Subgroups — #29650573

Order 15616.

598 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456405	C₂	1	1	yes	maximal, characteristic
2	#21454753	C₂	1	2	yes	maximal
3	#21444729	C₂	1	4	yes	maximal, complemented
4	#21444639	C₂	1	4	yes	maximal, complemented
5	#21444575	C₂	1	1	yes	maximal, characteristic
6	#21444275	C₂	1	2	yes	maximal
7	C₂×C₆₁×#322	C₂	1	1	yes	maximal, characteristic
8	#16840319	C₂²	1	2	yes	—
9	#16840167	C₄	1	1	yes	characteristic
10	#16840142	C₄	1	1	yes	characteristic
11	#16840113	C₄	1	2	yes	—
12	#16840003	C₄	1	4	yes	—
13	#16839892	—	2	2	no	—
14	#16839892	C₂²	1	4	yes	—
15	C₈×Dic₁₂₂	—	2	2	no	—
16	C₈×Dic₁₂₂	C₂²	1	4	yes	—
17	C₄×Dic₂₄₄	C₂²	1	4	yes	—
18	#16839788	—	2	2	no	—
19	#16839786	C₂²	1	2	yes	—
20	#16839717	—	2	2	no	—
21	#16839513	C₂²	1	4	yes	—
22	#16839494	C₂²	1	8	yes	—
23	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#322	C₂²	1	2	yes	—
27	C₆₁×#322	C₂²	1	2	yes	—
28	C₂×Dic₂₄₄	C₂×C₄	1	4	yes	—
29	C₂×#11758194	—	2	2	no	—
30	#14519777	C₂×C₄	1	2	yes	—
31	#14519777	C₂×C₄	1	2	yes	—
32	#14519769	—	2	2	no	—
33	#14519756	—	2	2	no	—
34	#14519756	C₂×C₄	1	4	yes	—
35	C₄×Dic₁₂₂	—	2	2	no	—
36	C₄×Dic₁₂₂	C₂³	1	4	yes	—
37	#14519736	—	2	2	no	—
38	#14519711	C₂×C₄	1	4	yes	—
39	C₄⋊#92418	—	2	2	no	—
40	C₄⋊#92418	C₂³	1	4	yes	—
41	#14519706	—	2	4	no	—
42	#14519706	—	2	4	no	—
43	#14519702	C₂×C₄	1	8	yes	—
44	#14519687	—	2	4	no	—
45	C₈×Dic₆₁	—	2	4	no	—
46	C₂²×C₈×C₆₁	—	2	1	no	—
47	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₆₁	D₄	1	2	yes	—
49	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₆₁	—	2	1	no	—
51	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₆₁	D₄	1	2	yes	—
54	C₄×C₈×C₆₁	C₂³	1	2	yes	—
55	C₄×C₈×C₆₁	C₂³	1	2	yes	—
56	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
61	C₂×Dic₁₂₂	—	2	2	no	—
62	C₂×Dic₁₂₂	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₆₁	—	2	4	no	—
64	C₂×C₈×C₆₁	—	2	4	no	—
65	C₂×C₈×C₆₁	—	2	2	no	—
66	C₂×C₈×C₆₁	—	4	2	no	—
67	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
71	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₆₁	#41	1	1	yes	characteristic
74	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
75	C₂²⋊Dic₆₁	—	2	4	no	—
76	C₄⋊Dic₆₁	—	2	2	no	—
77	C₄⋊Dic₆₁	—	2	2	no	—
78	C₄⋊Dic₆₁	—	2	2	no	—
79	C₄⋊Dic₆₁	C₂²×C₄	1	4	yes	—
80	#11758196	—	2	4	no	—
81	#11758196	—	2	4	no	—
82	C₄×Dic₆₁	—	2	4	no	—
83	#11758194	—	4	4	no	—
84	C₂×#92418	—	2	4	no	—
85	Dic₂₄₄	—	2	8	no	—
86	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₆₁	—	2	2	no	—
88	C₂×C₄×C₆₁	—	2	2	no	—
89	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₆₁	#137	1	4	yes	—
93	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
96	C₂×Dic₆₁	—	2	4	no	—
97	C₂×Dic₆₁	—	2	4	no	—
98	Dic₁₂₂	—	2	8	no	—
99	Dic₁₂₂	—	4	4	no	—
100	C₈×C₆₁	—	2	2	no	—
101	C₈×C₆₁	—	2	2	no	—
102	C₈×C₆₁	—	2	2	no	—
103	C₈×C₆₁	—	4	2	no	—
104	C₈×C₆₁	#131	1	4	yes	—
105	#92418	—	4	4	no	—
106	#17961	—	61	1	no	maximal, Sylow
107	C₂²×C₆₁	—	2	1	no	—
108	C₂²×C₆₁	—	2	1	no	—
109	C₂²×C₆₁	#441	1	2	yes	—
110	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
111	C₄×C₆₁	—	2	1	no	—
112	C₄×C₆₁	—	2	1	no	—
113	C₄×C₆₁	#441	1	2	yes	—
114	C₄×C₆₁	#444	1	2	yes	—
115	C₄×C₆₁	C₂×#131	1	2	yes	—
116	C₄×C₆₁	C₂×#137	1	2	yes	—
117	Dic₆₁	—	4	4	no	—
118	Dic₆₁	—	4	4	no	—
119	#3170	—	61	1	no	—
120	#2960	—	61	2	no	—
121	#1593	—	61	4	no	—
122	#1579	—	61	4	no	—
123	#1566	—	61	1	no	—
124	#1493	—	61	2	no	—
125	C₂×#322	D₆₁	1	1	yes	characteristic
126	C₂×C₆₁	—	2	2	no	—
127	C₂×C₆₁	#2931	1	1	yes	characteristic
128	C₂×C₆₁	C₂×#441	1	1	yes	characteristic
129	C₂×C₆₁	C₂×#444	1	1	yes	characteristic
130	#533	—	61	2	no	—
131	#500	—	61	1	no	—
132	#494	—	61	1	no	—
133	#487	—	61	2	no	—
134	#467	—	61	4	no	—
135	#446	—	61	4	no	—
136	#446	—	122	2	no	—
137	C₈×Q₈	—	61	4	no	—
138	C₈×Q₈	—	122	2	no	—
139	C₄×Q₁₆	—	61	4	no	—
140	#424	—	122	2	no	—
141	#423	—	61	2	no	—
142	#405	—	122	2	no	—
143	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
146	#334	—	61	4	no	—
147	Q₈⋊C₈	—	61	8	no	—
148	#322	C₂×D₆₁	1	2	yes	—
149	#322	C₂×D₆₁	1	2	yes	—
150	C₆₁	#17961	1	1	yes	characteristic, complemented, Sylow
151	C₂×Q₁₆	—	61	4	no	—
152	C₂×M₁₆	—	122	2	no	—
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	C₄×D₆₁	1	2	yes	—
155	C₂²×C₈	D₂₄₄	1	2	yes	—
156	C₄⋊Q₈	—	61	2	no	—
157	C₄⋊Q₈	—	61	2	no	—
158	#125	—	122	2	no	—
159	C₂²⋊Q₈	—	61	4	no	—
160	C₂²⋊Q₈	—	122	2	no	—
161	C₄×Q₈	—	61	4	no	—
162	C₄×Q₈	—	122	2	no	—
163	#117	—	122	2	no	—
164	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
165	#107	—	61	4	no	—
166	C₄⋊C₈	—	61	4	no	—
167	C₄⋊C₈	—	122	4	no	—
168	C₄⋊C₈	—	122	4	no	—
169	C₄⋊C₈	—	122	2	no	—
170	#103	—	61	8	no	—
171	#97	—	122	4	no	—
172	C₄×C₈	—	2	1	no	—
173	C₄×C₈	—	122	4	no	—
174	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
175	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
176	C₄×C₈	D₂₄₄	1	2	yes	—
177	C₄×C₈	C₂²×D₆₁	1	2	yes	—
178	C₄×C₈	C₂²×D₆₁	1	2	yes	—
179	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
180	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
181	C₂×Q₈	—	61	4	no	—
182	C₂×Q₈	—	122	2	no	—
183	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
185	C₂²×C₄	#11758222	1	1	yes	characteristic
186	Q₁₆	—	122	8	no	—
187	M₁₆	—	244	4	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	122	4	no	—
193	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
194	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
195	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
196	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
197	C₄⋊C₄	—	61	4	no	—
198	C₄⋊C₄	—	122	4	no	—
199	C₄⋊C₄	—	122	4	no	—
200	C₄⋊C₄	—	122	2	no	—
201	C₄⋊C₄	—	122	2	no	—
202	C₄⋊C₄	—	122	2	no	—
203	C₂²⋊C₄	—	122	4	no	—
204	C₄²	—	122	4	no	—
205	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
206	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
207	C₄²	#11758222	1	1	yes	characteristic
208	C₄²	C₂³×D₆₁	1	1	yes	characteristic
209	C₂³	C₄×D₂₄₄	1	1	yes	characteristic
210	Q₈	—	122	8	no	—
211	Q₈	—	244	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	122	4	no	—
215	C₂×C₄	—	122	4	no	—
216	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
217	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
218	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
219	C₂×C₄	#14519810	1	4	yes	—
220	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
222	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	4	2	no	—
227	C₈	—	244	4	no	—
228	C₈	#14519783	1	4	yes	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	#16839870	1	2	yes	—
232	C₂²	C₂×C₄×D₂₄₄	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	244	4	no	—
236	C₄	—	244	4	no	—
237	C₄	#16839870	1	2	yes	—
238	C₄	#16839885	1	2	yes	—
239	C₄	C₂×#14519783	1	2	yes	—
240	C₄	C₂×#14519810	1	2	yes	—
241	C₂	—	2	2	no	—
242	C₂	#21454532	1	1	yes	characteristic
243	C₂	C₂×#16839870	1	1	yes	characteristic
244	C₂	C₂×#16839885	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₄²	C₂³×D₆₁
Fitting	C₂×C₆₁×#322	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	C₂×#322	D₆₁
hypocenter	C₆₁	#17961
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
3. #21444729	C₂	semidirect
4. #21444639	C₂	semidirect
150. C₆₁	#17961	semidirect

Maximal subgroups

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

subgroup	index
#21456405	2
#21454753	2
#21444729	2
#21444639	2
#21444575	2
#21444275	2
C₂×C₆₁×#322	2
#17961	61