Subgroups — #29596210

Order 15616.

358 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839592	C₂	1	1	yes	maximal, characteristic, complemented
2	#21447603	C₂	1	1	yes	maximal, characteristic, complemented
3	#21445873	C₂	1	2	yes	maximal
4	#21443934	C₂	1	2	yes	maximal, complemented
5	C₂×C₆₁×#504	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519713	C₂²	1	2	yes	—
7	#16839800	C₂²	1	2	yes	—
8	#16839779	—	2	2	no	—
9	#16839592	—	2	1	no	—
10	#16839592	—	2	1	no	—
11	#16839590	—	2	1	no	—
12	#16839588	—	2	1	no	—
13	#16839534	—	2	2	no	—
14	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#504	—	2	1	no	—
16	C₆₁×#504	—	2	1	no	—
17	C₆₁×#504	C₄	1	2	yes	complemented
18	C₆₁×#504	C₄	1	2	yes	complemented
19	C₂×M₃₂×C₆₁	—	2	1	no	—
20	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₆₁	—	2	1	no	—
22	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×#11758194	—	2	2	no	—
24	#14519736	—	2	2	no	—
25	#14519713	—	2	2	no	—
26	#14519713	—	2	2	no	—
27	#14519711	—	2	2	no	—
28	#14519709	—	2	2	no	—
29	#14519704	—	4	4	no	—
30	C₂×#41×C₆₁	D₄	1	1	yes	characteristic
31	#131×C₆₁	—	2	1	no	—
32	#131×C₆₁	—	2	1	no	—
33	#131×C₆₁	C₂×C₄	1	2	yes	—
34	#131×C₆₁	Q₈	1	2	yes	—
35	C₂×M₁₆×C₆₁	—	2	1	no	—
36	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₁	—	2	1	no	—
38	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
39	M₃₂×C₆₁	—	2	1	no	—
40	M₃₂×C₆₁	—	2	2	no	—
41	M₃₂×C₆₁	—	2	2	no	—
42	M₃₂×C₆₁	C₂×C₄	1	2	yes	—
43	C₂×C₁₆×C₆₁	—	2	1	no	—
44	C₂×C₁₆×C₆₁	—	2	2	no	—
45	C₂×C₁₆×C₆₁	—	2	2	no	—
46	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
49	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
50	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
51	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
52	#41×C₆₁	—	2	1	no	—
53	#41×C₆₁	—	2	1	no	—
54	#41×C₆₁	C₄⋊C₄	1	2	yes	—
55	#41×C₆₁	D₈	1	2	yes	—
56	C₂×Q₈×C₆₁	Q₁₆	1	1	yes	characteristic
57	C₂×D₄×C₆₁	—	2	1	no	—
58	C₂×D₄×C₆₁	Q₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	—	2	1	no	—
60	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
61	M₁₆×C₆₁	—	2	1	no	—
62	M₁₆×C₆₁	—	2	2	no	—
63	M₁₆×C₆₁	—	2	2	no	—
64	M₁₆×C₆₁	C₄⋊C₄	1	2	yes	—
65	C₂×C₈×C₆₁	—	2	1	no	—
66	C₂×C₈×C₆₁	—	2	2	no	—
67	C₂×C₈×C₆₁	—	2	2	no	—
68	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
73	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
74	C₂²⋊Dic₆₁	—	4	2	no	—
75	C₄⋊Dic₆₁	—	4	2	no	—
76	C₄×Dic₆₁	—	4	2	no	—
77	#11758194	—	4	2	no	—
78	#11758194	—	4	2	no	—
79	C₂×#92418	—	4	2	no	—
80	C₁₆×C₆₁	—	2	2	no	—
81	C₁₆×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₁₆×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
83	C₁₆×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₁₆×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
85	C₂³×C₆₁	—	2	1	no	—
86	C₂³×C₆₁	#135	1	1	yes	characteristic
87	Q₈×C₆₁	—	2	1	no	—
88	Q₈×C₆₁	#107	1	2	yes	—
89	D₄×C₆₁	—	2	2	no	—
90	D₄×C₆₁	—	2	2	no	—
91	D₄×C₆₁	—	2	1	no	—
92	D₄×C₆₁	#107	1	2	yes	—
93	C₂×C₄×C₆₁	—	2	1	no	—
94	C₂×C₄×C₆₁	—	2	2	no	—
95	C₂×C₄×C₆₁	—	2	2	no	—
96	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
98	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
101	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
102	C₂×Dic₆₁	—	8	2	no	—
103	C₈×C₆₁	—	2	2	no	—
104	C₈×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₈×C₆₁	#117	1	1	yes	characteristic
106	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
107	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
108	#92418	—	8	2	no	—
109	#12841	—	61	1	no	maximal, Sylow
110	C₂²×C₆₁	—	2	2	no	—
111	C₂²×C₆₁	—	4	1	no	—
112	C₂²×C₆₁	—	4	2	no	—
113	C₂²×C₆₁	#427	1	1	yes	characteristic
114	C₂²×C₆₁	#466	1	1	yes	characteristic
115	C₂²×C₆₁	#467	1	1	yes	characteristic
116	C₄×C₆₁	—	2	2	no	—
117	C₄×C₆₁	#380	1	1	yes	characteristic
118	C₄×C₆₁	C₂×#107	1	1	yes	characteristic
119	C₄×C₆₁	#466	1	1	yes	characteristic
120	C₄×C₆₁	#467	1	1	yes	characteristic
121	Dic₆₁	—	16	2	no	—
122	C₂×#504	D₆₁	1	1	yes	characteristic
123	C₂×#368	—	61	1	no	—
124	#2158	—	61	1	no	—
125	#1814	—	61	2	no	—
126	#1389	—	61	2	no	—
127	C₂×C₆₁	—	2	1	no	—
128	C₂×C₆₁	—	2	1	no	—
129	C₂×C₆₁	—	2	1	no	—
130	C₂×C₆₁	—	4	2	no	—
131	C₂×C₆₁	#1808	1	1	yes	characteristic
132	C₂×#131	C₂×D₆₁	1	1	yes	characteristic
133	#504	—	2	1	no	—
134	#504	—	2	1	no	—
135	#504	Dic₆₁	1	2	yes	complemented
136	#504	Dic₆₁	1	2	yes	complemented
137	C₂×M₃₂	—	2	1	no	—
138	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
139	C₂²×C₁₆	—	2	1	no	—
140	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
141	C₂×#108	—	61	2	no	—
142	#427	—	61	2	no	—
143	#421	—	122	2	no	—
144	#368	—	122	1	no	—
145	#368	—	122	1	no	—
146	#367	—	122	1	no	—
147	#366	—	122	1	no	—
148	#344	—	122	2	no	—
149	C₆₁	#12841	1	1	yes	characteristic, complemented, Sylow
150	C₂×#41	D₂₄₄	1	1	yes	characteristic
151	#131	—	2	1	no	—
152	#131	—	2	1	no	—
153	#131	Dic₁₂₂	1	2	yes	—
154	#131	C₂×Dic₆₁	1	2	yes	—
155	C₂×M₁₆	—	2	1	no	—
156	C₂×M₁₆	—	122	2	no	—
157	C₂×M₁₆	D₂₄₄	1	1	yes	characteristic
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
160	#117	—	122	2	no	—
161	M₃₂	—	2	1	no	—
162	M₃₂	—	2	2	no	—
163	M₃₂	—	2	2	no	—
164	M₃₂	C₂×Dic₆₁	1	2	yes	—
165	C₂×C₁₆	—	2	1	no	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	2	2	no	—
168	C₂×C₁₆	C₂×Dic₆₁	1	1	yes	characteristic
169	C₂×C₁₆	C₂×Dic₆₁	1	1	yes	characteristic
170	C₂×C₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
171	C₂×C₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
172	C₂×C₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
173	C₂×C₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
174	#108	—	122	2	no	—
175	#108	—	122	2	no	—
176	#107	—	122	2	no	—
177	#106	—	122	2	no	—
178	#104	—	244	4	no	—
179	#41	—	2	1	no	—
180	#41	—	2	1	no	—
181	#41	D₄₈₈	1	2	yes	—
182	#41	C₄⋊Dic₆₁	1	2	yes	—
183	C₂×Q₈	Dic₂₄₄	1	1	yes	characteristic
184	C₂×D₄	—	2	1	no	—
185	C₂×D₄	Dic₂₄₄	1	1	yes	characteristic
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
188	M₁₆	—	2	1	no	—
189	M₁₆	—	2	2	no	—
190	M₁₆	—	2	2	no	—
191	M₁₆	—	244	2	no	—
192	M₁₆	—	244	2	no	—
193	M₁₆	C₄⋊Dic₆₁	1	2	yes	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	244	2	no	—
198	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
199	C₂×C₈	#11758222	1	1	yes	characteristic
200	C₂×C₈	#11758222	1	1	yes	characteristic
201	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
202	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
203	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
204	C₄⋊C₄	—	244	2	no	—
205	C₂²⋊C₄	—	244	2	no	—
206	C₄²	—	244	2	no	—
207	C₁₆	—	2	2	no	—
208	C₁₆	C₂²⋊Dic₆₁	1	1	yes	characteristic
209	C₁₆	C₂²⋊Dic₆₁	1	1	yes	characteristic
210	C₁₆	C₂²⋊Dic₆₁	1	1	yes	characteristic
211	C₁₆	C₂²⋊Dic₆₁	1	1	yes	characteristic
212	C₂³	—	2	1	no	—
213	C₂³	#14519799	1	1	yes	characteristic
214	Q₈	—	2	1	no	—
215	Q₈	#14519711	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	#14519711	1	2	yes	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	488	2	no	—
224	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
225	C₂×C₄	#14519751	1	1	yes	characteristic
226	C₂×C₄	#14519751	1	1	yes	characteristic
227	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
228	C₂×C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
229	C₂×C₄	#14519799	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	488	2	no	—
232	C₈	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
233	C₈	#14519736	1	1	yes	characteristic
234	C₈	#14519756	1	1	yes	characteristic
235	C₈	#14519756	1	1	yes	characteristic
236	C₂²	—	2	2	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	2	no	—
239	C₂²	#16839800	1	1	yes	characteristic
240	C₂²	#16839998	1	1	yes	characteristic
241	C₂²	#16840003	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	976	2	no	—
244	C₄	#16839622	1	1	yes	characteristic
245	C₄	C₂×#14519711	1	1	yes	characteristic
246	C₄	#16839998	1	1	yes	characteristic
247	C₄	#16840003	1	1	yes	characteristic
248	C₂	—	2	1	no	—
249	C₂	—	2	1	no	—
250	C₂	—	2	1	no	—
251	C₂	—	4	2	no	—
252	C₂	#21445848	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16839998
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×#504	C₂
socle	C₂×C₆₁	#1808
prime radical	C₂×#41×C₆₁	D₄
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×#504	D₆₁
hypocenter	C₆₁	#12841
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#16839592	C₂	semidirect
2. #21447603	C₂	semidirect
4. #21443934	C₂	semidirect
17. C₆₁×#504	C₄	semidirect
18. C₆₁×#504	C₄	semidirect
135. #504	Dic₆₁	semidirect
136. #504	Dic₆₁	semidirect
149. C₆₁	#12841	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839592	2
#21447603	2
#21445873	2
#21443934	2
C₂×C₆₁×#504	2
#12841	61