Subgroups — #29598577

Order 15616.

334 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	C₂×#16840164	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839583	C₂	1	1	yes	maximal, characteristic
3	#21443924	C₂	1	2	yes	maximal
4	#21443919	C₂	1	2	yes	maximal
5	C₆₁×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519769	—	2	1	no	—
7	C₂×#14519717	C₂²	1	2	yes	—
8	#16840164	—	2	2	no	—
9	#16840164	C₄	1	4	yes	complemented
10	C₂×#14519711	—	2	1	no	—
11	C₂×#14519709	C₂²	1	2	yes	—
12	#16839583	—	2	2	no	—
13	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#427	C₂²	1	1	yes	characteristic
15	C₆₁×#405	—	2	1	no	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#339	—	2	1	no	—
18	#14519769	—	2	4	no	—
19	#14519769	—	4	2	no	—
20	C₂×C₄⋊Dic₆₁	—	2	2	no	—
21	C₂×#11758196	—	4	2	no	—
22	#14519717	—	2	2	no	—
23	#14519717	—	2	2	no	—
24	C₂×#11758185	—	2	2	no	—
25	#14519711	—	2	4	no	—
26	#14519709	—	2	2	no	—
27	#14519709	C₂×C₄	1	4	yes	—
28	C₂×M₁₆×C₆₁	—	2	1	no	—
29	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
31	#117×C₆₁	—	2	1	no	—
32	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
33	#108×C₆₁	—	2	1	no	—
34	#108×C₆₁	Q₈	1	2	yes	—
35	#107×C₆₁	Q₈	1	2	yes	—
36	#106×C₆₁	—	2	1	no	—
37	#97×C₆₁	—	4	1	no	—
38	C₄×C₈×C₆₁	—	4	1	no	—
39	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₆₁	D₄	1	2	yes	—
41	C₂²×C₄×C₆₁	—	2	1	no	—
42	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₆₁	—	4	2	no	—
44	M₁₆×C₆₁	—	2	2	no	—
45	M₁₆×C₆₁	—	4	1	no	—
46	C₂×C₈×C₆₁	—	4	1	no	—
47	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
51	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
56	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
57	C₄⋊C₄×C₆₁	—	2	2	no	—
58	C₂²⋊C₄×C₆₁	—	4	1	no	—
59	C₄²×C₆₁	—	2	1	no	—
60	C₄²×C₆₁	—	4	1	no	—
61	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
62	C₄⋊Dic₆₁	—	2	4	no	—
63	C₄⋊Dic₆₁	—	4	2	no	—
64	#11758196	—	4	8	no	—
65	#11758185	—	4	2	no	—
66	C₂³×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂×C₄×C₆₁	—	2	2	no	—
68	C₂×C₄×C₆₁	—	4	2	no	—
69	C₂×C₄×C₆₁	—	4	2	no	—
70	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂²⋊Q₈	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	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₂×Dic₆₁	—	4	4	no	—
77	C₂×Dic₆₁	—	8	4	no	—
78	C₈×C₆₁	—	4	2	no	—
79	C₈×C₆₁	#103	1	2	yes	—
80	C₈×C₆₁	#103	1	2	yes	—
81	C₈×C₆₁	C₄×Q₈	1	1	yes	characteristic
82	C₈×C₆₁	C₄×Q₈	1	1	yes	characteristic
83	C₈×C₆₁	#124	1	1	yes	characteristic
84	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
85	#13273	—	61	1	no	maximal, Sylow
86	C₂²×C₆₁	—	2	2	no	—
87	C₂²×C₆₁	#418	1	1	yes	characteristic
88	C₂²×C₆₁	#476	1	1	yes	characteristic
89	C₂²×C₆₁	#477	1	1	yes	characteristic
90	C₄×C₆₁	—	4	2	no	—
91	C₄×C₆₁	—	8	2	no	—
92	C₄×C₆₁	#391	1	1	yes	characteristic
93	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
94	C₄×C₆₁	#476	1	1	yes	characteristic
95	C₄×C₆₁	#477	1	1	yes	characteristic
96	Dic₆₁	—	8	4	no	—
97	C₂×#499	—	61	1	no	—
98	C₂×#364	—	61	1	no	—
99	#1772	D₆₁	1	1	yes	characteristic
100	#1386	—	61	2	no	—
101	#1383	—	61	2	no	—
102	C₂×C₆₁	—	2	1	no	—
103	C₂×C₆₁	—	2	1	no	—
104	C₂×C₆₁	#1987	1	1	yes	characteristic
105	C₂×C₆₁	#2231	1	2	yes	—
106	C₂×#125	—	122	1	no	—
107	C₂×M₃₂	—	61	2	no	—
108	#499	—	61	4	no	—
109	#499	—	122	2	no	—
110	C₂×#108	C₂×D₆₁	1	1	yes	characteristic
111	#427	C₂×D₆₁	1	1	yes	characteristic
112	C₂×#107	—	122	1	no	—
113	C₂×#106	—	61	2	no	—
114	#405	—	2	1	no	—
115	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
116	#364	—	122	2	no	—
117	#339	—	2	1	no	—
118	C₆₁	#13273	1	1	yes	characteristic, complemented, Sylow
119	C₂×M₁₆	—	2	1	no	—
120	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
122	#125	—	122	4	no	—
123	#125	—	244	2	no	—
124	#117	—	2	1	no	—
125	C₂×C₄⋊C₄	—	122	2	no	—
126	C₂×C₄⋊C₄	—	244	2	no	—
127	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
128	M₃₂	—	122	2	no	—
129	M₃₂	—	122	2	no	—
130	C₂×C₁₆	—	122	2	no	—
131	#108	—	2	1	no	—
132	#108	Dic₁₂₂	1	2	yes	—
133	#107	—	122	4	no	—
134	#107	Dic₁₂₂	1	2	yes	—
135	#106	—	2	1	no	—
136	#106	—	61	4	no	—
137	#106	—	122	2	no	—
138	#97	—	4	1	no	—
139	C₄×C₈	—	4	1	no	—
140	C₄×C₈	C₄×D₆₁	1	2	yes	—
141	C₄×C₈	D₂₄₄	1	2	yes	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	244	2	no	—
144	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
145	M₁₆	—	2	2	no	—
146	M₁₆	—	4	1	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	#11758198	1	2	yes	—
149	C₂×C₈	#11758200	1	1	yes	characteristic
150	C₂×C₈	#11758200	1	1	yes	characteristic
151	C₂×C₈	#11758202	1	2	yes	—
152	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
153	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
154	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
155	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
156	C₂×C₈	#11758222	1	1	yes	characteristic
157	C₂×C₈	#11758222	1	1	yes	characteristic
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	122	4	no	—
160	C₄⋊C₄	—	244	8	no	—
161	C₄⋊C₄	—	244	2	no	—
162	C₂²⋊C₄	—	4	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	4	1	no	—
165	C₄²	#11758198	1	2	yes	—
166	C₁₆	—	244	2	no	—
167	C₂³	#14519802	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	488	4	no	—
172	C₂×C₄	—	244	4	no	—
173	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
174	C₂×C₄	#14519756	1	1	yes	characteristic
175	C₂×C₄	#14519756	1	1	yes	characteristic
176	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
177	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
178	C₂×C₄	#14519802	1	1	yes	characteristic
179	C₈	—	4	2	no	—
180	C₈	#14519700	1	2	yes	—
181	C₈	#14519700	1	2	yes	—
182	C₈	C₄×Dic₁₂₂	1	1	yes	characteristic
183	C₈	C₄×Dic₁₂₂	1	1	yes	characteristic
184	C₈	#14519765	1	1	yes	characteristic
185	C₈	#14519777	1	1	yes	characteristic
186	C₂²	—	2	2	no	—
187	C₂²	#16839766	1	1	yes	characteristic
188	C₂²	#16840052	1	1	yes	characteristic
189	C₂²	#16840057	1	1	yes	characteristic
190	C₄	—	4	2	no	—
191	C₄	—	8	2	no	—
192	C₄	—	488	4	no	—
193	C₄	#16839668	1	1	yes	characteristic
194	C₄	C₂×#14519700	1	1	yes	characteristic
195	C₄	#16840052	1	1	yes	characteristic
196	C₄	#16840057	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#21446767	1	1	yes	characteristic
200	C₂	#21447995	1	2	yes	—

	subgroup	G/H
center	C₂²	#16840052
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1772	C₂
socle	C₂²×C₆₁	#476
prime radical	C₂³×C₆₁	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1772	D₆₁
hypocenter	C₆₁	#13273
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
9. #16840164	C₄	semidirect
118. C₆₁	#13273	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840164	2
C₂×#16839583	2
#21443924	2
#21443919	2
C₆₁×#1772	2
#13273	61