Subgroups — #30624099

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840820	C₂	1	2	yes	maximal
2	#21462156	C₂	1	6	yes	maximal, complemented
3	#21462153	C₂	1	3	yes	maximal, complemented
4	C₂×#16840683	C₂	1	3	yes	maximal, complemented
5	#131×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	#16840820	—	2	8	no	—
7	C₂×#14519851	C₂²	1	6	yes	—
8	C₂²×#11758187	C₂²	1	6	yes	—
9	#16840683	C₂²	1	1	yes	characteristic
10	#16840683	C₂²	1	3	yes	—
11	#16840683	C₂²	1	6	yes	—
12	#16840683	C₂²	1	6	yes	—
13	C₂×#14519784	C₂²	1	1	yes	characteristic
14	#41×Dic₆₁	C₄	1	1	yes	characteristic
15	#16839720	C₂²	1	3	yes	—
16	#16839717	C₄	1	3	yes	—
17	M₁₆×Dic₆₁	C₄	1	3	yes	—
18	C₂×C₈×Dic₆₁	C₂²	1	3	yes	—
19	C₂×#131×C₆₁	C₄	1	1	yes	characteristic
20	#14519851	—	2	12	no	—
21	C₂×#11758187	—	2	12	no	—
22	C₂×#11758187	C₂³	1	2	yes	—
23	C₂×#11758187	C₂³	1	6	yes	—
24	#14519784	D₄	1	4	yes	—
25	#14519784	Q₈	1	4	yes	—
26	C₂×#11758194	C₂×C₄	1	3	yes	—
27	C₂²×#92418	C₂³	1	3	yes	—
28	Q₈×Dic₆₁	C₈	1	1	yes	characteristic
29	Q₈×Dic₆₁	C₈	1	1	yes	characteristic
30	D₄×Dic₆₁	C₈	1	3	yes	—
31	D₄×Dic₆₁	C₈	1	3	yes	—
32	#14519736	C₂×C₄	1	3	yes	—
33	C₂×C₄×Dic₆₁	C₂×C₄	1	3	yes	—
34	#14519687	C₂×C₄	1	6	yes	—
35	C₈×Dic₆₁	C₂×C₄	1	3	yes	—
36	C₈×Dic₆₁	C₂×C₄	1	3	yes	—
37	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
38	C₈×Dic₆₁	C₂³	1	3	yes	—
39	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
40	#131×C₆₁	—	2	4	no	—
41	C₂×M₁₆×C₆₁	C₂×C₄	1	3	yes	—
42	C₂²×C₈×C₆₁	C₂×C₄	1	3	yes	—
43	#41×C₆₁	C₄⋊C₄	1	4	yes	—
44	#41×C₆₁	M₁₆	1	4	yes	—
45	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
46	C₂×D₄×C₆₁	C₂×C₈	1	3	yes	—
47	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
48	C₂²×Dic₆₁	C₂×C₈	1	3	yes	—
49	C₂²×Dic₆₁	C₂×C₈	1	3	yes	—
50	M₁₆×C₆₁	—	2	6	no	—
51	C₂×C₈×C₆₁	—	2	6	no	—
52	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
54	C₂²⋊Dic₆₁	C₂×C₈	1	6	yes	—
55	C₄⋊Dic₆₁	C₂×C₈	1	3	yes	—
56	C₄⋊Dic₆₁	C₂×C₈	1	3	yes	—
57	C₄×Dic₆₁	C₂×C₈	1	3	yes	—
58	C₄×Dic₆₁	C₂×C₈	1	3	yes	—
59	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
60	C₄×Dic₆₁	C₂²×C₄	1	3	yes	—
61	#11758194	C₄⋊C₄	1	12	yes	—
62	C₂×#92418	C₂²×C₄	1	3	yes	—
63	C₂×#92418	C₂×D₄	1	6	yes	—
64	C₂×#92418	C₂×Q₈	1	6	yes	—
65	C₂×#92418	C₂⁴	1	1	yes	characteristic
66	#11758187	—	2	2	no	—
67	#11758187	—	2	6	no	—
68	C₂³×C₆₁	C₂²×C₈	1	3	yes	—
69	Q₈×C₆₁	C₄⋊C₈	1	4	yes	—
70	D₄×C₆₁	C₄⋊C₈	1	12	yes	—
71	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	6	yes	—
72	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
73	C₂×C₄×C₆₁	C₂×M₁₆	1	6	yes	—
74	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
75	C₂×Dic₆₁	—	2	6	no	—
76	C₂×Dic₆₁	—	4	3	no	—
77	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
78	C₂×Dic₆₁	C₂²×C₈	1	3	yes	—
79	C₂×Dic₆₁	C₂²×C₈	1	3	yes	—
80	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
81	C₈×C₆₁	—	2	1	no	—
82	C₈×C₆₁	—	2	3	no	—
83	#92418	C₂×C₄⋊C₄	1	6	yes	—
84	#92418	C₂²×D₄	1	1	yes	characteristic
85	#92418	C₂²×Q₈	1	1	yes	characteristic
86	#33398	—	61	1	no	maximal, Sylow
87	C₂²×C₆₁	—	2	3	no	—
88	C₂²×C₆₁	—	2	3	no	—
89	C₂²×C₆₁	C₂×C₄⋊C₈	1	6	yes	—
90	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
91	C₄×C₆₁	C₂×C₄⋊C₈	1	6	yes	—
92	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
93	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
94	Dic₆₁	—	2	3	no	—
95	Dic₆₁	—	2	1	no	—
96	Dic₆₁	—	2	3	no	—
97	Dic₆₁	#504	1	2	yes	complemented
98	C₂×#504	—	61	2	no	—
99	C₄×#131	—	61	1	no	—
100	#2151	—	61	6	no	—
101	#2150	—	61	3	no	—
102	C₂×C₄⋊C₁₆	—	61	3	no	—
103	C₂×C₆₁	—	2	6	no	—
104	C₂×C₆₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
105	C₂×C₆₁	C₂×#504	1	1	yes	characteristic
106	C₂×C₆₁	#3407	1	1	yes	characteristic
107	C₂×#131	—	61	1	no	—
108	C₂×#131	#6923	1	1	yes	characteristic
109	C₄×#41	—	61	1	no	—
110	#504	—	122	8	no	—
111	C₂×M₃₂	—	61	6	no	—
112	C₂²×C₁₆	—	61	6	no	—
113	#405	—	61	3	no	—
114	#405	—	61	3	no	—
115	C₄×M₁₆	—	61	3	no	—
116	C₂×C₄×C₈	—	61	3	no	—
117	C₄⋊C₁₆	—	61	1	no	—
118	C₄⋊C₁₆	—	61	3	no	—
119	C₄⋊C₁₆	—	61	6	no	—
120	C₄⋊C₁₆	—	61	6	no	—
121	C₆₁	#33398	1	1	yes	characteristic, complemented, Sylow
122	C₂×#41	C₂×#6923	1	1	yes	characteristic
123	#131	—	2	4	no	—
124	#131	—	61	4	no	—
125	#131	—	61	4	no	—
126	C₂×M₁₆	—	61	3	no	—
127	C₂×M₁₆	C₂×#6923	1	3	yes	—
128	C₂²×C₈	—	61	3	no	—
129	C₂²×C₈	C₂×#6923	1	3	yes	—
130	C₄×Q₈	—	61	1	no	—
131	C₄×Q₈	—	61	1	no	—
132	C₄×D₄	—	61	3	no	—
133	C₄×D₄	—	61	3	no	—
134	#117	—	61	3	no	—
135	C₂×C₄²	—	61	3	no	—
136	M₃₂	—	122	12	no	—
137	C₂×C₁₆	—	61	2	no	—
138	C₂×C₁₆	—	61	6	no	—
139	C₂×C₁₆	—	122	12	no	—
140	#97	—	61	6	no	—
141	C₄×C₈	—	61	1	no	—
142	C₄×C₈	—	61	3	no	—
143	C₄×C₈	—	61	3	no	—
144	C₄×C₈	—	61	3	no	—
145	#41	C₄⋊#6923	1	4	yes	—
146	#41	#11758217	1	4	yes	—
147	C₂×Q₈	D₆₁⋊C₈	1	1	yes	characteristic
148	C₂×D₄	D₆₁⋊C₈	1	3	yes	—
149	C₂²×C₄	—	61	3	no	—
150	C₂²×C₄	—	61	3	no	—
151	C₂²×C₄	C₂²×#6923	1	3	yes	—
152	M₁₆	—	2	6	no	—
153	M₁₆	—	61	12	no	—
154	C₂×C₈	—	2	6	no	—
155	C₂×C₈	—	61	6	no	—
156	C₂×C₈	—	61	6	no	—
157	C₂×C₈	—	61	1	no	—
158	C₂×C₈	—	61	3	no	—
159	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
160	C₂×C₈	C₂²×#6923	1	3	yes	—
161	C₄⋊C₄	—	61	3	no	—
162	C₄⋊C₄	—	61	3	no	—
163	C₂²⋊C₄	—	61	6	no	—
164	C₄²	—	61	3	no	—
165	C₄²	—	61	3	no	—
166	C₄²	—	61	1	no	—
167	C₄²	—	61	3	no	—
168	C₁₆	—	122	2	no	—
169	C₁₆	—	122	6	no	—
170	C₂³	C₂×D₆₁⋊C₈	1	3	yes	—
171	Q₈	#14519844	1	4	yes	—
172	D₄	#14519844	1	12	yes	—
173	C₂×C₄	—	61	1	no	—
174	C₂×C₄	—	61	1	no	—
175	C₂×C₄	—	61	3	no	—
176	C₂×C₄	—	61	3	no	—
177	C₂×C₄	—	122	6	no	—
178	C₂×C₄	—	244	3	no	—
179	C₂×C₄	C₂×C₄⋊#6923	1	6	yes	—
180	C₂×C₄	C₂×D₆₁⋊C₈	1	3	yes	—
181	C₂×C₄	C₂×#11758217	1	6	yes	—
182	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	3	no	—
185	C₈	—	61	6	no	—
186	C₈	—	61	1	no	—
187	C₈	—	61	1	no	—
188	C₂²	—	2	3	no	—
189	C₂²	—	2	3	no	—
190	C₂²	C₂×#14519844	1	6	yes	—
191	C₂²	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
192	C₄	—	61	2	no	—
193	C₄	—	122	3	no	—
194	C₄	—	122	3	no	—
195	C₄	—	122	1	no	—
196	C₄	C₂×#14519844	1	6	yes	—
197	C₄	C₂²×C₄⋊#6923	1	1	yes	characteristic
198	C₄	C₂²×#11758217	1	1	yes	characteristic
199	C₂	—	2	6	no	—
200	C₂	C₂²×#14519844	1	1	yes	characteristic
201	C₂	D₆₁⋊#504	1	1	yes	characteristic
202	C₂	#21462787	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₂×#131×C₆₁	C₄
prime radical	C₂×#41×C₆₁	C₂×C₄
cosocle	C₂×#92418	C₂⁴
hypercenter	C₂×#131	#6923
hypocenter	C₆₁	#33398
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #21462156	C₂	semidirect
3. #21462153	C₂	semidirect
4. C₂×#16840683	C₂	semidirect
97. Dic₆₁	#504	semidirect
121. C₆₁	#33398	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840820	2
#21462156	2
#21462153	2
C₂×#16840683	2
#131×Dic₆₁	2
#33398	61