Subgroups — #29568219

Order 15616.

462 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	#21445962	C₂	1	1	yes	maximal, characteristic
2	#21445672	C₂	1	1	yes	maximal, characteristic
3	#21443642	C₂	1	4	yes	maximal
4	C₄×C₄⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#92418	—	2	1	no	—
6	C₂×#14519688	—	2	1	no	—
7	#16839517	—	2	1	no	—
8	#16839517	—	2	1	no	—
9	#16839517	C₂²	1	2	yes	—
10	#16839517	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₆₁	—	2	1	no	—
13	C₆₁×#391	—	2	1	no	—
14	C₆₁×#389	—	2	1	no	—
15	C₆₁×#389	C₂²	1	2	yes	—
16	C₆₁×#384	C₄	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₆₁	—	2	1	no	—
18	C₄×C₄⋊C₄×C₆₁	C₄	1	2	yes	—
19	C₄³×C₆₁	C₂²	1	1	yes	characteristic
20	C₂²×#92418	—	2	2	no	—
21	C₂²×#92418	—	2	2	no	—
22	C₄⋊#92418	—	2	4	no	—
23	#14519688	—	4	2	no	—
24	C₂²×Q₈×C₆₁	—	2	1	no	—
25	C₄⋊Q₈×C₆₁	—	2	4	no	—
26	C₄×Q₈×C₆₁	—	4	4	no	—
27	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
28	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
29	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
30	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₆₁	—	2	2	no	—
34	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
41	#95×C₆₁	—	2	2	no	—
42	#95×C₆₁	—	2	2	no	—
43	C₂×Q₈×C₆₁	—	2	4	no	—
44	C₂×Q₈×C₆₁	—	4	2	no	—
45	C₂×Q₈×C₆₁	—	4	2	no	—
46	C₂²×C₄×C₆₁	—	2	2	no	—
47	C₂²×C₄×C₆₁	—	2	2	no	—
48	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₄⋊C₄×C₆₁	—	2	4	no	—
56	C₄⋊C₄×C₆₁	—	2	2	no	—
57	C₄⋊C₄×C₆₁	—	4	2	no	—
58	C₄⋊C₄×C₆₁	—	4	2	no	—
59	C₄⋊C₄×C₆₁	—	4	2	no	—
60	C₄⋊C₄×C₆₁	—	4	4	no	—
61	C₄⋊C₄×C₆₁	M₁₆	1	4	yes	—
62	C₄²×C₆₁	—	2	2	no	—
63	C₄²×C₆₁	—	2	2	no	—
64	C₄²×C₆₁	—	2	2	no	—
65	C₄²×C₆₁	—	2	2	no	—
66	C₄²×C₆₁	—	4	1	no	—
67	C₄²×C₆₁	—	4	1	no	—
68	C₄²×C₆₁	—	4	4	no	—
69	C₄²×C₆₁	SD₁₆	1	2	yes	—
70	C₄²×C₆₁	Q₁₆	1	2	yes	—
71	C₂×#92418	—	2	4	no	—
72	C₂×#92418	—	4	2	no	—
73	C₂×#92418	—	4	2	no	—
74	C₂×#92418	—	4	2	no	—
75	C₂×#92418	—	4	2	no	—
76	C₂×#92418	—	4	2	no	—
77	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	Q₈×C₆₁	—	4	8	no	—
79	C₂×C₄×C₆₁	—	2	1	no	—
80	C₂×C₄×C₆₁	—	2	1	no	—
81	C₂×C₄×C₆₁	—	2	1	no	—
82	C₂×C₄×C₆₁	—	2	1	no	—
83	C₂×C₄×C₆₁	—	2	1	no	—
84	C₂×C₄×C₆₁	—	2	1	no	—
85	C₂×C₄×C₆₁	—	2	1	no	—
86	C₂×C₄×C₆₁	—	2	1	no	—
87	C₂×C₄×C₆₁	—	2	1	no	—
88	C₂×C₄×C₆₁	—	2	1	no	—
89	C₂×C₄×C₆₁	—	2	1	no	—
90	C₂×C₄×C₆₁	—	2	1	no	—
91	C₂×C₄×C₆₁	—	2	1	no	—
92	C₂×C₄×C₆₁	—	2	1	no	—
93	C₂×C₄×C₆₁	—	2	4	no	—
94	C₂×C₄×C₆₁	—	4	2	no	—
95	C₂×C₄×C₆₁	—	4	2	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	#101	1	2	yes	—
100	C₂×C₄×C₆₁	#103	1	2	yes	—
101	C₂×C₄×C₆₁	#103	1	2	yes	—
102	C₂×C₄×C₆₁	#104	1	2	yes	—
103	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
104	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
106	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
107	#92418	—	4	4	no	—
108	#92418	—	8	2	no	—
109	#7622	—	61	1	no	maximal, Sylow
110	C₂²×C₆₁	#407	1	1	yes	characteristic
111	C₂²×C₆₁	#410	1	1	yes	characteristic
112	C₂²×C₆₁	C₂×#101	1	1	yes	characteristic
113	C₂²×C₆₁	C₂×#103	1	1	yes	characteristic
114	C₂²×C₆₁	#418	1	1	yes	characteristic
115	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
116	C₂²×C₆₁	#421	1	1	yes	characteristic
117	C₄×C₆₁	—	2	2	no	—
118	C₄×C₆₁	—	2	2	no	—
119	C₄×C₆₁	—	2	2	no	—
120	C₄×C₆₁	—	2	2	no	—
121	C₄×C₆₁	—	2	1	no	—
122	C₄×C₆₁	—	2	1	no	—
123	C₄×C₆₁	—	2	2	no	—
124	C₄×C₆₁	—	2	2	no	—
125	C₄×C₆₁	—	4	4	no	—
126	C₄×C₆₁	—	8	2	no	—
127	C₄×C₄⋊Q₈	D₆₁	1	1	yes	characteristic
128	#1839	—	61	1	no	—
129	#1767	—	61	1	no	—
130	#1271	—	61	4	no	—
131	C₂×C₆₁	#1467	1	1	yes	characteristic
132	C₂×C₆₁	#1487	1	1	yes	characteristic
133	C₂×C₆₁	#1490	1	1	yes	characteristic
134	C₂×C₆₁	#1503	1	1	yes	characteristic
135	C₂×C₆₁	#1514	1	1	yes	characteristic
136	C₂×C₆₁	#1531	1	1	yes	characteristic
137	C₂×C₆₁	#1549	1	1	yes	characteristic
138	C₂×C₄⋊Q₈	Dic₆₁	1	1	yes	characteristic
139	C₂×C₄×Q₈	—	2	1	no	—
140	C₂×C₄⋊C₈	—	122	1	no	—
141	C₂×#97	—	122	1	no	—
142	#391	—	2	1	no	—
143	#389	—	2	1	no	—
144	#389	C₂×D₆₁	1	2	yes	—
145	#384	Dic₆₁	1	1	yes	characteristic
146	C₄×C₄⋊C₄	—	2	1	no	—
147	C₄×C₄⋊C₄	Dic₆₁	1	2	yes	—
148	C₄³	C₂×D₆₁	1	1	yes	characteristic
149	#336	—	61	2	no	—
150	#336	—	61	2	no	—
151	#336	—	122	1	no	—
152	#336	—	122	1	no	—
153	C₆₁	#7622	1	1	yes	characteristic, complemented, Sylow
154	C₂²×Q₈	—	2	1	no	—
155	C₂²×C₈	—	122	2	no	—
156	C₂²×C₈	—	122	2	no	—
157	C₄⋊Q₈	—	2	4	no	—
158	C₄×Q₈	—	4	4	no	—
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
164	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
167	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
169	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
170	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
171	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
172	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
173	C₄⋊C₈	—	122	4	no	—
174	#97	—	244	2	no	—
175	#95	—	2	2	no	—
176	#95	—	2	2	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	4	2	no	—
179	C₂×Q₈	—	4	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
183	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
185	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
186	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
189	C₂×C₈	—	122	4	no	—
190	C₂×C₈	—	244	2	no	—
191	C₂×C₈	—	244	2	no	—
192	C₂×C₈	—	244	2	no	—
193	C₂×C₈	—	244	2	no	—
194	C₂×C₈	—	244	2	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	4	no	—
201	C₄⋊C₄	#11758194	1	4	yes	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	4	1	no	—
207	C₄²	—	4	1	no	—
208	C₄²	—	4	4	no	—
209	C₄²	#11758201	1	2	yes	—
210	C₄²	#11758202	1	2	yes	—
211	C₂³	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
212	Q₈	—	4	8	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	#14519696	1	2	yes	—
234	C₂×C₄	#14519701	1	2	yes	—
235	C₂×C₄	#14519701	1	2	yes	—
236	C₂×C₄	#14519704	1	2	yes	—
237	C₂×C₄	C₂×#11758194	1	2	yes	—
238	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
239	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
240	C₂×C₄	#14519803	1	2	yes	—
241	C₈	—	488	2	no	—
242	C₈	—	244	4	no	—
243	C₂²	#16839726	1	1	yes	characteristic
244	C₂²	#16839736	1	1	yes	characteristic
245	C₂²	C₂×#14519696	1	1	yes	characteristic
246	C₂²	C₂×#14519701	1	1	yes	characteristic
247	C₂²	#16839767	1	1	yes	characteristic
248	C₂²	C₂×#14519704	1	1	yes	characteristic
249	C₂²	#16839779	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	4	4	no	—
259	C₄	—	8	2	no	—
260	C₂	#21444163	1	1	yes	characteristic
261	C₂	#21444248	1	1	yes	characteristic
262	C₂	#21444262	1	1	yes	characteristic
263	C₂	#21444318	1	1	yes	characteristic
264	C₂	#21444366	1	1	yes	characteristic
265	C₂	#21444434	1	1	yes	characteristic
266	C₂	#21444504	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₁
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₄×C₄⋊Q₈×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₄×C₄⋊Q₈	D₆₁
hypocenter	C₆₁	#7622
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
153. C₆₁	#7622	semidirect

Maximal subgroups

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

subgroup	index
#21445962	2
#21445672	2
#21443642	2
C₄×C₄⋊Q₈×C₆₁	2
#7622	61