Subgroups — #29649488

Order 15616.

678 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454899	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic
3	#21444639	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₂₄₄	C₂	1	4	yes	maximal, complemented
5	#21444567	C₂	1	1	yes	maximal, characteristic
6	#21444215	C₂	1	2	yes	maximal
7	C₆₁×#1450	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₄₄	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₂₂	C₂²	1	2	yes	—
10	C₈×Dic₁₂₂	C₂²	1	8	yes	—
11	#16839886	—	2	4	no	—
12	C₄×Dic₂₄₄	C₄	1	2	yes	—
13	C₄×Dic₂₄₄	C₄	1	2	yes	—
14	C₄×Dic₂₄₄	C₂²	1	2	yes	—
15	C₄×Dic₂₄₄	C₂²	1	2	yes	—
16	#16839815	C₂²	1	2	yes	—
17	#16839812	—	2	4	no	—
18	C₂×#14519711	C₄	1	1	yes	characteristic
19	C₂×#14519702	C₄	1	2	yes	—
20	#16839515	C₂²	1	2	yes	—
21	#16839513	C₂²	1	2	yes	—
22	#16839494	C₂²	1	8	yes	—
23	C₆₁×#431	C₂²	1	1	yes	characteristic
24	C₆₁×#431	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#322	C₂²	1	2	yes	—
27	C₈²×C₆₁	C₂²	1	2	yes	—
28	C₂²×Dic₁₂₂	C₂×C₄	1	2	yes	—
29	C₂×Dic₂₄₄	C₈	1	8	yes	complemented
30	C₂×Dic₂₄₄	C₂×C₄	1	2	yes	—
31	C₂×Dic₂₄₄	C₂×C₄	1	2	yes	—
32	C₄×Dic₁₂₂	—	2	2	no	—
33	C₄×Dic₁₂₂	—	2	2	no	—
34	C₄×Dic₁₂₂	C₂×C₄	1	4	yes	—
35	C₄×Dic₁₂₂	C₂³	1	4	yes	—
36	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
37	C₂×#11758196	—	2	2	no	—
38	C₂×C₄×Dic₆₁	—	2	2	no	—
39	#14519711	C₂×C₄	1	2	yes	—
40	#14519711	C₂×C₄	1	2	yes	—
41	C₄⋊#92418	C₂³	1	4	yes	—
42	#14519706	—	2	8	no	—
43	#14519702	C₂×C₄	1	4	yes	—
44	#14519702	C₂×C₄	1	4	yes	—
45	C₂²⋊#92418	—	2	2	no	—
46	C₄×#92418	—	2	2	no	—
47	C₈×Dic₆₁	—	2	8	no	—
48	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₆₁	—	2	1	no	—
52	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
53	C₂²⋊C₈×C₆₁	—	2	1	no	—
54	C₂²⋊C₈×C₆₁	D₄	1	2	yes	—
55	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₆₁	C₂³	1	2	yes	—
58	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₆₁	C₂³	1	2	yes	—
61	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
64	C₂²×Dic₆₁	—	2	2	no	—
65	C₂²×Dic₆₁	—	2	2	no	—
66	C₂×Dic₁₂₂	—	2	2	no	—
67	C₂×Dic₁₂₂	—	2	4	no	—
68	C₂×Dic₁₂₂	C₂×C₈	1	8	yes	—
69	C₂×Dic₁₂₂	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₆₁	—	2	2	no	—
71	C₂×C₈×C₆₁	—	2	2	no	—
72	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
73	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
78	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₆₁	#41	1	1	yes	characteristic
81	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
82	C₄⋊Dic₆₁	—	2	2	no	—
83	C₄⋊Dic₆₁	C₂²×C₄	1	4	yes	—
84	#11758196	—	2	4	no	—
85	#11758196	—	2	4	no	—
86	C₄×Dic₆₁	—	2	4	no	—
87	C₄×Dic₆₁	—	2	4	no	—
88	C₂×#92418	—	2	4	no	—
89	Dic₂₄₄	C₂×C₈	1	8	yes	—
90	Dic₂₄₄	C₂×C₈	1	8	yes	—
91	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₆₁	—	2	2	no	—
93	C₂×C₄×C₆₁	—	4	1	no	—
94	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
98	C₂×C₄×C₆₁	#131	1	2	yes	—
99	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
102	C₂×Dic₆₁	—	2	4	no	—
103	C₂×Dic₆₁	—	2	8	no	—
104	C₂×Dic₆₁	—	2	4	no	—
105	C₂×Dic₆₁	—	8	2	no	—
106	Dic₁₂₂	—	2	4	no	—
107	Dic₁₂₂	—	2	8	no	—
108	Dic₁₂₂	C₂²×C₈	1	8	yes	—
109	C₈×C₆₁	—	2	1	no	—
110	C₈×C₆₁	—	2	1	no	—
111	C₈×C₆₁	—	2	4	no	—
112	C₈×C₆₁	—	4	2	no	—
113	C₈×C₆₁	C₂²×C₈	1	2	yes	—
114	C₈×C₆₁	C₂²×C₈	1	2	yes	—
115	#92418	—	4	4	no	—
116	#17856	—	61	1	no	maximal, Sylow
117	C₂²×C₆₁	—	2	1	no	—
118	C₂²×C₆₁	—	2	1	no	—
119	C₂²×C₆₁	C₈×D₄	1	2	yes	—
120	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
121	C₄×C₆₁	—	2	1	no	—
122	C₄×C₆₁	—	2	1	no	—
123	C₄×C₆₁	—	2	1	no	—
124	C₄×C₆₁	C₈×D₄	1	2	yes	—
125	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
126	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
127	C₄×C₆₁	#578	1	2	yes	—
128	Dic₆₁	—	2	8	no	—
129	Dic₆₁	—	4	4	no	—
130	#2982	—	61	2	no	—
131	C₂×C₄×Q₁₆	—	61	1	no	—
132	#1579	—	61	4	no	—
133	C₈×Q₁₆	—	61	4	no	—
134	#1564	—	61	1	no	—
135	#1479	—	61	2	no	—
136	#1450	D₆₁	1	1	yes	characteristic
137	C₂×C₆₁	—	2	2	no	—
138	C₂×C₆₁	C₂×C₈×D₄	1	1	yes	characteristic
139	C₂×C₆₁	#2950	1	1	yes	characteristic
140	C₂×C₆₁	#2957	1	1	yes	characteristic
141	C₂²×Q₁₆	—	61	1	no	—
142	C₂×C₄×Q₈	—	61	2	no	—
143	C₈×Q₈	—	61	8	no	—
144	C₈×Q₈	—	122	4	no	—
145	C₄×Q₁₆	—	61	2	no	—
146	C₄×Q₁₆	—	61	2	no	—
147	C₄×Q₁₆	—	61	2	no	—
148	C₄×Q₁₆	—	61	2	no	—
149	#431	—	61	2	no	—
150	#431	—	122	4	no	—
151	#431	C₂×D₆₁	1	1	yes	characteristic
152	#431	C₂×D₆₁	1	1	yes	characteristic
153	C₂×#107	—	61	1	no	—
154	C₂×#103	—	61	2	no	—
155	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
156	#335	—	61	2	no	—
157	#334	—	61	2	no	—
158	Q₈⋊C₈	—	61	8	no	—
159	#322	C₂×D₆₁	1	2	yes	—
160	C₈²	C₂×D₆₁	1	2	yes	—
161	C₆₁	#17856	1	1	yes	characteristic, complemented, Sylow
162	C₂²×Q₈	—	61	2	no	—
163	C₂×Q₁₆	—	61	8	no	—
164	C₂×Q₁₆	—	61	2	no	—
165	C₂×Q₁₆	—	61	2	no	—
166	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
167	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
168	C₄×Q₈	—	61	4	no	—
169	C₄×Q₈	—	61	4	no	—
170	C₄×Q₈	—	122	2	no	—
171	C₄×Q₈	—	122	2	no	—
172	C₂×C₄⋊C₄	—	61	2	no	—
173	C₂×C₄⋊C₄	—	122	2	no	—
174	C₂×C₄²	—	122	2	no	—
175	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
176	#107	—	61	2	no	—
177	#107	—	61	2	no	—
178	C₄⋊C₈	—	2	1	no	—
179	C₄⋊C₈	—	61	4	no	—
180	C₄⋊C₈	—	122	8	no	—
181	C₄⋊C₈	D₂₄₄	1	2	yes	—
182	#103	—	61	4	no	—
183	#103	—	61	4	no	—
184	C₂²⋊C₈	—	2	1	no	—
185	C₂²⋊C₈	—	122	2	no	—
186	C₂²⋊C₈	D₂₄₄	1	2	yes	—
187	C₄×C₈	—	122	8	no	—
188	C₄×C₈	—	122	2	no	—
189	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
190	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
191	C₄×C₈	C₂²×D₆₁	1	2	yes	—
192	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₆₁	1	2	yes	—
195	C₂×Q₈	—	61	4	no	—
196	C₂×Q₈	—	61	8	no	—
197	C₂×Q₈	—	122	2	no	—
198	C₂×Q₈	—	122	4	no	—
199	C₂²×C₄	—	122	2	no	—
200	C₂²×C₄	—	122	2	no	—
201	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
203	C₂²×C₄	#11758222	1	1	yes	characteristic
204	Q₁₆	—	61	8	no	—
205	Q₁₆	—	61	8	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	122	4	no	—
209	C₂×C₈	C₈×D₆₁	1	4	yes	—
210	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
211	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
212	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
213	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
214	C₂×C₈	C₂×D₂₄₄	1	4	yes	—
215	C₄⋊C₄	—	61	4	no	—
216	C₄⋊C₄	—	122	4	no	—
217	C₄⋊C₄	—	122	2	no	—
218	C₄⋊C₄	—	122	4	no	—
219	C₄²	—	122	4	no	—
220	C₄²	—	122	4	no	—
221	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
222	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
223	C₄²	#11758222	1	1	yes	characteristic
224	C₄²	C₂³×D₆₁	1	1	yes	characteristic
225	C₂³	C₄×D₂₄₄	1	1	yes	characteristic
226	Q₈	—	61	8	no	—
227	Q₈	—	122	4	no	—
228	Q₈	—	122	8	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	122	4	no	—
232	C₂×C₄	—	122	8	no	—
233	C₂×C₄	—	122	4	no	—
234	C₂×C₄	—	488	2	no	—
235	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
236	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
237	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
238	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
239	C₂×C₄	#14519783	1	2	yes	—
240	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
241	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
242	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	4	no	—
246	C₈	—	4	2	no	—
247	C₈	—	244	4	no	—
248	C₈	C₂×C₈×D₆₁	1	2	yes	—
249	C₈	C₂×C₈×D₆₁	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	C₈×D₂₄₄	1	2	yes	—
253	C₂²	C₂×C₄×D₂₄₄	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	122	8	no	—
258	C₄	—	244	4	no	—
259	C₄	C₈×D₂₄₄	1	2	yes	—
260	C₄	C₂²×C₈×D₆₁	1	1	yes	characteristic
261	C₄	C₂×#14519783	1	1	yes	characteristic
262	C₄	#16840579	1	2	yes	—
263	C₂	—	2	2	no	—
264	C₂	C₂×C₈×D₂₄₄	1	1	yes	characteristic
265	C₂	#21454688	1	1	yes	characteristic
266	C₂	#21454737	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₄₄
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₄²	C₂³×D₆₁
Fitting	C₆₁×#1450	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	#1450	D₆₁
hypocenter	C₆₁	#17856
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #21444639	C₂	semidirect
4. C₈×Dic₂₄₄	C₂	semidirect
29. C₂×Dic₂₄₄	C₈	semidirect
161. C₆₁	#17856	semidirect

Maximal subgroups

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

subgroup	index
#21454899	2
C₂×C₄×Dic₂₄₄	2
#21444639	2
C₈×Dic₂₄₄	2
#21444567	2
#21444215	2
C₆₁×#1450	2
#17856	61