Subgroups — #29833209

Order 15616.

674 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	C₂×#16840178	C₂	1	2	yes	maximal, complemented
2	#21454532	C₂	1	1	yes	maximal, characteristic
3	#21447646	C₂	1	8	yes	maximal, complemented
4	#21447575	C₂	1	1	yes	maximal, characteristic, complemented
5	#21447432	C₂	1	2	yes	maximal, complemented
6	C₂×C₆₁×#346	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14519783	C₂²	1	2	yes	—
8	#16840312	C₄	1	1	yes	characteristic
9	#16840178	—	2	8	no	—
10	C₂×#14519717	C₂²	1	2	yes	—
11	C₂×#14519716	C₂²	1	4	yes	—
12	C₂×C₁₆×D₆₁	C₂²	1	4	yes	—
13	#16839841	C₄	1	4	yes	—
14	C₈×D₂₄₄	C₂²	1	4	yes	—
15	#16839786	C₄	1	1	yes	characteristic
16	#16839728	C₄	1	2	yes	—
17	C₄⋊#11758185	C₂²	1	4	yes	—
18	#16839541	C₂²	1	8	yes	—
19	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₆₁×#346	C₂²	1	4	yes	—
23	C₂×#11758222	C₂×C₄	1	2	yes	—
24	#14519783	—	2	4	no	—
25	#14519783	—	2	4	no	—
26	C₂×#11758194	C₂×C₄	1	2	yes	—
27	C₂×#11758189	C₂×C₄	1	4	yes	—
28	C₂×C₈×D₆₁	C₂³	1	4	yes	—
29	#14519777	C₈	1	1	yes	characteristic
30	#14519774	C₈	1	1	yes	characteristic
31	#14519765	C₈	1	2	yes	—
32	#14519751	C₈	1	4	yes	—
33	C₄×D₂₄₄	C₂×C₄	1	4	yes	—
34	#14519717	—	2	4	no	—
35	#14519716	—	2	8	no	—
36	C₂×#11758185	C₂³	1	4	yes	—
37	C₁₆×D₆₁	—	2	8	no	—
38	C₄⋊#92418	C₂×C₄	1	4	yes	—
39	#14519689	C₂×C₄	1	8	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	—	2	2	no	—
44	C₂×C₁₆×C₆₁	D₄	1	4	yes	complemented
45	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
46	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
47	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₆₁	C₂³	1	2	yes	—
49	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊D₆₁	C₂×C₈	1	4	yes	—
52	#11758222	—	2	4	no	—
53	#11758222	—	2	4	no	—
54	C₂×D₂₄₄	C₂×C₈	1	2	yes	—
55	C₂×D₂₄₄	C₂×C₈	1	4	yes	—
56	C₂×C₄×D₆₁	C₂²×C₄	1	4	yes	—
57	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₆₁	—	2	2	no	—
59	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₆₁	#41	1	2	yes	—
63	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
64	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
65	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
66	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
67	#11758198	C₂×C₈	1	8	yes	—
68	C₄⋊Dic₆₁	C₂×C₈	1	4	yes	—
69	#11758194	—	2	4	no	—
70	C₂×#92418	C₂²×C₄	1	4	yes	—
71	#11758189	—	2	8	no	—
72	C₈×D₆₁	—	2	4	no	—
73	C₈×D₆₁	—	2	4	no	—
74	C₁₆×C₆₁	—	2	2	no	—
75	C₁₆×C₆₁	C₂×D₄	1	4	yes	—
76	#11758185	—	2	4	no	—
77	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
78	C₂²×D₆₁	C₂²×C₈	1	4	yes	—
79	C₂×C₄×C₆₁	—	4	2	no	—
80	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
82	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
83	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₆₁	#131	1	2	yes	—
85	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
86	C₂²⋊D₆₁	—	2	8	no	—
87	C₂×Dic₆₁	C₂²×C₈	1	4	yes	—
88	D₂₄₄	—	2	4	no	—
89	D₂₄₄	—	4	4	no	—
90	C₄×D₆₁	—	2	4	no	—
91	C₄×D₆₁	—	2	4	no	—
92	Dic₁₂₂	—	2	4	no	—
93	C₈×C₆₁	—	2	2	no	—
94	C₈×C₆₁	C₄×D₄	1	2	yes	—
95	C₈×C₆₁	C₂²×D₄	1	1	yes	characteristic
96	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
97	#92418	—	2	4	no	—
98	#33435	—	61	1	no	maximal, Sylow
99	C₂²×C₆₁	—	2	2	no	—
100	C₂²×C₆₁	C₈×D₄	1	2	yes	—
101	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
102	C₂×D₆₁	—	2	8	no	—
103	C₂×D₆₁	—	2	4	no	—
104	C₄×C₆₁	—	2	2	no	—
105	C₄×C₆₁	—	2	2	no	—
106	C₄×C₆₁	C₈×D₄	1	2	yes	—
107	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
108	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
109	Dic₆₁	—	2	4	no	—
110	C₂×#504	—	61	2	no	—
111	#2931	—	61	1	no	—
112	#2168	—	61	8	no	—
113	#2150	—	61	1	no	—
114	#2113	—	61	2	no	—
115	C₂×#346	D₆₁	1	1	yes	characteristic, complemented
116	C₂×C₆₁	—	2	2	no	—
117	C₂×C₆₁	C₂×C₈×D₄	1	1	yes	characteristic
118	C₂×C₆₁	#3407	1	2	yes	—
119	D₆₁	—	4	4	no	—
120	C₂×#131	—	61	2	no	—
121	#532	—	61	1	no	—
122	#504	—	122	8	no	—
123	C₂×M₃₂	—	61	2	no	—
124	C₂×M₃₂	—	61	4	no	—
125	C₂²×C₁₆	—	61	4	no	—
126	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
127	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
128	#436	—	61	4	no	—
129	C₈×D₄	—	61	4	no	—
130	#423	—	61	1	no	—
131	#408	—	61	2	no	—
132	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
133	C₄⋊C₁₆	—	61	4	no	—
134	C₂²⋊C₁₆	—	61	8	no	—
135	#346	C₂×D₆₁	1	4	yes	—
136	C₆₁	#33435	1	1	yes	characteristic, complemented, Sylow
137	C₂×#41	—	61	2	no	—
138	#131	—	122	4	no	—
139	#131	—	122	4	no	—
140	C₂×M₁₆	—	61	2	no	—
141	C₂×M₁₆	—	61	4	no	—
142	C₂²×C₈	—	61	4	no	—
143	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
144	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
145	C₄⋊Q₈	—	61	1	no	—
146	#127	—	61	1	no	—
147	#124	—	61	2	no	—
148	#121	—	61	4	no	—
149	C₄×D₄	—	61	4	no	—
150	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
151	M₃₂	—	122	4	no	—
152	M₃₂	—	122	8	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	—	61	4	no	—
155	C₂×C₁₆	—	122	8	no	—
156	C₂×C₁₆	D₂₄₄	1	4	yes	complemented
157	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
158	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
159	C₄⋊C₈	—	61	4	no	—
160	C₂²⋊C₈	—	61	8	no	—
161	C₄×C₈	C₄×D₆₁	1	2	yes	—
162	C₄×C₈	C₂²×D₆₁	1	2	yes	—
163	#41	—	122	4	no	—
164	#41	—	122	4	no	—
165	C₂×Q₈	—	61	2	no	—
166	C₂×D₄	—	61	4	no	—
167	C₂×D₄	—	61	2	no	—
168	C₂×D₄	—	61	4	no	—
169	C₂²×C₄	—	61	4	no	—
170	C₂²×C₄	C₈×D₆₁	1	2	yes	—
171	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
172	M₁₆	—	122	4	no	—
173	M₁₆	—	122	8	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	61	4	no	—
176	C₂×C₈	—	122	4	no	—
177	C₂×C₈	—	122	4	no	—
178	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
179	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
180	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
181	C₂×C₈	#11758222	1	2	yes	—
182	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
183	C₄⋊C₄	—	61	4	no	—
184	C₂²⋊C₄	—	61	8	no	—
185	C₄²	C₈×D₆₁	1	1	yes	characteristic
186	C₄²	C₈×D₆₁	1	1	yes	characteristic
187	C₄²	C₂×C₄×D₆₁	1	2	yes	—
188	C₁₆	—	2	2	no	—
189	C₁₆	—	122	4	no	—
190	C₁₆	C₂×D₂₄₄	1	4	yes	—
191	C₂³	—	61	4	no	—
192	C₂³	C₂×C₈×D₆₁	1	1	yes	characteristic
193	Q₈	—	122	4	no	—
194	D₄	—	122	8	no	—
195	D₄	—	122	4	no	—
196	D₄	—	244	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	61	4	no	—
199	C₂×C₄	—	122	4	no	—
200	C₂×C₄	—	122	4	no	—
201	C₂×C₄	C₄×D₂₄₄	1	2	yes	—
202	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
203	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
204	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
205	C₂×C₄	#14519783	1	2	yes	—
206	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	122	4	no	—
209	C₈	C₄×D₂₄₄	1	2	yes	—
210	C₈	C₂²×D₂₄₄	1	1	yes	characteristic
211	C₈	C₂×#11758222	1	1	yes	characteristic
212	C₂²	—	2	2	no	—
213	C₂²	—	122	4	no	—
214	C₂²	—	122	8	no	—
215	C₂²	C₈×D₂₄₄	1	2	yes	—
216	C₂²	C₂²×C₈×D₆₁	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	122	4	no	—
220	C₄	C₈×D₂₄₄	1	2	yes	—
221	C₄	C₂×C₄×D₂₄₄	1	1	yes	characteristic
222	C₄	C₂×#14519783	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	—	244	4	no	—
225	C₂	C₂×C₈×D₂₄₄	1	1	yes	characteristic
226	C₂	#21458921	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₂×C₆₁×#346	C₂
prime radical	#16840312	C₄
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	C₂×#346	D₆₁
hypocenter	C₆₁	#33435
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
1. C₂×#16840178	C₂	semidirect
3. #21447646	C₂	semidirect
4. #21447575	C₂	semidirect
5. #21447432	C₂	semidirect
6. C₂×C₆₁×#346	C₂	semidirect
44. C₂×C₁₆×C₆₁	D₄	semidirect
115. C₂×#346	D₆₁	semidirect
136. C₆₁	#33435	semidirect
156. C₂×C₁₆	D₂₄₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840178	2
#21454532	2
#21447646	2
#21447575	2
#21447432	2
C₂×C₆₁×#346	2
#33435	61