Subgroups — #29573147

Order 15616.

466 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445957	C₂	1	1	yes	maximal, characteristic, complemented
2	#21444567	C₂	1	1	yes	maximal, characteristic, complemented
3	#21444244	C₂	1	2	yes	maximal
4	#21443716	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₄×C₆₁	C₂	1	1	yes	maximal, characteristic
6	#16839815	C₂²	1	2	yes	—
7	#16839786	C₂²	1	2	yes	—
8	#16839527	—	2	1	no	—
9	#16839527	—	2	1	no	—
10	#16839515	—	2	1	no	—
11	#16839513	—	2	1	no	—
12	D₄⋊#92418	—	2	4	no	—
13	C₂³×C₈×C₆₁	—	2	1	no	—
14	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₆₁	—	2	1	no	—
16	C₈×D₄×C₆₁	—	2	1	no	—
17	C₈×D₄×C₆₁	C₄	1	2	yes	—
18	C₈×D₄×C₆₁	C₄	1	2	yes	—
19	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₆₁	—	2	1	no	—
21	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×#11758194	—	2	2	no	—
23	C₄⋊#92418	—	2	2	no	—
24	C₄⋊#92418	—	2	2	no	—
25	C₄⋊#92418	—	2	2	no	—
26	C₂²⋊#92418	—	2	2	no	—
27	C₄×#92418	—	2	2	no	—
28	C₂²×D₄×C₆₁	Q₈	1	1	yes	characteristic
29	C₂³×C₄×C₆₁	—	2	1	no	—
30	C₂²×C₈×C₆₁	—	2	1	no	—
31	C₂²×C₈×C₆₁	—	2	2	no	—
32	C₂²×C₈×C₆₁	—	2	2	no	—
33	C₂²×C₈×C₆₁	—	4	4	no	—
34	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
36	C₄×D₄×C₆₁	—	2	1	no	—
37	C₄×D₄×C₆₁	—	2	1	no	—
38	C₄×D₄×C₆₁	C₂×C₄	1	2	yes	—
39	C₄×D₄×C₆₁	D₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₆₁	Q₈	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
42	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₆₁	—	2	1	no	—
44	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
45	C₂²⋊C₈×C₆₁	—	2	2	no	—
46	C₂²⋊C₈×C₆₁	—	2	2	no	—
47	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
50	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
51	C₂⁴×C₆₁	—	2	1	no	—
52	C₂×D₄×C₆₁	—	2	4	no	—
53	C₂×D₄×C₆₁	—	2	1	no	—
54	C₂×D₄×C₆₁	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₆₁	—	2	1	no	—
56	C₂²×C₄×C₆₁	—	2	2	no	—
57	C₂²×C₄×C₆₁	—	2	2	no	—
58	C₂²×C₄×C₆₁	—	4	2	no	—
59	C₂²×C₄×C₆₁	—	4	2	no	—
60	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
63	C₂×C₈×C₆₁	—	2	2	no	—
64	C₂×C₈×C₆₁	—	2	2	no	—
65	C₂×C₈×C₆₁	—	2	2	no	—
66	C₂×C₈×C₆₁	—	4	4	no	—
67	C₂×C₈×C₆₁	—	4	4	no	—
68	C₂×C₈×C₆₁	—	4	2	no	—
69	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₄⋊C₄×C₆₁	—	2	1	no	—
74	C₄⋊C₄×C₆₁	C₄⋊C₄	1	2	yes	—
75	C₂²⋊C₄×C₆₁	—	2	2	no	—
76	C₂²⋊C₄×C₆₁	—	2	2	no	—
77	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₆₁	#41	1	1	yes	characteristic
81	#11758194	—	4	2	no	—
82	#11758194	—	4	2	no	—
83	C₂×#92418	—	4	2	no	—
84	C₂×#92418	—	4	2	no	—
85	C₂³×C₆₁	—	2	4	no	—
86	C₂³×C₆₁	—	2	2	no	—
87	C₂³×C₆₁	—	4	2	no	—
88	C₂³×C₆₁	—	4	2	no	—
89	C₂³×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
90	D₄×C₆₁	—	2	4	no	—
91	D₄×C₆₁	—	2	2	no	—
92	D₄×C₆₁	#108	1	4	yes	—
93	C₂×C₄×C₆₁	—	2	2	no	—
94	C₂×C₄×C₆₁	—	2	2	no	—
95	C₂×C₄×C₆₁	—	4	1	no	—
96	C₂×C₄×C₆₁	—	4	2	no	—
97	C₂×C₄×C₆₁	—	4	2	no	—
98	C₂×C₄×C₆₁	—	4	2	no	—
99	C₂×C₄×C₆₁	—	4	4	no	—
100	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
103	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
104	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₆₁	#135	1	2	yes	—
107	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
108	C₈×C₆₁	—	2	1	no	—
109	C₈×C₆₁	—	2	1	no	—
110	C₈×C₆₁	—	2	1	no	—
111	C₈×C₆₁	—	2	1	no	—
112	C₈×C₆₁	—	4	2	no	—
113	#92418	—	8	2	no	—
114	#92418	—	8	2	no	—
115	#8660	—	61	1	no	maximal, Sylow
116	C₂²×C₆₁	—	2	1	no	—
117	C₂²×C₆₁	—	2	1	no	—
118	C₂²×C₆₁	—	2	4	no	—
119	C₂²×C₆₁	—	4	8	no	—
120	C₂²×C₆₁	—	4	2	no	—
121	C₂²×C₆₁	—	4	2	no	—
122	C₂²×C₆₁	#380	1	1	yes	characteristic
123	C₂²×C₆₁	#478	1	2	yes	—
124	C₄×C₆₁	—	2	1	no	—
125	C₄×C₆₁	—	2	1	no	—
126	C₄×C₆₁	—	2	1	no	—
127	C₄×C₆₁	—	4	2	no	—
128	C₄×C₆₁	#416	1	1	yes	characteristic
129	C₄×C₆₁	#418	1	1	yes	characteristic
130	C₄×C₆₁	C₂×#108	1	2	yes	—
131	C₄×C₆₁	#463	1	2	yes	—
132	C₂×C₈×D₄	D₆₁	1	1	yes	characteristic
133	#1837	—	61	1	no	—
134	#1564	—	61	1	no	—
135	#1486	—	61	2	no	—
136	#1301	—	61	2	no	—
137	C₂×C₆₁	—	2	2	no	—
138	C₂×C₆₁	—	4	4	no	—
139	C₂×C₆₁	#1804	1	1	yes	characteristic
140	C₂×C₆₁	#1810	1	1	yes	characteristic
141	C₂×C₆₁	#1814	1	1	yes	characteristic
142	C₂³×C₈	—	2	1	no	—
143	C₂×C₄×D₄	C₂×D₆₁	1	1	yes	characteristic
144	C₈×D₄	—	2	1	no	—
145	C₈×D₄	—	2	1	no	—
146	C₈×D₄	Dic₆₁	1	2	yes	—
147	C₈×D₄	Dic₆₁	1	2	yes	—
148	#431	—	61	2	no	—
149	#423	—	61	2	no	—
150	C₂×C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
151	C₂×C₂²⋊C₈	—	2	1	no	—
152	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
153	#341	—	122	1	no	—
154	#341	—	122	1	no	—
155	#335	—	122	1	no	—
156	#334	—	122	1	no	—
157	D₄⋊C₈	—	122	4	no	—
158	C₆₁	#8660	1	1	yes	characteristic, complemented, Sylow
159	C₂²×D₄	Dic₁₂₂	1	1	yes	characteristic
160	C₂³×C₄	—	2	1	no	—
161	C₂×M₁₆	—	122	2	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	4	4	no	—
166	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
167	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	D₂₄₄	1	2	yes	—
171	C₄×D₄	C₂×Dic₆₁	1	2	yes	—
172	C₂×C₄⋊C₄	Dic₁₂₂	1	1	yes	characteristic
173	C₂×C₂²⋊C₄	—	2	1	no	—
174	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
175	C₄⋊C₈	—	2	1	no	—
176	C₄⋊C₈	—	122	2	no	—
177	C₄⋊C₈	—	122	2	no	—
178	C₄⋊C₈	—	122	2	no	—
179	C₄⋊C₈	C₂×Dic₆₁	1	2	yes	—
180	C₂²⋊C₈	—	2	2	no	—
181	C₂²⋊C₈	—	2	2	no	—
182	C₂²⋊C₈	—	122	2	no	—
183	C₄×C₈	—	122	2	no	—
184	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
185	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
186	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
187	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
188	C₂⁴	—	2	1	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	C₄⋊Dic₆₁	1	2	yes	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
198	C₂²×C₄	#11758222	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
200	M₁₆	—	244	2	no	—
201	M₁₆	—	244	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	4	4	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	244	2	no	—
209	C₂×C₈	—	244	2	no	—
210	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
211	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
212	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
213	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	C₄⋊Dic₆₁	1	2	yes	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
219	C₄²	#11758222	1	1	yes	characteristic
220	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
221	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
222	C₂³	—	2	4	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	4	2	no	—
226	C₂³	#14519756	1	1	yes	characteristic
227	D₄	—	2	4	no	—
228	D₄	—	2	2	no	—
229	D₄	#14519713	1	4	yes	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
239	C₂×C₄	#14519736	1	1	yes	characteristic
240	C₂×C₄	#14519751	1	1	yes	characteristic
241	C₂×C₄	#14519751	1	1	yes	characteristic
242	C₂×C₄	#14519756	1	1	yes	characteristic
243	C₂×C₄	#14519797	1	2	yes	—
244	C₂×C₄	#14519803	1	2	yes	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	2	1	no	—
249	C₈	—	4	2	no	—
250	C₈	—	488	2	no	—
251	C₈	—	488	2	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	8	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	#16839622	1	1	yes	characteristic
259	C₂²	#16840062	1	2	yes	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	2	no	—
264	C₄	#16839758	1	1	yes	characteristic
265	C₄	#16839767	1	1	yes	characteristic
266	C₄	C₂×#14519713	1	2	yes	—
267	C₄	#16839977	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	4	4	no	—
270	C₂	#21445829	1	1	yes	characteristic
271	C₂	#21445857	1	1	yes	characteristic
272	C₂	#21445873	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14519751
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₈×D₄×C₆₁	C₂
socle	C₂²×C₆₁	#380
prime radical	C₂²×D₄×C₆₁	Q₈
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×C₈×D₄	D₆₁
hypocenter	C₆₁	#8660
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #21445957	C₂	semidirect
2. #21444567	C₂	semidirect
4. #21443716	C₂	semidirect
158. C₆₁	#8660	semidirect

Maximal subgroups

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

subgroup	index
#21445957	2
#21444567	2
#21444244	2
#21443716	2
C₂×C₈×D₄×C₆₁	2
#8660	61