Subgroups — C₄×#16840821 #30623829

Order 15616.

834 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840821	C₂	1	2	yes	maximal
2	#21462030	C₂	1	2	yes	maximal, complemented
3	#21462029	C₂	1	4	yes	maximal, complemented
4	C₄×#14519852	C₂	1	2	yes	maximal, complemented
5	C₄×#14519851	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₆₁⋊C₁₆	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×#11758187	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#14519785	C₂	1	1	yes	maximal, characteristic
9	#16840821	C₄	1	16	yes	direct product factor
10	C₂×#14519852	C₂²	1	4	yes	—
11	C₂×#14519851	C₂²	1	2	yes	—
12	C₂×D₆₁⋊C₁₆	C₂²	1	4	yes	—
13	C₂²×#11758187	C₂²	1	2	yes	—
14	#16840651	C₂²	1	4	yes	—
15	#16840650	C₂²	1	2	yes	—
16	#16840648	C₂²	1	4	yes	—
17	C₄×#11758187	C₂²	1	1	yes	characteristic
18	C₄×#11758187	C₂²	1	2	yes	—
19	C₄×#11758187	C₂²	1	1	yes	characteristic
20	C₄×#11758187	C₂²	1	2	yes	—
21	C₂×#14519785	C₄	1	1	yes	characteristic
22	C₂×#14519785	C₂²	1	1	yes	characteristic
23	C₄×#11758222	C₄	1	1	yes	characteristic
24	#16839720	C₄	1	1	yes	characteristic
25	#16839720	C₂²	1	1	yes	characteristic
26	#16839719	C₄	1	2	yes	—
27	#16839719	C₂²	1	2	yes	—
28	C₄×#11758189	C₄	1	2	yes	—
29	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₆₁	C₂²	1	2	yes	—
31	C₄×M₁₆×C₆₁	C₄	1	1	yes	characteristic
32	#14519852	C₂×C₄	1	16	yes	complemented
33	#14519851	C₂×C₄	1	8	yes	complemented
34	D₆₁⋊C₁₆	C₂×C₄	1	16	yes	complemented
35	C₂×#11758187	C₂×C₄	1	8	yes	complemented
36	C₂×#11758187	C₂³	1	2	yes	—
37	C₂×#11758187	C₂³	1	2	yes	—
38	C₂×#11758187	C₂³	1	4	yes	—
39	C₂×#11758222	C₂×C₄	1	1	yes	characteristic
40	#14519785	C₈	1	8	yes	—
41	#14519785	C₂×C₄	1	8	yes	—
42	C₂×#11758189	C₂×C₄	1	2	yes	—
43	C₂×#11758189	C₂×C₄	1	2	yes	—
44	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
45	C₂²×#92418	C₂³	1	1	yes	characteristic
46	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₆₁	C₂³	1	2	yes	—
48	C₄×Dic₁₂₂	C₈	1	2	yes	—
49	C₄×D₂₄₄	C₈	1	2	yes	—
50	C₄×C₂²⋊D₆₁	C₈	1	4	yes	—
51	#14519736	C₂×C₄	1	1	yes	characteristic
52	#14519734	C₂×C₄	1	2	yes	—
53	C₄²×D₆₁	C₂×C₄	1	2	yes	—
54	#14519687	C₂×C₄	1	2	yes	—
55	#14519687	C₂×C₄	1	2	yes	—
56	C₄×#92418	C₂×C₄	1	1	yes	characteristic
57	C₄×#92418	C₂×C₄	1	1	yes	characteristic
58	C₄×#92418	C₂³	1	1	yes	characteristic
59	C₄×#92418	C₂³	1	1	yes	characteristic
60	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
61	C₈×Dic₆₁	C₂³	1	2	yes	—
62	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
63	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
65	#97×C₆₁	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
67	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
68	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₆₁	C₂×C₈	1	2	yes	—
70	#11758222	C₄²	1	4	yes	—
71	#11758222	C₂×C₈	1	4	yes	—
72	C₂×D₂₄₄	C₂×C₈	1	1	yes	characteristic
73	C₂×C₄×D₆₁	C₂×C₈	1	4	yes	—
74	C₂×C₄×D₆₁	C₂²×C₄	1	2	yes	—
75	C₂×Dic₁₂₂	C₂×C₈	1	1	yes	characteristic
76	M₁₆×C₆₁	C₄²	1	4	yes	—
77	M₁₆×C₆₁	C₂×C₈	1	4	yes	—
78	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
80	C₄²×C₆₁	C₂×C₈	1	2	yes	—
81	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
83	C₂²⋊Dic₆₁	C₂×C₈	1	2	yes	—
84	#11758198	C₂×C₈	1	4	yes	—
85	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
86	#11758196	C₂×C₈	1	4	yes	—
87	C₄×Dic₆₁	C₂×C₈	1	4	yes	—
88	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
89	C₂×#92418	C₂×C₈	1	4	yes	—
90	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
91	C₂×#92418	C₂²×C₄	1	4	yes	—
92	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
93	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
94	C₂×#92418	C₂⁴	1	1	yes	characteristic
95	#11758189	C₄²	1	8	yes	—
96	#11758189	C₂×C₈	1	8	yes	—
97	C₈×D₆₁	C₂×C₈	1	8	yes	—
98	C₈×D₆₁	C₂²×C₄	1	8	yes	—
99	#11758187	C₂²×C₄	1	4	yes	—
100	#11758187	C₂²×C₄	1	4	yes	—
101	#11758187	C₂²×C₄	1	8	yes	—
102	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
103	C₂²×D₆₁	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₆₁	—	2	4	no	—
105	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
111	C₂²⋊D₆₁	C₄×C₈	1	8	yes	—
112	C₂×Dic₆₁	C₂²×C₈	1	4	yes	—
113	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
114	D₂₄₄	C₄×C₈	1	4	yes	—
115	C₄×D₆₁	—	2	8	no	—
116	C₄×D₆₁	C₂×C₄²	1	4	yes	—
117	C₄×D₆₁	C₂²×C₈	1	4	yes	—
118	Dic₁₂₂	C₄×C₈	1	4	yes	—
119	C₈×C₆₁	C₂×C₄²	1	4	yes	—
120	C₈×C₆₁	C₂²×C₈	1	4	yes	—
121	#92418	C₂×C₄²	1	2	yes	—
122	#92418	C₂²×C₈	1	2	yes	—
123	#92418	C₂²×C₈	1	2	yes	—
124	#92418	C₂³×C₄	1	2	yes	—
125	C₄×#504	—	61	1	no	maximal, Sylow
126	C₂²×C₆₁	—	2	2	no	—
127	C₂²×C₆₁	C₂×C₄×C₈	1	2	yes	—
128	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
129	C₂×D₆₁	—	2	4	no	—
130	C₂×D₆₁	C₂×C₄×C₈	1	4	yes	—
131	C₄×C₆₁	—	2	2	no	—
132	C₄×C₆₁	C₂×C₄×C₈	1	2	yes	—
133	C₄×C₆₁	#504	1	4	yes	complemented
134	C₄×C₆₁	C₂²×C₄²	1	1	yes	characteristic
135	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
136	Dic₆₁	—	2	4	no	—
137	Dic₆₁	C₂×C₄×C₈	1	4	yes	—
138	C₂×#504	—	61	2	no	—
139	C₄×#131	—	61	1	no	—
140	#2108	—	61	4	no	—
141	#2108	—	61	2	no	—
142	C₄×M₃₂	—	61	1	no	—
143	C₄×M₃₂	—	61	2	no	—
144	C₂×C₄×C₁₆	—	61	2	no	—
145	C₂×C₄×C₁₆	—	61	1	no	—
146	C₂×C₆₁	—	2	2	no	—
147	C₂×C₆₁	C₂²×C₄×C₈	1	1	yes	characteristic
148	C₂×C₆₁	C₂×#504	1	2	yes	—
149	D₆₁	—	2	4	no	—
150	C₂×#131	—	61	1	no	—
151	C₂×#131	—	61	1	no	—
152	C₄×#41	—	61	1	no	—
153	#504	—	61	16	no	—
154	C₂×M₃₂	—	61	2	no	—
155	C₂×M₃₂	—	61	4	no	—
156	C₂²×C₁₆	—	61	2	no	—
157	C₂²×C₁₆	—	61	4	no	—
158	#405	—	61	1	no	—
159	#405	—	61	1	no	—
160	#405	—	61	2	no	—
161	#405	—	61	2	no	—
162	C₄×M₁₆	—	61	2	no	—
163	C₄×M₁₆	#6923	1	1	yes	characteristic
164	C₂×C₄×C₈	—	61	1	no	—
165	C₂×C₄×C₈	—	61	2	no	—
166	#346	—	61	2	no	—
167	#346	—	61	4	no	—
168	C₄×C₁₆	—	61	1	no	—
169	C₄×C₁₆	—	61	2	no	—
170	C₄×C₁₆	—	61	1	no	—
171	C₄×C₁₆	—	61	2	no	—
172	C₄×C₁₆	—	61	4	no	—
173	C₆₁	C₄×#504	1	1	yes	characteristic, complemented, Sylow
174	C₂×#41	—	61	1	no	—
175	#131	—	61	8	no	—
176	#131	—	61	8	no	—
177	C₂×M₁₆	—	61	2	no	—
178	C₂×M₁₆	—	61	2	no	—
179	C₂×M₁₆	C₂×#6923	1	1	yes	characteristic
180	C₂×M₁₆	C₂×#6923	1	1	yes	characteristic
181	C₂²×C₈	—	61	1	no	—
182	C₂²×C₈	—	61	1	no	—
183	C₂²×C₈	—	61	2	no	—
184	C₂²×C₈	—	61	2	no	—
185	C₄×Q₈	—	61	2	no	—
186	C₄×D₄	—	61	4	no	—
187	C₄×D₄	—	61	2	no	—
188	#117	—	61	1	no	—
189	#117	—	61	2	no	—
190	C₂×C₄²	—	61	2	no	—
191	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
192	M₃₂	—	61	8	no	—
193	M₃₂	—	61	16	no	—
194	C₂×C₁₆	—	61	8	no	—
195	C₂×C₁₆	—	61	2	no	—
196	C₂×C₁₆	—	61	2	no	—
197	C₂×C₁₆	—	61	4	no	—
198	C₂×C₁₆	—	61	16	no	—
199	#97	—	61	2	no	—
200	#97	—	61	2	no	—
201	#97	C₂×#6923	1	2	yes	—
202	C₄×C₈	—	61	2	no	—
203	C₄×C₈	—	61	2	no	—
204	C₄×C₈	—	61	1	no	—
205	C₄×C₈	—	61	1	no	—
206	C₄×C₈	—	61	1	no	—
207	C₄×C₈	—	61	1	no	—
208	C₄×C₈	C₂×#6923	1	2	yes	—
209	#41	—	61	4	no	—
210	#41	—	61	4	no	—
211	C₂×Q₈	—	61	1	no	—
212	C₂×D₄	—	61	2	no	—
213	C₂×D₄	—	61	1	no	—
214	C₂²×C₄	—	61	2	no	—
215	C₂²×C₄	—	61	4	no	—
216	C₂²×C₄	D₆₁⋊C₈	1	2	yes	—
217	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
218	M₁₆	—	61	8	no	—
219	M₁₆	—	61	8	no	—
220	M₁₆	D₆₁⋊C₈	1	4	yes	—
221	M₁₆	C₄×#6923	1	4	yes	—
222	C₂×C₈	—	61	4	no	—
223	C₂×C₈	—	61	4	no	—
224	C₂×C₈	—	61	1	no	—
225	C₂×C₈	—	61	1	no	—
226	C₂×C₈	—	61	1	no	—
227	C₂×C₈	—	61	1	no	—
228	C₂×C₈	—	61	8	no	—
229	C₂×C₈	—	61	8	no	—
230	C₂×C₈	C₂²×#6923	1	2	yes	—
231	C₂×C₈	C₂²×#6923	1	2	yes	—
232	C₄⋊C₄	—	61	4	no	—
233	C₄⋊C₄	—	61	2	no	—
234	C₂²⋊C₄	—	61	2	no	—
235	C₂²⋊C₄	—	61	4	no	—
236	C₄²	—	61	4	no	—
237	C₄²	—	61	2	no	—
238	C₄²	D₆₁⋊C₈	1	2	yes	—
239	C₄²	C₂²×#6923	1	1	yes	characteristic
240	C₄²	C₂²×#6923	1	1	yes	characteristic
241	C₁₆	—	61	4	no	—
242	C₁₆	—	61	4	no	—
243	C₁₆	—	61	8	no	—
244	C₂³	—	61	2	no	—
245	C₂³	C₂×D₆₁⋊C₈	1	1	yes	characteristic
246	Q₈	—	61	4	no	—
247	D₄	—	61	8	no	—
248	D₄	—	61	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	61	2	no	—
251	C₂×C₄	—	61	4	no	—
252	C₂×C₄	—	61	4	no	—
253	C₂×C₄	—	61	4	no	—
254	C₂×C₄	—	122	8	no	—
255	C₂×C₄	C₂×C₄×#6923	1	2	yes	—
256	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
257	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
258	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
259	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
260	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
261	C₈	—	61	2	no	—
262	C₈	—	61	2	no	—
263	C₈	—	61	2	no	—
264	C₈	—	61	2	no	—
265	C₈	C₂×C₄×#6923	1	4	yes	—
266	C₈	C₂×D₆₁⋊C₈	1	4	yes	—
267	C₂²	—	2	2	no	—
268	C₂²	—	61	4	no	—
269	C₂²	—	122	4	no	—
270	C₂²	C₄×D₆₁⋊C₈	1	2	yes	—
271	C₂²	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	61	4	no	—
274	C₄	—	122	4	no	—
275	C₄	C₄×D₆₁⋊C₈	1	2	yes	—
276	C₄	#16840821	1	4	yes	direct product factor
277	C₄	C₂²×C₄×#6923	1	1	yes	characteristic
278	C₄	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	122	4	no	—
281	C₂	C₂×C₄×D₆₁⋊C₈	1	1	yes	characteristic
282	C₂	C₂×#16840821	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²×#6923
commutator	C₂×C₆₁	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₄×M₁₆×C₆₁	C₄
socle	C₂²×C₆₁	C₂³×C₈
prime radical	C₂×#11758222	C₂×C₄
cosocle	C₂×#92418	C₂⁴
hypercenter	C₄×M₁₆	#6923
hypocenter	C₆₁	C₄×#504
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #21462030	C₂	semidirect
3. #21462029	C₂	semidirect
4. C₄×#14519852	C₂	semidirect
5. C₄×#14519851	C₂	semidirect
6. C₄×D₆₁⋊C₁₆	C₂	semidirect
7. C₂×C₄×#11758187	C₂	semidirect
9. #16840821	C₄	direct (N × H)
32. #14519852	C₂×C₄	semidirect
33. #14519851	C₂×C₄	semidirect
34. D₆₁⋊C₁₆	C₂×C₄	semidirect
35. C₂×#11758187	C₂×C₄	semidirect
133. C₄×C₆₁	#504	semidirect
173. C₆₁	C₄×#504	semidirect
276. C₄	#16840821	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16840821	2
#21462030	2
#21462029	2
C₄×#14519852	2
C₄×#14519851	2
C₄×D₆₁⋊C₁₆	2
C₂×C₄×#11758187	2
C₄×#14519785	2
C₄×#504	61

Subgroups — C4×#16840821 #30623829

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16840821 #30623829