Finite Group Databasebeta

← C61×#34187 #29537620

Subgroups — C61×#34187 #29537620

Order 15616.

2250 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Q8×C61C211yesmaximal, characteristic
2C61×#2356C218yesmaximal, complemented
3C61×#2355C218yesmaximal, complemented
4C61×#2313C214yesmaximal
5C61×#2310C212yesmaximal
6C61×#2285C212yesmaximal
7C61×#2283C214yesmaximal
8C61×#2274C212yesmaximal
9C23×Q8×C61C2211yescharacteristic
10C2×C4×Q8×C61C4116yescomplemented
11C2×C4×Q8×C61C2214yes—
12C2×C4×Q8×C61C2214yes—
13C2×#117×C61C2214yes—
14C22×C4⋊C4×C61C2211yescharacteristic
15C22×C4⋊C4×C61C2212yes—
16C22×C42×C61C2212yes—
17C22×C42×C61C2211yescharacteristic
18C61×#391C22116yes—
19C61×#387C22132yes—
20C61×#384C2214yes—
21C61×#383C2218yes—
22C61×#382C2218yes—
23C61×#382C2214yes—
24C61×#381C2218yes—
25C61×#380C2214yes—
26C61×#380C2214yes—
27C4×C4⋊C4×C61C22116yes—
28C4×C4⋊C4×C61C2218yes—
29C4×C4⋊C4×C61C2214yes—
30C4×C4⋊C4×C61C2214yes—
31C4×C22⋊C4×C61C2218yes—
32C61×#376C2218yes—
33C22×Q8×C61—28no—
34C22×Q8×C61C2×C418yes—
35C22×Q8×C61C2314yes—
36C23×C4×C61C2312yes—
37C23×C4×C61C2311yescharacteristic
38C23×C4×C61C2312yes—
39C23×C4×C61C2311yescharacteristic
40C23×C4×C61C2311yescharacteristic
41C4×Q8×C61C2×C4132yes—
42C4×Q8×C61C2×C4132yes—
43#117×C61—216no—
44C2×C4⋊C4×C61C2×C4116yes—
45C2×C4⋊C4×C61C2×C418yes—
46C2×C4⋊C4×C61C2318yes—
47C2×C4⋊C4×C61C2318yes—
48C2×C4⋊C4×C61C2318yes—
49C2×C4⋊C4×C61C2314yes—
50C2×C4⋊C4×C61C2318yes—
51C2×C22⋊C4×C61C2318yes—
52C2×C42×C61C2×C4116yes—
53C2×C42×C61C2×C418yes—
54C2×C42×C61C2318yes—
55C2×C42×C61C23116yes—
56C2×C42×C61C2314yes—
57C2×C42×C61C2312yes—
58C2×C42×C61C2314yes—
59C2×C42×C61C2312yes—
60#95×C61C2318yes—
61#95×C61C23116yes—
62#95×C61C23116yes—
63#95×C61C2318yes—
64#95×C61C23116yes—
65C24×C61C2411yescharacteristic
66C2×Q8×C61—232no—
67C2×Q8×C61—28no—
68C2×Q8×C61—28no—
69C2×Q8×C61C22×C4116yes—
70C22×C4×C61—28no—
71C22×C4×C61—24no—
72C22×C4×C61—44no—
73C22×C4×C61—42no—
74C22×C4×C61C22×C414yes—
75C22×C4×C61C22×C418yes—
76C22×C4×C61C22×C418yes—
77C22×C4×C61C22×C414yes—
78C22×C4×C61C22×C414yes—
79C22×C4×C61#4118yes—
80C22×C4×C61C24116yes—
81C22×C4×C61C2411yescharacteristic
82C22×C4×C61C2414yes—
83C22×C4×C61C2414yes—
84C22×C4×C61C2412yes—
85C22×C4×C61C2412yes—
86C22×C4×C61C2411yescharacteristic
87C4⋊C4×C61—216no—
88C4⋊C4×C61—216no—
89C4⋊C4×C61—48no—
90C4⋊C4×C61C22×C4132yes—
91C4⋊C4×C61C22×C4116yes—
92C22⋊C4×C61—216no—
93C42×C61—216no—
94C42×C61—416no—
95C42×C61C22×C4116yes—
96C42×C61C22×C418yes—
97C42×C61C22×C4116yes—
98C42×C61C22×C418yes—
99C23×C61—22no—
100C23×C61—22no—
101C23×C61C23×C412yes—
102C23×C61C2×#4114yes—
103C23×C61C2511yescharacteristic
104Q8×C61—232no—
105C2×C4×C61—216no—
106C2×C4×C61—28no—
107C2×C4×C61—22no—
108C2×C4×C61—28no—
109C2×C4×C61—24no—
110C2×C4×C61—28no—
111C2×C4×C61—216no—
112C2×C4×C61—216no—
113C2×C4×C61—216no—
114C2×C4×C61—48no—
115C2×C4×C61—44no—
116C2×C4×C61#117116yes—
117C2×C4×C61C23×C412yes—
118C2×C4×C61C23×C418yes—
119C2×C4×C61C23×C414yes—
120C2×C4×C61C23×C418yes—
121C2×C4×C61C23×C414yes—
122C2×C4×C61C23×C412yes—
123C2×C4×C61C2×#4114yes—
124C2×C4×C61C2×#4114yes—
125C2×C4×C612−1+414yes—
126#34187C6111yesmaximal, characteristic, direct product factor, Sylow
127C22×C61—22no—
128C22×C61—22no—
129C22×C61—28no—
130C22×C61C2×#11714yes—
131C22×C61C24×C411yescharacteristic
132C22×C61C22×#4112yes—
133C22×C61C2×2−1+412yes—
134C22×C61#58512yes—
135C4×C61—28no—
136C4×C61—22no—
137C4×C61—28no—
138C4×C61—24no—
139C4×C61—416no—
140C4×C61C2×#11718yes—
141C4×C61#51914yes—
142C22×C4×Q8C2×C6111yescharacteristic
143#2356C2×C6118yescomplemented
144#2355C2×C6118yescomplemented
145#2313C2×C6114yes—
146#2310C2×C6112yes—
147#2285C2×C6112yes—
148#2283C2×C6114yes—
149#2274C2×C6112yes—
150C2×C61—24no—
151C2×C61C22×#11711yescharacteristic
152C2×C61C2×#51911yescharacteristic
153C2×C61#342811yescharacteristic
154C2×C61#350714yes—
155C23×Q8C22×C6111yescharacteristic
156C2×C4×Q8C4×C61116yescomplemented
157C2×C4×Q8C22×C6114yes—
158C2×C4×Q8C22×C6114yes—
159C2×#117C22×C6114yes—
160C22×C4⋊C4C22×C6112yes—
161C22×C4⋊C4C22×C6111yescharacteristic
162C22×C42C22×C6112yes—
163C22×C42C22×C6111yescharacteristic
164#391C22×C61116yes—
165#387C22×C61132yes—
166#384C22×C6114yes—
167#383C22×C6118yes—
168#382C22×C6118yes—
169#382C22×C6114yes—
170#381C22×C6118yes—
171#380C22×C6114yes—
172#380C22×C6114yes—
173C4×C4⋊C4C22×C6118yes—
174C4×C4⋊C4C22×C6114yes—
175C4×C4⋊C4C22×C61116yes—
176C4×C4⋊C4C22×C6114yes—
177C4×C22⋊C4C22×C6118yes—
178#376C22×C6118yes—
179C61#3418711yescharacteristic, direct product factor, Sylow
180C22×Q8—28no—
181C22×Q8C2×C4×C6118yes—
182C22×Q8C23×C6114yes—
183C23×C4C23×C6112yes—
184C23×C4C23×C6111yescharacteristic
185C23×C4C23×C6112yes—
186C23×C4C23×C6111yescharacteristic
187C23×C4C23×C6111yescharacteristic
188C4×Q8C2×C4×C61132yes—
189C4×Q8C2×C4×C61132yes—
190#117—216no—
191C2×C4⋊C4C2×C4×C61116yes—
192C2×C4⋊C4C2×C4×C6118yes—
193C2×C4⋊C4C23×C6118yes—
194C2×C4⋊C4C23×C6118yes—
195C2×C4⋊C4C23×C6118yes—
196C2×C4⋊C4C23×C6114yes—
197C2×C4⋊C4C23×C6118yes—
198C2×C22⋊C4C23×C6118yes—
199C2×C42C2×C4×C61116yes—
200C2×C42C2×C4×C6118yes—
201C2×C42C23×C6118yes—
202C2×C42C23×C61116yes—
203C2×C42C23×C6114yes—
204C2×C42C23×C6112yes—
205C2×C42C23×C6114yes—
206C2×C42C23×C6112yes—
207#95C23×C6118yes—
208#95C23×C61116yes—
209#95C23×C61116yes—
210#95C23×C6118yes—
211#95C23×C61116yes—
212C24C24×C6111yescharacteristic
213C2×Q8—232no—
214C2×Q8—28no—
215C2×Q8—28no—
216C2×Q8C22×C4×C61116yes—
217C22×C4—28no—
218C22×C4—24no—
219C22×C4—44no—
220C22×C4—42no—
221C22×C4C22×C4×C6114yes—
222C22×C4C22×C4×C6118yes—
223C22×C4C22×C4×C6118yes—
224C22×C4C22×C4×C6114yes—
225C22×C4C22×C4×C6114yes—
226C22×C4#41×C6118yes—
227C22×C4C24×C61116yes—
228C22×C4C24×C6111yescharacteristic
229C22×C4C24×C6114yes—
230C22×C4C24×C6114yes—
231C22×C4C24×C6112yes—
232C22×C4C24×C6112yes—
233C22×C4C24×C6111yescharacteristic
234C4⋊C4—216no—
235C4⋊C4—216no—
236C4⋊C4—48no—
237C4⋊C4C22×C4×C61132yes—
238C4⋊C4C22×C4×C61116yes—
239C22⋊C4—216no—
240C42—216no—
241C42—416no—
242C42C22×C4×C61116yes—
243C42C22×C4×C6118yes—
244C42C22×C4×C61116yes—
245C42C22×C4×C6118yes—
246C23—22no—
247C23—22no—
248C23C23×C4×C6112yes—
249C23C2×#41×C6114yes—
250C23C25×C6111yescharacteristic
251Q8—232no—
252C2×C4—28no—
253C2×C4—22no—
254C2×C4—28no—
255C2×C4—24no—
256C2×C4—216no—
257C2×C4—28no—
258C2×C4—216no—
259C2×C4—216no—
260C2×C4—216no—
261C2×C4—48no—
262C2×C4—44no—
263C2×C4#117×C61116yes—
264C2×C4C23×C4×C6112yes—
265C2×C4C23×C4×C6118yes—
266C2×C4C23×C4×C6114yes—
267C2×C4C23×C4×C6118yes—
268C2×C4C23×C4×C6114yes—
269C2×C4C23×C4×C6112yes—
270C2×C4C2×#41×C6114yes—
271C2×C4C2×#41×C6114yes—
272C2×C42−1+4×C6114yes—
273C22—22no—
274C22—22no—
275C22—28no—
276C22C2×#117×C6114yes—
277C22C24×C4×C6111yescharacteristic
278C22C22×#41×C6112yes—
279C22C2×2−1+4×C6112yes—
280C22C61×#58512yes—
281C4—28no—
282C4—22no—
283C4—28no—
284C4—24no—
285C4—416no—
286C4C2×#117×C6118yes—
287C4C61×#51914yes—
288C2—24no—
289C2C22×#117×C6111yescharacteristic
290C2C2×C61×#51911yescharacteristic
291C2C61×#342811yescharacteristic
292C2C61×#350714yes—
subgroupG/H
center, socleC23×C61C25
commutatorC22C24×C4×C61
Frattini, cosocleC23C25×C61
prime radicalC24×C61C24
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C61×#2356C2semidirect
3. C61×#2355C2semidirect
10. C2×C4×Q8×C61C4semidirect
126. #34187C61direct (N × H)
143. #2356C2×C61semidirect
144. #2355C2×C61semidirect
156. C2×C4×Q8C4×C61semidirect
179. C61#34187direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×Q8×C612
C61×#23562
C61×#23552
C61×#23132
C61×#23102
C61×#22852
C61×#22832
C61×#22742
#3418761