Subgroups — #29566902

Order 15616.

200 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839583	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519717	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#364	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519717	—	2	1	no	—
5	C₂×#14519717	C₄	1	1	yes	characteristic
6	C₂×#14519717	C₄	1	1	yes	characteristic
7	#16839583	—	2	1	no	—
8	#16839583	C₄	1	2	yes	—
9	#16839538	—	2	1	no	—
10	C₄×#11758185	—	2	1	no	—
11	C₂×M₃₂×C₆₁	—	2	1	no	—
12	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×#364	—	2	1	no	—
14	C₆₁×#364	C₄	1	2	yes	—
15	#14519717	—	2	1	no	—
16	#14519717	—	2	1	no	—
17	#14519717	—	2	2	no	—
18	#14519717	—	4	1	no	—
19	#14519717	C₈	1	2	yes	—
20	#14519717	C₈	1	2	yes	—
21	C₂×#11758185	—	2	1	no	—
22	C₂×#11758185	—	2	1	no	—
23	C₂×#11758185	—	4	1	no	—
24	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₆₁	Q₈	1	1	yes	characteristic
27	M₃₂×C₆₁	—	2	2	no	—
28	M₃₂×C₆₁	—	4	1	no	—
29	C₂×C₁₆×C₆₁	—	4	1	no	—
30	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
33	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	—	2	1	no	—
35	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₆₁	M₁₆	1	1	yes	characteristic
43	C₂×C₈×C₆₁	M₁₆	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	—	2	1	no	—
49	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
51	C₁₆×C₆₁	—	4	2	no	—
52	#11758185	—	2	2	no	—
53	#11758185	—	2	2	no	—
54	#11758185	—	4	2	no	—
55	C₂³×C₆₁	#108	1	1	yes	characteristic
56	C₂×C₄×C₆₁	—	2	1	no	—
57	C₂×C₄×C₆₁	—	2	1	no	—
58	C₂×C₄×C₆₁	—	4	2	no	—
59	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
63	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
64	C₂×C₄×C₆₁	#108	1	1	yes	characteristic
65	C₈×C₆₁	C₄×C₈	1	1	yes	characteristic
66	C₈×C₆₁	#97	1	1	yes	characteristic
67	C₈×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
68	C₈×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
69	C₈×C₆₁	#102	1	1	yes	characteristic
70	C₈×C₆₁	#102	1	1	yes	characteristic
71	C₈×C₆₁	#103	1	1	yes	characteristic
72	C₈×C₆₁	#103	1	1	yes	characteristic
73	#7162	—	61	1	no	maximal, Sylow
74	C₂²×C₆₁	—	4	1	no	—
75	C₂²×C₆₁	#334	1	1	yes	characteristic
76	C₂²×C₆₁	#335	1	1	yes	characteristic
77	C₂²×C₆₁	#341	1	1	yes	characteristic
78	C₄×C₆₁	—	2	2	no	—
79	C₄×C₆₁	—	4	1	no	—
80	C₄×C₆₁	#334	1	1	yes	characteristic
81	C₄×C₆₁	#335	1	1	yes	characteristic
82	C₄×C₆₁	#336	1	1	yes	characteristic
83	C₄×C₆₁	#340	1	1	yes	characteristic
84	C₂×#364	—	61	1	no	—
85	C₂×#364	D₆₁	1	1	yes	characteristic
86	C₄×M₃₂	—	61	1	no	—
87	C₂×C₆₁	—	2	1	no	—
88	C₂×C₆₁	—	2	1	no	—
89	C₂×C₆₁	—	2	1	no	—
90	C₂×C₆₁	#1277	1	1	yes	characteristic
91	C₂×M₃₂	—	2	1	no	—
92	C₂×M₃₂	—	61	1	no	—
93	C₂×M₃₂	—	61	1	no	—
94	C₂×M₃₂	—	122	1	no	—
95	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
96	#364	—	2	1	no	—
97	#364	—	61	2	no	—
98	#364	—	122	1	no	—
99	#364	Dic₆₁	1	2	yes	—
100	#346	—	122	1	no	—
101	C₄×C₁₆	—	122	1	no	—
102	C₆₁	#7162	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
104	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
105	C₂×C₄²	Dic₁₂₂	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	—	4	1	no	—
108	M₃₂	—	61	2	no	—
109	M₃₂	—	61	2	no	—
110	M₃₂	—	122	1	no	—
111	M₃₂	—	122	1	no	—
112	M₃₂	—	122	2	no	—
113	M₃₂	—	244	1	no	—
114	C₂×C₁₆	—	4	1	no	—
115	C₂×C₁₆	—	122	1	no	—
116	C₂×C₁₆	—	122	1	no	—
117	C₂×C₁₆	—	244	1	no	—
118	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
119	C₄×C₈	D₂₄₄	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
121	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	#11758196	1	1	yes	characteristic
124	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
125	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
126	C₂×C₈	#11758189	1	1	yes	characteristic
127	C₂×C₈	#11758189	1	1	yes	characteristic
128	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
129	C₂×C₈	#11758198	1	1	yes	characteristic
130	C₂×C₈	#11758198	1	1	yes	characteristic
131	C₂×C₈	#11758199	1	1	yes	characteristic
132	C₂×C₈	#11758200	1	1	yes	characteristic
133	C₂×C₈	#11758201	1	1	yes	characteristic
134	C₂×C₈	#11758202	1	1	yes	characteristic
135	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
136	C₄²	—	2	1	no	—
137	C₄²	#11758196	1	1	yes	characteristic
138	C₄²	C₄⋊Dic₆₁	1	1	yes	characteristic
139	C₁₆	—	4	2	no	—
140	C₁₆	—	122	2	no	—
141	C₁₆	—	122	2	no	—
142	C₁₆	—	244	2	no	—
143	C₂³	#14519712	1	1	yes	characteristic
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	#14519684	1	1	yes	characteristic
148	C₂×C₄	#14519706	1	1	yes	characteristic
149	C₂×C₄	#14519706	1	1	yes	characteristic
150	C₂×C₄	#14519708	1	1	yes	characteristic
151	C₂×C₄	#14519710	1	1	yes	characteristic
152	C₂×C₄	#14519712	1	1	yes	characteristic
153	C₈	C₈×Dic₆₁	1	1	yes	characteristic
154	C₈	#14519687	1	1	yes	characteristic
155	C₈	#14519689	1	1	yes	characteristic
156	C₈	#14519689	1	1	yes	characteristic
157	C₈	#14519697	1	1	yes	characteristic
158	C₈	#14519698	1	1	yes	characteristic
159	C₈	#14519700	1	1	yes	characteristic
160	C₈	#14519701	1	1	yes	characteristic
161	C₂²	—	4	1	no	—
162	C₂²	#16839512	1	1	yes	characteristic
163	C₂²	#16839514	1	1	yes	characteristic
164	C₂²	#16839526	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	4	1	no	—
167	C₄	#16839512	1	1	yes	characteristic
168	C₄	#16839514	1	1	yes	characteristic
169	C₄	#16839516	1	1	yes	characteristic
170	C₄	#16839524	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	#21443653	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14519697
commutator	C₈×C₆₁	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₆₁
Fitting	C₂×C₆₁×#364	C₂
socle	C₂×C₆₁	#1277
prime radical	C₂³×C₆₁	#108
cosocle	C₂×C₄×C₈×C₆₁	C₂²
hypercenter	C₂×#364	D₆₁
hypocenter	C₆₁	#7162
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
102. C₆₁	#7162	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839583	2
C₄×#14519717	2
C₂×C₆₁×#364	2
#7162	61