Subgroups — #30606664

Order 15616.

398 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840685	C₂	1	2	yes	maximal
2	#21462025	C₂	1	2	yes	maximal
3	#21460864	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519851	C₂²	1	4	yes	—
6	#16840685	C₄	1	8	yes	complemented
7	#16840651	—	2	2	no	—
8	#16840648	—	2	2	no	—
9	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
14	C₄²×Dic₆₁	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
16	#14519851	—	2	4	no	—
17	#14519851	C₂×C₄	1	8	yes	—
18	C₂×#11758187	—	2	4	no	—
19	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
20	C₂²×#92418	C₂³	1	1	yes	characteristic
21	C₂×C₄×Dic₆₁	—	2	2	no	—
22	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
24	C₄×#92418	C₂×C₄	1	4	yes	—
25	C₈×Dic₆₁	—	2	2	no	—
26	C₈×Dic₆₁	C₈	1	4	yes	—
27	C₈×Dic₆₁	C₈	1	4	yes	—
28	C₈×Dic₆₁	D₄	1	1	yes	characteristic
29	C₈×Dic₆₁	D₄	1	1	yes	characteristic
30	C₈×Dic₆₁	Q₈	1	1	yes	characteristic
31	C₈×Dic₆₁	Q₈	1	1	yes	characteristic
32	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₆₁	—	2	2	no	—
36	C₂²×C₄×C₆₁	—	2	1	no	—
37	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₂²×Dic₆₁	—	2	1	no	—
39	C₂²×Dic₆₁	M₁₆	1	2	yes	—
40	C₂×C₈×C₆₁	—	2	1	no	—
41	C₂×C₈×C₆₁	—	2	1	no	—
42	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
47	C₄²×C₆₁	—	2	1	no	—
48	C₄²×C₆₁	C₄²	1	2	yes	—
49	C₄×Dic₆₁	—	2	8	no	—
50	C₄×Dic₆₁	—	2	1	no	—
51	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
52	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
53	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
54	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
55	C₂×#92418	C₄²	1	2	yes	—
56	C₂×#92418	C₂×C₈	1	2	yes	—
57	C₂×#92418	C₂×C₈	1	2	yes	—
58	C₂×#92418	C₂²×C₄	1	2	yes	—
59	C₂×#92418	C₂×D₄	1	1	yes	characteristic
60	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
61	C₂×#92418	#41	1	2	yes	—
62	#11758187	—	2	8	no	—
63	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
64	C₂×C₄×C₆₁	—	2	4	no	—
65	C₂×C₄×C₆₁	—	2	2	no	—
66	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
72	C₂×Dic₆₁	—	2	4	no	—
73	C₂×Dic₆₁	—	2	2	no	—
74	C₂×Dic₆₁	—	2	4	no	—
75	C₂×Dic₆₁	C₄⋊C₈	1	2	yes	—
76	C₂×Dic₆₁	C₄⋊C₈	1	2	yes	—
77	C₈×C₆₁	—	2	2	no	—
78	C₈×C₆₁	C₄⋊C₈	1	2	yes	—
79	C₈×C₆₁	C₄⋊C₈	1	2	yes	—
80	#92418	C₄×C₈	1	4	yes	—
81	#92418	C₄×D₄	1	2	yes	—
82	#92418	C₄×Q₈	1	2	yes	—
83	C₄×#364	—	61	1	no	maximal, Sylow
84	C₂²×C₆₁	—	2	2	no	—
85	C₂²×C₆₁	C₄×M₁₆	1	1	yes	characteristic
86	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₂²×C₆₁	#431	1	1	yes	characteristic
88	C₄×C₆₁	—	2	4	no	—
89	C₄×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
90	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
91	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
92	C₄×C₆₁	#431	1	1	yes	characteristic
93	Dic₆₁	—	2	4	no	—
94	Dic₆₁	—	2	2	no	—
95	Dic₆₁	#364	1	4	yes	complemented
96	C₂×#364	—	61	2	no	—
97	C₄×M₃₂	—	61	2	no	—
98	C₄²×C₈	—	61	1	no	—
99	#1376	—	61	2	no	—
100	C₂×C₆₁	—	2	1	no	—
101	C₂×C₆₁	—	2	1	no	—
102	C₂×C₆₁	C₄×C₄⋊C₈	1	1	yes	characteristic
103	C₂×C₆₁	C₂×#364	1	2	yes	—
104	C₂×M₃₂	—	61	4	no	—
105	C₂×C₄×C₈	—	61	1	no	—
106	C₂×C₄×C₈	—	61	1	no	—
107	C₂×C₄×C₈	—	61	1	no	—
108	C₂×C₄×C₈	—	61	1	no	—
109	C₂×C₄×C₈	—	61	1	no	—
110	C₂×C₄×C₈	#6923	1	1	yes	characteristic
111	C₄³	—	61	1	no	—
112	#364	—	61	8	no	—
113	#346	—	122	2	no	—
114	C₄×C₁₆	—	122	2	no	—
115	C₆₁	C₄×#364	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₈	—	61	1	no	—
117	C₂²×C₈	—	61	1	no	—
118	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
119	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
120	C₂×C₄²	—	61	1	no	—
121	C₂×C₄²	—	61	1	no	—
122	C₂×C₄²	—	122	2	no	—
123	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
124	M₃₂	—	61	8	no	—
125	M₃₂	—	122	4	no	—
126	C₂×C₁₆	—	122	4	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	61	4	no	—
129	C₄×C₈	—	61	4	no	—
130	C₄×C₈	—	61	1	no	—
131	C₄×C₈	—	61	1	no	—
132	C₄×C₈	—	61	1	no	—
133	C₄×C₈	—	61	1	no	—
134	C₄×C₈	—	61	4	no	—
135	C₄×C₈	—	122	2	no	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	61	2	no	—
138	C₂²×C₄	—	122	1	no	—
139	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	61	2	no	—
143	C₂×C₈	—	61	2	no	—
144	C₂×C₈	—	61	1	no	—
145	C₂×C₈	—	61	1	no	—
146	C₂×C₈	—	61	2	no	—
147	C₂×C₈	—	61	2	no	—
148	C₂×C₈	—	61	2	no	—
149	C₂×C₈	D₆₁⋊C₈	1	2	yes	—
150	C₂×C₈	D₆₁⋊C₈	1	2	yes	—
151	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
152	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
153	C₂×C₈	#11758217	1	2	yes	—
154	C₄²	—	2	1	no	—
155	C₄²	—	61	1	no	—
156	C₄²	—	61	1	no	—
157	C₄²	—	61	2	no	—
158	C₄²	—	61	2	no	—
159	C₄²	—	122	8	no	—
160	C₄²	—	122	1	no	—
161	C₄²	C₄×#6923	1	2	yes	—
162	C₁₆	—	122	8	no	—
163	C₂³	D₆₁⋊M₁₆	1	1	yes	characteristic
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	61	2	no	—
167	C₂×C₄	—	61	2	no	—
168	C₂×C₄	—	122	4	no	—
169	C₂×C₄	—	122	4	no	—
170	C₂×C₄	—	122	2	no	—
171	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
173	C₂×C₄	#14519856	1	1	yes	characteristic
174	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
175	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
176	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
177	C₈	—	2	2	no	—
178	C₈	—	61	4	no	—
179	C₈	—	61	2	no	—
180	C₈	—	61	2	no	—
181	C₈	#14519844	1	2	yes	—
182	C₈	#14519844	1	2	yes	—
183	C₂²	—	2	2	no	—
184	C₂²	C₄×#11758217	1	1	yes	characteristic
185	C₂²	D₆₁⋊(C₄⋊C₈)	1	1	yes	characteristic
186	C₂²	#16840780	1	1	yes	characteristic
187	C₄	—	2	4	no	—
188	C₄	—	61	4	no	—
189	C₄	—	122	4	no	—
190	C₄	—	122	2	no	—
191	C₄	C₄×C₄⋊#6923	1	1	yes	characteristic
192	C₄	C₄×D₆₁⋊C₈	1	1	yes	characteristic
193	C₄	C₂×#14519844	1	1	yes	characteristic
194	C₄	#16840779	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	C₄×#14519844	1	1	yes	characteristic
198	C₂	D₆₁⋊#364	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄⋊#6923
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×C₄×C₈×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₄⋊C₈
prime radical	C₂³×C₆₁	C₂×M₁₆
cosocle	C₂²×#92418	C₂³
hypercenter	C₂×C₄×C₈	#6923
hypocenter	C₆₁	C₄×#364
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #16840685	C₄	semidirect
95. Dic₆₁	#364	semidirect
115. C₆₁	C₄×#364	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840685	2
#21462025	2
#21460864	2
C₄×C₈×Dic₆₁	2
C₄×#364	61