Subgroups — #29571215

Order 15616.

414 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445672	C₂	1	2	yes	maximal
2	C₂×#16839515	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839513	C₂	1	1	yes	maximal, characteristic
4	#21443654	C₂	1	2	yes	maximal
5	C₄×#97×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92418	C₂²	1	2	yes	—
7	C₂×C₄⋊#92418	C₂²	1	2	yes	—
8	C₂×#14519688	—	2	2	no	—
9	#16839517	—	2	2	no	—
10	#16839515	C₄	1	4	yes	complemented
11	#16839513	C₄	1	4	yes	complemented
12	C₂×#97×C₆₁	C₄	1	2	yes	—
13	C₂×#97×C₆₁	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₄³×C₆₁	C₂²	1	1	yes	characteristic
17	C₂²×#92418	—	2	2	no	—
18	C₂²×#92418	—	2	2	no	—
19	C₄⋊#92418	—	2	2	no	—
20	C₄⋊#92418	—	2	2	no	—
21	C₄⋊#92418	C₂×C₄	1	4	yes	—
22	C₄⋊#92418	C₂×C₄	1	4	yes	—
23	#14519688	—	4	4	no	—
24	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₆₁	D₄	1	2	yes	—
29	C₂×C₄²×C₆₁	Q₈	1	2	yes	—
30	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
31	#97×C₆₁	—	2	4	no	—
32	#97×C₆₁	—	2	4	no	—
33	C₄×C₈×C₆₁	—	2	1	no	—
34	C₄×C₈×C₆₁	—	2	1	no	—
35	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
38	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₆₁	—	2	2	no	—
42	C₂×C₈×C₆₁	—	2	2	no	—
43	C₂×C₈×C₆₁	—	2	2	no	—
44	C₂×C₈×C₆₁	C₄²	1	2	yes	—
45	C₂×C₈×C₆₁	C₄²	1	2	yes	—
46	C₂×C₈×C₆₁	M₁₆	1	4	yes	—
47	C₂×C₈×C₆₁	M₁₆	1	4	yes	—
48	C₄²×C₆₁	—	2	2	no	—
49	C₄²×C₆₁	—	2	2	no	—
50	C₄²×C₆₁	—	2	2	no	—
51	C₄²×C₆₁	—	2	2	no	—
52	C₄²×C₆₁	—	4	2	no	—
53	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₆₁	#41	1	1	yes	characteristic
56	C₄²×C₆₁	#41	1	1	yes	characteristic
57	C₂×#92418	—	2	4	no	—
58	C₂×#92418	—	2	4	no	—
59	C₂×#92418	—	4	2	no	—
60	C₂×#92418	—	4	2	no	—
61	C₂×#92418	—	4	2	no	—
62	C₂×#92418	—	4	2	no	—
63	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₁	—	2	2	no	—
65	C₂×C₄×C₆₁	—	2	4	no	—
66	C₂×C₄×C₆₁	—	2	4	no	—
67	C₂×C₄×C₆₁	—	2	4	no	—
68	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
75	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
76	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×C₄×C₆₁	#137	1	2	yes	—
78	C₈×C₆₁	—	2	2	no	—
79	C₈×C₆₁	—	2	4	no	—
80	C₈×C₆₁	#97	1	4	yes	complemented
81	#92418	—	4	4	no	—
82	#92418	—	4	4	no	—
83	#8276	—	61	1	no	maximal, Sylow
84	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₆₁	#423	1	2	yes	—
86	C₂²×C₆₁	#428	1	2	yes	—
87	C₂²×C₆₁	#430	1	2	yes	—
88	C₄×C₆₁	—	2	1	no	—
89	C₄×C₆₁	—	2	1	no	—
90	C₄×C₆₁	—	4	4	no	—
91	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
92	C₄×C₆₁	C₄×M₁₆	1	1	yes	characteristic
93	C₄×C₆₁	#432	1	2	yes	—
94	C₄×C₆₁	#440	1	2	yes	—
95	C₄×C₆₁	#441	1	1	yes	characteristic
96	C₄×C₆₁	#442	1	1	yes	characteristic
97	#1767	—	61	2	no	—
98	C₄×#97	D₆₁	1	1	yes	characteristic
99	C₂×#335	—	61	1	no	—
100	C₂×#334	—	61	1	no	—
101	#1277	—	61	2	no	—
102	C₂×C₆₁	#1569	1	4	yes	—
103	C₂×C₆₁	#1767	1	1	yes	characteristic
104	C₂×C₆₁	#1776	1	1	yes	characteristic
105	C₂×C₆₁	#1778	1	1	yes	characteristic
106	C₂×C₄⋊C₈	—	61	2	no	—
107	C₂×C₄⋊C₈	—	61	2	no	—
108	C₂×#97	—	122	2	no	—
109	C₂×#97	Dic₆₁	1	2	yes	—
110	C₂×#97	Dic₆₁	1	2	yes	—
111	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
112	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
113	C₄³	C₂×D₆₁	1	1	yes	characteristic
114	#336	—	122	2	no	—
115	#335	—	61	4	no	—
116	#334	—	61	4	no	—
117	C₆₁	#8276	1	1	yes	characteristic, complemented, Sylow
118	C₂²×C₈	—	122	2	no	—
119	C₂²×C₈	—	122	2	no	—
120	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
121	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
122	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
123	C₂×C₄²	Dic₁₂₂	1	2	yes	—
124	C₂×C₄²	D₂₄₄	1	2	yes	—
125	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
126	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
127	C₄⋊C₈	—	61	4	no	—
128	C₄⋊C₈	—	61	4	no	—
129	C₄⋊C₈	—	122	2	no	—
130	C₄⋊C₈	—	122	2	no	—
131	#97	—	2	4	no	—
132	#97	—	2	4	no	—
133	#97	—	244	4	no	—
134	C₄×C₈	—	2	1	no	—
135	C₄×C₈	—	2	1	no	—
136	C₄×C₈	C₄×D₆₁	1	2	yes	—
137	C₄×C₈	C₄×D₆₁	1	2	yes	—
138	C₂²×C₄	C₄⋊Dic₆₁	1	4	yes	—
139	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
141	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	122	4	no	—
146	C₂×C₈	—	122	4	no	—
147	C₂×C₈	—	244	2	no	—
148	C₂×C₈	—	244	2	no	—
149	C₂×C₈	—	244	2	no	—
150	C₂×C₈	—	244	2	no	—
151	C₂×C₈	#11758194	1	4	yes	—
152	C₂×C₈	#11758194	1	4	yes	—
153	C₂×C₈	C₄×Dic₆₁	1	2	yes	—
154	C₂×C₈	C₄×Dic₆₁	1	2	yes	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	2	2	no	—
159	C₄²	—	4	2	no	—
160	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
161	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
162	C₄²	#11758222	1	1	yes	characteristic
163	C₄²	#11758222	1	1	yes	characteristic
164	C₂³	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
170	C₂×C₄	#14519736	1	1	yes	characteristic
171	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
172	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
173	C₂×C₄	C₄×Dic₁₂₂	1	1	yes	characteristic
174	C₂×C₄	C₄×Dic₁₂₂	1	1	yes	characteristic
175	C₂×C₄	C₂×#11758194	1	2	yes	—
176	C₂×C₄	C₂×#11758194	1	2	yes	—
177	C₂×C₄	#14519805	1	2	yes	—
178	C₂×C₄	#14519810	1	2	yes	—
179	C₈	—	2	2	no	—
180	C₈	—	2	4	no	—
181	C₈	—	244	4	no	—
182	C₈	—	244	4	no	—
183	C₈	#14519688	1	4	yes	complemented
184	C₂²	C₄×C₄⋊Dic₆₁	1	1	yes	characteristic
185	C₂²	#16839786	1	2	yes	—
186	C₂²	#16839804	1	2	yes	—
187	C₂²	#16839811	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	4	4	no	—
191	C₄	C₂×#14519688	1	1	yes	characteristic
192	C₄	C₄×#11758194	1	1	yes	characteristic
193	C₄	#16839819	1	2	yes	—
194	C₄	#16839864	1	2	yes	—
195	C₄	#16839870	1	1	yes	characteristic
196	C₄	#16839875	1	1	yes	characteristic
197	C₂	#21444586	1	4	yes	—
198	C₂	#21445672	1	1	yes	characteristic
199	C₂	#21445716	1	1	yes	characteristic
200	C₂	#21445725	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₄×#97×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₄×#97	D₆₁
hypocenter	C₆₁	#8276
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
10. #16839515	C₄	semidirect
11. #16839513	C₄	semidirect
80. C₈×C₆₁	#97	semidirect
117. C₆₁	#8276	semidirect
183. C₈	#14519688	semidirect

Maximal subgroups

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

subgroup	index
#21445672	2
C₂×#16839515	2
C₂×#16839513	2
#21443654	2
C₄×#97×C₆₁	2
#8276	61