Subgroups — #29598572

Order 15616.

366 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840160	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×#14519777	—	2	1	no	—
7	C₂×#14519717	C₂²	1	2	yes	—
8	#16840160	—	2	2	no	—
9	#16840160	C₄	1	4	yes	complemented
10	C₂×#14519709	—	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	C₂²×Dic₁₂₂	—	4	1	no	—
19	#14519777	—	2	4	no	—
20	#14519777	—	4	2	no	—
21	C₂×C₄⋊Dic₆₁	—	2	2	no	—
22	C₂×C₄⋊Dic₆₁	—	4	1	no	—
23	#14519717	—	2	2	no	—
24	#14519717	—	2	2	no	—
25	C₂×#11758185	—	2	2	no	—
26	#14519709	—	2	4	no	—
27	#14519709	—	2	2	no	—
28	#14519709	C₂×C₄	1	4	yes	—
29	C₂×M₁₆×C₆₁	—	2	1	no	—
30	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
32	#117×C₆₁	—	2	1	no	—
33	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
34	#108×C₆₁	—	2	1	no	—
35	#108×C₆₁	Q₈	1	2	yes	complemented
36	#107×C₆₁	Q₈	1	2	yes	complemented
37	#106×C₆₁	—	2	1	no	—
38	#97×C₆₁	—	4	1	no	—
39	C₄×C₈×C₆₁	—	4	1	no	—
40	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₆₁	D₄	1	2	yes	—
42	C₂²×C₄×C₆₁	—	2	1	no	—
43	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₆₁	—	4	2	no	—
45	C₂×Dic₁₂₂	—	4	4	no	—
46	C₂×Dic₁₂₂	—	8	4	no	—
47	M₁₆×C₆₁	—	2	2	no	—
48	M₁₆×C₆₁	—	4	1	no	—
49	C₂×C₈×C₆₁	—	4	1	no	—
50	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
54	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
59	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
60	C₄⋊C₄×C₆₁	—	2	2	no	—
61	C₂²⋊C₄×C₆₁	—	4	1	no	—
62	C₄²×C₆₁	—	2	1	no	—
63	C₄²×C₆₁	—	4	1	no	—
64	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
65	C₄⋊Dic₆₁	—	2	4	no	—
66	C₄⋊Dic₆₁	—	4	4	no	—
67	C₄⋊Dic₆₁	—	4	2	no	—
68	#11758185	—	4	2	no	—
69	C₂³×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₂×C₄×C₆₁	—	2	4	no	—
71	C₂×C₄×C₆₁	—	2	2	no	—
72	C₂×C₄×C₆₁	—	4	2	no	—
73	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×Dic₆₁	—	4	4	no	—
80	C₂×Dic₆₁	—	8	4	no	—
81	Dic₁₂₂	—	8	8	no	—
82	C₈×C₆₁	—	4	2	no	—
83	C₈×C₆₁	#103	1	2	yes	—
84	C₈×C₆₁	#103	1	2	yes	—
85	C₈×C₆₁	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₆₁	C₄×Q₈	1	1	yes	characteristic
87	C₈×C₆₁	#124	1	1	yes	characteristic
88	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
89	#13272	—	61	1	no	maximal, Sylow
90	C₂²×C₆₁	—	2	2	no	—
91	C₂²×C₆₁	#418	1	1	yes	characteristic
92	C₂²×C₆₁	#476	1	1	yes	characteristic
93	C₂²×C₆₁	#477	1	1	yes	characteristic
94	C₄×C₆₁	—	2	4	no	—
95	C₄×C₆₁	—	8	2	no	—
96	C₄×C₆₁	#391	1	1	yes	characteristic
97	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
98	C₄×C₆₁	#476	1	1	yes	characteristic
99	C₄×C₆₁	#477	1	1	yes	characteristic
100	Dic₆₁	—	8	4	no	—
101	C₂×#498	—	61	1	no	—
102	C₂×#364	—	61	1	no	—
103	#1772	D₆₁	1	1	yes	characteristic
104	#1386	—	61	2	no	—
105	#1383	—	61	2	no	—
106	C₂×C₆₁	—	2	1	no	—
107	C₂×C₆₁	—	2	1	no	—
108	C₂×C₆₁	#1987	1	1	yes	characteristic
109	C₂×C₆₁	#2230	1	2	yes	—
110	C₂×C₄⋊Q₈	—	122	1	no	—
111	C₂×M₃₂	—	61	2	no	—
112	#498	—	61	4	no	—
113	#498	—	122	2	no	—
114	C₂×#108	C₂×D₆₁	1	1	yes	characteristic
115	#427	C₂×D₆₁	1	1	yes	characteristic
116	C₂×#106	—	61	2	no	—
117	C₂×#106	—	122	1	no	—
118	#405	—	2	1	no	—
119	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
120	#364	—	122	2	no	—
121	#339	—	2	1	no	—
122	C₆₁	#13272	1	1	yes	characteristic, complemented, Sylow
123	C₂²×Q₈	—	244	1	no	—
124	C₂×M₁₆	—	2	1	no	—
125	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
127	C₄⋊Q₈	—	122	4	no	—
128	C₄⋊Q₈	—	244	2	no	—
129	#117	—	2	1	no	—
130	C₂×C₄⋊C₄	—	122	2	no	—
131	C₂×C₄⋊C₄	—	244	1	no	—
132	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
133	M₃₂	—	122	2	no	—
134	M₃₂	—	122	2	no	—
135	C₂×C₁₆	—	122	2	no	—
136	#108	—	2	1	no	—
137	#108	Dic₁₂₂	1	2	yes	complemented
138	#107	Dic₁₂₂	1	2	yes	complemented
139	#106	—	2	1	no	—
140	#106	—	61	4	no	—
141	#106	—	122	4	no	—
142	#106	—	122	2	no	—
143	#97	—	4	1	no	—
144	C₄×C₈	—	4	1	no	—
145	C₄×C₈	C₄×D₆₁	1	2	yes	—
146	C₄×C₈	D₂₄₄	1	2	yes	—
147	C₂×Q₈	—	488	4	no	—
148	C₂×Q₈	—	244	4	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	244	2	no	—
151	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	4	1	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	#11758198	1	2	yes	—
156	C₂×C₈	#11758200	1	1	yes	characteristic
157	C₂×C₈	#11758200	1	1	yes	characteristic
158	C₂×C₈	#11758202	1	2	yes	—
159	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
160	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
161	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
162	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
163	C₂×C₈	#11758222	1	1	yes	characteristic
164	C₂×C₈	#11758222	1	1	yes	characteristic
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	122	4	no	—
167	C₄⋊C₄	—	244	4	no	—
168	C₄⋊C₄	—	244	2	no	—
169	C₂²⋊C₄	—	4	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	4	1	no	—
172	C₄²	#11758198	1	2	yes	—
173	C₁₆	—	244	2	no	—
174	C₂³	#14519802	1	1	yes	characteristic
175	Q₈	—	488	8	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	488	4	no	—
180	C₂×C₄	—	244	4	no	—
181	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
182	C₂×C₄	#14519756	1	1	yes	characteristic
183	C₂×C₄	#14519756	1	1	yes	characteristic
184	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
185	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
186	C₂×C₄	#14519802	1	1	yes	characteristic
187	C₈	—	4	2	no	—
188	C₈	#14519700	1	2	yes	—
189	C₈	#14519700	1	2	yes	—
190	C₈	C₄×Dic₁₂₂	1	1	yes	characteristic
191	C₈	C₄×Dic₁₂₂	1	1	yes	characteristic
192	C₈	#14519765	1	1	yes	characteristic
193	C₈	#14519777	1	1	yes	characteristic
194	C₂²	—	2	2	no	—
195	C₂²	#16839766	1	1	yes	characteristic
196	C₂²	#16840052	1	1	yes	characteristic
197	C₂²	#16840057	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	8	2	no	—
200	C₄	—	488	4	no	—
201	C₄	#16839668	1	1	yes	characteristic
202	C₄	C₂×#14519700	1	1	yes	characteristic
203	C₄	#16840052	1	1	yes	characteristic
204	C₄	#16840057	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	#21446767	1	1	yes	characteristic
208	C₂	#21447990	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	#13272
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. #16840160	C₄	semidirect
35. #108×C₆₁	Q₈	semidirect
36. #107×C₆₁	Q₈	semidirect
122. C₆₁	#13272	semidirect
137. #108	Dic₁₂₂	semidirect
138. #107	Dic₁₂₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840160	2
C₂×#16839583	2
#21443924	2
#21443919	2
C₆₁×#1772	2
#13272	61