Subgroups — #29589366

Order 15616.

292 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21447622	C₂	1	1	yes	maximal, characteristic
2	#21447618	C₂	1	1	yes	maximal, characteristic
3	#21443761	C₂	1	4	yes	maximal
4	C₆₁×#2960	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11758185	—	2	1	no	—
6	C₄⋊#11758185	—	2	1	no	—
7	C₂²⋊#11758185	C₂²	1	2	yes	—
8	C₂²⋊#11758185	C₂²	1	2	yes	—
9	#16839538	—	2	1	no	—
10	C₄×#11758185	—	2	1	no	—
11	C₆₁×#533	C₄	1	1	yes	characteristic
12	C₆₁×#446	—	2	1	no	—
13	C₆₁×#446	—	2	1	no	—
14	C₈×Q₈×C₆₁	—	2	1	no	—
15	C₈×Q₈×C₆₁	—	2	1	no	—
16	C₆₁×#424	C₂²	1	2	yes	—
17	C₆₁×#423	C₄	1	1	yes	characteristic
18	C₆₁×#405	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×#11758185	—	2	2	no	—
21	C₂×#11758185	—	2	2	no	—
22	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
25	C₄⋊Q₈×C₆₁	C₈	1	1	yes	characteristic
26	C₄⋊Q₈×C₆₁	C₈	1	1	yes	characteristic
27	#125×C₆₁	C₈	1	2	yes	—
28	C₂²⋊Q₈×C₆₁	—	2	1	no	—
29	C₂²⋊Q₈×C₆₁	—	2	1	no	—
30	C₄×Q₈×C₆₁	—	2	1	no	—
31	C₄×Q₈×C₆₁	—	2	1	no	—
32	#117×C₆₁	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₄⋊C₈×C₆₁	—	2	2	no	—
35	C₄⋊C₈×C₆₁	—	2	2	no	—
36	C₄⋊C₈×C₆₁	—	2	1	no	—
37	C₄⋊C₈×C₆₁	—	2	1	no	—
38	#97×C₆₁	—	2	2	no	—
39	C₄×C₈×C₆₁	—	2	2	no	—
40	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
42	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
43	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₆₁	—	2	1	no	—
45	C₂×Q₈×C₆₁	—	2	1	no	—
46	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
49	M₁₆×C₆₁	—	4	2	no	—
50	C₂×C₈×C₆₁	—	2	2	no	—
51	C₂×C₈×C₆₁	—	2	2	no	—
52	C₂×C₈×C₆₁	—	4	1	no	—
53	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₄⋊C₄×C₆₁	—	2	2	no	—
58	C₄⋊C₄×C₆₁	—	2	2	no	—
59	C₄⋊C₄×C₆₁	—	2	1	no	—
60	C₄⋊C₄×C₆₁	—	2	1	no	—
61	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
62	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
63	C₂²⋊C₄×C₆₁	—	2	2	no	—
64	C₄²×C₆₁	—	2	2	no	—
65	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
68	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
69	#11758185	—	4	2	no	—
70	#11758185	—	4	2	no	—
71	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
72	Q₈×C₆₁	—	4	2	no	—
73	Q₈×C₆₁	—	4	2	no	—
74	C₂×C₄×C₆₁	—	2	2	no	—
75	C₂×C₄×C₆₁	—	2	1	no	—
76	C₂×C₄×C₆₁	—	2	1	no	—
77	C₂×C₄×C₆₁	—	2	2	no	—
78	C₂×C₄×C₆₁	—	2	2	no	—
79	C₂×C₄×C₆₁	—	4	1	no	—
80	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
86	C₈×C₆₁	—	2	1	no	—
87	C₈×C₆₁	—	2	1	no	—
88	C₈×C₆₁	—	2	1	no	—
89	C₈×C₆₁	—	4	2	no	—
90	C₈×C₆₁	#101	1	2	yes	—
91	#11498	—	61	1	no	maximal, Sylow
92	C₂²×C₆₁	—	2	1	no	—
93	C₂²×C₆₁	—	2	1	no	—
94	C₂²×C₆₁	—	2	1	no	—
95	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
96	C₄×C₆₁	—	2	1	no	—
97	C₄×C₆₁	—	2	1	no	—
98	C₄×C₆₁	—	2	1	no	—
99	C₄×C₆₁	—	2	1	no	—
100	C₄×C₆₁	—	4	2	no	—
101	C₄×C₆₁	—	4	2	no	—
102	C₄×C₆₁	#350	1	2	yes	—
103	C₄×C₆₁	#410	1	1	yes	characteristic
104	C₄×C₆₁	C₂×#101	1	1	yes	characteristic
105	#2960	D₆₁	1	1	yes	characteristic
106	#2163	—	61	1	no	—
107	#2162	—	61	1	no	—
108	#1323	—	61	4	no	—
109	C₂×C₆₁	—	4	1	no	—
110	C₂×C₆₁	#1461	1	1	yes	characteristic
111	C₂×C₆₁	C₂×#350	1	1	yes	characteristic
112	C₂×C₆₁	#2117	1	1	yes	characteristic
113	#533	Dic₆₁	1	1	yes	characteristic
114	#446	—	2	1	no	—
115	#446	—	2	1	no	—
116	C₈×Q₈	—	2	1	no	—
117	C₈×Q₈	—	2	1	no	—
118	#424	C₂×D₆₁	1	2	yes	—
119	#423	Dic₆₁	1	1	yes	characteristic
120	#405	Dic₆₁	1	2	yes	—
121	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
122	C₄⋊C₁₆	—	122	1	no	—
123	C₄⋊C₁₆	—	122	1	no	—
124	C₂²⋊C₁₆	—	61	2	no	—
125	C₂²⋊C₁₆	—	61	2	no	—
126	#346	—	122	1	no	—
127	C₄×C₁₆	—	122	1	no	—
128	C₆₁	#11498	1	1	yes	characteristic, complemented, Sylow
129	C₂×M₁₆	C₂×Dic₆₁	1	2	yes	—
130	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
131	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
132	C₄⋊Q₈	#92418	1	1	yes	characteristic
133	C₄⋊Q₈	#92418	1	1	yes	characteristic
134	#125	#92418	1	2	yes	—
135	C₂²⋊Q₈	—	2	1	no	—
136	C₂²⋊Q₈	—	2	1	no	—
137	C₄×Q₈	—	2	1	no	—
138	C₄×Q₈	—	2	1	no	—
139	#117	C₂×Dic₆₁	1	2	yes	—
140	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
141	C₂×C₁₆	—	122	2	no	—
142	C₂×C₁₆	—	122	2	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	2	1	no	—
146	C₄⋊C₈	—	2	1	no	—
147	#97	—	2	2	no	—
148	C₄×C₈	—	2	2	no	—
149	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
151	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
152	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
153	C₂×Q₈	—	2	1	no	—
154	C₂×Q₈	—	2	1	no	—
155	C₂²×C₄	#11758194	1	1	yes	characteristic
156	C₂²×C₄	#11758194	1	1	yes	characteristic
157	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
158	M₁₆	—	4	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	4	1	no	—
162	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
163	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
164	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
165	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	C₂×#92418	1	2	yes	—
171	C₄⋊C₄	C₂×#92418	1	2	yes	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	C₂×#92418	1	1	yes	characteristic
175	C₄²	C₂×#92418	1	1	yes	characteristic
176	C₄²	C₂²⋊Dic₆₁	1	1	yes	characteristic
177	C₄²	C₂²⋊Dic₆₁	1	1	yes	characteristic
178	C₁₆	—	244	2	no	—
179	C₁₆	—	244	2	no	—
180	C₂³	C₂×#11758194	1	1	yes	characteristic
181	Q₈	—	4	2	no	—
182	Q₈	—	4	2	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	C₂²⋊#92418	1	1	yes	characteristic
190	C₂×C₄	C₂²⋊#92418	1	1	yes	characteristic
191	C₂×C₄	C₂²⋊#92418	1	1	yes	characteristic
192	C₂×C₄	C₂²⋊#92418	1	1	yes	characteristic
193	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
194	C₂×C₄	C₂²×#92418	1	1	yes	characteristic
195	C₈	—	2	1	no	—
196	C₈	—	2	1	no	—
197	C₈	—	2	1	no	—
198	C₈	—	4	2	no	—
199	C₈	#14519696	1	2	yes	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	C₂×C₂²⋊#92418	1	1	yes	characteristic
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	2	no	—
210	C₄	#16839546	1	2	yes	—
211	C₄	#16839736	1	1	yes	characteristic
212	C₄	C₂×#14519696	1	1	yes	characteristic
213	C₂	—	4	1	no	—
214	C₂	#21444143	1	1	yes	characteristic
215	C₂	C₂×#16839546	1	1	yes	characteristic
216	C₂	#21447447	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#2960	C₂
socle	C₂²×C₆₁	C₂×C₂²⋊C₈
prime radical	C₂³×C₆₁	C₂×M₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#2960	D₆₁
hypocenter	C₆₁	#11498
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
128. C₆₁	#11498	semidirect

Maximal subgroups

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

subgroup	index
#21447622	2
#21447618	2
#21443761	2
C₆₁×#2960	2
#11498	61