Finite Group Databasebeta

← C61×#32926 #29536359

Subgroups — C61×#32926 #29536359

Order 15616.

1922 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 1922 conjugacy classes of subgroups — too many to draw. Only its 298 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
1C4×H(5,𝔽2)×C61C211yesmaximal, characteristic, complemented
2C2×C61×B(4,𝔽2)C212yesmaximal, complemented
3C61×#2408C213yesmaximal, complemented
4C61×#1889C216yesmaximal, complemented
5C61×#1799C213yesmaximal, complemented
6C2×H(5,𝔽2)×C61C2211yescharacteristic
7C23×D4×C61C2211yescharacteristic, complemented
8C2×#121×C61—23no—
9C2×#120×C61C2216yes—
10C61×#520—23no—
11C61×#518—23no—
12C61×#518C2213yes—
13C4×#41×C61—23no—
14C2×C4×D4×C61—23no—
15C2×C4×D4×C61C2213yes—
16C61×B(4,𝔽2)—28no—
17C2×#99×C61C2216yes—
18C61×#394—23no—
19C61×#392—23no—
20C61×#390—23no—
21C61×#380C2213yescomplemented
22C61×#379C2216yescomplemented
23C61×#342C2216yescomplemented
24C25×C61C2312yes—
25H(5,𝔽2)×C61—26no—
26H(5,𝔽2)×C61D414yescomplemented
27C2×#41×C61—23no—
28C22×D4×C61—23no—
29C22×D4×C61—26no—
30C22×D4×C61—23no—
31C22×D4×C61—23no—
32C22×D4×C61—44no—
33C22×D4×C61C2313yes—
34C23×C4×C61—21no—
35C23×C4×C61D413yescomplemented
36C23×C4×C61C2311yescharacteristic
37#121×C61—412no—
38#120×C61—212no—
39#120×C61—212no—
40C4×Q8×C61—21no—
41C4×Q8×C61—23no—
42C4×D4×C61—23no—
43C4×D4×C61—26no—
44C4×D4×C61—23no—
45C4×D4×C61—26no—
46C4×D4×C61—26no—
47C4×D4×C61—26no—
48C4×D4×C61—26no—
49#117×C61—26no—
50#117×C61—23no—
51C2×C4⋊C4×C61—23no—
52C2×C4⋊C4×C61—23no—
53C2×C4⋊C4×C61D413yescomplemented
54C2×C22⋊C4×C61—23no—
55C2×C22⋊C4×C61—23no—
56C2×C22⋊C4×C61—26no—
57C2×C22⋊C4×C61—26no—
58C2×C22⋊C4×C61—26no—
59C2×C22⋊C4×C61D413yescomplemented
60C2×C22⋊C4×C61C2316yes—
61C2×C22⋊C4×C61C2313yes—
62C2×C42×C61—23no—
63C2×C42×C61D413yescomplemented
64#99×C61—212no—
65#95×C61—23no—
66C24×C61—21no—
67C24×C61—26no—
68C24×C61—22no—
69C24×C61—26no—
70C24×C61—44no—
71C24×C61—82no—
72C24×C61C2×D413yes—
73C24×C61C2411yescharacteristic
74#41×C61—212no—
75#41×C61—212no—
76C2×Q8×C61—21no—
77C2×D4×C61—23no—
78C2×D4×C61—212no—
79C2×D4×C61—212no—
80C2×D4×C61—26no—
81C2×D4×C61—26no—
82C2×D4×C61—212no—
83C2×D4×C61—46no—
84C2×D4×C61—412no—
85C2×D4×C61—412no—
86C2×D4×C61—412no—
87C2×D4×C61—43no—
88C2×D4×C61—41no—
89C2×D4×C61—86no—
90C2×D4×C61C2×D416yes—
91C2×D4×C61#4116yes—
92C22×C4×C61—23no—
93C22×C4×C61—23no—
94C22×C4×C61—23no—
95C22×C4×C61—23no—
96C22×C4×C61—23no—
97C22×C4×C61—26no—
98C22×C4×C61—42no—
99C22×C4×C61—46no—
100C22×C4×C61—82no—
101C22×C4×C61C2×D413yes—
102C22×C4×C61C2×D413yes—
103C22×C4×C61C2×D413yes—
104C22×C4×C61C2×D413yes—
105C22×C4×C61C2×D413yes—
106C4⋊C4×C61—23no—
107C4⋊C4×C61—23no—
108C4⋊C4×C61—26no—
109C4⋊C4×C61—46no—
110C22⋊C4×C61—26no—
111C22⋊C4×C61—26no—
112C22⋊C4×C61—26no—
113C22⋊C4×C61—26no—
114C22⋊C4×C61—26no—
115C22⋊C4×C61—412no—
116C22⋊C4×C61—412no—
117C22⋊C4×C61—86no—
118C42×C61—26no—
119C42×C61—23no—
120C42×C61—23no—
121C23×C61—24no—
122C23×C61—23no—
123C23×C61—23no—
124C23×C61—212no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61—43no—
128C23×C61—43no—
129C23×C61—46no—
130C23×C61—46no—
131C23×C61—46no—
132C23×C61—42no—
133C23×C61—46no—
134C23×C61—46no—
135C23×C61—84no—
136C23×C61—812no—
137C23×C61—81no—
138C23×C61—81no—
139C23×C61#12116yescomplemented
140C23×C61C22×D413yes—
141C23×C61C22×D411yescharacteristic
142C23×C61C2×#4113yes—
143Q8×C61—24no—
144D4×C61—212no—
145D4×C61—212no—
146D4×C61—212no—
147D4×C61—412no—
148D4×C61—44no—
149D4×C61—824no—
150C2×C4×C61—212no—
151C2×C4×C61—23no—
152C2×C4×C61—23no—
153C2×C4×C61—26no—
154C2×C4×C61—23no—
155C2×C4×C61—23no—
156C2×C4×C61—43no—
157C2×C4×C61—46no—
158C2×C4×C61—43no—
159C2×C4×C61—43no—
160C2×C4×C61—46no—
161C2×C4×C61—46no—
162C2×C4×C61—46no—
163C2×C4×C61—83no—
164C2×C4×C61—83no—
165C2×C4×C61—83no—
166C2×C4×C61#12013yes—
167C2×C4×C61#12011yescharacteristic
168C2×C4×C61#12116yescomplemented
169#32926C6111yesmaximal, characteristic, direct product factor, Sylow
170C22×C61—23no—
171C22×C61—23no—
172C22×C61—26no—
173C22×C61—41no—
174C22×C61—43no—
175C22×C61—46no—
176C22×C61—412no—
177C22×C61—412no—
178C22×C61—43no—
179C22×C61—43no—
180C22×C61—44no—
181C22×C61—83no—
182C22×C61—86no—
183C22×C61—86no—
184C22×C61—84no—
185C22×C61—82no—
186C22×C61—82no—
187C22×C61C2×#12011yescharacteristic
188C22×C61C2×#12113yes—
189C22×C61#52513yes—
190C4×C61—26no—
191C4×C61—23no—
192C4×C61—43no—
193C4×C61—43no—
194C4×C61—86no—
195C4×C61B(4,𝔽2)12yescomplemented
196C4×H(5,𝔽2)C2×C6111yescharacteristic, complemented
197C2×B(4,𝔽2)C2×C6112yescomplemented
198#2408C2×C6113yescomplemented
199#1889C2×C6116yescomplemented
200#1799C2×C6113yescomplemented
201C2×C61—23no—
202C2×C61—23no—
203C2×C61—46no—
204C2×C61—84no—
205C2×C61#284711yescharacteristic
206C2×C61C2×B(4,𝔽2)11yescharacteristic
207C2×C61#302711yescharacteristic
208C2×H(5,𝔽2)C22×C6111yescharacteristic
209C23×D4C22×C6111yescharacteristic, complemented
210C2×#121—23no—
211C2×#120C22×C6116yes—
212#520—23no—
213#518—23no—
214#518C22×C6113yes—
215C4×#41—23no—
216C2×C4×D4—23no—
217C2×C4×D4C22×C6113yes—
218B(4,𝔽2)—28no—
219C2×#99C22×C6116yes—
220#394—23no—
221#392—23no—
222#390—23no—
223#380C22×C6113yescomplemented
224#379C22×C6116yescomplemented
225#342C22×C6116yescomplemented
226C61#3292611yescharacteristic, direct product factor, Sylow
227C25C23×C6112yes—
228H(5,𝔽2)—26no—
229H(5,𝔽2)D4×C6114yescomplemented
230C2×#41—23no—
231C22×D4—23no—
232C22×D4—26no—
233C22×D4—23no—
234C22×D4—23no—
235C22×D4—44no—
236C22×D4C23×C6113yes—
237C23×C4—21no—
238C23×C4D4×C6113yescomplemented
239C23×C4C23×C6111yescharacteristic
240#121—412no—
241#120—212no—
242#120—212no—
243C4×Q8—21no—
244C4×Q8—23no—
245C4×D4—23no—
246C4×D4—26no—
247C4×D4—23no—
248C4×D4—26no—
249C4×D4—26no—
250C4×D4—26no—
251C4×D4—26no—
252#117—26no—
253#117—23no—
254C2×C4⋊C4—23no—
255C2×C4⋊C4—23no—
256C2×C4⋊C4D4×C6113yescomplemented
257C2×C22⋊C4—23no—
258C2×C22⋊C4—23no—
259C2×C22⋊C4—26no—
260C2×C22⋊C4—26no—
261C2×C22⋊C4—26no—
262C2×C22⋊C4D4×C6113yescomplemented
263C2×C22⋊C4C23×C6116yes—
264C2×C22⋊C4C23×C6113yes—
265C2×C42—23no—
266C2×C42D4×C6113yescomplemented
267#99—212no—
268#95—23no—
269C24—21no—
270C24—26no—
271C24—22no—
272C24—26no—
273C24—44no—
274C24—82no—
275C24C2×D4×C6113yes—
276C24C24×C6111yescharacteristic
277#41—212no—
278#41—212no—
279C2×Q8—21no—
280C2×D4—23no—
281C2×D4—212no—
282C2×D4—212no—
283C2×D4—26no—
284C2×D4—26no—
285C2×D4—212no—
286C2×D4—46no—
287C2×D4—412no—
288C2×D4—412no—
289C2×D4—412no—
290C2×D4—43no—
291C2×D4—41no—
292C2×D4—86no—
293C2×D4C2×D4×C6116yes—
294C2×D4#41×C6116yes—
295C22×C4—23no—
296C22×C4—23no—
297C22×C4—23no—
298C22×C4—23no—
299C22×C4—23no—
300C22×C4—26no—
301C22×C4—42no—
302C22×C4—46no—
303C22×C4—82no—
304C22×C4C2×D4×C6113yes—
305C22×C4C2×D4×C6113yes—
306C22×C4C2×D4×C6113yes—
307C22×C4C2×D4×C6113yes—
308C22×C4C2×D4×C6113yes—
309C4⋊C4—23no—
310C4⋊C4—23no—
311C4⋊C4—26no—
312C4⋊C4—46no—
313C22⋊C4—26no—
314C22⋊C4—26no—
315C22⋊C4—26no—
316C22⋊C4—26no—
317C22⋊C4—26no—
318C22⋊C4—412no—
319C22⋊C4—412no—
320C22⋊C4—86no—
321C42—26no—
322C42—23no—
323C42—23no—
324C23—24no—
325C23—23no—
326C23—23no—
327C23—212no—
328C23—41no—
329C23—41no—
330C23—43no—
331C23—43no—
332C23—46no—
333C23—46no—
334C23—46no—
335C23—42no—
336C23—46no—
337C23—46no—
338C23—84no—
339C23—812no—
340C23—81no—
341C23—81no—
342C23#121×C6116yescomplemented
343C23C22×D4×C6113yes—
344C23C22×D4×C6111yescharacteristic
345C23C2×#41×C6113yes—
346Q8—24no—
347D4—212no—
348D4—212no—
349D4—212no—
350D4—412no—
351D4—44no—
352D4—824no—
353C2×C4—212no—
354C2×C4—23no—
355C2×C4—23no—
356C2×C4—26no—
357C2×C4—23no—
358C2×C4—23no—
359C2×C4—43no—
360C2×C4—46no—
361C2×C4—43no—
362C2×C4—43no—
363C2×C4—46no—
364C2×C4—46no—
365C2×C4—46no—
366C2×C4—83no—
367C2×C4—83no—
368C2×C4—83no—
369C2×C4#120×C6113yes—
370C2×C4#120×C6111yescharacteristic
371C2×C4#121×C6116yescomplemented
372C22—23no—
373C22—23no—
374C22—26no—
375C22—41no—
376C22—43no—
377C22—46no—
378C22—412no—
379C22—412no—
380C22—43no—
381C22—43no—
382C22—44no—
383C22—83no—
384C22—86no—
385C22—86no—
386C22—84no—
387C22—82no—
388C22—82no—
389C22C2×#120×C6111yescharacteristic
390C22C2×#121×C6113yes—
391C22C61×#52513yes—
392C4—26no—
393C4—23no—
394C4—43no—
395C4—43no—
396C4—86no—
397C4C61×B(4,𝔽2)12yescomplemented
398C2—23no—
399C2—23no—
400C2—46no—
401C2—84no—
402C2C61×#284711yescharacteristic
403C2C2×C61×B(4,𝔽2)11yescharacteristic
404C2C61×#302711yescharacteristic
subgroupG/H
center, socleC22×C61C2×#120
commutator, Frattini, cosocleC24C24×C61
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C23×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C4×H(5,𝔽2)×C61C2semidirect
2. C2×C61×B(4,𝔽2)C2semidirect
3. C61×#2408C2semidirect
4. C61×#1889C2semidirect
5. C61×#1799C2semidirect
7. C23×D4×C61C22semidirect
21. C61×#380C22semidirect
22. C61×#379C22semidirect
23. C61×#342C22semidirect
26. H(5,𝔽2)×C61D4semidirect
35. C23×C4×C61D4semidirect
53. C2×C4⋊C4×C61D4semidirect
59. C2×C22⋊C4×C61D4semidirect
63. C2×C42×C61D4semidirect
139. C23×C61#121semidirect
168. C2×C4×C61#121semidirect
169. #32926C61direct (N × H)
195. C4×C61B(4,𝔽2)semidirect
196. C4×H(5,𝔽2)C2×C61semidirect
197. C2×B(4,𝔽2)C2×C61semidirect
198. #2408C2×C61semidirect
199. #1889C2×C61semidirect
200. #1799C2×C61semidirect
209. C23×D4C22×C61semidirect
223. #380C22×C61semidirect
224. #379C22×C61semidirect
225. #342C22×C61semidirect
226. C61#32926direct (N × H)
229. H(5,𝔽2)D4×C61semidirect
238. C23×C4D4×C61semidirect
256. C2×C4⋊C4D4×C61semidirect
262. C2×C22⋊C4D4×C61semidirect
266. C2×C42D4×C61semidirect
342. C23#121×C61semidirect
371. C2×C4#121×C61semidirect
397. C4C61×B(4,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
C4×H(5,𝔽2)×C612
C2×C61×B(4,𝔽2)2
C61×#24082
C61×#18892
C61×#17992
#3292661