Subgroups — #30606656

Order 15616.

414 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₆₁⋊(C₄⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×#16840649	C₂	1	2	yes	maximal
3	C₄×C₈×#6923	C₂	1	1	yes	maximal, characteristic
4	(C₂×#6923)⋊C₁₆	C₂	1	2	yes	maximal
5	C₄⋊C₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×D₆₁⋊C₁₆	C₂²	1	2	yes	—
7	C₄×D₆₁⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₈×#6923	C₄	1	1	yes	characteristic
9	C₂×C₈×#6923	C₄	1	1	yes	characteristic
10	C₂×C₈×#6923	C₂²	1	1	yes	characteristic
11	C₂×C₈×#6923	C₂²	1	1	yes	characteristic
12	C₄²×#6923	C₄	1	1	yes	characteristic
13	#16840683	C₄	1	2	yes	complemented
14	C₄⋊#11758187	C₄	1	2	yes	complemented
15	#16840649	—	2	4	no	—
16	C₂×C₁₆×D₆₁	C₂²	1	2	yes	—
17	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
18	C₄⋊#11758185	C₄	1	1	yes	characteristic, complemented
19	#16839580	C₄	1	2	yes	complemented
20	C₆₁×C₄⋊C₁₆	C₄	1	1	yes	characteristic, complemented
21	C₂×D₆₁⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×D₆₁⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×#6923	C₈	1	2	yes	—
24	C₂×C₄×#6923	C₈	1	2	yes	—
25	C₂×C₄×#6923	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×#6923	C₂×C₄	1	1	yes	characteristic
27	D₆₁⋊C₁₆	—	2	2	no	—
28	D₆₁⋊C₁₆	—	2	2	no	—
29	C₂×#11758187	C₂×C₄	1	4	yes	—
30	C₄×#92420	C₈	1	2	yes	—
31	C₄×#92420	C₈	1	2	yes	—
32	C₈×#6923	—	2	2	no	—
33	C₈×#6923	—	2	2	no	—
34	C₈×#6923	—	2	1	no	—
35	C₈×#6923	—	2	1	no	—
36	C₈×#6923	D₄	1	2	yes	—
37	C₈×#6923	Q₈	1	2	yes	—
38	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
40	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
41	C₂×#11758185	C₂×C₄	1	2	yes	—
42	C₁₆×D₆₁	—	2	2	no	—
43	C₁₆×D₆₁	—	2	2	no	—
44	C₄×#92418	C₂×C₄	1	1	yes	characteristic
45	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
46	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
49	C₂²×#6923	C₂×C₈	1	2	yes	—
50	C₂²×#6923	C₂×C₈	1	2	yes	—
51	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
52	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
53	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×#92420	C₂×C₈	1	2	yes	—
55	C₂×#92420	C₂×C₈	1	2	yes	—
56	C₄×#6923	—	2	4	no	—
57	C₄×#6923	—	2	4	no	—
58	C₄×#6923	—	2	1	no	—
59	C₄×#6923	—	2	1	no	—
60	C₄×#6923	C₄⋊C₄	1	2	yes	—
61	C₄×#6923	M₁₆	1	2	yes	—
62	D₆₁⋊C₈	—	2	1	no	—
63	D₆₁⋊C₈	—	2	1	no	—
64	D₆₁⋊C₈	C₄⋊C₄	1	2	yes	—
65	D₆₁⋊C₈	M₁₆	1	2	yes	—
66	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
67	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₆₁	C₄²	1	1	yes	characteristic
69	C₄×Dic₆₁	C₄²	1	1	yes	characteristic
70	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
71	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
72	C₂×#92418	C₄²	1	1	yes	characteristic
73	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
74	C₈×D₆₁	—	2	1	no	—
75	C₈×D₆₁	—	2	1	no	—
76	C₈×D₆₁	C₂×D₄	1	1	yes	characteristic
77	C₈×D₆₁	C₂×Q₈	1	1	yes	characteristic
78	C₈×D₆₁	#41	1	1	yes	characteristic
79	C₈×D₆₁	#41	1	1	yes	characteristic
80	#11758187	—	2	4	no	—
81	C₁₆×C₆₁	—	2	2	no	—
82	#11758185	—	2	2	no	—
83	C₂²×D₆₁	C₂²×C₈	1	1	yes	characteristic
84	C₂×#6923	—	2	2	no	—
85	C₂×#6923	—	2	2	no	—
86	C₂×#6923	—	2	2	no	—
87	C₂×#6923	—	2	2	no	—
88	C₂×#6923	C₄⋊C₈	1	4	yes	—
89	C₂×#6923	M₃₂	1	4	yes	—
90	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
91	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
92	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
93	C₂×Dic₆₁	C₄×C₈	1	1	yes	characteristic
94	C₂×Dic₆₁	C₄×C₈	1	1	yes	characteristic
95	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
96	C₄×D₆₁	—	2	2	no	—
97	C₄×D₆₁	M₃₂	1	4	yes	—
98	C₄×D₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₄×D₆₁	#117	1	1	yes	characteristic
100	C₄×D₆₁	C₂×M₁₆	1	1	yes	characteristic
101	C₄×D₆₁	C₂×M₁₆	1	1	yes	characteristic
102	#92420	—	2	2	no	—
103	#92420	C₄⋊C₈	1	4	yes	—
104	C₈×C₆₁	—	2	1	no	—
105	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
106	C₈×C₆₁	C₄×Q₈	1	1	yes	characteristic
107	#92418	—	2	1	no	—
108	#92418	C₄×D₄	1	1	yes	characteristic
109	#92418	C₄×Q₈	1	1	yes	characteristic
110	#12653	—	61	1	no	maximal, Sylow
111	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
112	C₂×D₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
113	C₂×D₆₁	#431	1	1	yes	characteristic
114	C₂×D₆₁	C₂×M₃₂	1	2	yes	—
115	C₂×D₆₁	C₂×M₃₂	1	2	yes	—
116	#6923	—	2	4	no	—
117	#6923	—	2	4	no	—
118	C₄×C₆₁	—	2	1	no	—
119	C₄×C₆₁	#346	1	2	yes	complemented
120	C₄×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
121	C₄×C₆₁	C₄×M₁₆	1	1	yes	characteristic
122	Dic₆₁	—	2	1	no	—
123	Dic₆₁	#346	1	2	yes	complemented
124	Dic₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
125	Dic₆₁	#431	1	1	yes	characteristic
126	C₂×C₄⋊C₁₆	—	61	1	no	—
127	C₂×C₄⋊C₁₆	—	61	1	no	—
128	C₂×#346	—	61	2	no	—
129	C₄²×C₈	—	61	1	no	—
130	#1374	—	61	2	no	—
131	C₂×C₆₁	C₄×C₄⋊C₈	1	1	yes	characteristic
132	C₂×C₆₁	C₂×#346	1	1	yes	characteristic
133	C₂×C₆₁	C₄×M₃₂	1	1	yes	characteristic
134	D₆₁	#2150	1	2	yes	—
135	D₆₁	#2163	1	2	yes	—
136	C₂²×C₁₆	—	61	2	no	—
137	C₂²×C₁₆	—	61	2	no	—
138	C₂×C₄×C₈	—	61	1	no	—
139	C₂×C₄×C₈	—	61	1	no	—
140	C₂×C₄×C₈	—	61	1	no	—
141	C₂×C₄×C₈	—	61	1	no	—
142	C₂×C₄×C₈	—	61	1	no	—
143	C₂×C₄×C₈	—	61	1	no	—
144	C₄³	—	61	1	no	—
145	C₄⋊C₁₆	—	61	2	no	—
146	C₄⋊C₁₆	—	61	2	no	—
147	C₄⋊C₁₆	—	61	2	no	—
148	C₄⋊C₁₆	—	61	1	no	—
149	C₄⋊C₁₆	#6923	1	1	yes	characteristic, complemented
150	#346	—	122	4	no	—
151	C₆₁	#12653	1	1	yes	characteristic, complemented, Sylow
152	C₂²×C₈	—	61	1	no	—
153	C₂²×C₈	—	61	1	no	—
154	C₂²×C₈	—	61	1	no	—
155	C₂²×C₈	—	61	1	no	—
156	C₂×C₄²	—	61	1	no	—
157	C₂×C₄²	—	61	1	no	—
158	C₂×C₄²	—	61	2	no	—
159	C₂×C₄²	—	61	2	no	—
160	C₂×C₄²	—	61	1	no	—
161	C₂×C₁₆	—	61	4	no	—
162	C₂×C₁₆	—	61	2	no	—
163	C₂×C₁₆	—	122	2	no	—
164	C₂×C₁₆	—	122	2	no	—
165	C₂×C₁₆	—	122	2	no	—
166	C₂×C₁₆	—	122	2	no	—
167	C₂×C₁₆	C₂×#6923	1	2	yes	—
168	C₄×C₈	—	61	2	no	—
169	C₄×C₈	—	61	2	no	—
170	C₄×C₈	—	61	2	no	—
171	C₄×C₈	—	61	2	no	—
172	C₄×C₈	—	61	1	no	—
173	C₄×C₈	—	61	1	no	—
174	C₄×C₈	—	61	1	no	—
175	C₄×C₈	—	122	2	no	—
176	C₄×C₈	—	122	2	no	—
177	C₄×C₈	—	122	1	no	—
178	C₄×C₈	—	122	1	no	—
179	C₄×C₈	C₂×#6923	1	1	yes	characteristic
180	C₂²×C₄	—	61	1	no	—
181	C₂²×C₄	—	61	1	no	—
182	C₂²×C₄	—	61	2	no	—
183	C₂²×C₄	—	61	2	no	—
184	C₂²×C₄	—	61	1	no	—
185	C₂×C₈	—	61	2	no	—
186	C₂×C₈	—	61	2	no	—
187	C₂×C₈	—	61	2	no	—
188	C₂×C₈	—	61	1	no	—
189	C₂×C₈	—	61	1	no	—
190	C₂×C₈	—	61	1	no	—
191	C₂×C₈	—	61	1	no	—
192	C₂×C₈	—	61	1	no	—
193	C₂×C₈	—	61	2	no	—
194	C₂×C₈	—	61	1	no	—
195	C₂×C₈	—	122	1	no	—
196	C₂×C₈	—	122	1	no	—
197	C₂×C₈	—	122	1	no	—
198	C₂×C₈	—	122	1	no	—
199	C₂×C₈	C₄×#6923	1	1	yes	characteristic
200	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
201	C₄²	—	61	2	no	—
202	C₄²	—	61	2	no	—
203	C₄²	—	61	1	no	—
204	C₄²	—	61	1	no	—
205	C₄²	—	61	1	no	—
206	C₄²	—	122	1	no	—
207	C₄²	—	122	1	no	—
208	C₄²	—	122	4	no	—
209	C₄²	—	122	4	no	—
210	C₄²	C₄×#6923	1	1	yes	characteristic
211	C₁₆	—	2	2	no	—
212	C₁₆	—	122	4	no	—
213	C₁₆	—	122	2	no	—
214	C₂³	—	61	1	no	—
215	C₂×C₄	—	61	1	no	—
216	C₂×C₄	—	61	1	no	—
217	C₂×C₄	—	61	1	no	—
218	C₂×C₄	—	61	1	no	—
219	C₂×C₄	—	61	4	no	—
220	C₂×C₄	—	61	1	no	—
221	C₂×C₄	—	61	1	no	—
222	C₂×C₄	—	61	4	no	—
223	C₂×C₄	—	61	4	no	—
224	C₂×C₄	—	61	1	no	—
225	C₂×C₄	—	122	2	no	—
226	C₂×C₄	—	122	2	no	—
227	C₂×C₄	—	122	2	no	—
228	C₂×C₄	—	122	2	no	—
229	C₂×C₄	—	122	2	no	—
230	C₂×C₄	C₈×#6923	1	1	yes	characteristic
231	C₂×C₄	C₈×#6923	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
233	C₈	—	2	1	no	—
234	C₈	—	61	4	no	—
235	C₈	—	61	1	no	—
236	C₈	—	61	1	no	—
237	C₈	—	122	2	no	—
238	C₈	—	122	1	no	—
239	C₈	D₄×#6923	1	1	yes	characteristic
240	C₈	Q₈×#6923	1	1	yes	characteristic
241	C₂²	—	61	2	no	—
242	C₂²	—	61	2	no	—
243	C₂²	—	61	1	no	—
244	C₂²	—	61	1	no	—
245	C₂²	C₂×C₈×#6923	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	61	2	no	—
248	C₄	—	61	1	no	—
249	C₄	—	61	1	no	—
250	C₄	—	122	4	no	—
251	C₄	—	122	4	no	—
252	C₄	—	122	1	no	—
253	C₄	#16840649	1	2	yes	complemented
254	C₄	C₄⋊C₄×#6923	1	1	yes	characteristic
255	C₄	M₁₆×#6923	1	1	yes	characteristic
256	C₂	—	61	2	no	—
257	C₂	—	61	2	no	—
258	C₂	C₄⋊C₈×#6923	1	1	yes	characteristic
259	C₂	C₂×#16840649	1	1	yes	characteristic
260	C₂	M₃₂×#6923	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#6923
commutator, socle	C₂²×C₆₁	C₂×C₄×C₈
Fitting	C₆₁×C₄⋊C₁₆	C₄
prime radical	C₂²×D₆₁	C₂²×C₈
cosocle	C₂×C₈×D₆₁	C₂³
hypercenter	C₄⋊C₁₆	#6923
hypocenter	C₆₁	#12653
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
13. #16840683	C₄	semidirect
14. C₄⋊#11758187	C₄	semidirect
18. C₄⋊#11758185	C₄	semidirect
19. #16839580	C₄	semidirect
20. C₆₁×C₄⋊C₁₆	C₄	semidirect
119. C₄×C₆₁	#346	semidirect
123. Dic₆₁	#346	semidirect
149. C₄⋊C₁₆	#6923	semidirect
151. C₆₁	#12653	semidirect
253. C₄	#16840649	semidirect

Maximal subgroups

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

subgroup	index
D₆₁⋊(C₄⋊C₁₆)	2
C₂×#16840649	2
C₄×C₈×#6923	2
(C₂×#6923)⋊C₁₆	2
C₄⋊C₁₆×D₆₁	2
#12653	61