Subgroups — #29833678

Order 15616.

444 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454753	C₂	1	1	yes	maximal, characteristic
2	#21447726	C₂	1	4	yes	maximal, complemented
3	#21447725	C₂	1	4	yes	maximal, complemented
4	#21447623	C₂	1	4	yes	maximal
5	#21447622	C₂	1	1	yes	maximal, characteristic
6	C₆₁×#2163	C₂	1	1	yes	maximal, characteristic
7	#16840319	C₄	1	1	yes	characteristic
8	#16839892	C₄	1	2	yes	—
9	#16839892	C₄	1	2	yes	—
10	C₈×Dic₁₂₂	C₂²	1	2	yes	—
11	C₈×Dic₁₂₂	C₂²	1	2	yes	—
12	#16839788	C₄	1	2	yes	—
13	#16839786	C₄	1	1	yes	characteristic
14	#16839717	C₂²	1	2	yes	—
15	C₄⋊#11758185	C₂²	1	2	yes	—
16	#16839580	C₂²	1	8	yes	—
17	C₂²⋊#11758185	C₂²	1	2	yes	—
18	#16839538	C₂²	1	2	yes	—
19	#16839537	C₂²	1	4	yes	—
20	C₁₆×Dic₆₁	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
23	C₆₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
24	C₆₁×#346	C₂²	1	2	yes	—
25	C₂×#11758194	C₂×C₄	1	2	yes	—
26	#14519777	C₈	1	1	yes	characteristic
27	#14519777	C₈	1	1	yes	characteristic
28	#14519769	C₈	1	2	yes	—
29	#14519756	C₈	1	2	yes	—
30	#14519756	C₈	1	2	yes	—
31	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
32	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
33	#14519736	C₂×C₄	1	2	yes	—
34	C₂×#11758185	C₂³	1	4	yes	—
35	C₄⋊#92418	C₂×C₄	1	2	yes	—
36	C₄⋊#92418	C₂×C₄	1	2	yes	—
37	#14519706	C₂×C₄	1	4	yes	—
38	#14519706	C₂×C₄	1	4	yes	—
39	#14519687	C₂×C₄	1	4	yes	—
40	C₈×Dic₆₁	C₂³	1	4	yes	—
41	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₆×C₆₁	C₂³	1	4	yes	—
45	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
53	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
54	C₂×C₈×C₆₁	—	2	2	no	—
55	C₂×C₈×C₆₁	—	4	1	no	—
56	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
60	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
61	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²⋊Dic₆₁	C₂×C₈	1	4	yes	—
65	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
66	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
67	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
68	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
69	#11758196	C₂×C₈	1	4	yes	—
70	#11758196	C₂×C₈	1	4	yes	—
71	C₄×Dic₆₁	C₂²×C₄	1	4	yes	—
72	#11758194	—	4	2	no	—
73	C₂×#92418	C₂²×C₄	1	4	yes	—
74	C₁₆×C₆₁	—	2	4	no	—
75	#11758185	—	2	4	no	—
76	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₆₁	—	2	1	no	—
78	C₂×C₄×C₆₁	—	2	1	no	—
79	C₂×C₄×C₆₁	—	4	1	no	—
80	C₂×C₄×C₆₁	—	4	1	no	—
81	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
87	C₂×Dic₆₁	C₂²×C₈	1	4	yes	—
88	C₂×Dic₆₁	C₂²×C₈	1	4	yes	—
89	Dic₁₂₂	—	4	2	no	—
90	Dic₁₂₂	—	4	2	no	—
91	C₈×C₆₁	—	2	1	no	—
92	C₈×C₆₁	—	2	1	no	—
93	C₈×C₆₁	—	2	1	no	—
94	C₈×C₆₁	2₋¹⁺⁴	1	2	yes	—
95	#92418	—	2	4	no	—
96	#33501	—	61	1	no	maximal, Sylow
97	C₂²×C₆₁	—	2	1	no	—
98	C₂²×C₆₁	—	2	1	no	—
99	C₂²×C₆₁	—	2	1	no	—
100	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
101	C₄×C₆₁	—	2	1	no	—
102	C₄×C₆₁	—	2	1	no	—
103	C₄×C₆₁	—	2	1	no	—
104	C₄×C₆₁	—	2	1	no	—
105	C₄×C₆₁	—	2	1	no	—
106	C₄×C₆₁	#519	1	1	yes	characteristic
107	C₄×C₆₁	#568	1	1	yes	characteristic
108	Dic₆₁	—	2	4	no	—
109	Dic₆₁	—	2	4	no	—
110	#2960	—	61	1	no	—
111	#2183	—	61	4	no	—
112	#2183	—	61	4	no	—
113	#2164	—	61	4	no	—
114	#2163	—	61	1	no	—
115	#2163	D₆₁	1	1	yes	characteristic
116	C₂×C₆₁	—	4	1	no	—
117	C₂×C₆₁	#2982	1	1	yes	characteristic
118	C₂×C₆₁	#3407	1	1	yes	characteristic
119	C₂×C₆₁	#3407	1	1	yes	characteristic
120	#533	—	61	1	no	—
121	#446	—	61	2	no	—
122	#446	—	61	2	no	—
123	C₈×Q₈	—	61	2	no	—
124	C₈×Q₈	—	61	2	no	—
125	#424	—	61	2	no	—
126	#423	—	61	1	no	—
127	#405	—	61	2	no	—
128	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
129	C₄⋊C₁₆	—	61	8	no	—
130	C₄⋊C₁₆	—	61	2	no	—
131	C₄⋊C₁₆	C₂×D₆₁	1	2	yes	—
132	C₂²⋊C₁₆	—	61	2	no	—
133	C₂²⋊C₁₆	C₂×D₆₁	1	2	yes	—
134	#346	—	61	4	no	—
135	#346	—	61	2	no	—
136	#346	C₂×D₆₁	1	2	yes	—
137	C₄×C₁₆	—	61	4	no	—
138	C₆₁	#33501	1	1	yes	characteristic, complemented, Sylow
139	C₂×M₁₆	—	61	2	no	—
140	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
141	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
142	C₄⋊Q₈	—	61	1	no	—
143	C₄⋊Q₈	—	61	1	no	—
144	#125	—	61	2	no	—
145	C₂²⋊Q₈	—	61	2	no	—
146	C₂²⋊Q₈	—	61	2	no	—
147	C₄×Q₈	—	61	2	no	—
148	C₄×Q₈	—	61	2	no	—
149	#117	—	61	2	no	—
150	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
151	C₂×C₁₆	—	61	4	no	—
152	C₂×C₁₆	C₂²×D₆₁	1	4	yes	—
153	C₄⋊C₈	—	61	4	no	—
154	C₄⋊C₈	—	61	4	no	—
155	C₄⋊C₈	—	61	2	no	—
156	C₄⋊C₈	—	61	2	no	—
157	#97	—	61	4	no	—
158	C₄×C₈	—	61	4	no	—
159	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
160	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
161	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
162	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
163	C₂×Q₈	—	61	2	no	—
164	C₂×Q₈	—	61	2	no	—
165	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
166	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
168	M₁₆	—	244	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	—	61	4	no	—
172	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
173	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
174	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
175	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
176	C₄⋊C₄	—	61	4	no	—
177	C₄⋊C₄	—	61	4	no	—
178	C₄⋊C₄	—	61	2	no	—
179	C₄⋊C₄	—	61	2	no	—
180	C₄⋊C₄	—	61	2	no	—
181	C₄⋊C₄	—	61	2	no	—
182	C₂²⋊C₄	—	61	4	no	—
183	C₄²	—	61	4	no	—
184	C₄²	C₈×D₆₁	1	1	yes	characteristic
185	C₄²	C₈×D₆₁	1	1	yes	characteristic
186	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
187	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
188	C₁₆	—	2	4	no	—
189	C₁₆	—	122	4	no	—
190	C₂³	C₂×C₈×D₆₁	1	1	yes	characteristic
191	Q₈	—	244	2	no	—
192	Q₈	—	244	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	61	4	no	—
198	C₂×C₄	—	61	4	no	—
199	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
200	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
201	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
202	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
203	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
204	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	122	4	no	—
209	C₈	#14519824	1	2	yes	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	C₂²×C₈×D₆₁	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	122	4	no	—
220	C₄	—	122	4	no	—
221	C₄	#16840239	1	1	yes	characteristic
222	C₄	#16840528	1	1	yes	characteristic
223	C₂	—	4	1	no	—
224	C₂	#21454899	1	1	yes	characteristic
225	C₂	#21458921	1	1	yes	characteristic
226	C₂	#21458921	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₆₁×#2163	C₂
prime radical	C₂³×C₆₁	C₂²×C₈
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	#2163	D₆₁
hypocenter	C₆₁	#33501
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
2. #21447726	C₂	semidirect
3. #21447725	C₂	semidirect
138. C₆₁	#33501	semidirect

Maximal subgroups

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

subgroup	index
#21454753	2
#21447726	2
#21447725	2
#21447623	2
#21447622	2
C₆₁×#2163	2
#33501	61