Subgroups — #29573681

Order 15616.

378 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445936	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519688	C₂	1	1	yes	maximal, characteristic
3	#21443653	C₂	1	4	yes	maximal
4	C₆₁×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#92418	C₂²	1	2	yes	—
6	C₂×#14519688	C₄	1	2	yes	—
7	C₂×#14519688	C₄	1	1	yes	characteristic
8	C₂×#14519688	C₄	1	1	yes	characteristic
9	C₂×C₄×#92418	C₂²	1	2	yes	—
10	#16839517	—	2	2	no	—
11	C₂×C₄⋊C₈×C₆₁	—	2	2	no	—
12	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
13	C₄³×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²×#92418	—	2	2	no	—
15	C₂²×#92418	C₂×C₄	1	2	yes	—
16	C₂²×#92418	C₂×C₄	1	2	yes	—
17	C₄⋊#92418	—	2	4	no	—
18	#14519688	—	2	2	no	—
19	#14519688	—	2	4	no	—
20	#14519688	C₈	1	4	yes	complemented
21	C₄×#92418	—	2	2	no	—
22	C₄×#92418	—	2	2	no	—
23	C₂²×C₈×C₆₁	—	2	2	no	—
24	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₆₁	D₄	1	2	yes	—
27	C₂×C₄²×C₆₁	Q₈	1	2	yes	—
28	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₆₁	—	2	4	no	—
30	C₄⋊C₈×C₆₁	—	4	4	no	—
31	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
33	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₆₁	—	4	2	no	—
37	C₂×C₈×C₆₁	—	4	4	no	—
38	C₄²×C₆₁	—	2	2	no	—
39	C₄²×C₆₁	—	2	2	no	—
40	C₄²×C₆₁	—	2	2	no	—
41	C₄²×C₆₁	—	4	2	no	—
42	C₄²×C₆₁	C₂×C₈	1	2	yes	—
43	C₄²×C₆₁	M₁₆	1	2	yes	—
44	C₄²×C₆₁	#41	1	2	yes	—
45	C₄²×C₆₁	#41	1	2	yes	—
46	C₂×#92418	—	2	2	no	—
47	C₂×#92418	—	2	2	no	—
48	C₂×#92418	—	2	2	no	—
49	C₂×#92418	—	2	2	no	—
50	C₂×#92418	—	4	2	no	—
51	C₂×#92418	—	4	2	no	—
52	C₂×#92418	—	4	2	no	—
53	C₂×#92418	C₂×C₈	1	4	yes	—
54	C₂×#92418	M₁₆	1	4	yes	—
55	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₁	—	2	2	no	—
57	C₂×C₄×C₆₁	—	2	2	no	—
58	C₂×C₄×C₆₁	—	2	2	no	—
59	C₂×C₄×C₆₁	—	2	2	no	—
60	C₂×C₄×C₆₁	—	2	2	no	—
61	C₂×C₄×C₆₁	—	2	2	no	—
62	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
63	C₂×C₄×C₆₁	#117	1	2	yes	—
64	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
65	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
66	C₂×C₄×C₆₁	#125	1	2	yes	—
67	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×C₄×C₆₁	#137	1	2	yes	—
72	C₈×C₆₁	—	8	2	no	—
73	#92418	—	2	4	no	—
74	#92418	—	2	4	no	—
75	#92418	—	8	2	no	—
76	#8786	—	61	1	no	maximal, Sylow
77	C₂²×C₆₁	#382	1	1	yes	characteristic
78	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
79	C₂²×C₆₁	#423	1	1	yes	characteristic
80	C₂²×C₆₁	#428	1	2	yes	—
81	C₂²×C₆₁	#430	1	2	yes	—
82	C₄×C₆₁	—	2	4	no	—
83	C₄×C₆₁	—	2	2	no	—
84	C₄×C₆₁	—	4	2	no	—
85	C₄×C₆₁	#431	1	2	yes	—
86	C₄×C₆₁	#432	1	2	yes	—
87	C₄×C₆₁	#489	1	2	yes	—
88	C₄×C₆₁	#490	1	1	yes	characteristic
89	C₄×C₆₁	#491	1	1	yes	characteristic
90	#1842	D₆₁	1	1	yes	characteristic
91	#1831	—	61	1	no	—
92	C₄×#97	—	61	1	no	—
93	#1277	—	61	4	no	—
94	C₂×C₆₁	#1565	1	2	yes	—
95	C₂×C₆₁	#1569	1	2	yes	—
96	C₂×C₆₁	#1831	1	1	yes	characteristic
97	C₂×C₆₁	#1836	1	1	yes	characteristic
98	C₂×C₆₁	#1838	1	1	yes	characteristic
99	C₂×C₄⋊C₈	—	2	2	no	—
100	C₂×C₄⋊C₈	—	61	2	no	—
101	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
102	C₂×#97	—	61	1	no	—
103	C₂×#97	—	61	2	no	—
104	C₂×#97	—	61	1	no	—
105	C₂×C₄×C₈	—	61	2	no	—
106	C₄³	C₂×D₆₁	1	1	yes	characteristic
107	#336	—	122	2	no	—
108	C₆₁	#8786	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	—	61	2	no	—
111	C₂²×C₈	—	61	2	no	—
112	C₂²×C₈	—	122	2	no	—
113	C₂×C₄²	Dic₁₂₂	1	2	yes	—
114	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
115	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
116	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
117	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
118	C₄⋊C₈	—	2	4	no	—
119	C₄⋊C₈	—	4	4	no	—
120	C₄⋊C₈	—	122	4	no	—
121	#97	—	61	4	no	—
122	#97	—	122	4	no	—
123	#97	—	122	2	no	—
124	C₄×C₈	—	122	2	no	—
125	C₄×C₈	—	122	2	no	—
126	C₂²×C₄	#11758196	1	2	yes	—
127	C₂²×C₄	#11758196	1	2	yes	—
128	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
129	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
130	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	—	4	4	no	—
133	C₂×C₈	—	61	4	no	—
134	C₂×C₈	—	61	4	no	—
135	C₂×C₈	—	122	2	no	—
136	C₂×C₈	—	122	2	no	—
137	C₂×C₈	—	122	2	no	—
138	C₂×C₈	—	122	2	no	—
139	C₂×C₈	—	244	2	no	—
140	C₂×C₈	—	244	2	no	—
141	C₂×C₈	—	244	2	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	C₈×D₆₁	1	2	yes	—
147	C₄²	#11758189	1	2	yes	—
148	C₄²	#11758224	1	2	yes	—
149	C₄²	#11758226	1	2	yes	—
150	C₂³	C₂×#11758196	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	#14519706	1	4	yes	—
158	C₂×C₄	#14519735	1	2	yes	—
159	C₂×C₄	#14519764	1	1	yes	characteristic
160	C₂×C₄	#14519766	1	1	yes	characteristic
161	C₂×C₄	#14519768	1	2	yes	—
162	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
163	C₂×C₄	M₁₆×D₆₁	1	2	yes	—
164	C₂×C₄	C₂×#11758189	1	1	yes	characteristic
165	C₂×C₄	#14519803	1	2	yes	—
166	C₂×C₄	#14519808	1	2	yes	—
167	C₈	—	8	2	no	—
168	C₈	—	122	4	no	—
169	C₈	—	122	4	no	—
170	C₈	—	488	2	no	—
171	C₂²	#16839627	1	1	yes	characteristic
172	C₂²	C₂×#14519706	1	1	yes	characteristic
173	C₂²	#16839785	1	1	yes	characteristic
174	C₂²	#16839802	1	2	yes	—
175	C₂²	#16839810	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	4	no	—
178	C₄	—	4	2	no	—
179	C₄	#16839814	1	2	yes	—
180	C₄	#16839818	1	2	yes	—
181	C₄	#16840124	1	2	yes	—
182	C₄	#16840126	1	1	yes	characteristic
183	C₄	#16840130	1	1	yes	characteristic
184	C₂	#21444569	1	2	yes	—
185	C₂	#21444584	1	2	yes	—
186	C₂	#21445935	1	1	yes	characteristic
187	C₂	#21445954	1	1	yes	characteristic
188	C₂	#21445959	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1842	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1842	D₆₁
hypocenter	C₆₁	#8786
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
20. #14519688	C₈	semidirect
108. C₆₁	#8786	semidirect

Maximal subgroups

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

subgroup	index
#21445936	2
C₄×#14519688	2
#21443653	2
C₆₁×#1842	2
#8786	61