Subgroups — #29680249

Order 15616.

778 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	#21454383	C₂	1	2	yes	maximal
2	#21445960	C₂	1	4	yes	maximal, complemented
3	#21445957	C₂	1	4	yes	maximal, complemented
4	#21445546	C₂	1	2	yes	maximal
5	C₂×#16839527	C₂	1	2	yes	maximal
6	C₂×C₆₁×#405	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11758194	C₂²	1	4	yes	—
8	#16839789	—	2	8	no	—
9	#16839786	C₂²	1	8	yes	—
10	#16839527	C₂²	1	4	yes	—
11	#16839527	C₂²	1	8	yes	—
12	#16839527	C₂²	1	4	yes	—
13	C₂²×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#405	C₄	1	8	yes	—
17	C₂×#97×C₆₁	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
19	C₂×#11758194	—	2	4	no	—
20	C₂×#11758194	—	2	8	no	—
21	C₂×#11758194	—	2	8	no	—
22	C₂×#11758194	C₂³	1	8	yes	—
23	C₂²×#92418	—	2	4	no	—
24	C₄⋊#92418	—	2	16	no	—
25	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₆₁	—	2	4	no	—
27	C₂×M₁₆×C₆₁	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₆₁	—	2	4	no	—
29	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
33	#117×C₆₁	C₂×C₄	1	4	yes	—
34	#117×C₆₁	D₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₆₁	Q₈	1	2	yes	—
36	C₂×C₂²⋊C₄×C₆₁	Q₈	1	2	yes	—
37	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
38	#97×C₆₁	C₂×C₄	1	8	yes	—
39	C₄×C₈×C₆₁	C₂×C₄	1	8	yes	—
40	C₂⁴×C₆₁	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
42	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₆₁	C₂×Q₈	1	4	yes	—
44	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	#41	1	4	yes	—
46	C₂²×C₄×C₆₁	#41	1	4	yes	—
47	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
48	M₁₆×C₆₁	—	2	8	no	—
49	C₂×C₈×C₆₁	—	2	8	no	—
50	C₂×C₈×C₆₁	—	2	2	no	—
51	C₂×C₈×C₆₁	—	2	2	no	—
52	C₂×C₈×C₆₁	—	2	4	no	—
53	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
55	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
56	C₄⋊C₄×C₆₁	C₄⋊C₄	1	8	yes	—
57	C₂²⋊C₄×C₆₁	C₄⋊C₄	1	8	yes	—
58	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
59	C₄²×C₆₁	C₂×D₄	1	4	yes	—
60	#11758194	—	2	8	no	—
61	#11758194	—	2	8	no	—
62	#11758194	—	4	8	no	—
63	C₂×#92418	—	2	8	no	—
64	C₂×#92418	—	4	8	no	—
65	C₂³×C₆₁	—	4	2	no	—
66	C₂³×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
67	C₂³×C₆₁	#125	1	4	yes	—
68	C₂³×C₆₁	C₂²×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₆₁	—	4	4	no	—
70	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	8	yes	—
71	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₆₁	#117	1	4	yes	—
73	C₂×C₄×C₆₁	#117	1	4	yes	—
74	C₂×C₄×C₆₁	#124	1	8	yes	—
75	C₂×C₄×C₆₁	#125	1	4	yes	—
76	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
79	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
80	C₈×C₆₁	—	2	2	no	—
81	C₈×C₆₁	—	2	2	no	—
82	C₈×C₆₁	—	2	4	no	—
83	#92418	—	4	8	no	—
84	#20768	—	61	1	no	maximal, Sylow
85	C₂²×C₆₁	—	2	2	no	—
86	C₂²×C₆₁	—	2	4	no	—
87	C₂²×C₆₁	—	4	4	no	—
88	C₂²×C₆₁	#382	1	4	yes	—
89	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₆₁	C₂×#125	1	2	yes	—
91	C₄×C₆₁	—	2	8	no	—
92	C₄×C₆₁	#382	1	4	yes	—
93	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
94	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
95	C₄×C₆₁	C₂×#124	1	2	yes	—
96	#2905	—	61	2	no	—
97	C₂×#405	D₆₁	1	1	yes	characteristic
98	#1838	—	61	4	no	—
99	#1837	—	61	4	no	—
100	#1738	—	61	2	no	—
101	C₂×#341	—	61	2	no	—
102	C₂×C₆₁	—	2	2	no	—
103	C₂×C₆₁	—	2	4	no	—
104	C₂×C₆₁	C₂×#382	1	1	yes	characteristic
105	C₂×C₆₁	#2916	1	2	yes	—
106	C₂²×M₁₆	—	61	4	no	—
107	C₂²×M₁₆	C₂×D₆₁	1	1	yes	characteristic
108	C₂³×C₈	C₂×D₆₁	1	1	yes	characteristic
109	C₂×#117	C₂×D₆₁	1	1	yes	characteristic
110	#424	—	122	8	no	—
111	#423	—	61	8	no	—
112	#405	Dic₆₁	1	8	yes	—
113	C₂×#97	C₂×D₆₁	1	2	yes	—
114	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
115	#341	—	61	4	no	—
116	#341	—	61	8	no	—
117	#341	—	61	4	no	—
118	C₆₁	#20768	1	1	yes	characteristic, complemented, Sylow
119	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
120	C₂×M₁₆	—	2	4	no	—
121	C₂×M₁₆	—	61	8	no	—
122	C₂×M₁₆	—	122	4	no	—
123	C₂×M₁₆	—	122	8	no	—
124	C₂×M₁₆	—	122	8	no	—
125	C₂×M₁₆	C₂×Dic₆₁	1	4	yes	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	—	122	4	no	—
128	C₂²×C₈	C₂×Dic₆₁	1	4	yes	—
129	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
131	C₂²×C₈	C₂²×D₆₁	1	2	yes	—
132	#117	D₂₄₄	1	4	yes	—
133	#117	C₂×Dic₆₁	1	4	yes	—
134	C₂×C₄⋊C₄	Dic₁₂₂	1	2	yes	—
135	C₂×C₂²⋊C₄	Dic₁₂₂	1	2	yes	—
136	C₂×C₄²	C₂²×D₆₁	1	2	yes	—
137	C₄⋊C₈	—	122	16	no	—
138	#97	C₂×Dic₆₁	1	8	yes	—
139	C₄×C₈	C₂×Dic₆₁	1	8	yes	—
140	C₂⁴	C₂×Dic₁₂₂	1	1	yes	characteristic
141	C₂²×C₄	C₂×Dic₁₂₂	1	4	yes	—
142	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
143	C₂²×C₄	C₂×D₂₄₄	1	2	yes	—
144	C₂²×C₄	#11758222	1	4	yes	—
145	C₂²×C₄	#11758222	1	4	yes	—
146	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
147	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
148	M₁₆	—	2	8	no	—
149	M₁₆	—	122	8	no	—
150	M₁₆	—	122	8	no	—
151	M₁₆	—	244	8	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	2	8	no	—
156	C₂×C₈	—	122	8	no	—
157	C₂×C₈	—	244	8	no	—
158	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
159	C₂×C₈	C₂²×Dic₆₁	1	4	yes	—
160	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
161	C₄⋊C₄	C₄⋊Dic₆₁	1	8	yes	—
162	C₂²⋊C₄	C₄⋊Dic₆₁	1	8	yes	—
163	C₄²	C₂×D₂₄₄	1	4	yes	—
164	C₄²	C₂²×Dic₆₁	1	4	yes	—
165	C₂³	—	4	2	no	—
166	C₂³	C₂×C₄⋊Dic₆₁	1	2	yes	—
167	C₂³	#14519769	1	4	yes	—
168	C₂³	C₂²×Dic₁₂₂	1	1	yes	characteristic
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	C₂×C₄⋊Dic₆₁	1	8	yes	—
171	C₂×C₄	C₂×C₄⋊Dic₆₁	1	2	yes	—
172	C₂×C₄	#14519736	1	4	yes	—
173	C₂×C₄	#14519736	1	4	yes	—
174	C₂×C₄	#14519765	1	8	yes	—
175	C₂×C₄	#14519769	1	4	yes	—
176	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
177	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
178	C₂×C₄	C₂×#11758222	1	2	yes	—
179	C₂×C₄	C₂×#11758222	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	2	2	no	—
182	C₈	—	2	4	no	—
183	C₈	—	244	8	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	4	4	no	—
187	C₂²	#16839628	1	4	yes	—
188	C₂²	C₂²×C₄⋊Dic₆₁	1	1	yes	characteristic
189	C₂²	C₂×#14519769	1	2	yes	—
190	C₄	—	2	8	no	—
191	C₄	#16839628	1	4	yes	—
192	C₄	C₂×#14519736	1	1	yes	characteristic
193	C₄	C₂×#14519736	1	1	yes	characteristic
194	C₄	C₂×#14519765	1	2	yes	—
195	C₂	—	2	2	no	—
196	C₂	—	2	4	no	—
197	C₂	C₂×#16839628	1	1	yes	characteristic
198	C₂	#21454430	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₂×C₆₁×#405	C₂
socle	C₂²×C₆₁	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₆₁	C₂×Q₈
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	C₂×#405	D₆₁
hypocenter	C₆₁	#20768
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #21445960	C₂	semidirect
3. #21445957	C₂	semidirect
118. C₆₁	#20768	semidirect

Maximal subgroups

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

subgroup	index
#21454383	2
#21445960	2
#21445957	2
#21445546	2
C₂×#16839527	2
C₂×C₆₁×#405	2
#20768	61