Subgroups — #29833656

Order 15616.

486 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
2	#21447727	C₂	1	4	yes	maximal, complemented
3	#21447724	C₂	1	4	yes	maximal, complemented
4	#21447619	C₂	1	4	yes	maximal
5	#21447618	C₂	1	1	yes	maximal, characteristic
6	C₆₁×#2162	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Dic₁₂₂	C₄	1	1	yes	characteristic
8	C₈×Dic₁₂₂	C₄	1	2	yes	—
9	C₈×Dic₁₂₂	C₄	1	2	yes	—
10	C₈×Dic₁₂₂	C₂²	1	2	yes	—
11	C₈×Dic₁₂₂	C₂²	1	2	yes	—
12	C₂×C₄⋊#92418	C₄	1	1	yes	characteristic
13	C₂×#14519706	C₄	1	2	yes	—
14	C₂×C₈×Dic₆₁	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	#16839537	C₂²	1	8	yes	—
19	C₄×#11758185	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
22	C₆₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
23	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
24	C₂²×Dic₁₂₂	C₈	1	1	yes	characteristic
25	C₂²×#92418	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₂₂	C₈	1	2	yes	—
27	C₄×Dic₁₂₂	C₈	1	2	yes	—
28	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
29	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊Dic₆₁	C₈	1	1	yes	characteristic
31	C₂×#11758196	C₈	1	2	yes	—
32	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
33	C₂×#11758185	C₂³	1	4	yes	—
34	C₄⋊#92418	C₂×C₄	1	2	yes	—
35	C₄⋊#92418	C₂×C₄	1	2	yes	—
36	#14519706	C₂×C₄	1	4	yes	—
37	#14519706	C₂×C₄	1	4	yes	—
38	C₈×Dic₆₁	C₂×C₄	1	4	yes	—
39	C₈×Dic₆₁	C₂³	1	4	yes	—
40	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₁₆×C₆₁	C₂³	1	4	yes	—
44	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
47	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
52	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
53	C₂×Dic₁₂₂	—	2	4	no	—
54	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
55	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
56	C₂×C₈×C₆₁	—	2	2	no	—
57	C₂×C₈×C₆₁	—	4	1	no	—
58	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
62	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
63	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
64	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
66	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
67	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
68	#11758196	C₂×C₈	1	4	yes	—
69	#11758196	C₂×C₈	1	4	yes	—
70	C₄×Dic₆₁	C₂×C₈	1	4	yes	—
71	C₄×Dic₆₁	C₂²×C₄	1	4	yes	—
72	C₂×#92418	—	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	M₃₂	1	4	yes	—
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	—	2	4	no	—
88	C₂×Dic₆₁	—	4	2	no	—
89	C₂×Dic₆₁	C₂²×C₈	1	4	yes	—
90	C₂×Dic₆₁	C₂²×C₈	1	4	yes	—
91	Dic₁₂₂	—	2	4	no	—
92	Dic₁₂₂	—	2	4	no	—
93	C₈×C₆₁	—	2	1	no	—
94	C₈×C₆₁	—	2	1	no	—
95	C₈×C₆₁	—	2	1	no	—
96	C₈×C₆₁	2₋¹⁺⁴	1	2	yes	—
97	#92418	—	2	4	no	—
98	#33497	—	61	1	no	maximal, Sylow
99	C₂²×C₆₁	—	2	1	no	—
100	C₂²×C₆₁	—	2	1	no	—
101	C₂²×C₆₁	C₂×M₃₂	1	2	yes	—
102	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
103	C₄×C₆₁	—	2	1	no	—
104	C₄×C₆₁	—	2	1	no	—
105	C₄×C₆₁	—	2	1	no	—
106	C₄×C₆₁	C₂×M₃₂	1	2	yes	—
107	C₄×C₆₁	C₂×M₃₂	1	2	yes	—
108	C₄×C₆₁	#519	1	1	yes	characteristic
109	C₄×C₆₁	#568	1	1	yes	characteristic
110	Dic₆₁	—	2	4	no	—
111	Dic₆₁	—	2	4	no	—
112	C₂×C₈×Q₈	—	61	1	no	—
113	#2183	—	61	4	no	—
114	#2183	—	61	4	no	—
115	#2163	—	61	4	no	—
116	#2162	—	61	1	no	—
117	#2162	D₆₁	1	1	yes	characteristic
118	C₂×C₆₁	—	2	2	no	—
119	C₂×C₆₁	#2982	1	1	yes	characteristic
120	C₂×C₆₁	C₂²×M₃₂	1	1	yes	characteristic
121	C₂×C₆₁	#3407	1	1	yes	characteristic
122	C₂×C₄×Q₈	—	61	1	no	—
123	C₈×Q₈	—	61	2	no	—
124	C₈×Q₈	—	61	2	no	—
125	C₈×Q₈	—	61	2	no	—
126	C₈×Q₈	—	61	2	no	—
127	C₂×C₄⋊C₈	—	61	2	no	—
128	C₂×C₄⋊C₈	—	61	1	no	—
129	C₂×C₄×C₈	—	61	2	no	—
130	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
131	C₄⋊C₁₆	—	61	8	no	—
132	C₄⋊C₁₆	—	61	2	no	—
133	C₄⋊C₁₆	C₂×D₆₁	1	2	yes	—
134	C₂²⋊C₁₆	—	61	2	no	—
135	C₂²⋊C₁₆	C₂×D₆₁	1	2	yes	—
136	#346	—	61	8	no	—
137	C₄×C₁₆	—	61	2	no	—
138	C₄×C₁₆	C₂×D₆₁	1	2	yes	—
139	C₆₁	#33497	1	1	yes	characteristic, complemented, Sylow
140	C₂²×Q₈	—	61	1	no	—
141	C₂²×C₈	—	61	2	no	—
142	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
143	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
144	C₄×Q₈	—	61	2	no	—
145	C₄×Q₈	—	61	2	no	—
146	C₄×Q₈	—	61	2	no	—
147	C₄×Q₈	—	61	2	no	—
148	C₂×C₄⋊C₄	—	61	2	no	—
149	C₂×C₄⋊C₄	—	61	1	no	—
150	C₂×C₄²	—	61	2	no	—
151	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
152	C₂×C₁₆	—	61	4	no	—
153	C₂×C₁₆	C₂²×D₆₁	1	4	yes	—
154	C₄⋊C₈	—	61	4	no	—
155	C₄⋊C₈	—	61	4	no	—
156	C₄⋊C₈	—	61	2	no	—
157	C₄⋊C₈	—	61	2	no	—
158	C₄×C₈	—	61	4	no	—
159	C₄×C₈	—	61	4	no	—
160	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
161	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
162	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
163	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
164	C₂×Q₈	—	61	2	no	—
165	C₂×Q₈	—	61	2	no	—
166	C₂×Q₈	—	122	4	no	—
167	C₂²×C₄	—	61	2	no	—
168	C₂²×C₄	—	61	2	no	—
169	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
170	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	4	1	no	—
174	C₂×C₈	—	61	4	no	—
175	C₂×C₈	—	244	2	no	—
176	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
177	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
178	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
179	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
180	C₄⋊C₄	—	61	4	no	—
181	C₄⋊C₄	—	61	4	no	—
182	C₄⋊C₄	—	61	2	no	—
183	C₄⋊C₄	—	61	2	no	—
184	C₄²	—	61	4	no	—
185	C₄²	—	61	4	no	—
186	C₄²	C₈×D₆₁	1	1	yes	characteristic
187	C₄²	C₈×D₆₁	1	1	yes	characteristic
188	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
189	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
190	C₁₆	—	2	4	no	—
191	C₁₆	—	122	4	no	—
192	C₂³	C₂×C₈×D₆₁	1	1	yes	characteristic
193	Q₈	—	122	4	no	—
194	Q₈	—	122	4	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	61	4	no	—
199	C₂×C₄	—	61	4	no	—
200	C₂×C₄	—	122	4	no	—
201	C₂×C₄	—	244	2	no	—
202	C₂×C₄	#14519716	1	4	yes	—
203	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
204	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
205	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
206	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
207	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
208	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	122	4	no	—
213	C₈	#14519824	1	2	yes	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	C₂×#14519716	1	2	yes	—
217	C₂²	C₂²×C₈×D₆₁	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	122	4	no	—
222	C₄	—	122	4	no	—
223	C₄	C₂×#14519716	1	2	yes	—
224	C₄	C₂×#14519716	1	2	yes	—
225	C₄	#16840239	1	1	yes	characteristic
226	C₄	#16840528	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	#21454899	1	1	yes	characteristic
229	C₂	C₂²×#14519716	1	1	yes	characteristic
230	C₂	#21458921	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₆₁×#2162	C₂
prime radical	C₂³×C₆₁	C₂²×C₈
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	#2162	D₆₁
hypocenter	C₆₁	#33497
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
2. #21447727	C₂	semidirect
3. #21447724	C₂	semidirect
139. C₆₁	#33497	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₂₂	2
#21447727	2
#21447724	2
#21447619	2
#21447618	2
C₆₁×#2162	2
#33497	61