Subgroups — #29578717

Order 15616.

298 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21446594	C₂	1	2	yes	maximal
2	#21444578	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839515	C₂	1	1	yes	maximal, characteristic
4	#21443715	C₂	1	2	yes	maximal
5	C₈×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519712	—	2	2	no	—
7	#16839786	C₂²	1	2	yes	—
8	C₂×C₄⋊#92418	C₂²	1	2	yes	—
9	#16839526	—	2	2	no	—
10	#16839515	—	2	1	no	—
11	#16839515	—	2	1	no	—
12	#16839515	C₄	1	2	yes	—
13	#16839515	C₄	1	2	yes	—
14	C₆₁×#431	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#322	—	2	1	no	—
18	C₈²×C₆₁	—	2	1	no	—
19	C₂×#11758194	—	2	2	no	—
20	C₂²×#92418	—	2	2	no	—
21	#14519712	—	4	4	no	—
22	C₄⋊#92418	—	2	2	no	—
23	C₄⋊#92418	—	2	2	no	—
24	C₄⋊#92418	—	2	2	no	—
25	C₄⋊#92418	C₂×C₄	1	4	yes	—
26	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
27	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₁	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₆₁	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₆₁	—	2	1	no	—
32	C₂²⋊C₈×C₆₁	Q₈	1	2	yes	—
33	#97×C₆₁	—	2	1	no	—
34	C₄×C₈×C₆₁	—	2	1	no	—
35	C₄×C₈×C₆₁	—	2	1	no	—
36	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
39	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
43	M₁₆×C₆₁	—	2	1	no	—
44	M₁₆×C₆₁	—	2	1	no	—
45	M₁₆×C₆₁	D₈	1	2	yes	—
46	M₁₆×C₆₁	Q₁₆	1	2	yes	—
47	C₂×C₈×C₆₁	—	2	1	no	—
48	C₂×C₈×C₆₁	—	2	1	no	—
49	C₂×C₈×C₆₁	—	2	2	no	—
50	C₂×C₈×C₆₁	—	2	2	no	—
51	C₂×C₈×C₆₁	—	4	1	no	—
52	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
56	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₆₁	#41	1	1	yes	characteristic
59	C₄²×C₆₁	#41	1	1	yes	characteristic
60	#11758194	—	4	2	no	—
61	#11758194	—	4	2	no	—
62	C₂×#92418	—	2	4	no	—
63	C₂×#92418	—	4	2	no	—
64	C₂×#92418	—	4	2	no	—
65	C₂×#92418	—	4	2	no	—
66	C₂³×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₆₁	—	2	1	no	—
68	C₂×C₄×C₆₁	—	2	1	no	—
69	C₂×C₄×C₆₁	—	4	1	no	—
70	C₂×C₄×C₆₁	C₂×C₄⋊C₄	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	#121	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₆₁	#125	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
79	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
80	C₈×C₆₁	—	2	1	no	—
81	C₈×C₆₁	—	2	1	no	—
82	C₈×C₆₁	—	2	1	no	—
83	C₈×C₆₁	—	2	2	no	—
84	C₈×C₆₁	—	2	2	no	—
85	C₈×C₆₁	—	4	2	no	—
86	C₈×C₆₁	#108	1	2	yes	complemented
87	#92418	—	4	4	no	—
88	#92418	—	8	2	no	—
89	#9624	—	61	1	no	maximal, Sylow
90	C₂²×C₆₁	—	2	1	no	—
91	C₂²×C₆₁	—	2	1	no	—
92	C₂²×C₆₁	#389	1	1	yes	characteristic
93	C₂²×C₆₁	#499	1	1	yes	characteristic
94	C₂²×C₆₁	#500	1	1	yes	characteristic
95	C₄×C₆₁	—	2	1	no	—
96	C₄×C₆₁	—	2	1	no	—
97	C₄×C₆₁	—	2	1	no	—
98	C₄×C₆₁	C₂×#108	1	1	yes	characteristic
99	C₄×C₆₁	#430	1	1	yes	characteristic
100	C₄×C₆₁	C₄×D₈	1	1	yes	characteristic
101	C₄×C₆₁	C₄×Q₁₆	1	1	yes	characteristic
102	C₄×C₆₁	#466	1	1	yes	characteristic
103	C₄×C₆₁	#467	1	1	yes	characteristic
104	#1950	—	61	2	no	—
105	#1567	—	61	1	no	—
106	C₂×#335	—	61	1	no	—
107	C₈×M₁₆	D₆₁	1	1	yes	characteristic
108	#1301	—	61	2	no	—
109	C₂×C₆₁	—	2	1	no	—
110	C₂×C₆₁	—	2	1	no	—
111	C₂×C₆₁	#1942	1	1	yes	characteristic
112	C₂×C₆₁	#1950	1	1	yes	characteristic
113	C₂×C₆₁	#1953	1	1	yes	characteristic
114	#431	C₂×D₆₁	1	1	yes	characteristic
115	C₂×#108	—	122	2	no	—
116	#423	—	61	2	no	—
117	C₂×C₄⋊C₈	—	61	2	no	—
118	C₄×M₁₆	C₂×D₆₁	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
120	#341	—	122	2	no	—
121	#335	—	61	2	no	—
122	#335	—	61	2	no	—
123	#335	—	122	1	no	—
124	#335	—	122	1	no	—
125	#322	—	2	1	no	—
126	C₈²	—	2	1	no	—
127	C₆₁	#9624	1	1	yes	characteristic, complemented, Sylow
128	C₂×M₁₆	—	122	2	no	—
129	C₂×M₁₆	D₂₄₄	1	1	yes	characteristic
130	C₂×M₁₆	D₂₄₄	1	1	yes	characteristic
131	C₂²×C₈	—	122	2	no	—
132	C₂²×C₈	Dic₁₂₂	1	1	yes	characteristic
133	C₂²×C₈	Dic₁₂₂	1	1	yes	characteristic
134	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
135	#108	—	244	4	no	—
136	C₄⋊C₈	—	2	1	no	—
137	C₄⋊C₈	—	61	4	no	—
138	C₄⋊C₈	—	122	2	no	—
139	C₄⋊C₈	—	122	2	no	—
140	C₄⋊C₈	—	122	2	no	—
141	C₂²⋊C₈	Dic₁₂₂	1	2	yes	—
142	#97	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
146	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
148	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
149	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
150	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
152	M₁₆	—	2	1	no	—
153	M₁₆	—	2	1	no	—
154	M₁₆	—	244	2	no	—
155	M₁₆	—	244	2	no	—
156	M₁₆	D₄₈₈	1	2	yes	—
157	M₁₆	Dic₂₄₄	1	2	yes	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	4	1	no	—
163	C₂×C₈	—	122	4	no	—
164	C₂×C₈	—	244	2	no	—
165	C₂×C₈	—	244	2	no	—
166	C₂×C₈	—	244	2	no	—
167	C₂×C₈	#11758196	1	1	yes	characteristic
168	C₂×C₈	#11758196	1	1	yes	characteristic
169	C₂×C₈	#11758196	1	1	yes	characteristic
170	C₂×C₈	#11758196	1	1	yes	characteristic
171	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
172	C₄²	#11758222	1	1	yes	characteristic
173	C₄²	#11758222	1	1	yes	characteristic
174	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
175	C₂³	#14519777	1	1	yes	characteristic
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
180	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
181	C₂×C₄	C₄×Dic₁₂₂	1	1	yes	characteristic
182	C₂×C₄	#14519751	1	1	yes	characteristic
183	C₂×C₄	#14519756	1	1	yes	characteristic
184	C₂×C₄	#14519769	1	1	yes	characteristic
185	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
186	C₂×C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
187	C₂×C₄	#14519799	1	1	yes	characteristic
188	C₂×C₄	#14519799	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	—	2	2	no	—
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	2	1	no	—
194	C₈	—	4	2	no	—
195	C₈	—	488	2	no	—
196	C₈	—	244	4	no	—
197	C₈	#14519712	1	2	yes	complemented
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	#16839660	1	1	yes	characteristic
201	C₂²	#16840164	1	1	yes	characteristic
202	C₂²	#16840167	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	C₂×#14519712	1	1	yes	characteristic
207	C₄	#16839810	1	1	yes	characteristic
208	C₄	C₄×D₄₈₈	1	1	yes	characteristic
209	C₄	C₄×Dic₂₄₄	1	1	yes	characteristic
210	C₄	#16839998	1	1	yes	characteristic
211	C₄	#16840003	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#21446552	1	1	yes	characteristic
215	C₂	#21446594	1	1	yes	characteristic
216	C₂	#21446609	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14519751
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₈×M₁₆×C₆₁	C₂
socle	C₂²×C₆₁	#389
prime radical	C₂³×C₆₁	C₄⋊Q₈
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₈×M₁₆	D₆₁
hypocenter	C₆₁	#9624
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
86. C₈×C₆₁	#108	semidirect
127. C₆₁	#9624	semidirect
197. C₈	#14519712	semidirect

Maximal subgroups

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

subgroup	index
#21446594	2
#21444578	2
C₂×#16839515	2
#21443715	2
C₈×M₁₆×C₆₁	2
#9624	61