Subgroups — #29833262

Order 15616.

642 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21458921	C₂	1	2	yes	maximal, complemented
2	#21454723	C₂	1	1	yes	maximal, characteristic
3	#21447662	C₂	1	4	yes	maximal, complemented
4	#21447657	C₂	1	4	yes	maximal, complemented
5	#21447585	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16839545	C₂	1	2	yes	maximal, complemented
7	C₆₁×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	#16840562	C₄	1	1	yes	characteristic
9	C₂×#14519783	C₂²	1	2	yes	—
10	#16840180	—	2	4	no	—
11	#16840178	C₂²	1	8	yes	complemented
12	C₂×#14519717	C₂²	1	2	yes	—
13	M₃₂×D₆₁	—	2	4	no	—
14	#16839885	C₄	1	4	yes	—
15	#16839880	C₂²	1	4	yes	—
16	#16839811	C₄	1	1	yes	characteristic
17	#16839780	C₄	1	2	yes	—
18	#16839583	C₂²	1	4	yes	complemented
19	#16839545	C₂²	1	4	yes	complemented
20	#16839545	C₂²	1	4	yes	complemented
21	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
23	C₆₁×#405	C₂²	1	1	yes	characteristic
24	C₆₁×#346	C₂²	1	2	yes	complemented
25	C₄×C₁₆×C₆₁	C₂²	1	2	yes	complemented
26	C₂×#11758222	C₂×C₄	1	2	yes	—
27	#14519810	C₈	1	4	yes	—
28	#14519805	C₈	1	4	yes	—
29	#14519799	C₂×C₄	1	4	yes	—
30	#14519783	—	2	2	no	—
31	#14519783	—	2	2	no	—
32	#14519783	C₂×C₄	1	4	yes	—
33	#14519783	C₂³	1	4	yes	—
34	C₂×#11758194	C₂×C₄	1	2	yes	—
35	C₂×#11758189	—	2	2	no	—
36	C₂×C₈×D₆₁	—	2	2	no	—
37	#14519717	—	2	2	no	—
38	#14519717	C₂³	1	4	yes	—
39	#14519716	—	2	8	no	—
40	C₂×#11758185	—	2	2	no	—
41	C₁₆×D₆₁	—	2	8	no	—
42	#14519713	C₂×C₄	1	4	yes	—
43	#14519705	C₂×C₄	1	8	yes	—
44	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
46	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
47	M₃₂×C₆₁	D₄	1	4	yes	—
48	C₂×C₁₆×C₆₁	—	2	2	no	—
49	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
50	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
51	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
52	#97×C₆₁	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₆₁	C₂³	1	2	yes	—
54	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₂²⋊D₆₁	—	2	2	no	—
56	#11758222	—	2	2	no	—
57	#11758222	—	2	2	no	—
58	#11758222	C₂×C₈	1	4	yes	—
59	#11758222	C₂²×C₄	1	4	yes	—
60	C₂×D₂₄₄	C₂×C₈	1	2	yes	—
61	C₂×C₄×D₆₁	—	2	2	no	—
62	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
63	M₁₆×C₆₁	C₂×C₈	1	4	yes	—
64	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₆₁	#41	1	2	yes	—
68	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₆₁	—	2	1	no	—
70	C₂²⋊C₄×C₆₁	—	2	1	no	—
71	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
72	#11758194	—	2	2	no	—
73	#11758194	C₂²×C₄	1	4	yes	—
74	C₂×#92418	—	2	2	no	—
75	Dic₂₄₄	C₂×C₈	1	4	yes	—
76	D₄₈₈	C₂×C₈	1	4	yes	—
77	#11758190	C₂×C₈	1	8	yes	—
78	#11758189	—	2	4	no	—
79	#11758189	—	2	4	no	—
80	C₈×D₆₁	—	2	4	no	—
81	C₈×D₆₁	—	2	4	no	—
82	C₁₆×C₆₁	—	2	1	no	—
83	C₁₆×C₆₁	—	2	1	no	—
84	C₁₆×C₆₁	C₂×D₄	1	4	yes	—
85	#11758185	—	2	4	no	—
86	C₂³×C₆₁	#131	1	1	yes	characteristic
87	C₂²×D₆₁	—	2	2	no	—
88	C₂×C₄×C₆₁	—	2	2	no	—
89	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
91	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
92	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
93	C₂²⋊D₆₁	—	2	4	no	—
94	C₂²⋊D₆₁	—	2	4	no	—
95	C₂×Dic₆₁	—	2	2	no	—
96	D₂₄₄	—	2	2	no	—
97	D₂₄₄	C₂²×C₈	1	4	yes	—
98	C₄×D₆₁	—	2	4	no	—
99	C₄×D₆₁	—	2	4	no	—
100	Dic₁₂₂	—	2	2	no	—
101	Dic₁₂₂	C₂²×C₈	1	4	yes	—
102	C₈×C₆₁	C₄×D₄	1	2	yes	—
103	C₈×C₆₁	C₂²×C₈	1	4	yes	—
104	C₈×C₆₁	C₂²×D₄	1	1	yes	characteristic
105	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
106	#92418	—	2	4	no	—
107	#33442	—	61	1	no	maximal, Sylow
108	C₂²×C₆₁	—	4	1	no	—
109	C₂²×C₆₁	C₈×D₄	1	2	yes	—
110	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
111	C₂×D₆₁	—	2	4	no	—
112	C₂×D₆₁	—	8	2	no	—
113	C₄×C₆₁	—	4	2	no	—
114	C₄×C₆₁	C₈×D₄	1	2	yes	—
115	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
116	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
117	Dic₆₁	—	2	4	no	—
118	#3407	—	61	2	no	—
119	#2955	—	61	1	no	—
120	#2171	—	61	4	no	—
121	#2170	—	61	4	no	—
122	#2153	—	61	1	no	—
123	C₂×#350	—	61	2	no	—
124	#2108	D₆₁	1	1	yes	characteristic, complemented
125	C₂×C₆₁	—	2	1	no	—
126	C₂×C₆₁	—	2	2	no	—
127	C₂×C₆₁	C₂×C₈×D₄	1	1	yes	characteristic
128	D₆₁	—	4	4	no	—
129	#575	—	61	1	no	—
130	C₂×#131	—	61	2	no	—
131	#504	—	61	8	no	—
132	#504	—	122	4	no	—
133	C₂×M₃₂	—	61	2	no	—
134	C₂×M₃₂	—	122	4	no	—
135	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
136	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
137	#444	—	61	4	no	—
138	#443	—	61	4	no	—
139	#430	—	61	1	no	—
140	#421	—	61	2	no	—
141	#405	C₂×D₆₁	1	1	yes	characteristic
142	#364	—	61	4	no	—
143	#350	—	61	4	no	—
144	#350	—	61	4	no	—
145	#346	C₂×D₆₁	1	2	yes	complemented
146	C₄×C₁₆	C₂×D₆₁	1	2	yes	complemented
147	C₆₁	#33442	1	1	yes	characteristic, complemented, Sylow
148	C₂×#41	—	61	2	no	—
149	#137	—	61	4	no	—
150	AGL(1,ℤ/8)	—	61	4	no	—
151	#135	—	61	4	no	—
152	#131	—	61	4	no	—
153	#131	—	61	4	no	—
154	#131	—	122	2	no	—
155	#131	—	122	2	no	—
156	C₂×M₁₆	—	61	2	no	—
157	C₂×M₁₆	—	122	2	no	—
158	C₂×M₁₆	C₄×D₆₁	1	1	yes	characteristic
159	C₂²×C₈	—	122	2	no	—
160	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
161	#117	C₄×D₆₁	1	1	yes	characteristic
162	M₃₂	—	61	4	no	—
163	M₃₂	—	122	2	no	—
164	M₃₂	—	122	8	no	—
165	M₃₂	D₂₄₄	1	4	yes	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	122	2	no	—
168	C₂×C₁₆	—	122	8	no	—
169	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
170	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
171	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
172	#108	—	61	4	no	—
173	#104	—	61	8	no	—
174	#97	C₄×D₆₁	1	2	yes	—
175	C₄×C₈	C₂²×D₆₁	1	2	yes	—
176	#41	—	61	4	no	—
177	#41	—	61	4	no	—
178	#41	—	122	2	no	—
179	#41	—	122	2	no	—
180	C₂×Q₈	—	61	2	no	—
181	C₂×D₄	—	61	2	no	—
182	C₂×D₄	—	122	2	no	—
183	C₂²×C₄	—	122	2	no	—
184	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
185	Q₁₆	—	61	4	no	—
186	SD₁₆	—	61	8	no	—
187	D₈	—	61	4	no	—
188	M₁₆	—	61	4	no	—
189	M₁₆	—	122	2	no	—
190	M₁₆	—	122	4	no	—
191	M₁₆	—	122	4	no	—
192	M₁₆	C₈×D₆₁	1	4	yes	—
193	C₂×C₈	—	122	2	no	—
194	C₂×C₈	—	122	4	no	—
195	C₂×C₈	—	122	4	no	—
196	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
197	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
198	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
199	C₂×C₈	#11758222	1	2	yes	—
200	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
201	C₄⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₄²	C₂×C₄×D₆₁	1	2	yes	—
204	C₁₆	—	2	1	no	—
205	C₁₆	—	2	1	no	—
206	C₁₆	—	122	4	no	—
207	C₁₆	C₂×D₂₄₄	1	4	yes	—
208	C₂³	—	122	2	no	—
209	C₂³	#14519783	1	1	yes	characteristic
210	Q₈	—	61	4	no	—
211	Q₈	—	122	2	no	—
212	D₄	—	61	4	no	—
213	D₄	—	122	4	no	—
214	D₄	—	122	4	no	—
215	D₄	—	122	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	122	2	no	—
218	C₂×C₄	—	122	4	no	—
219	C₂×C₄	—	122	4	no	—
220	C₂×C₄	C₄×D₂₄₄	1	2	yes	—
221	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
222	C₂×C₄	#14519783	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
224	C₈	—	122	4	no	—
225	C₈	C₄×D₂₄₄	1	2	yes	—
226	C₈	C₂×C₈×D₆₁	1	4	yes	—
227	C₈	C₂²×D₂₄₄	1	1	yes	characteristic
228	C₈	C₂×#11758222	1	1	yes	characteristic
229	C₂²	—	4	1	no	—
230	C₂²	—	122	4	no	—
231	C₂²	—	488	2	no	—
232	C₂²	C₈×D₂₄₄	1	2	yes	—
233	C₂²	C₂×#14519783	1	1	yes	characteristic
234	C₄	—	4	2	no	—
235	C₄	—	122	4	no	—
236	C₄	C₈×D₂₄₄	1	2	yes	—
237	C₄	C₂×C₄×D₂₄₄	1	1	yes	characteristic
238	C₄	C₂²×C₈×D₆₁	1	1	yes	characteristic
239	C₂	—	2	1	no	—
240	C₂	—	2	2	no	—
241	C₂	—	244	4	no	—
242	C₂	C₂×C₈×D₂₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×D₂₄₄
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#2108	C₂
socle	C₂×C₆₁	C₂×C₈×D₄
prime radical	#16840562	C₄
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	#2108	D₆₁
hypocenter	C₆₁	#33442
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ #2108

Complements

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

N	H ≅ G/N	type
1. #21458921	C₂	semidirect
3. #21447662	C₂	semidirect
4. #21447657	C₂	semidirect
5. #21447585	C₂	semidirect
6. C₂×#16839545	C₂	semidirect
7. C₆₁×#2108	C₂	semidirect
11. #16840178	C₂²	semidirect
18. #16839583	C₂²	semidirect
19. #16839545	C₂²	semidirect
20. #16839545	C₂²	semidirect
24. C₆₁×#346	C₂²	semidirect
25. C₄×C₁₆×C₆₁	C₂²	semidirect
124. #2108	D₆₁	semidirect
145. #346	C₂×D₆₁	semidirect
146. C₄×C₁₆	C₂×D₆₁	semidirect
147. C₆₁	#33442	semidirect

Maximal subgroups

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

subgroup	index
#21458921	2
#21454723	2
#21447662	2
#21447657	2
#21447585	2
C₂×#16839545	2
C₆₁×#2108	2
#33442	61