C22⋊Q8×C61—22no—
8C2×C22⋊Q8×C61C2212yes—
9C61×#484—22no—
10C61×#467—22no—
11C61×#451—22no—
12C2×C22⋊C8×C61—22no—
13C2×C22⋊C8×C61C2211yescharacteristic
14C2×C22⋊C8×C61C2211yescharacteristic
15C61×#385—22no—
16C61×#385—24no—
17C61×#380—22no—
18C61×#380C2212yes—
19C61×#379—22no—
20C61×#328—24no—
21C25×C61D411yescharacteristic
22C22×Q8×C61D412yes—
23C23×C4×C61—22no—
24C23×C4×C61—21no—
25C23×C4×C61—22no—
26C23×C4×C61—21no—
27C23×C4×C61—44no—
28C23×C4×C61D411yescharacteristic
29C23×C4×C61C2311yescharacteristic
30C2×Q16×C61—22no—
31C22×C8×C61—21no—
32C22×C8×C61D412yes—
33C22⋊Q8×C61—22no—
34C22⋊Q8×C61—22no—
35C22⋊Q8×C61—22no—
36C22⋊Q8×C61—22no—
37C22⋊Q8×C61—48no—
38C2×C4⋊C4×C61—24no—
39C2×C4⋊C4×C61—22no—
40C2×C4⋊C4×C61—22no—
41C2×C4⋊C4×C61D412yes—
42C2×C22⋊C4×C61—24no—
43C2×C22⋊C4×C61—24no—
44C2×C22⋊C4×C61—22no—
45C2×C22⋊C4×C61—22no—
46#107×C61—22no—
47#103×C61—24no—
48C22⋊C8×C61—22no—
49C22⋊C8×C61—22no—
50C22⋊C8×C61—44no—
51#95×C61—24no—
52#95×C61D414yes—
53C24×C61—22no—
54C24×C61—21no—
55C24×C61—21no—
56C24×C61—22no—
57C24×C61—42no—
58C24×C61—42no—
59C24×C61Q1612yescomplemented
60C24×C61C2×D411yescharacteristic
61C2×Q8×C61—22no—
62C2×Q8×C61—22no—
63C2×Q8×C61—44no—
64C22×C4×C61—24no—
65C22×C4×C61—28no—
66C22×C4×C61—22no—
67C22×C4×C61—21no—
68C22×C4×C61—22no—
69C22×C4×C61—24no—
70C22×C4×C61—22no—
71C22×C4×C61—42no—
72C22×C4×C61—42no—
73C22×C4×C61—44no—
74C22×C4×C61—42no—
75C22×C4×C61—44no—
76C22×C4×C61—42no—
77C22×C4×C61—44no—
78C22×C4×C61—42no—
79C22×C4×C61—41no—
80C22×C4×C61—42no—
81C22×C4×C61—41no—
82C22×C4×C61C2×D412yes—
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D412yes—
85C22×C4×C61#4111yescharacteristic
86Q16×C61—82no—
87C2×C8×C61—22no—
88C2×C8×C61—42no—
89C2×C8×C61—42no—
90C2×C8×C61—41no—
91C4⋊C4×C61—22no—
92C4⋊C4×C61—22no—
93C4⋊C4×C61—44no—
94C4⋊C4×C61—48no—
95C4⋊C4×C61—42no—
96C4⋊C4×C61—42no—
97C22⋊C4×C61—48no—
98C22⋊C4×C61—48no—
99C22⋊C4×C61—42no—
100C22⋊C4×C61—42no—
101C22⋊C4×C61—42no—
102C22⋊C4×C61—42no—
103C23×C61—24no—
104C23×C61—24no—
105C23×C61—22no—
106C23×C61—22no—
107C23×C61—24no—
108C23×C61—22no—
109C23×C61—41no—
110C23×C61—44no—
111C23×C61—41no—
112C23×C61—42no—
113C23×C61—41no—
114C23×C61—42no—
115C23×C61—41no—
116C23×C61—44no—
117C23×C61—48no—
118C23×C61—41no—
119C23×C61—42no—
120C23×C61—41no—
121C23×C61#12012yes—
122C23×C61#12111yescharacteristic
123C23×C61C2×Q1611yescharacteristic
124C23×C61#13511yescharacteristic
125C23×C61#13712yes—
126Q8×C61—44no—
127Q8×C61—82no—
128C2×C4×C61—24no—
129C2×C4×C61—22no—
130C2×C4×C61—24no—
131C2×C4×C61—22no—
132C2×C4×C61—42no—
133C2×C4×C61—41no—
134C2×C4×C61—44no—
135C2×C4×C61—48no—
136C2×C4×C61—42no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—44no—
140C2×C4×C61—44no—
141C2×C4×C61—44no—
142C2×C4×C61—44no—
143C2×C4×C61—44no—
144C2×C4×C61—42no—
145C2×C4×C61—42no—
146C2×C4×C61—44no—
147C2×C4×C61#12011yescharacteristic
148C2×C4×C61#12112yes—
149C2×C4×C61#12711yescharacteristic
150C8×C61—42no—
151C8×C61—81no—
152#10324C6111yesmaximal, characteristic, direct product factor, Sylow
153C22×C61—22no—
154C22×C61—21no—
155C22×C61—22no—
156C22×C61—21no—
157C22×C61—28no—
158C22×C61—24no—
159C22×C61—41no—
160C22×C61—41no—
161C22×C61—42no—
162C22×C61—44no—
163C22×C61—44no—
164C22×C61—44no—
165C22×C61—42no—
166C22×C61—42no—
167C22×C61—42no—
168C22×C61—42no—
169C22×C61—42no—
170C22×C61—42no—
171C22×C61#39211yescharacteristic
172C22×C61#45112yes—
173C22×C61#45212yes—
174C22×C61#46711yescharacteristic
175C22×C61#47011yescharacteristic
176C4×C61—21no—
177C4×C61—22no—
178C4×C61—42no—
179C4×C61—84no—
180C4×C61—82no—
181C4×C61—82no—
182C4×C61B(4,𝔽2)12yes—
183#3189C2×C6111yescharacteristic, complemented
184#2848C2×C6112yes—
185#1779C2×C6111yescharacteristic
186#1520C2×C6112yescomplemented
187#1289C2×C6111yescharacteristic, complemented
188C2×C61—22no—
189C2×C61—22no—
190C2×C61—21no—
191C2×C61—21no—
192C2×C61—44no—
193C2×C61#200111yescharacteristic
194C2×C61#201011yescharacteristic
195C2×C61#202111yescharacteristic
196C24×C4C22×C6111yescharacteristic
197C2×C22⋊Q8—22no—
198C2×C22⋊Q8C22×C6112yes—
199#484—22no—
200#467—22no—
201#451—22no—
202C2×C22⋊C8—22no—
203C2×C22⋊C8C22×C6111yescharacteristic
204C2×C22⋊C8C22×C6111yescharacteristic
205#385—22no—
206#385—24no—
207#380—22no—
208#380C22×C6112yes—
209#379—22no—
210#328—24no—
211C61#1032411yescharacteristic, direct product factor, Sylow
212C25D4×C6111yescharacteristic
213C22×Q8D4×C6112yes—
214C23×C4—22no—
215C23×C4—21no—
216C23×C4—22no—
217C23×C4—21no—
218C23×C4—44no—
219C23×C4D4×C6111yescharacteristic
220C23×C4C23×C6111yescharacteristic
221C2×Q16—22no—
222C22×C8—21no—
223C22×C8D4×C6112yes—
224C22⋊Q8—22no—
225C22⋊Q8—22no—
226C22⋊Q8—22no—
227C22⋊Q8—22no—
228C22⋊Q8—48no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4D4×C6112yes—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237#107—22no—
238#103—24no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241C22⋊C8—44no—
242#95—24no—
243#95D4×C6114yes—
244C24—22no—
245C24—21no—
246C24—21no—
247C24—22no—
248C24—42no—
249C24—42no—
250C24Q16×C6112yescomplemented
251C24C2×D4×C6111yescharacteristic
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—44no—
255C22×C4—24no—
256C22×C4—28no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—42no—
266C22×C4—44no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4C2×D4×C6112yes—
274C22×C4C2×D4×C6111yescharacteristic
275C22×C4C2×D4×C6112yes—
276C22×C4#41×C6111yescharacteristic
277Q16—82no—
278C2×C8—22no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—41no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—44no—
285C4⋊C4—48no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C22⋊C4—48no—
289C22⋊C4—48no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C23—24no—
295C23—24no—
296C23—22no—
297C23—22no—
298C23—24no—
299C23—22no—
300C23—41no—
301C23—44no—
302C23—41no—
303C23—42no—
304C23—41no—
305C23—42no—
306C23—41no—
307C23—44no—
308C23—48no—
309C23—41no—
310C23—42no—
311C23—41no—
312C23#120×C6112yes—
313C23#121×C6111yescharacteristic
314C23C2×Q16×C6111yescharacteristic
315C23#135×C6111yescharacteristic
316C23#137×C6112yes—
317Q8—44no—
318Q8—82no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—22no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—44no—
326C2×C4—48no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4#120×C6111yescharacteristic
339C2×C4#121×C6112yes—
340C2×C4#127×C6111yescharacteristic
341C8—42no—
342C8—81no—
343C22—22no—
344C22—21no—
345C22—22no—
346C22—21no—
347C22—28no—
348C22—24no—
349C22—41no—
350C22—41no—
351C22—42no—
352C22—44no—
353C22—44no—
354C22—44no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22C61×#39211yescharacteristic
362C22C61×#45112yes—
363C22C61×#45212yes—
364C22C61×#46711yescharacteristic
365C22C61×#47011yescharacteristic
366C4—21no—
367C4—22no—
368C4—42no—
369C4—84no—
370C4—82no—
371C4—82no—
372C4C61×B(4,𝔽2)12yes—
373C2—22no—
374C2—22no—
375C2—21no—
376C2—21no—
377C2—44no—
378C2C61×#200111yescharacteristic
379C2C61×#201011yescharacteristic
380C2C61×#202111yescharacteristic
subgroupG/H
center, socleC22×C61#392
commutator, Frattini, cosocleC23×C4C23×C61
prime radicalC25×C61D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C61 ◁ C23×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C61×#3189C2semidirect
4. C61×#1520C2semidirect
5. C61×#1289C2semidirect
59. C24×C61Q16semidirect
152. #10324C61direct (N × H)
183. #3189C2×C61semidirect
186. #1520C2×C61semidirect
187. #1289C2×C61semidirect
211. C61#10324direct (N × H)
250. C24Q16×C61semidirect

Maximal subgroups

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

subgroupindex
C61×#31892
C61×#28482
C61×#17792
C61×#15202
C61×#12892
#1032461