Subgroups — #29572945

Order 15616.

398 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445815	C₂	1	2	yes	maximal
2	#21444578	C₂	1	2	yes	maximal
3	#21443694	C₂	1	2	yes	maximal
4	C₂×C₈×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
5	#16839786	C₂²	1	4	yes	—
6	C₂×#14519696	—	2	2	no	—
7	#16839531	—	2	2	no	—
8	#16839515	—	2	2	no	—
9	#16839515	—	2	2	no	—
10	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₈×Q₈×C₆₁	—	2	2	no	—
12	C₈×Q₈×C₆₁	—	2	2	no	—
13	C₂×C₄⋊C₈×C₆₁	C₄	1	2	yes	—
14	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₁	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×#11758194	—	2	4	no	—
18	C₄⋊#92418	—	2	4	no	—
19	C₄⋊#92418	—	2	4	no	—
20	#14519696	—	4	4	no	—
21	C₂²×Q₈×C₆₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₄×Q₈×C₆₁	—	2	2	no	—
26	C₄×Q₈×C₆₁	—	2	2	no	—
27	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₆₁	Q₈	1	2	yes	—
29	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₆₁	—	2	4	no	—
32	C₄⋊C₈×C₆₁	—	2	2	no	—
33	C₄×C₈×C₆₁	—	2	2	no	—
34	C₄×C₈×C₆₁	—	2	2	no	—
35	C₄×C₈×C₆₁	D₄	1	2	yes	—
36	C₄×C₈×C₆₁	D₄	1	2	yes	—
37	C₂×Q₈×C₆₁	—	2	2	no	—
38	C₂×Q₈×C₆₁	—	2	2	no	—
39	C₂×Q₈×C₆₁	D₈	1	2	yes	—
40	C₂×Q₈×C₆₁	Q₁₆	1	2	yes	—
41	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₆₁	—	2	2	no	—
47	C₂×C₈×C₆₁	—	2	2	no	—
48	C₂×C₈×C₆₁	—	4	1	no	—
49	C₂×C₈×C₆₁	—	4	2	no	—
50	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
52	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₄⋊C₄×C₆₁	—	2	2	no	—
54	C₄⋊C₄×C₆₁	—	2	4	no	—
55	C₄²×C₆₁	—	2	2	no	—
56	C₄²×C₆₁	—	2	2	no	—
57	C₄²×C₆₁	C₂×D₄	1	2	yes	—
58	C₄²×C₆₁	#41	1	2	yes	—
59	#11758194	—	4	4	no	—
60	#11758194	—	4	4	no	—
61	C₂×#92418	—	4	4	no	—
62	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
63	Q₈×C₆₁	—	2	4	no	—
64	Q₈×C₆₁	—	2	4	no	—
65	C₂×C₄×C₆₁	—	2	2	no	—
66	C₂×C₄×C₆₁	—	2	2	no	—
67	C₂×C₄×C₆₁	—	2	1	no	—
68	C₂×C₄×C₆₁	—	2	1	no	—
69	C₂×C₄×C₆₁	—	2	2	no	—
70	C₂×C₄×C₆₁	—	4	1	no	—
71	C₂×C₄×C₆₁	—	4	2	no	—
72	C₂×C₄×C₆₁	#107	1	4	yes	—
73	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
75	C₂×C₄×C₆₁	#121	1	2	yes	—
76	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
77	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₆₁	#135	1	2	yes	—
80	C₈×C₆₁	—	2	1	no	—
81	C₈×C₆₁	—	2	2	no	—
82	C₈×C₆₁	—	4	2	no	—
83	C₈×C₆₁	#101	1	2	yes	complemented
84	#92418	—	8	4	no	—
85	#8617	—	61	1	no	maximal, Sylow
86	C₂²×C₆₁	—	2	1	no	—
87	C₂²×C₆₁	—	2	1	no	—
88	C₂²×C₆₁	#380	1	1	yes	characteristic
89	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
90	C₂²×C₆₁	#427	1	1	yes	characteristic
91	C₄×C₆₁	—	2	1	no	—
92	C₄×C₆₁	—	2	4	no	—
93	C₄×C₆₁	—	2	2	no	—
94	C₄×C₆₁	—	4	2	no	—
95	C₄×C₆₁	C₂×#101	1	1	yes	characteristic
96	C₄×C₆₁	#413	1	1	yes	characteristic
97	C₄×C₆₁	#466	1	2	yes	—
98	C₄×C₆₁	#467	1	2	yes	—
99	C₂×C₈×Q₈	D₆₁	1	1	yes	characteristic
100	#1801	—	61	2	no	—
101	#1567	—	61	2	no	—
102	#1292	—	61	2	no	—
103	C₂×C₆₁	—	2	1	no	—
104	C₂×C₆₁	—	2	1	no	—
105	C₂×C₆₁	#1801	1	1	yes	characteristic
106	C₂×C₆₁	#1808	1	1	yes	characteristic
107	C₂×C₆₁	#1813	1	1	yes	characteristic
108	C₂×C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
109	C₈×Q₈	—	2	2	no	—
110	C₈×Q₈	—	2	2	no	—
111	#423	—	61	4	no	—
112	C₂×C₄⋊C₈	Dic₆₁	1	2	yes	—
113	C₂×C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
114	C₂×#101	—	122	2	no	—
115	C₂×C₄×C₈	Dic₆₁	1	2	yes	—
116	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
117	#343	—	122	2	no	—
118	#335	—	122	2	no	—
119	#335	—	122	2	no	—
120	C₆₁	#8617	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	D₂₄₄	1	1	yes	characteristic
122	C₂×M₁₆	—	122	4	no	—
123	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
124	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
125	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
126	C₄×Q₈	—	2	2	no	—
127	C₄×Q₈	—	2	2	no	—
128	C₂×C₄⋊C₄	Dic₁₂₂	1	2	yes	—
129	C₂×C₄⋊C₄	D₂₄₄	1	1	yes	characteristic
130	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
131	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
132	C₄⋊C₈	—	2	4	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	122	4	no	—
135	C₄⋊C₈	—	122	4	no	—
136	#101	—	244	4	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
140	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂×Q₈	D₄₈₈	1	2	yes	—
144	C₂×Q₈	Dic₂₄₄	1	2	yes	—
145	C₂²×C₄	C₄⋊Dic₆₁	1	2	yes	—
146	C₂²×C₄	C₄⋊Dic₆₁	1	2	yes	—
147	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
148	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
149	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
150	M₁₆	—	244	4	no	—
151	M₁₆	—	244	4	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	—	244	4	no	—
157	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
158	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
159	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	#11758222	1	2	yes	—
165	C₄²	C₂×C₂²⋊D₆₁	1	2	yes	—
166	C₂³	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
167	Q₈	—	2	4	no	—
168	Q₈	—	2	4	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	#14519711	1	4	yes	—
177	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
178	C₂×C₄	#14519736	1	1	yes	characteristic
179	C₂×C₄	#14519751	1	2	yes	—
180	C₂×C₄	#14519756	1	2	yes	—
181	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
182	C₂×C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
183	C₂×C₄	#14519799	1	2	yes	—
184	C₈	—	2	1	no	—
185	C₈	—	2	2	no	—
186	C₈	—	4	2	no	—
187	C₈	—	488	4	no	—
188	C₈	#14519696	1	2	yes	complemented
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	#16839622	1	1	yes	characteristic
192	C₂²	C₂×#14519711	1	1	yes	characteristic
193	C₂²	#16839800	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	4	no	—
196	C₄	—	2	2	no	—
197	C₄	—	4	2	no	—
198	C₄	C₂×#14519696	1	1	yes	characteristic
199	C₄	#16839746	1	1	yes	characteristic
200	C₄	#16839998	1	2	yes	—
201	C₄	#16840003	1	2	yes	—
202	C₂	—	2	1	no	—
203	C₂	—	2	1	no	—
204	C₂	#21445815	1	1	yes	characteristic
205	C₂	#21445848	1	1	yes	characteristic
206	C₂	#21445869	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16839622
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₈×Q₈×C₆₁	C₂
socle	C₂²×C₆₁	#380
prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×C₈×Q₈	D₆₁
hypocenter	C₆₁	#8617
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
83. C₈×C₆₁	#101	semidirect
120. C₆₁	#8617	semidirect
188. C₈	#14519696	semidirect

Maximal subgroups

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

subgroup	index
#21445815	2
#21444578	2
#21443694	2
C₂×C₈×Q₈×C₆₁	2
#8617	61