Subgroups — #29833253

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840180	C₂	1	2	yes	maximal, complemented
2	#21454723	C₂	1	1	yes	maximal, characteristic
3	#21447660	C₂	1	4	yes	maximal, complemented
4	#21447655	C₂	1	4	yes	maximal, complemented
5	#21447448	C₂	1	2	yes	maximal, complemented
6	#21447416	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2153	C₂	1	1	yes	maximal, characteristic, complemented
8	#16840562	C₄	1	1	yes	characteristic
9	C₂×#14519783	C₂²	1	2	yes	—
10	#16840180	—	2	2	no	—
11	#16840180	—	2	2	no	—
12	#16840180	C₂²	1	4	yes	complemented
13	#16840180	C₂²	1	4	yes	complemented
14	C₂×#14519717	C₂²	1	1	yes	characteristic
15	C₂×#14519716	—	2	2	no	—
16	C₂²×#11758185	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₆₁	—	2	2	no	—
18	#16839885	C₄	1	4	yes	—
19	#16839880	C₂²	1	4	yes	—
20	#16839811	C₄	1	1	yes	characteristic
21	#16839780	C₄	1	2	yes	—
22	#16839547	C₂²	1	8	yes	complemented
23	#16839538	C₂²	1	2	yes	complemented
24	C₄×#11758185	C₂²	1	2	yes	complemented
25	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
26	C₆₁×#405	C₂²	1	1	yes	characteristic
27	C₆₁×#364	C₂²	1	4	yes	complemented
28	C₂×#11758222	C₂×C₄	1	2	yes	—
29	#14519810	C₈	1	4	yes	—
30	#14519805	C₈	1	4	yes	—
31	#14519799	C₂×C₄	1	4	yes	—
32	#14519783	—	2	2	no	—
33	#14519783	—	2	2	no	—
34	#14519783	C₂×C₄	1	4	yes	—
35	#14519783	C₂³	1	4	yes	—
36	C₂×#11758194	C₂×C₄	1	2	yes	—
37	C₂×#11758189	—	2	2	no	—
38	C₂×C₈×D₆₁	—	2	2	no	—
39	#14519717	—	2	2	no	—
40	#14519716	—	2	4	no	—
41	#14519716	—	2	4	no	—
42	C₂×#11758185	D₄	1	2	yes	—
43	C₂×#11758185	D₄	1	2	yes	—
44	C₂×#11758185	C₂³	1	2	yes	—
45	C₂×#11758185	C₂³	1	1	yes	characteristic
46	C₂×#11758185	C₂³	1	1	yes	characteristic
47	C₁₆×D₆₁	—	2	4	no	—
48	C₁₆×D₆₁	—	2	4	no	—
49	#14519713	C₂×C₄	1	4	yes	—
50	#14519705	C₂×C₄	1	8	yes	—
51	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
53	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
54	M₃₂×C₆₁	—	2	2	no	—
55	M₃₂×C₆₁	C₂³	1	4	yes	—
56	C₂×C₁₆×C₆₁	—	2	2	no	—
57	#97×C₆₁	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₆₁	C₂³	1	2	yes	—
59	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₆₁	—	2	2	no	—
61	#11758222	—	2	2	no	—
62	#11758222	—	2	2	no	—
63	#11758222	C₂×C₈	1	4	yes	—
64	#11758222	C₂²×C₄	1	4	yes	—
65	C₂×D₂₄₄	C₂×C₈	1	2	yes	—
66	C₂×C₄×D₆₁	—	2	2	no	—
67	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
68	M₁₆×C₆₁	C₂×C₈	1	4	yes	—
69	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₆₁	#41	1	2	yes	—
73	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₆₁	—	2	1	no	—
75	C₂²⋊C₄×C₆₁	—	2	1	no	—
76	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
77	#11758194	—	2	2	no	—
78	#11758194	C₂²×C₄	1	4	yes	—
79	C₂×#92418	—	2	2	no	—
80	Dic₂₄₄	C₂×C₈	1	4	yes	—
81	D₄₈₈	C₂×C₈	1	4	yes	—
82	#11758190	C₂×C₈	1	8	yes	—
83	#11758189	—	2	4	no	—
84	#11758189	—	2	4	no	—
85	C₈×D₆₁	—	2	4	no	—
86	C₈×D₆₁	—	2	4	no	—
87	C₁₆×C₆₁	—	2	4	no	—
88	#11758185	—	2	2	no	—
89	#11758185	C₂×D₄	1	1	yes	characteristic
90	#11758185	C₂×D₄	1	2	yes	—
91	#11758185	C₂×D₄	1	1	yes	characteristic
92	C₂³×C₆₁	#131	1	1	yes	characteristic
93	C₂²×D₆₁	—	2	2	no	—
94	C₂×C₄×C₆₁	—	2	2	no	—
95	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
97	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
98	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
99	C₂²⋊D₆₁	—	2	4	no	—
100	C₂²⋊D₆₁	—	2	4	no	—
101	C₂×Dic₆₁	—	2	2	no	—
102	D₂₄₄	—	2	2	no	—
103	D₂₄₄	C₂²×C₈	1	4	yes	—
104	C₄×D₆₁	—	2	4	no	—
105	C₄×D₆₁	—	2	4	no	—
106	Dic₁₂₂	—	2	2	no	—
107	Dic₁₂₂	C₂²×C₈	1	4	yes	—
108	C₈×C₆₁	C₄×D₄	1	2	yes	—
109	C₈×C₆₁	C₂²×C₈	1	4	yes	—
110	C₈×C₆₁	C₂²×D₄	1	1	yes	characteristic
111	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
112	#92418	—	2	4	no	—
113	#33441	—	61	1	no	maximal, Sylow
114	C₂²×C₆₁	—	4	1	no	—
115	C₂²×C₆₁	C₈×D₄	1	2	yes	—
116	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
117	C₂×D₆₁	—	2	4	no	—
118	C₂×D₆₁	—	4	4	no	—
119	C₄×C₆₁	—	4	2	no	—
120	C₄×C₆₁	C₈×D₄	1	2	yes	—
121	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
122	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
123	Dic₆₁	—	2	4	no	—
124	C₂×#504	—	61	2	no	—
125	#2955	—	61	1	no	—
126	#2171	—	61	4	no	—
127	#2170	—	61	4	no	—
128	#2153	D₆₁	1	1	yes	characteristic, complemented
129	#2117	—	61	2	no	—
130	#2108	—	61	1	no	—
131	C₂×C₆₁	—	2	1	no	—
132	C₂×C₆₁	—	2	2	no	—
133	C₂×C₆₁	C₂×C₈×D₄	1	1	yes	characteristic
134	D₆₁	—	4	4	no	—
135	#575	—	61	1	no	—
136	C₂×#131	—	61	2	no	—
137	#504	—	61	4	no	—
138	#504	—	61	4	no	—
139	#504	—	122	2	no	—
140	#504	—	122	2	no	—
141	C₂×M₃₂	—	61	1	no	—
142	C₂×M₃₂	—	122	2	no	—
143	C₂×M₃₂	C₂×D₆₁	1	2	yes	—
144	C₂²×C₁₆	—	61	1	no	—
145	C₂²×C₁₆	—	122	2	no	—
146	#444	—	61	4	no	—
147	#443	—	61	4	no	—
148	#430	—	61	1	no	—
149	#421	—	61	2	no	—
150	#405	C₂×D₆₁	1	1	yes	characteristic
151	#364	C₂×D₆₁	1	4	yes	complemented
152	#350	—	61	8	no	—
153	#346	—	61	2	no	—
154	C₄×C₁₆	—	61	2	no	—
155	C₆₁	#33441	1	1	yes	characteristic, complemented, Sylow
156	C₂×#41	—	61	2	no	—
157	#137	—	61	4	no	—
158	AGL(1,ℤ/8)	—	61	4	no	—
159	#135	—	61	4	no	—
160	#131	—	61	4	no	—
161	#131	—	61	4	no	—
162	#131	—	122	2	no	—
163	#131	—	122	2	no	—
164	C₂×M₁₆	—	61	2	no	—
165	C₂×M₁₆	—	122	2	no	—
166	C₂×M₁₆	C₄×D₆₁	1	1	yes	characteristic
167	C₂²×C₈	—	122	2	no	—
168	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
169	#117	C₄×D₆₁	1	1	yes	characteristic
170	M₃₂	—	2	2	no	—
171	M₃₂	—	122	4	no	—
172	M₃₂	—	122	4	no	—
173	M₃₂	—	122	2	no	—
174	M₃₂	C₂²×D₆₁	1	4	yes	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	61	2	no	—
177	C₂×C₁₆	—	61	2	no	—
178	C₂×C₁₆	—	61	1	no	—
179	C₂×C₁₆	—	61	1	no	—
180	C₂×C₁₆	—	61	2	no	—
181	C₂×C₁₆	—	122	4	no	—
182	C₂×C₁₆	—	122	4	no	—
183	#108	—	61	4	no	—
184	#104	—	61	8	no	—
185	#97	C₄×D₆₁	1	2	yes	—
186	C₄×C₈	C₂²×D₆₁	1	2	yes	—
187	#41	—	61	4	no	—
188	#41	—	61	4	no	—
189	#41	—	122	2	no	—
190	#41	—	122	2	no	—
191	C₂×Q₈	—	61	2	no	—
192	C₂×D₄	—	61	2	no	—
193	C₂×D₄	—	122	2	no	—
194	C₂²×C₄	—	122	2	no	—
195	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
196	Q₁₆	—	61	4	no	—
197	SD₁₆	—	61	8	no	—
198	D₈	—	61	4	no	—
199	M₁₆	—	61	4	no	—
200	M₁₆	—	122	2	no	—
201	M₁₆	—	122	4	no	—
202	M₁₆	—	122	4	no	—
203	M₁₆	C₈×D₆₁	1	4	yes	—
204	C₂×C₈	—	122	2	no	—
205	C₂×C₈	—	122	4	no	—
206	C₂×C₈	—	122	4	no	—
207	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
208	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
209	C₂×C₈	D₄×D₆₁	1	2	yes	—
210	C₂×C₈	#11758226	1	2	yes	—
211	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
212	C₄⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₄²	C₂×C₄×D₆₁	1	2	yes	—
215	C₁₆	—	2	4	no	—
216	C₁₆	—	61	1	no	—
217	C₁₆	—	61	2	no	—
218	C₁₆	—	61	1	no	—
219	C₁₆	—	122	2	no	—
220	C₂³	—	122	2	no	—
221	C₂³	#14519785	1	1	yes	characteristic
222	Q₈	—	61	4	no	—
223	Q₈	—	122	2	no	—
224	D₄	—	61	4	no	—
225	D₄	—	122	4	no	—
226	D₄	—	122	4	no	—
227	D₄	—	122	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	122	2	no	—
230	C₂×C₄	—	122	4	no	—
231	C₂×C₄	—	122	4	no	—
232	C₂×C₄	#14519739	1	2	yes	—
233	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
234	C₂×C₄	#14519785	1	1	yes	characteristic
235	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
236	C₈	—	122	4	no	—
237	C₈	#14519739	1	2	yes	—
238	C₈	C₂×C₈×D₆₁	1	4	yes	—
239	C₈	C₂×D₄×D₆₁	1	1	yes	characteristic
240	C₈	C₂×#11758226	1	1	yes	characteristic
241	C₂²	—	4	1	no	—
242	C₂²	—	122	4	no	—
243	C₂²	—	244	4	no	—
244	C₂²	#16839827	1	2	yes	—
245	C₂²	C₂×#14519785	1	1	yes	characteristic
246	C₄	—	4	2	no	—
247	C₄	—	122	4	no	—
248	C₄	#16839827	1	2	yes	—
249	C₄	C₂×#14519739	1	1	yes	characteristic
250	C₄	C₂²×C₈×D₆₁	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	2	no	—
253	C₂	—	244	4	no	—
254	C₂	C₂×#16839827	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₆₁
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#2153	C₂
socle	C₂×C₆₁	C₂×C₈×D₄
prime radical	#16840562	C₄
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	#2153	D₆₁
hypocenter	C₆₁	#33441
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

Complements

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

N	H ≅ G/N	type
1. C₂×#16840180	C₂	semidirect
3. #21447660	C₂	semidirect
4. #21447655	C₂	semidirect
5. #21447448	C₂	semidirect
6. #21447416	C₂	semidirect
7. C₆₁×#2153	C₂	semidirect
12. #16840180	C₂²	semidirect
13. #16840180	C₂²	semidirect
22. #16839547	C₂²	semidirect
23. #16839538	C₂²	semidirect
24. C₄×#11758185	C₂²	semidirect
27. C₆₁×#364	C₂²	semidirect
128. #2153	D₆₁	semidirect
151. #364	C₂×D₆₁	semidirect
155. C₆₁	#33441	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840180	2
#21454723	2
#21447660	2
#21447655	2
#21447448	2
#21447416	2
C₆₁×#2153	2
#33441	61