Subgroups — #29833254

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×#16840179	C₂	1	2	yes	maximal, complemented
2	#21447661	C₂	1	4	yes	maximal, complemented
3	#21447656	C₂	1	4	yes	maximal, complemented
4	#21447585	C₂	1	1	yes	maximal, characteristic, complemented
5	#21447447	C₂	1	2	yes	maximal, complemented
6	#21447416	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2955	C₂	1	1	yes	maximal, characteristic
8	#16840179	—	2	2	no	—
9	#16840179	—	2	2	no	—
10	#16840179	C₂²	1	4	yes	complemented
11	#16840179	C₂²	1	4	yes	complemented
12	C₂×#14519717	—	2	2	no	—
13	C₂×#14519717	C₂²	1	2	yes	—
14	C₂×#14519717	C₂²	1	1	yes	characteristic
15	C₂²×#11758185	—	2	2	no	—
16	C₂²×#11758185	C₂²	1	1	yes	characteristic
17	#16839583	C₂²	1	4	yes	complemented
18	#16839546	C₂²	1	8	yes	complemented
19	#16839538	C₂²	1	2	yes	complemented
20	C₄×#11758185	C₂²	1	2	yes	complemented
21	C₆₁×#575	C₄	1	1	yes	characteristic
22	C₂×#131×C₆₁	C₂²	1	2	yes	—
23	C₆₁×#444	C₄	1	4	yes	—
24	C₆₁×#443	C₂²	1	4	yes	—
25	C₆₁×#430	C₄	1	1	yes	characteristic
26	C₆₁×#421	C₄	1	2	yes	—
27	C₆₁×#405	C₂²	1	1	yes	characteristic
28	#14519717	—	2	2	no	—
29	#14519717	—	2	2	no	—
30	#14519717	—	2	4	no	—
31	#14519717	—	2	4	no	—
32	#14519717	C₂³	1	4	yes	—
33	C₂×#11758185	—	2	2	no	—
34	C₂×#11758185	—	2	4	no	—
35	C₂×#11758185	—	2	4	no	—
36	C₂×#11758185	D₄	1	2	yes	—
37	C₂×#11758185	D₄	1	2	yes	—
38	C₂×#11758185	C₂³	1	2	yes	—
39	C₂×#11758185	C₂³	1	1	yes	characteristic
40	C₂×#11758185	C₂³	1	1	yes	characteristic
41	C₂×#41×C₆₁	C₂×C₄	1	2	yes	—
42	#137×C₆₁	C₈	1	4	yes	—
43	AGL(1,ℤ/8)×C₆₁	C₈	1	4	yes	—
44	#135×C₆₁	C₂×C₄	1	4	yes	—
45	#131×C₆₁	—	2	2	no	—
46	#131×C₆₁	—	2	2	no	—
47	#131×C₆₁	C₂×C₄	1	4	yes	—
48	#131×C₆₁	C₂³	1	4	yes	—
49	C₂×M₁₆×C₆₁	—	2	2	no	—
50	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
51	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₁	—	2	2	no	—
53	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
54	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
55	#108×C₆₁	C₂×C₄	1	4	yes	—
56	#104×C₆₁	C₂×C₄	1	8	yes	—
57	#97×C₆₁	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₆₁	C₂³	1	2	yes	—
59	#41×C₆₁	—	2	2	no	—
60	#41×C₆₁	—	2	2	no	—
61	#41×C₆₁	C₂×C₈	1	4	yes	—
62	#41×C₆₁	C₂²×C₄	1	4	yes	—
63	C₂×Q₈×C₆₁	C₂×C₈	1	2	yes	—
64	C₂×D₄×C₆₁	—	2	2	no	—
65	C₂×D₄×C₆₁	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₆₁	—	2	2	no	—
67	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
68	Q₁₆×C₆₁	C₂×C₈	1	4	yes	—
69	SD₁₆×C₆₁	C₂×C₈	1	8	yes	—
70	D₈×C₆₁	C₂×C₈	1	4	yes	—
71	M₁₆×C₆₁	—	2	2	no	—
72	M₁₆×C₆₁	—	2	4	no	—
73	M₁₆×C₆₁	—	2	4	no	—
74	M₁₆×C₆₁	C₂×C₈	1	4	yes	—
75	M₁₆×C₆₁	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₆₁	—	2	2	no	—
77	C₂×C₈×C₆₁	—	2	4	no	—
78	C₂×C₈×C₆₁	—	2	4	no	—
79	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₆₁	#41	1	2	yes	—
83	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄×C₆₁	—	2	1	no	—
85	C₂²⋊C₄×C₆₁	—	2	1	no	—
86	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
87	#11758185	—	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	—	2	2	no	—
93	C₂³×C₆₁	#131	1	1	yes	characteristic
94	Q₈×C₆₁	—	2	2	no	—
95	Q₈×C₆₁	C₂²×C₈	1	4	yes	—
96	D₄×C₆₁	—	2	4	no	—
97	D₄×C₆₁	—	2	4	no	—
98	D₄×C₆₁	—	2	2	no	—
99	D₄×C₆₁	C₂²×C₈	1	4	yes	—
100	C₂×C₄×C₆₁	—	2	2	no	—
101	C₂×C₄×C₆₁	—	2	4	no	—
102	C₂×C₄×C₆₁	—	2	4	no	—
103	C₂×C₄×C₆₁	—	2	2	no	—
104	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
108	C₈×C₆₁	—	2	4	no	—
109	C₈×C₆₁	C₄×D₄	1	2	yes	—
110	C₈×C₆₁	C₂²×C₈	1	4	yes	—
111	C₈×C₆₁	C₂²×D₄	1	1	yes	characteristic
112	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
113	#33441	—	61	1	no	maximal, Sylow
114	C₂²×C₆₁	—	2	4	no	—
115	C₂²×C₆₁	—	4	1	no	—
116	C₂²×C₆₁	—	4	4	no	—
117	C₂²×C₆₁	C₈×D₄	1	2	yes	—
118	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
119	C₄×C₆₁	—	2	4	no	—
120	C₄×C₆₁	—	4	2	no	—
121	C₄×C₆₁	C₈×D₄	1	2	yes	—
122	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
123	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
124	C₂×#504	—	61	2	no	—
125	#2955	D₆₁	1	1	yes	characteristic
126	#2171	—	61	4	no	—
127	#2170	—	61	4	no	—
128	#2153	—	61	1	no	—
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	—	4	4	no	—
134	C₂×C₆₁	C₂×C₈×D₄	1	1	yes	characteristic
135	#575	Dic₆₁	1	1	yes	characteristic
136	C₂×#131	C₂×D₆₁	1	2	yes	—
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	—	61	2	no	—
143	C₂×M₃₂	—	122	2	no	—
144	C₂²×C₁₆	—	61	1	no	—
145	C₂²×C₁₆	—	122	2	no	—
146	#444	Dic₆₁	1	4	yes	—
147	#443	C₂×D₆₁	1	4	yes	—
148	#430	Dic₆₁	1	1	yes	characteristic
149	#421	Dic₆₁	1	2	yes	—
150	#405	C₂×D₆₁	1	1	yes	characteristic
151	#364	—	61	4	no	—
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	C₂×Dic₆₁	1	2	yes	—
157	#137	#92418	1	4	yes	—
158	AGL(1,ℤ/8)	#92418	1	4	yes	—
159	#135	C₂×Dic₆₁	1	4	yes	—
160	#131	—	2	2	no	—
161	#131	—	2	2	no	—
162	#131	C₂×Dic₆₁	1	4	yes	—
163	#131	C₂²×D₆₁	1	4	yes	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	C₂×Dic₆₁	1	2	yes	—
166	C₂×M₁₆	C₂×Dic₆₁	1	1	yes	characteristic
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
169	#117	C₂×Dic₆₁	1	1	yes	characteristic
170	M₃₂	—	61	4	no	—
171	M₃₂	—	122	4	no	—
172	M₃₂	—	122	4	no	—
173	M₃₂	—	122	2	no	—
174	M₃₂	—	122	2	no	—
175	C₂×C₁₆	—	61	2	no	—
176	C₂×C₁₆	—	61	2	no	—
177	C₂×C₁₆	—	61	1	no	—
178	C₂×C₁₆	—	61	1	no	—
179	C₂×C₁₆	—	61	2	no	—
180	C₂×C₁₆	—	122	4	no	—
181	C₂×C₁₆	—	122	4	no	—
182	C₂×C₁₆	—	122	2	no	—
183	#108	C₂×Dic₆₁	1	4	yes	—
184	#104	C₂×Dic₆₁	1	8	yes	—
185	#97	C₂×Dic₆₁	1	2	yes	—
186	C₄×C₈	C₂²×D₆₁	1	2	yes	—
187	#41	—	2	2	no	—
188	#41	—	2	2	no	—
189	#41	C₂×#92418	1	4	yes	—
190	#41	C₂²×Dic₆₁	1	4	yes	—
191	C₂×Q₈	C₂×#92418	1	2	yes	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	C₂×#92418	1	2	yes	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
196	Q₁₆	C₂×#92418	1	4	yes	—
197	SD₁₆	C₂×#92418	1	8	yes	—
198	D₈	C₂×#92418	1	4	yes	—
199	M₁₆	—	2	2	no	—
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	C₂×#92418	1	4	yes	—
203	M₁₆	C₂²×Dic₆₁	1	4	yes	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	D₄×D₆₁	1	2	yes	—
208	C₂×C₈	#11758224	1	2	yes	—
209	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
210	C₂×C₈	C₂²×Dic₆₁	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	C₂²×Dic₆₁	1	2	yes	—
215	C₁₆	—	61	1	no	—
216	C₁₆	—	61	2	no	—
217	C₁₆	—	61	1	no	—
218	C₁₆	—	122	4	no	—
219	C₁₆	—	122	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	#14519784	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	Q₈	C₂²×#92418	1	4	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	4	no	—
226	D₄	—	2	2	no	—
227	D₄	C₂²×#92418	1	4	yes	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
233	C₂×C₄	C₂²×#92418	1	2	yes	—
234	C₂×C₄	#14519784	1	1	yes	characteristic
235	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
236	C₈	—	2	4	no	—
237	C₈	D₄×Dic₆₁	1	2	yes	—
238	C₈	C₂²×#92418	1	4	yes	—
239	C₈	C₂×D₄×D₆₁	1	1	yes	characteristic
240	C₈	C₂×#11758224	1	1	yes	characteristic
241	C₂²	—	2	4	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	4	no	—
244	C₂²	D₄×#92418	1	2	yes	—
245	C₂²	C₂×#14519784	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	4	2	no	—
248	C₄	D₄×#92418	1	2	yes	—
249	C₄	C₂×D₄×Dic₆₁	1	1	yes	characteristic
250	C₄	C₂³×#92418	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	2	no	—
253	C₂	—	4	4	no	—
254	C₂	C₂×D₄×#92418	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₆₁
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#2955	C₂
socle	C₂×C₆₁	C₂×C₈×D₄
prime radical	C₆₁×#575	C₄
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	#2955	D₆₁
hypocenter	C₆₁	#33441
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2955

Complements

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

N	H ≅ G/N	type
1. C₂×#16840179	C₂	semidirect
2. #21447661	C₂	semidirect
3. #21447656	C₂	semidirect
4. #21447585	C₂	semidirect
5. #21447447	C₂	semidirect
6. #21447416	C₂	semidirect
10. #16840179	C₂²	semidirect
11. #16840179	C₂²	semidirect
17. #16839583	C₂²	semidirect
18. #16839546	C₂²	semidirect
19. #16839538	C₂²	semidirect
20. C₄×#11758185	C₂²	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×#16840179	2
#21447661	2
#21447656	2
#21447585	2
#21447447	2
#21447416	2
C₆₁×#2955	2
#33441	61