Subgroups — #29567211

Order 15616.

246 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	#21445976	C₂	1	1	yes	maximal, characteristic
2	#21444567	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2150	C₂	1	1	yes	maximal, characteristic
4	#16839815	—	2	1	no	—
5	C₂×#14519709	—	2	1	no	—
6	C₂×#14519709	C₄	1	2	yes	—
7	#16839632	—	2	1	no	—
8	#16839515	C₄	1	2	yes	complemented
9	#16839513	—	2	1	no	—
10	C₂×M₃₂×C₆₁	—	2	1	no	—
11	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×C₄⋊C₁₆	—	2	1	no	—
13	C₆₁×C₄⋊C₁₆	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₆₁	—	2	2	no	—
15	C₂×C₄⋊Dic₆₁	—	4	1	no	—
16	C₂×C₄⋊Dic₆₁	—	4	1	no	—
17	#14519709	—	4	1	no	—
18	#14519709	—	4	1	no	—
19	#14519709	—	4	2	no	—
20	C₄⋊#92418	—	2	2	no	—
21	C₂²⋊#92418	—	4	1	no	—
22	C₄×#92418	—	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	M₃₂×C₆₁	—	4	1	no	—
27	M₃₂×C₆₁	—	4	1	no	—
28	C₂×C₁₆×C₆₁	—	2	2	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	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
44	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₆₁	Q₁₆	1	1	yes	characteristic
47	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
50	C₄²×C₆₁	Q₁₆	1	1	yes	characteristic
51	C₄⋊Dic₆₁	—	4	2	no	—
52	C₄⋊Dic₆₁	—	4	2	no	—
53	C₄⋊Dic₆₁	—	8	2	no	—
54	C₄⋊Dic₆₁	—	8	2	no	—
55	C₂×#92418	—	4	2	no	—
56	C₁₆×C₆₁	—	4	2	no	—
57	C₂³×C₆₁	#102	1	1	yes	characteristic
58	C₂×C₄×C₆₁	—	2	2	no	—
59	C₂×C₄×C₆₁	—	2	2	no	—
60	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
61	C₂×C₄×C₆₁	#100	1	1	yes	characteristic
62	C₂×C₄×C₆₁	#101	1	1	yes	characteristic
63	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
64	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
65	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
66	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
67	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
68	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
69	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
70	C₂×Dic₆₁	—	8	2	no	—
71	C₂×Dic₆₁	—	8	2	no	—
72	C₂×Dic₆₁	—	8	2	no	—
73	C₈×C₆₁	—	2	1	no	—
74	C₈×C₆₁	—	2	1	no	—
75	C₈×C₆₁	#102	1	1	yes	characteristic
76	C₈×C₆₁	#103	1	1	yes	characteristic
77	C₈×C₆₁	#106	1	1	yes	characteristic
78	C₈×C₆₁	#107	1	1	yes	characteristic
79	#92418	—	8	2	no	—
80	#7278	—	61	1	no	maximal, Sylow
81	C₂²×C₆₁	—	2	1	no	—
82	C₂²×C₆₁	—	2	1	no	—
83	C₂²×C₆₁	#331	1	1	yes	characteristic
84	C₂²×C₆₁	#332	1	1	yes	characteristic
85	C₂²×C₆₁	#340	1	1	yes	characteristic
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	C₄×C₆₁	#360	1	2	yes	—
93	C₄×C₆₁	#365	1	2	yes	—
94	Dic₆₁	—	16	2	no	—
95	#2150	D₆₁	1	1	yes	characteristic
96	#1844	—	61	1	no	—
97	#1564	—	61	1	no	—
98	C₂×C₆₁	—	2	1	no	—
99	C₂×C₆₁	—	2	1	no	—
100	C₂×C₆₁	#1296	1	1	yes	characteristic
101	C₂×C₆₁	#1383	1	1	yes	characteristic
102	C₂×C₆₁	#1386	1	1	yes	characteristic
103	C₂×M₃₂	—	2	1	no	—
104	#431	—	122	1	no	—
105	C₂×#106	—	61	2	no	—
106	C₂×#106	—	122	1	no	—
107	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
108	#384	—	122	1	no	—
109	C₄⋊C₁₆	—	2	1	no	—
110	C₄⋊C₁₆	Dic₆₁	1	2	yes	complemented
111	#335	—	61	2	no	—
112	#334	—	122	1	no	—
113	C₆₁	#7278	1	1	yes	characteristic, complemented, Sylow
114	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
115	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	122	2	no	—
117	C₂×C₄⋊C₄	—	244	1	no	—
118	C₂×C₄⋊C₄	—	244	1	no	—
119	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
120	M₃₂	—	4	1	no	—
121	M₃₂	—	4	1	no	—
122	C₂×C₁₆	—	2	2	no	—
123	#106	—	244	1	no	—
124	#106	—	244	1	no	—
125	#106	—	244	2	no	—
126	C₄⋊C₈	—	122	2	no	—
127	C₂²⋊C₈	—	244	1	no	—
128	C₄×C₈	—	244	1	no	—
129	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
130	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
131	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
132	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
133	C₂²×C₄	—	244	2	no	—
134	C₂²×C₄	#11758190	1	1	yes	characteristic
135	C₂²×C₄	D₄₈₈	1	1	yes	characteristic
136	C₂²×C₄	#11758198	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	244	2	no	—
140	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
141	C₂×C₈	#11758196	1	1	yes	characteristic
142	C₂×C₈	#11758196	1	1	yes	characteristic
143	C₂×C₈	#11758199	1	1	yes	characteristic
144	C₂×C₈	#11758200	1	1	yes	characteristic
145	C₂×C₈	#11758201	1	1	yes	characteristic
146	C₂×C₈	#11758202	1	1	yes	characteristic
147	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
148	C₄⋊C₄	—	488	2	no	—
149	C₄⋊C₄	—	488	2	no	—
150	C₄⋊C₄	—	244	2	no	—
151	C₄⋊C₄	—	244	2	no	—
152	C₄²	#11758190	1	1	yes	characteristic
153	C₄²	Dic₂₄₄	1	1	yes	characteristic
154	C₄²	C₄⋊Dic₆₁	1	1	yes	characteristic
155	C₄²	#11758198	1	1	yes	characteristic
156	C₁₆	—	4	2	no	—
157	C₂³	#14519699	1	1	yes	characteristic
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	488	2	no	—
161	C₂×C₄	—	488	2	no	—
162	C₂×C₄	—	488	2	no	—
163	C₂×C₄	#14519684	1	1	yes	characteristic
164	C₂×C₄	#14519693	1	1	yes	characteristic
165	C₂×C₄	#14519695	1	1	yes	characteristic
166	C₂×C₄	#14519697	1	1	yes	characteristic
167	C₂×C₄	#14519699	1	1	yes	characteristic
168	C₂×C₄	#14519700	1	1	yes	characteristic
169	C₂×C₄	#14519702	1	1	yes	characteristic
170	C₂×C₄	#14519702	1	1	yes	characteristic
171	C₂×C₄	#14519709	1	1	yes	characteristic
172	C₂×C₄	#14519711	1	1	yes	characteristic
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	488	2	no	—
176	C₈	#14519698	1	1	yes	characteristic
177	C₈	#14519701	1	1	yes	characteristic
178	C₈	#14519708	1	1	yes	characteristic
179	C₈	#14519710	1	1	yes	characteristic
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	#16839505	1	1	yes	characteristic
183	C₂²	#16839507	1	1	yes	characteristic
184	C₂²	#16839525	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	976	2	no	—
188	C₄	#16839509	1	1	yes	characteristic
189	C₄	#16839510	1	1	yes	characteristic
190	C₄	#16839524	1	1	yes	characteristic
191	C₄	#16839530	1	1	yes	characteristic
192	C₄	#16839573	1	2	yes	—
193	C₄	#16839585	1	2	yes	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	#21443703	1	1	yes	characteristic
197	C₂	#21443918	1	1	yes	characteristic
198	C₂	#21443925	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16839525
commutator	C₂×C₈×C₆₁	C₄²
Frattini	C₂×C₄×C₈	C₂×D₆₁
Fitting	C₆₁×#2150	C₂
socle	C₂²×C₆₁	#340
prime radical	C₂³×C₆₁	#102
cosocle	C₂×C₄×C₈×C₆₁	C₂²
hypercenter	#2150	D₆₁
hypocenter	C₆₁	#7278
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
8. #16839515	C₄	semidirect
13. C₆₁×C₄⋊C₁₆	C₄	semidirect
110. C₄⋊C₁₆	Dic₆₁	semidirect
113. C₆₁	#7278	semidirect

Maximal subgroups

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

subgroup	index
#21445976	2
#21444567	2
C₆₁×#2150	2
#7278	61