Subgroups — #29578905

Order 15616.

482 conjugacy classes of subgroups.

The subgroup lattice has 248 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×C₆₁×#322	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	C₆₁×#322	—	2	2	no	—
20	C₂²×Dic₁₂₂	C₂×C₄	1	2	yes	—
21	C₂×Dic₂₄₄	—	2	4	no	—
22	C₂×Dic₂₄₄	—	2	2	no	—
23	C₂²×#92418	—	2	2	no	—
24	C₄×Dic₁₂₂	—	2	2	no	—
25	C₄×Dic₁₂₂	—	2	2	no	—
26	C₄×Dic₁₂₂	—	2	2	no	—
27	C₄×Dic₁₂₂	—	2	2	no	—
28	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
29	C₂×#11758196	—	2	2	no	—
30	C₂×C₄×Dic₆₁	—	2	2	no	—
31	#14519711	—	2	2	no	—
32	C₄⋊#92418	—	2	2	no	—
33	C₄⋊#92418	—	2	2	no	—
34	#14519702	—	2	4	no	—
35	C₂²×C₈×C₆₁	—	2	1	no	—
36	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₁	D₄	1	2	yes	—
39	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₆₁	—	2	1	no	—
41	C₄×C₈×C₆₁	—	2	1	no	—
42	C₄×C₈×C₆₁	—	2	1	no	—
43	C₄×C₈×C₆₁	D₄	1	2	yes	—
44	C₄×C₈×C₆₁	D₄	1	2	yes	—
45	C₄×C₈×C₆₁	D₄	1	2	yes	—
46	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
49	C₂²×Dic₆₁	—	2	2	no	—
50	C₂²×Dic₆₁	—	2	2	no	—
51	C₂×Dic₁₂₂	—	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	M₁₆	1	4	yes	—
57	C₂×C₈×C₆₁	—	2	2	no	—
58	C₂×C₈×C₆₁	—	2	2	no	—
59	C₂×C₈×C₆₁	—	2	2	no	—
60	C₂×C₈×C₆₁	—	2	2	no	—
61	C₂×C₈×C₆₁	—	2	2	no	—
62	C₂×C₈×C₆₁	—	4	2	no	—
63	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₆₁	Q₁₆	1	4	yes	complemented
66	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₆₁	#41	1	1	yes	characteristic
70	C₄⋊Dic₆₁	—	2	2	no	—
71	C₄⋊Dic₆₁	—	2	2	no	—
72	#11758196	—	4	2	no	—
73	#11758196	—	4	2	no	—
74	C₄×Dic₆₁	—	4	2	no	—
75	C₄×Dic₆₁	—	4	2	no	—
76	C₂×#92418	—	4	2	no	—
77	C₂×#92418	—	4	4	no	—
78	Dic₂₄₄	—	2	8	no	—
79	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₆₁	—	2	2	no	—
81	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
84	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
85	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
86	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
87	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₆₁	#131	1	2	yes	—
89	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
90	C₂×C₄×C₆₁	#135	1	2	yes	—
91	C₂×Dic₆₁	—	2	4	no	—
92	C₂×Dic₆₁	—	4	2	no	—
93	C₂×Dic₆₁	—	4	2	no	—
94	C₂×Dic₆₁	—	4	2	no	—
95	C₂×Dic₆₁	—	4	4	no	—
96	Dic₁₂₂	—	2	4	no	—
97	Dic₁₂₂	—	2	4	no	—
98	Dic₁₂₂	—	4	2	no	—
99	Dic₁₂₂	—	4	2	no	—
100	C₈×C₆₁	—	2	2	no	—
101	C₈×C₆₁	—	2	2	no	—
102	C₈×C₆₁	—	2	4	no	—
103	C₈×C₆₁	—	4	2	no	—
104	#92418	—	8	2	no	—
105	#9662	—	61	1	no	maximal, Sylow
106	C₂²×C₆₁	#390	1	1	yes	characteristic
107	C₂²×C₆₁	#435	1	2	yes	—
108	C₂²×C₆₁	C₄×Q₁₆	1	2	yes	—
109	C₂²×C₆₁	#444	1	2	yes	—
110	C₄×C₆₁	—	2	1	no	—
111	C₄×C₆₁	—	2	1	no	—
112	C₄×C₆₁	#407	1	1	yes	characteristic
113	C₄×C₆₁	#408	1	1	yes	characteristic
114	C₄×C₆₁	#435	1	2	yes	—
115	C₄×C₆₁	#467	1	2	yes	—
116	C₄×C₆₁	#494	1	1	yes	characteristic
117	C₄×C₆₁	#495	1	1	yes	characteristic
118	Dic₆₁	—	4	4	no	—
119	Dic₆₁	—	8	2	no	—
120	C₂×C₄×Q₁₆	—	61	1	no	—
121	#1984	—	61	2	no	—
122	C₂×#335	—	61	1	no	—
123	C₂×Q₈⋊C₈	—	61	2	no	—
124	C₂×#322	D₆₁	1	1	yes	characteristic
125	C₂×C₆₁	#1584	1	4	yes	—
126	C₂×C₆₁	#1955	1	1	yes	characteristic
127	C₂×C₆₁	#1969	1	1	yes	characteristic
128	C₂×C₆₁	#1975	1	1	yes	characteristic
129	C₂²×Q₁₆	—	61	1	no	—
130	C₂×C₄×Q₈	—	61	2	no	—
131	C₄×Q₁₆	—	122	2	no	—
132	C₄×Q₁₆	—	122	2	no	—
133	C₂×#107	—	61	1	no	—
134	C₂×C₄⋊C₈	—	61	2	no	—
135	C₂×#103	—	61	2	no	—
136	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
139	#336	—	122	4	no	—
140	#335	—	122	2	no	—
141	Q₈⋊C₈	—	122	4	no	—
142	#322	—	2	2	no	—
143	C₆₁	#9662	1	1	yes	characteristic, complemented, Sylow
144	C₂²×Q₈	—	61	2	no	—
145	C₂×Q₁₆	—	122	4	no	—
146	C₂×Q₁₆	—	122	2	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	—	122	2	no	—
149	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
150	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
151	C₂²×C₈	D₂₄₄	1	2	yes	—
152	C₄×Q₈	—	122	2	no	—
153	C₄×Q₈	—	122	2	no	—
154	C₄×Q₈	—	122	2	no	—
155	C₄×Q₈	—	122	2	no	—
156	C₂×C₄⋊C₄	—	61	2	no	—
157	C₂×C₄⋊C₄	—	122	2	no	—
158	C₂×C₄²	—	122	2	no	—
159	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
160	#107	—	122	2	no	—
161	C₄⋊C₈	—	122	2	no	—
162	C₄⋊C₈	—	122	2	no	—
163	#103	—	122	4	no	—
164	C₄×C₈	—	2	1	no	—
165	C₄×C₈	—	2	1	no	—
166	C₄×C₈	—	2	1	no	—
167	C₄×C₈	D₂₄₄	1	2	yes	—
168	C₄×C₈	D₂₄₄	1	2	yes	—
169	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
170	C₂×Q₈	—	61	4	no	—
171	C₂×Q₈	—	122	2	no	—
172	C₂×Q₈	—	122	2	no	—
173	C₂×Q₈	—	122	2	no	—
174	C₂×Q₈	—	122	2	no	—
175	C₂×Q₈	—	122	2	no	—
176	C₂²×C₄	—	122	2	no	—
177	C₂²×C₄	—	122	2	no	—
178	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
180	C₂²×C₄	#11758222	1	1	yes	characteristic
181	Q₁₆	—	122	8	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	244	2	no	—
189	C₂×C₈	—	244	4	no	—
190	C₂×C₈	Dic₂₄₄	1	4	yes	complemented
191	C₂×C₈	#11758198	1	2	yes	—
192	C₂×C₈	#11758198	1	2	yes	—
193	C₄⋊C₄	—	122	2	no	—
194	C₄⋊C₄	—	122	2	no	—
195	C₄⋊C₄	—	244	2	no	—
196	C₄⋊C₄	—	244	2	no	—
197	C₄²	—	244	2	no	—
198	C₄²	—	244	2	no	—
199	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
200	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
201	C₄²	#11758222	1	1	yes	characteristic
202	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
203	C₂³	C₄×D₂₄₄	1	1	yes	characteristic
204	Q₈	—	122	4	no	—
205	Q₈	—	122	4	no	—
206	Q₈	—	244	2	no	—
207	Q₈	—	244	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	122	4	no	—
210	C₂×C₄	—	244	4	no	—
211	C₂×C₄	—	244	2	no	—
212	C₂×C₄	—	244	2	no	—
213	C₂×C₄	—	244	2	no	—
214	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
215	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
216	C₂×C₄	#14519751	1	1	yes	characteristic
217	C₂×C₄	#14519751	1	1	yes	characteristic
218	C₂×C₄	#14519765	1	1	yes	characteristic
219	C₂×C₄	#14519774	1	1	yes	characteristic
220	C₂×C₄	M₁₆×D₆₁	1	2	yes	—
221	C₂×C₄	#14519785	1	2	yes	—
222	C₂×C₄	C₂×Dic₂₄₄	1	2	yes	—
223	C₂×C₄	#14519799	1	2	yes	—
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	2	4	no	—
227	C₈	—	4	2	no	—
228	C₈	—	488	2	no	—
229	C₂²	#16839664	1	1	yes	characteristic
230	C₂²	#16839835	1	2	yes	—
231	C₂²	C₄×Dic₂₄₄	1	2	yes	—
232	C₂²	#16839885	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	488	2	no	—
236	C₄	—	244	4	no	—
237	C₄	#16839725	1	1	yes	characteristic
238	C₄	#16839730	1	1	yes	characteristic
239	C₄	#16839835	1	2	yes	—
240	C₄	#16840003	1	2	yes	—
241	C₄	#16840142	1	1	yes	characteristic
242	C₄	#16840146	1	1	yes	characteristic
243	C₂	#21444671	1	4	yes	—
244	C₂	#21446617	1	1	yes	characteristic
245	C₂	#21446680	1	1	yes	characteristic
246	C₂	#21446712	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₄₄
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#322	C₂
socle, prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#322	D₆₁
hypocenter	C₆₁	#9662
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
65. C₂×C₈×C₆₁	Q₁₆	semidirect
143. C₆₁	#9662	semidirect
190. 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×C₆₁×#322	2
#9662	61