Subgroups — #29578871

Order 15616.

502 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic
2	#21446755	C₂	1	2	yes	maximal
3	C₂×#16839515	C₂	1	1	yes	maximal, characteristic
4	C₂×#16839494	C₂	1	2	yes	maximal
5	C₂×C₈²×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₄₄	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₂₂	C₂²	1	2	yes	—
8	C₄×Dic₂₄₄	—	2	2	no	—
9	C₄×Dic₂₄₄	—	2	2	no	—
10	C₂×#14519711	C₄	1	1	yes	characteristic
11	C₂×C₄⋊#92418	C₂²	1	2	yes	—
12	C₂×#14519702	C₄	1	2	yes	—
13	#16839516	—	2	4	no	—
14	#16839515	—	2	2	no	—
15	#16839494	—	2	4	no	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₈²×C₆₁	—	2	2	no	—
20	C₂²×Dic₁₂₂	C₂×C₄	1	2	yes	—
21	C₂×Dic₂₄₄	—	2	2	no	—
22	C₂×Dic₂₄₄	—	2	2	no	—
23	C₂×Dic₂₄₄	C₈	1	4	yes	complemented
24	C₂²×#92418	—	2	2	no	—
25	C₄×Dic₁₂₂	—	2	2	no	—
26	C₄×Dic₁₂₂	—	2	2	no	—
27	C₄×Dic₁₂₂	—	2	2	no	—
28	C₄×Dic₁₂₂	—	2	2	no	—
29	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
30	C₂×#11758196	—	2	2	no	—
31	C₂×C₄×Dic₆₁	—	2	2	no	—
32	#14519711	—	2	2	no	—
33	C₄⋊#92418	—	2	2	no	—
34	C₄⋊#92418	—	2	2	no	—
35	#14519702	—	2	4	no	—
36	C₂²×C₈×C₆₁	—	2	1	no	—
37	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₁	D₄	1	2	yes	—
40	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
41	C₄×C₈×C₆₁	—	2	1	no	—
42	C₄×C₈×C₆₁	—	2	1	no	—
43	C₄×C₈×C₆₁	—	2	1	no	—
44	C₄×C₈×C₆₁	D₄	1	2	yes	—
45	C₄×C₈×C₆₁	D₄	1	2	yes	—
46	C₄×C₈×C₆₁	D₄	1	2	yes	—
47	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
50	C₂²×Dic₆₁	—	2	2	no	—
51	C₂²×Dic₆₁	—	2	2	no	—
52	C₂×Dic₁₂₂	—	2	2	no	—
53	C₂×Dic₁₂₂	—	2	2	no	—
54	C₂×Dic₁₂₂	—	2	2	no	—
55	C₂×Dic₁₂₂	—	2	2	no	—
56	C₂×Dic₁₂₂	—	2	2	no	—
57	C₂×Dic₁₂₂	C₂×C₈	1	4	yes	—
58	C₂×C₈×C₆₁	—	2	4	no	—
59	C₂×C₈×C₆₁	—	2	2	no	—
60	C₂×C₈×C₆₁	—	2	2	no	—
61	C₂×C₈×C₆₁	—	2	1	no	—
62	C₂×C₈×C₆₁	—	2	1	no	—
63	C₂×C₈×C₆₁	—	2	2	no	—
64	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
67	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
68	C₂×C₈×C₆₁	Q₁₆	1	4	yes	complemented
69	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₆₁	#41	1	1	yes	characteristic
73	C₄⋊Dic₆₁	—	2	2	no	—
74	C₄⋊Dic₆₁	—	2	2	no	—
75	#11758196	—	4	2	no	—
76	#11758196	—	4	2	no	—
77	C₄×Dic₆₁	—	4	2	no	—
78	C₄×Dic₆₁	—	4	2	no	—
79	C₂×#92418	—	4	2	no	—
80	C₂×#92418	—	4	4	no	—
81	Dic₂₄₄	—	2	8	no	—
82	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₆₁	—	2	2	no	—
84	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
87	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
88	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
89	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
90	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
93	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
94	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
95	C₂×C₄×C₆₁	#135	1	2	yes	—
96	C₂×Dic₆₁	—	2	4	no	—
97	C₂×Dic₆₁	—	4	2	no	—
98	C₂×Dic₆₁	—	4	2	no	—
99	C₂×Dic₆₁	—	4	2	no	—
100	C₂×Dic₆₁	—	4	4	no	—
101	Dic₁₂₂	—	2	4	no	—
102	Dic₁₂₂	—	2	4	no	—
103	Dic₁₂₂	—	4	2	no	—
104	Dic₁₂₂	—	4	2	no	—
105	C₈×C₆₁	—	2	4	no	—
106	C₈×C₆₁	—	2	2	no	—
107	C₈×C₆₁	—	2	4	no	—
108	C₈×C₆₁	C₂²⋊C₈	1	4	yes	—
109	#92418	—	8	2	no	—
110	#9656	—	61	1	no	maximal, Sylow
111	C₂²×C₆₁	#390	1	1	yes	characteristic
112	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
113	C₂²×C₆₁	#436	1	1	yes	characteristic
114	C₂²×C₆₁	C₄×Q₁₆	1	2	yes	—
115	C₂²×C₆₁	#443	1	2	yes	—
116	C₄×C₆₁	—	2	1	no	—
117	C₄×C₆₁	—	2	1	no	—
118	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₄×C₆₁	#408	1	1	yes	characteristic
120	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
121	C₄×C₆₁	#436	1	1	yes	characteristic
122	C₄×C₆₁	#467	1	2	yes	—
123	C₄×C₆₁	#494	1	1	yes	characteristic
124	C₄×C₆₁	#495	1	1	yes	characteristic
125	Dic₆₁	—	4	4	no	—
126	Dic₆₁	—	8	2	no	—
127	C₂×C₄×Q₁₆	—	61	1	no	—
128	#1984	—	61	2	no	—
129	C₂×#335	—	61	1	no	—
130	C₂×Q₈⋊C₈	—	61	2	no	—
131	C₂×C₈²	D₆₁	1	1	yes	characteristic
132	C₂×C₆₁	C₈×Q₁₆	1	2	yes	—
133	C₂×C₆₁	#1591	1	2	yes	—
134	C₂×C₆₁	#1954	1	1	yes	characteristic
135	C₂×C₆₁	#1969	1	1	yes	characteristic
136	C₂×C₆₁	#1974	1	1	yes	characteristic
137	C₂²×Q₁₆	—	61	1	no	—
138	C₂×C₄×Q₈	—	61	2	no	—
139	C₄×Q₁₆	—	122	2	no	—
140	C₄×Q₁₆	—	122	2	no	—
141	C₂×#107	—	61	1	no	—
142	C₂×C₄⋊C₈	—	61	2	no	—
143	C₂×#103	—	61	2	no	—
144	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
147	#336	—	122	4	no	—
148	#335	—	122	2	no	—
149	Q₈⋊C₈	—	122	4	no	—
150	C₈²	—	2	2	no	—
151	C₆₁	#9656	1	1	yes	characteristic, complemented, Sylow
152	C₂²×Q₈	—	61	2	no	—
153	C₂×Q₁₆	—	61	4	no	—
154	C₂×Q₁₆	—	122	2	no	—
155	C₂×Q₁₆	—	122	2	no	—
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	—	122	2	no	—
158	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
159	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
160	C₂²×C₈	D₂₄₄	1	2	yes	—
161	C₄×Q₈	—	122	2	no	—
162	C₄×Q₈	—	122	2	no	—
163	C₄×Q₈	—	122	2	no	—
164	C₄×Q₈	—	122	2	no	—
165	C₂×C₄⋊C₄	—	61	2	no	—
166	C₂×C₄⋊C₄	—	122	2	no	—
167	C₂×C₄²	—	122	2	no	—
168	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
169	#107	—	122	2	no	—
170	C₄⋊C₈	—	122	2	no	—
171	C₄⋊C₈	—	122	2	no	—
172	#103	—	122	4	no	—
173	C₄×C₈	—	2	1	no	—
174	C₄×C₈	—	2	1	no	—
175	C₄×C₈	—	2	1	no	—
176	C₄×C₈	D₂₄₄	1	2	yes	—
177	C₄×C₈	D₂₄₄	1	2	yes	—
178	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
179	C₂×Q₈	—	61	4	no	—
180	C₂×Q₈	—	122	2	no	—
181	C₂×Q₈	—	122	2	no	—
182	C₂×Q₈	—	122	2	no	—
183	C₂×Q₈	—	122	2	no	—
184	C₂×Q₈	—	122	2	no	—
185	C₂²×C₄	—	122	2	no	—
186	C₂²×C₄	—	122	2	no	—
187	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
189	C₂²×C₄	#11758222	1	1	yes	characteristic
190	Q₁₆	—	122	8	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	244	2	no	—
198	C₂×C₈	—	244	4	no	—
199	C₂×C₈	C₈×D₆₁	1	2	yes	—
200	C₂×C₈	#11758189	1	2	yes	—
201	C₂×C₈	Dic₂₄₄	1	4	yes	complemented
202	C₂×C₈	#11758198	1	2	yes	—
203	C₂×C₈	#11758198	1	2	yes	—
204	C₄⋊C₄	—	122	2	no	—
205	C₄⋊C₄	—	122	2	no	—
206	C₄⋊C₄	—	244	2	no	—
207	C₄⋊C₄	—	244	2	no	—
208	C₄²	—	244	2	no	—
209	C₄²	—	244	2	no	—
210	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
211	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
212	C₄²	#11758222	1	1	yes	characteristic
213	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
214	C₂³	C₄×D₂₄₄	1	1	yes	characteristic
215	Q₈	—	122	4	no	—
216	Q₈	—	122	4	no	—
217	Q₈	—	244	2	no	—
218	Q₈	—	244	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	122	4	no	—
221	C₂×C₄	—	244	4	no	—
222	C₂×C₄	—	244	2	no	—
223	C₂×C₄	—	244	2	no	—
224	C₂×C₄	—	244	2	no	—
225	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
226	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
227	C₂×C₄	#14519751	1	1	yes	characteristic
228	C₂×C₄	#14519751	1	1	yes	characteristic
229	C₂×C₄	#14519765	1	1	yes	characteristic
230	C₂×C₄	#14519774	1	1	yes	characteristic
231	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
232	C₂×C₄	C₂×#11758189	1	1	yes	characteristic
233	C₂×C₄	#14519783	1	1	yes	characteristic
234	C₂×C₄	#14519783	1	1	yes	characteristic
235	C₂×C₄	C₂×Dic₂₄₄	1	2	yes	—
236	C₂×C₄	#14519799	1	2	yes	—
237	C₈	—	2	4	no	—
238	C₈	—	2	2	no	—
239	C₈	—	2	4	no	—
240	C₈	—	488	2	no	—
241	C₈	#14519689	1	4	yes	—
242	C₂²	#16839664	1	1	yes	characteristic
243	C₂²	C₈×D₂₄₄	1	1	yes	characteristic
244	C₂²	#16839841	1	1	yes	characteristic
245	C₂²	C₄×Dic₂₄₄	1	2	yes	—
246	C₂²	#16839880	1	2	yes	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	488	2	no	—
250	C₄	—	244	4	no	—
251	C₄	C₂×#14519689	1	1	yes	characteristic
252	C₄	#16839728	1	1	yes	characteristic
253	C₄	C₈×D₂₄₄	1	1	yes	characteristic
254	C₄	#16839841	1	1	yes	characteristic
255	C₄	#16840003	1	2	yes	—
256	C₄	#16840142	1	1	yes	characteristic
257	C₄	#16840146	1	1	yes	characteristic
258	C₂	C₈×Dic₂₄₄	1	2	yes	—
259	C₂	#21444717	1	2	yes	—
260	C₂	#21446613	1	1	yes	characteristic
261	C₂	#21446680	1	1	yes	characteristic
262	C₂	#21446707	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₈²×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×C₈²	D₆₁
hypocenter	C₆₁	#9656
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
23. C₂×Dic₂₄₄	C₈	semidirect
68. C₂×C₈×C₆₁	Q₁₆	semidirect
151. C₆₁	#9656	semidirect
201. C₂×C₈	Dic₂₄₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₄₄	2
#21446755	2
C₂×#16839515	2
C₂×#16839494	2
C₂×C₈²×C₆₁	2
#9656	61