Subgroups — #29579387

Order 15616.

394 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	#21456333	C₂	1	1	yes	maximal, characteristic
2	#21445983	C₂	1	1	yes	maximal, characteristic
3	#21444384	C₂	1	4	yes	maximal
4	C₆₁×#1450	C₂	1	1	yes	maximal, characteristic
5	#16840319	C₂²	1	2	yes	—
6	#16840167	C₄	1	2	yes	complemented
7	#16840164	—	2	1	no	—
8	#16840164	—	2	1	no	—
9	#16840160	C₄	1	2	yes	complemented
10	#16839800	—	2	1	no	—
11	#16839800	—	2	1	no	—
12	C₂×#14519711	—	2	1	no	—
13	C₂×#14519709	—	2	1	no	—
14	#16839628	C₂²	1	2	yes	—
15	#16839500	—	2	8	no	—
16	C₆₁×#431	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#322	—	2	1	no	—
19	C₈²×C₆₁	—	2	1	no	—
20	#14519777	C₂×C₄	1	4	yes	—
21	#14519769	—	2	2	no	—
22	#14519756	—	2	4	no	—
23	C₄×Dic₁₂₂	—	2	4	no	—
24	#14519736	—	2	2	no	—
25	C₂×C₄⋊Dic₆₁	—	2	2	no	—
26	#14519711	—	2	2	no	—
27	#14519711	—	2	2	no	—
28	#14519711	—	4	2	no	—
29	#14519709	—	2	2	no	—
30	#14519709	—	2	2	no	—
31	#14519709	—	4	2	no	—
32	#14519684	—	2	4	no	—
33	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₆₁	—	2	2	no	—
37	C₂²⋊C₈×C₆₁	D₄	1	4	yes	—
38	C₄×C₈×C₆₁	—	2	2	no	—
39	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
42	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₆₁	—	4	2	no	—
46	C₂×Dic₁₂₂	—	2	4	no	—
47	C₂×C₈×C₆₁	—	2	4	no	—
48	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
53	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
54	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
55	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
56	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₆₁	#41	1	2	yes	—
59	C₂²⋊Dic₆₁	—	4	2	no	—
60	C₄⋊Dic₆₁	—	2	4	no	—
61	C₄⋊Dic₆₁	—	2	4	no	—
62	C₄⋊Dic₆₁	—	4	2	no	—
63	C₄⋊Dic₆₁	—	4	2	no	—
64	#11758196	—	4	4	no	—
65	C₄×Dic₆₁	—	4	2	no	—
66	C₂³×C₆₁	#127	1	1	yes	characteristic
67	C₂×C₄×C₆₁	—	4	2	no	—
68	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
70	C₂×C₄×C₆₁	#121	1	2	yes	—
71	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×Dic₆₁	—	4	4	no	—
77	C₂×Dic₆₁	—	8	4	no	—
78	C₂×Dic₆₁	—	8	2	no	—
79	Dic₁₂₂	—	8	4	no	—
80	C₈×C₆₁	—	4	4	no	—
81	C₈×C₆₁	#103	1	2	yes	—
82	C₈×C₆₁	#103	1	2	yes	—
83	C₈×C₆₁	#103	1	2	yes	—
84	C₈×C₆₁	#103	1	2	yes	—
85	#9755	—	61	1	no	maximal, Sylow
86	C₂²×C₆₁	—	2	2	no	—
87	C₂²×C₆₁	#390	1	1	yes	characteristic
88	C₂²×C₆₁	#492	1	1	yes	characteristic
89	C₂²×C₆₁	#494	1	1	yes	characteristic
90	C₄×C₆₁	—	2	2	no	—
91	C₄×C₆₁	—	2	2	no	—
92	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
93	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
94	C₄×C₆₁	#487	1	1	yes	characteristic
95	C₄×C₆₁	#487	1	1	yes	characteristic
96	Dic₆₁	—	8	4	no	—
97	Dic₆₁	—	16	2	no	—
98	#3160	—	61	1	no	—
99	#1846	—	61	1	no	—
100	#1518	—	61	4	no	—
101	#1450	D₆₁	1	1	yes	characteristic
102	C₂×C₆₁	—	2	1	no	—
103	C₂×C₆₁	—	2	1	no	—
104	C₂×C₆₁	#1956	1	1	yes	characteristic
105	C₂×C₆₁	#1976	1	2	yes	—
106	#533	—	61	2	no	—
107	#500	—	61	2	no	—
108	#499	—	122	1	no	—
109	#499	—	122	1	no	—
110	#498	—	61	2	no	—
111	#431	C₂×D₆₁	1	2	yes	—
112	#427	—	122	1	no	—
113	#427	—	122	1	no	—
114	C₂×#107	—	122	1	no	—
115	C₂×#106	—	122	1	no	—
116	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
117	#382	—	61	2	no	—
118	#328	—	122	8	no	—
119	#322	—	2	1	no	—
120	C₈²	—	2	1	no	—
121	C₆₁	#9755	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
123	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
124	C₄⋊Q₈	—	61	4	no	—
125	#125	—	122	2	no	—
126	C₂²⋊Q₈	—	122	4	no	—
127	C₄×Q₈	—	122	4	no	—
128	#117	—	122	2	no	—
129	C₂×C₄⋊C₄	—	122	2	no	—
130	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
131	#107	—	122	2	no	—
132	#107	—	122	2	no	—
133	#107	—	244	2	no	—
134	#106	—	122	2	no	—
135	#106	—	122	2	no	—
136	#106	—	244	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₂²⋊C₈	D₂₄₄	1	4	yes	—
139	C₄×C₈	—	2	2	no	—
140	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
141	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
143	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
144	#95	—	122	4	no	—
145	C₂×Q₈	—	122	4	no	—
146	C₂²×C₄	—	244	2	no	—
147	C₂²×C₄	C₂×D₂₄₄	1	2	yes	—
148	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	#11758190	1	2	yes	—
151	C₂×C₈	#11758190	1	2	yes	—
152	C₂×C₈	Dic₂₄₄	1	2	yes	—
153	C₂×C₈	Dic₂₄₄	1	2	yes	—
154	C₂×C₈	#11758198	1	1	yes	characteristic
155	C₂×C₈	#11758198	1	1	yes	characteristic
156	C₂×C₈	#11758198	1	1	yes	characteristic
157	C₂×C₈	#11758198	1	1	yes	characteristic
158	C₄⋊C₄	—	122	4	no	—
159	C₄⋊C₄	—	122	4	no	—
160	C₄⋊C₄	—	244	2	no	—
161	C₄⋊C₄	—	244	2	no	—
162	C₄⋊C₄	—	244	4	no	—
163	C₂²⋊C₄	—	244	2	no	—
164	C₄²	—	244	2	no	—
165	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
166	C₄²	#11758222	1	2	yes	—
167	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
168	C₂³	#14519774	1	1	yes	characteristic
169	Q₈	—	488	4	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	488	4	no	—
172	C₂×C₄	—	488	2	no	—
173	C₂×C₄	—	244	4	no	—
174	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
175	C₂×C₄	C₄×D₂₄₄	1	2	yes	—
176	C₂×C₄	#14519751	1	2	yes	—
177	C₂×C₄	#14519765	1	1	yes	characteristic
178	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
179	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
180	C₂×C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
181	C₂×C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
182	C₈	—	4	4	no	—
183	C₈	#14519702	1	2	yes	—
184	C₈	#14519702	1	2	yes	—
185	C₈	#14519702	1	2	yes	—
186	C₈	#14519702	1	2	yes	—
187	C₂²	—	2	2	no	—
188	C₂²	#16839664	1	1	yes	characteristic
189	C₂²	#16840135	1	1	yes	characteristic
190	C₂²	#16840142	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	976	2	no	—
194	C₄	—	488	4	no	—
195	C₄	C₂×#14519702	1	1	yes	characteristic
196	C₄	C₂×#14519702	1	1	yes	characteristic
197	C₄	#16840113	1	1	yes	characteristic
198	C₄	#16840113	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#21446621	1	1	yes	characteristic
202	C₂	#21446717	1	2	yes	—

	subgroup	G/H
center	C₂²	#16839664
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1450	C₂
socle	C₂²×C₆₁	#390
prime radical	C₂³×C₆₁	#127
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1450	D₆₁
hypocenter	C₆₁	#9755
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

Complements

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

N	H ≅ G/N	type
6. #16840167	C₄	semidirect
9. #16840160	C₄	semidirect
121. C₆₁	#9755	semidirect

Maximal subgroups

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

subgroup	index
#21456333	2
#21445983	2
#21444384	2
C₆₁×#1450	2
#9755	61