Subgroups — #29835371

Order 15616.

578 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456333	C₂	1	1	yes	maximal, characteristic
2	#21448064	C₂	1	4	yes	maximal, complemented
3	#21448046	C₂	1	4	yes	maximal, complemented
4	#21447539	C₂	1	4	yes	maximal
5	C₄×#14519717	C₂	1	1	yes	maximal, characteristic
6	C₂×C₆₁×#494	C₂	1	1	yes	maximal, characteristic
7	#16840319	—	2	1	no	—
8	C₂×#14519717	C₂²	1	2	yes	—
9	#16840167	C₂²	1	2	yes	—
10	#16840164	—	2	1	no	—
11	#16840164	C₂²	1	2	yes	—
12	#16840160	—	2	1	no	—
13	#16839800	—	2	1	no	—
14	#16839800	C₂²	1	2	yes	—
15	#16839566	C₂²	1	16	yes	—
16	#16839538	C₂²	1	2	yes	—
17	C₄×#11758185	C₂²	1	2	yes	—
18	C₂²×Q₁₆×C₆₁	—	2	1	no	—
19	C₂²×Q₁₆×C₆₁	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₆₁	—	2	1	no	—
21	C₆₁×#494	—	2	1	no	—
22	C₆₁×#494	—	2	1	no	—
23	C₆₁×#494	C₂²	1	2	yes	—
24	C₆₁×#494	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
26	#14519777	—	2	2	no	—
27	#14519769	—	2	2	no	—
28	#14519756	—	4	2	no	—
29	C₄×Dic₁₂₂	—	4	2	no	—
30	#14519736	—	2	2	no	—
31	#14519717	—	2	4	no	—
32	C₂×#11758185	C₂³	1	4	yes	—
33	#14519711	—	2	2	no	—
34	#14519711	C₂³	1	4	yes	—
35	#14519709	—	2	2	no	—
36	#14519709	—	2	2	no	—
37	C₂²×Q₈×C₆₁	—	2	2	no	—
38	C₂×Q₁₆×C₆₁	—	2	2	no	—
39	C₂×Q₁₆×C₆₁	—	2	2	no	—
40	C₂×Q₁₆×C₆₁	—	2	2	no	—
41	C₂×Q₁₆×C₆₁	—	2	8	no	—
42	C₂×Q₁₆×C₆₁	—	4	4	no	—
43	C₂×Q₁₆×C₆₁	C₂³	1	4	yes	—
44	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
46	C₄⋊Q₈×C₆₁	—	2	2	no	—
47	C₄⋊Q₈×C₆₁	—	2	2	no	—
48	C₄⋊Q₈×C₆₁	—	4	2	no	—
49	C₂×C₄⋊C₄×C₆₁	—	4	2	no	—
50	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
52	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
53	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
55	C₂×Q₈×C₆₁	—	2	4	no	—
56	C₂×Q₈×C₆₁	—	4	2	no	—
57	C₂×Q₈×C₆₁	—	4	4	no	—
58	C₂×Q₈×C₆₁	—	4	4	no	—
59	C₂²×C₄×C₆₁	—	4	2	no	—
60	C₂²×C₄×C₆₁	D₈	1	2	yes	—
61	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂×Dic₁₂₂	—	4	2	no	—
63	Q₁₆×C₆₁	—	2	8	no	—
64	Q₁₆×C₆₁	—	4	4	no	—
65	Q₁₆×C₆₁	—	4	4	no	—
66	Q₁₆×C₆₁	—	4	4	no	—
67	C₂×C₈×C₆₁	SD₁₆	1	4	yes	—
68	C₂×C₈×C₆₁	C₂×D₄	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	#41	1	4	yes	—
72	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₆₁	—	4	4	no	—
74	C₄⋊C₄×C₆₁	—	4	4	no	—
75	C₄²×C₆₁	D₈	1	2	yes	—
76	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
78	C₂²⋊Dic₆₁	—	4	2	no	—
79	C₄⋊Dic₆₁	—	2	4	no	—
80	C₄⋊Dic₆₁	—	4	2	no	—
81	#11758196	—	4	4	no	—
82	C₄×Dic₆₁	—	4	2	no	—
83	#11758185	—	2	4	no	—
84	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
85	Q₈×C₆₁	—	4	4	no	—
86	Q₈×C₆₁	—	4	4	no	—
87	Q₈×C₆₁	—	8	4	no	—
88	C₂×C₄×C₆₁	—	2	4	no	—
89	C₂×C₄×C₆₁	—	4	4	no	—
90	C₂×C₄×C₆₁	—	8	4	no	—
91	C₂×C₄×C₆₁	#124	1	2	yes	—
92	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
93	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
96	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
97	C₂×Dic₆₁	—	4	4	no	—
98	Dic₁₂₂	—	8	4	no	—
99	C₈×C₆₁	#124	1	2	yes	—
100	C₈×C₆₁	C₂×SD₁₆	1	2	yes	—
101	C₈×C₆₁	C₂×SD₁₆	1	2	yes	—
102	C₈×C₆₁	C₂×#41	1	2	yes	—
103	#33705	—	61	1	no	maximal, Sylow
104	C₂²×C₆₁	—	2	2	no	—
105	C₂²×C₆₁	#486	1	2	yes	—
106	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
107	C₄×C₆₁	—	2	2	no	—
108	C₄×C₆₁	—	8	4	no	—
109	C₄×C₆₁	#486	1	2	yes	—
110	C₄×C₆₁	#510	1	4	yes	—
111	C₄×C₆₁	C₂×#124	1	1	yes	characteristic
112	C₄×C₆₁	C₂²×SD₁₆	1	1	yes	characteristic
113	Dic₆₁	—	8	4	no	—
114	#3160	—	61	1	no	—
115	C₂×#494	D₆₁	1	1	yes	characteristic
116	#2245	—	61	4	no	—
117	#2241	—	61	4	no	—
118	#2141	—	61	4	no	—
119	C₄×M₃₂	—	61	1	no	—
120	C₂×C₆₁	—	2	2	no	—
121	C₂×C₆₁	C₂×#486	1	1	yes	characteristic
122	C₂×C₆₁	C₂×#510	1	2	yes	—
123	C₂²×Q₁₆	—	2	1	no	—
124	C₂²×Q₁₆	C₂×D₆₁	1	2	yes	—
125	#533	—	122	1	no	—
126	C₂×C₄⋊Q₈	—	2	1	no	—
127	C₂×M₃₂	—	61	2	no	—
128	#500	—	61	2	no	—
129	#499	—	61	2	no	—
130	#499	—	122	1	no	—
131	#498	—	122	1	no	—
132	#494	—	2	1	no	—
133	#494	—	2	1	no	—
134	#494	C₂×D₆₁	1	2	yes	—
135	#494	C₂×D₆₁	1	2	yes	—
136	#427	—	61	2	no	—
137	#427	—	122	1	no	—
138	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
139	#358	—	61	16	no	—
140	#346	—	61	2	no	—
141	C₄×C₁₆	—	61	2	no	—
142	C₆₁	#33705	1	1	yes	characteristic, complemented, Sylow
143	C₂²×Q₈	—	2	2	no	—
144	C₂×Q₁₆	—	2	8	no	—
145	C₂×Q₁₆	—	2	2	no	—
146	C₂×Q₁₆	—	2	2	no	—
147	C₂×Q₁₆	—	2	2	no	—
148	C₂×Q₁₆	—	4	4	no	—
149	C₂×Q₁₆	C₂²×D₆₁	1	4	yes	—
150	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
151	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	—	2	2	no	—
154	C₄⋊Q₈	—	4	2	no	—
155	C₄⋊Q₈	—	122	2	no	—
156	#125	—	122	2	no	—
157	C₂²⋊Q₈	—	244	2	no	—
158	C₄×Q₈	—	244	2	no	—
159	#117	—	122	2	no	—
160	C₂×C₄⋊C₄	—	4	2	no	—
161	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
162	M₃₂	—	122	4	no	—
163	C₂×C₁₆	—	61	4	no	—
164	#107	—	61	4	no	—
165	#107	—	122	2	no	—
166	#106	—	122	2	no	—
167	#106	—	122	2	no	—
168	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
169	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
170	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
171	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	4	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂×Q₈	—	244	2	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	#11758199	1	2	yes	—
179	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
180	Q₁₆	—	2	8	no	—
181	Q₁₆	—	4	4	no	—
182	Q₁₆	—	4	4	no	—
183	Q₁₆	—	4	4	no	—
184	C₂×C₈	#11758200	1	4	yes	—
185	C₂×C₈	#11758224	1	4	yes	—
186	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
187	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
188	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
189	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
190	C₄⋊C₄	—	4	4	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	—	122	4	no	—
193	C₄⋊C₄	—	244	4	no	—
194	C₄⋊C₄	—	244	2	no	—
195	C₂²⋊C₄	—	244	2	no	—
196	C₄²	—	244	2	no	—
197	C₄²	#11758199	1	2	yes	—
198	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
199	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
200	C₁₆	—	122	4	no	—
201	C₂³	C₂×#11758199	1	1	yes	characteristic
202	Q₈	—	4	4	no	—
203	Q₈	—	4	4	no	—
204	Q₈	—	8	4	no	—
205	Q₈	—	488	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	8	4	no	—
209	C₂×C₄	—	244	4	no	—
210	C₂×C₄	#14519764	1	2	yes	—
211	C₂×C₄	C₂×#11758199	1	2	yes	—
212	C₂×C₄	C₂×#11758199	1	2	yes	—
213	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
214	C₂×C₄	C₂×#11758200	1	2	yes	—
215	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
216	C₈	#14519764	1	2	yes	—
217	C₈	C₂×#11758200	1	2	yes	—
218	C₈	C₂×#11758200	1	2	yes	—
219	C₈	C₂×#11758224	1	2	yes	—
220	C₂²	—	2	2	no	—
221	C₂²	#16840108	1	2	yes	—
222	C₂²	C₂²×#11758199	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	8	4	no	—
225	C₄	—	488	4	no	—
226	C₄	#16840108	1	2	yes	—
227	C₄	#16840203	1	4	yes	—
228	C₄	C₂×#14519764	1	1	yes	characteristic
229	C₄	C₂²×#11758200	1	1	yes	characteristic
230	C₂	—	2	2	no	—
231	C₂	C₂×#16840108	1	1	yes	characteristic
232	C₂	C₂×#16840203	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#11758199
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂×C₆₁×#494	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	C₂³×C₆₁	C₂×D₈
hypercenter	C₂×#494	D₆₁
hypocenter	C₆₁	#33705
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
2. #21448064	C₂	semidirect
3. #21448046	C₂	semidirect
142. C₆₁	#33705	semidirect

Maximal subgroups

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

subgroup	index
#21456333	2
#21448064	2
#21448046	2
#21447539	2
C₄×#14519717	2
C₂×C₆₁×#494	2
#33705	61