Subgroups — #29567215

Order 15616.

230 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21447575	C₂	1	1	yes	maximal, characteristic
2	#21445983	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1564	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519717	—	2	1	no	—
5	C₂×#14519711	—	2	1	no	—
6	C₂×#14519709	C₄	1	2	yes	—
7	#16839628	—	2	1	no	—
8	C₄⋊#11758185	—	2	1	no	—
9	C₄⋊#11758185	C₄	1	2	yes	complemented
10	C₆₁×#431	—	2	1	no	—
11	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#335	—	2	1	no	—
13	C₆₁×#334	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₆₁	—	2	2	no	—
15	#14519717	—	4	1	no	—
16	#14519717	—	4	1	no	—
17	C₂×#11758185	—	2	2	no	—
18	#14519711	—	4	1	no	—
19	#14519711	—	4	1	no	—
20	#14519709	—	4	2	no	—
21	#14519684	—	4	1	no	—
22	#14519684	—	4	1	no	—
23	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
26	C₄⋊C₈×C₆₁	—	2	2	no	—
27	C₂²⋊C₈×C₆₁	—	4	1	no	—
28	C₄×C₈×C₆₁	—	4	1	no	—
29	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
32	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
36	C₂²×Dic₆₁	—	4	2	no	—
37	C₂×C₈×C₆₁	—	2	1	no	—
38	C₂×C₈×C₆₁	—	2	1	no	—
39	C₂×C₈×C₆₁	—	4	2	no	—
40	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
45	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₆₁	Q₁₆	1	1	yes	characteristic
48	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
51	C₄²×C₆₁	Q₁₆	1	1	yes	characteristic
52	C₄⋊Dic₆₁	—	4	2	no	—
53	C₄⋊Dic₆₁	—	4	2	no	—
54	#11758185	—	4	2	no	—
55	C₂³×C₆₁	#102	1	1	yes	characteristic
56	C₂×C₄×C₆₁	—	4	1	no	—
57	C₂×C₄×C₆₁	—	4	1	no	—
58	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
59	C₂×C₄×C₆₁	#100	1	1	yes	characteristic
60	C₂×C₄×C₆₁	#101	1	1	yes	characteristic
61	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
62	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
63	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
64	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
65	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
66	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
67	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
68	C₂×Dic₆₁	—	8	2	no	—
69	C₂×Dic₆₁	—	8	2	no	—
70	C₂×Dic₆₁	—	8	2	no	—
71	C₈×C₆₁	—	2	1	no	—
72	C₈×C₆₁	—	2	1	no	—
73	C₈×C₆₁	—	8	2	no	—
74	C₈×C₆₁	#102	1	1	yes	characteristic
75	C₈×C₆₁	#103	1	1	yes	characteristic
76	C₈×C₆₁	#106	1	1	yes	characteristic
77	C₈×C₆₁	#107	1	1	yes	characteristic
78	#7279	—	61	1	no	maximal, Sylow
79	C₂²×C₆₁	—	2	1	no	—
80	C₂²×C₆₁	—	2	1	no	—
81	C₂²×C₆₁	#331	1	1	yes	characteristic
82	C₂²×C₆₁	#332	1	1	yes	characteristic
83	C₂²×C₆₁	#340	1	1	yes	characteristic
84	C₄×C₆₁	—	2	1	no	—
85	C₄×C₆₁	—	2	1	no	—
86	C₄×C₆₁	—	2	1	no	—
87	C₄×C₆₁	—	2	1	no	—
88	C₄×C₆₁	#332	1	1	yes	characteristic
89	C₄×C₆₁	#333	1	1	yes	characteristic
90	C₄×C₆₁	#340	1	1	yes	characteristic
91	C₄×C₆₁	#343	1	1	yes	characteristic
92	Dic₆₁	—	16	2	no	—
93	#2150	—	61	1	no	—
94	#1846	—	61	1	no	—
95	#1564	D₆₁	1	1	yes	characteristic
96	C₂×C₆₁	—	2	1	no	—
97	C₂×C₆₁	—	2	1	no	—
98	C₂×C₆₁	#1296	1	1	yes	characteristic
99	C₂×C₆₁	#1384	1	1	yes	characteristic
100	C₂×C₆₁	#1389	1	1	yes	characteristic
101	C₂×M₃₂	—	122	1	no	—
102	#431	—	2	1	no	—
103	C₂×#107	—	122	1	no	—
104	C₂×#106	—	61	2	no	—
105	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
106	#382	—	122	1	no	—
107	C₄⋊C₁₆	—	61	2	no	—
108	C₄⋊C₁₆	—	122	1	no	—
109	#335	—	2	1	no	—
110	#334	Dic₆₁	1	2	yes	complemented
111	C₆₁	#7279	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
113	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	122	2	no	—
115	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
116	M₃₂	—	244	1	no	—
117	M₃₂	—	244	1	no	—
118	C₂×C₁₆	—	122	2	no	—
119	#107	—	244	1	no	—
120	#107	—	244	1	no	—
121	#106	—	244	2	no	—
122	C₄⋊C₈	—	2	2	no	—
123	C₂²⋊C₈	—	4	1	no	—
124	C₄×C₈	—	4	1	no	—
125	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
126	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
127	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
128	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
129	#95	—	244	1	no	—
130	#95	—	244	1	no	—
131	C₂²×C₄	—	244	2	no	—
132	C₂²×C₄	#11758198	1	1	yes	characteristic
133	C₂²×C₄	#11758199	1	1	yes	characteristic
134	C₂²×C₄	#11758201	1	1	yes	characteristic
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	#11758190	1	1	yes	characteristic
139	C₂×C₈	#11758190	1	1	yes	characteristic
140	C₂×C₈	D₄₈₈	1	1	yes	characteristic
141	C₂×C₈	Dic₂₄₄	1	1	yes	characteristic
142	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
143	C₂×C₈	#11758196	1	1	yes	characteristic
144	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
145	C₂×C₈	#11758198	1	1	yes	characteristic
146	C₄⋊C₄	—	244	2	no	—
147	C₄⋊C₄	—	244	2	no	—
148	C₄²	#11758196	1	1	yes	characteristic
149	C₄²	#11758200	1	1	yes	characteristic
150	C₄²	#11758202	1	1	yes	characteristic
151	C₄²	C₂²⋊Dic₆₁	1	1	yes	characteristic
152	C₁₆	—	244	2	no	—
153	C₂³	#14519697	1	1	yes	characteristic
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	488	2	no	—
157	C₂×C₄	—	488	2	no	—
158	C₂×C₄	—	488	2	no	—
159	C₂×C₄	#14519684	1	1	yes	characteristic
160	C₂×C₄	#14519693	1	1	yes	characteristic
161	C₂×C₄	#14519695	1	1	yes	characteristic
162	C₂×C₄	#14519698	1	1	yes	characteristic
163	C₂×C₄	#14519699	1	1	yes	characteristic
164	C₂×C₄	#14519700	1	1	yes	characteristic
165	C₂×C₄	#14519701	1	1	yes	characteristic
166	C₂×C₄	#14519702	1	1	yes	characteristic
167	C₂×C₄	#14519708	1	1	yes	characteristic
168	C₂×C₄	#14519710	1	1	yes	characteristic
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	8	2	no	—
172	C₈	#14519699	1	1	yes	characteristic
173	C₈	#14519702	1	1	yes	characteristic
174	C₈	#14519709	1	1	yes	characteristic
175	C₈	#14519711	1	1	yes	characteristic
176	C₂²	—	2	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	#16839505	1	1	yes	characteristic
179	C₂²	#16839509	1	1	yes	characteristic
180	C₂²	#16839524	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	976	2	no	—
186	C₄	#16839507	1	1	yes	characteristic
187	C₄	#16839510	1	1	yes	characteristic
188	C₄	#16839525	1	1	yes	characteristic
189	C₄	#16839530	1	1	yes	characteristic
190	C₂	—	2	1	no	—
191	C₂	—	2	1	no	—
192	C₂	#21443703	1	1	yes	characteristic
193	C₂	#21443920	1	1	yes	characteristic
194	C₂	#21443933	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16839524
commutator	C₂×C₈×C₆₁	C₄²
Frattini	C₂×C₄×C₈	C₂×D₆₁
Fitting	C₆₁×#1564	C₂
socle	C₂²×C₆₁	#340
prime radical	C₂³×C₆₁	#102
cosocle	C₂×C₄×C₈×C₆₁	C₂²
hypercenter	#1564	D₆₁
hypocenter	C₆₁	#7279
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

Complements

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

N	H ≅ G/N	type
9. C₄⋊#11758185	C₄	semidirect
13. C₆₁×#334	C₄	semidirect
110. #334	Dic₆₁	semidirect
111. C₆₁	#7279	semidirect

Maximal subgroups

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

subgroup	index
#21447575	2
#21445983	2
C₆₁×#1564	2
#7279	61