Subgroups — C₂×#21443958 #29600213

Order 15616.

306 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14519716	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839596	C₂	1	1	yes	maximal, characteristic, complemented
3	#21443958	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₆₁×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₈×D₆₁	C₄	1	1	yes	characteristic
6	C₂×#14519716	—	2	1	no	—
7	C₂×#14519716	—	2	1	no	—
8	C₂×#14519716	—	2	1	no	—
9	C₂×#14519716	—	2	1	no	—
10	C₂×#14519716	C₄	1	2	yes	—
11	C₂×#14519716	C₂²	1	2	yes	—
12	C₂²×#11758185	C₄	1	1	yes	characteristic
13	#16839596	—	2	1	no	—
14	#16839596	C₂²	1	2	yes	complemented
15	C₂×#14519683	—	2	1	no	—
16	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×M₆₄	—	2	1	no	—
18	C₆₁×M₆₄	C₂²	1	2	yes	complemented
19	C₂×C₃₂×C₆₁	—	2	1	no	—
20	C₂²×C₄×D₆₁	C₈	1	1	yes	characteristic
21	C₂²×#92418	C₈	1	1	yes	characteristic
22	C₂×C₈×D₆₁	—	2	1	no	—
23	C₂×C₈×D₆₁	—	2	1	no	—
24	C₂×C₈×D₆₁	—	2	1	no	—
25	C₂×C₈×D₆₁	—	2	1	no	—
26	C₂×C₈×D₆₁	C₈	1	2	yes	—
27	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
28	#14519716	—	2	2	no	—
29	#14519716	—	2	2	no	—
30	#14519716	—	2	2	no	—
31	#14519716	—	2	2	no	—
32	C₂×#11758185	—	2	1	no	—
33	C₂×#11758185	—	2	1	no	—
34	C₂×#11758185	C₂×C₄	1	2	yes	—
35	#14519683	—	2	2	no	—
36	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
40	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
43	C₃₂×C₆₁	—	2	2	no	—
44	C₂³×D₆₁	C₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
46	C₂²×Dic₆₁	C₁₆	1	1	yes	characteristic
47	C₂×C₄×D₆₁	—	2	1	no	—
48	C₂×C₄×D₆₁	—	2	1	no	—
49	C₂×C₄×D₆₁	—	2	1	no	—
50	C₂×C₄×D₆₁	—	2	1	no	—
51	C₂×C₄×D₆₁	C₁₆	1	2	yes	—
52	C₂×C₄×D₆₁	C₂×C₈	1	2	yes	—
53	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
56	C₂×C₈×C₆₁	M₁₆	1	1	yes	characteristic
57	C₂×C₈×C₆₁	M₁₆	1	1	yes	characteristic
58	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×#92418	—	2	1	no	—
60	C₂×#92418	—	2	1	no	—
61	C₂×#92418	C₂×C₈	1	2	yes	—
62	C₈×D₆₁	—	2	2	no	—
63	C₈×D₆₁	—	2	2	no	—
64	C₈×D₆₁	—	2	2	no	—
65	C₈×D₆₁	—	2	2	no	—
66	C₁₆×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₁₆×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₁₆×C₆₁	C₂×D₄	1	1	yes	characteristic
69	C₁₆×C₆₁	C₂×D₄	1	1	yes	characteristic
70	#11758185	—	2	2	no	—
71	C₂³×C₆₁	C₂×C₁₆	1	1	yes	characteristic
72	C₂²×D₆₁	—	2	1	no	—
73	C₂²×D₆₁	—	2	1	no	—
74	C₂²×D₆₁	—	4	2	no	—
75	C₂²×D₆₁	C₂×C₁₆	1	2	yes	—
76	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₂×C₁₆	1	1	yes	characteristic
79	C₂×C₄×C₆₁	M₃₂	1	1	yes	characteristic
80	C₂×C₄×C₆₁	M₃₂	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
82	C₂×Dic₆₁	—	2	1	no	—
83	C₂×Dic₆₁	—	2	1	no	—
84	C₂×Dic₆₁	C₂×C₁₆	1	2	yes	—
85	C₄×D₆₁	—	2	2	no	—
86	C₄×D₆₁	—	2	2	no	—
87	C₄×D₆₁	—	2	2	no	—
88	C₄×D₆₁	—	2	2	no	—
89	C₈×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
90	C₈×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
91	C₈×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₈×C₆₁	C₂×M₁₆	1	1	yes	characteristic
93	#92418	—	2	2	no	—
94	C₂×#1400	—	61	1	no	maximal, Sylow
95	C₂²×C₆₁	—	2	2	no	—
96	C₂²×C₆₁	C₂²⋊C₁₆	1	1	yes	characteristic
97	C₂²×C₆₁	C₂²⋊C₁₆	1	1	yes	characteristic
98	C₂²×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
99	C₂×D₆₁	—	2	2	no	—
100	C₂×D₆₁	—	4	2	no	—
101	C₂×D₆₁	—	4	2	no	—
102	C₂×D₆₁	—	4	2	no	—
103	C₄×C₆₁	C₂²⋊C₁₆	1	1	yes	characteristic
104	C₄×C₆₁	C₂²⋊C₁₆	1	1	yes	characteristic
105	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
106	C₄×C₆₁	C₂×M₃₂	1	1	yes	characteristic
107	Dic₆₁	—	2	2	no	—
108	C₂²×M₃₂	—	61	1	no	—
109	C₂×M₆₄	—	61	1	no	—
110	C₂×M₆₄	D₆₁	1	1	yes	characteristic, complemented
111	#1400	—	61	4	no	—
112	C₂×C₆₁	—	2	1	no	—
113	C₂×C₆₁	—	2	1	no	—
114	C₂×C₆₁	#1400	1	2	yes	complemented
115	C₂×C₆₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
116	D₆₁	—	4	2	no	—
117	C₂³×C₈	—	61	1	no	—
118	C₂×M₃₂	—	61	2	no	—
119	C₂×M₃₂	—	61	2	no	—
120	C₂×M₃₂	—	122	1	no	—
121	C₂×M₃₂	—	122	1	no	—
122	C₂×M₃₂	—	122	1	no	—
123	C₂×M₃₂	—	122	1	no	—
124	C₂²×C₁₆	—	61	1	no	—
125	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
126	M₆₄	—	2	1	no	—
127	M₆₄	—	61	2	no	—
128	M₆₄	—	122	1	no	—
129	M₆₄	C₂×D₆₁	1	2	yes	complemented
130	C₂×C₃₂	—	2	1	no	—
131	C₂×C₃₂	—	122	1	no	—
132	C₆₁	C₂×#1400	1	1	yes	characteristic, complemented, Sylow
133	C₂³×C₄	—	61	1	no	—
134	C₂²×C₈	—	61	1	no	—
135	C₂²×C₈	—	61	2	no	—
136	C₂²×C₈	—	61	2	no	—
137	C₂²×C₈	—	122	1	no	—
138	C₂²×C₈	—	122	1	no	—
139	C₂²×C₈	—	122	1	no	—
140	C₂²×C₈	—	122	1	no	—
141	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
142	M₃₂	—	122	2	no	—
143	M₃₂	—	122	2	no	—
144	M₃₂	—	122	2	no	—
145	M₃₂	—	122	2	no	—
146	C₂×C₁₆	—	61	2	no	—
147	C₂×C₁₆	—	122	1	no	—
148	C₂×C₁₆	—	122	1	no	—
149	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
150	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
151	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
152	C₂×C₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
153	C₂×C₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
154	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
155	C₃₂	—	2	2	no	—
156	C₃₂	—	122	2	no	—
157	C₂⁴	—	61	1	no	—
158	C₂²×C₄	—	61	1	no	—
159	C₂²×C₄	—	61	2	no	—
160	C₂²×C₄	—	61	2	no	—
161	C₂²×C₄	—	122	1	no	—
162	C₂²×C₄	—	122	1	no	—
163	C₂²×C₄	—	122	1	no	—
164	C₂²×C₄	—	122	1	no	—
165	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
166	C₂×C₈	—	61	2	no	—
167	C₂×C₈	—	122	1	no	—
168	C₂×C₈	—	122	1	no	—
169	C₂×C₈	—	122	2	no	—
170	C₂×C₈	—	122	2	no	—
171	C₂×C₈	—	122	2	no	—
172	C₂×C₈	—	122	2	no	—
173	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
174	C₂×C₈	#11758189	1	1	yes	characteristic
175	C₂×C₈	#11758189	1	1	yes	characteristic
176	C₂×C₈	#11758198	1	1	yes	characteristic
177	C₂×C₈	#11758198	1	1	yes	characteristic
178	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
179	C₁₆	—	122	2	no	—
180	C₁₆	#11758198	1	1	yes	characteristic
181	C₁₆	#11758198	1	1	yes	characteristic
182	C₁₆	C₂×D₂₄₄	1	1	yes	characteristic
183	C₁₆	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
184	C₂³	—	61	2	no	—
185	C₂³	—	122	1	no	—
186	C₂³	—	122	1	no	—
187	C₂³	—	244	2	no	—
188	C₂³	C₁₆×D₆₁	1	1	yes	characteristic
189	C₂×C₄	—	61	2	no	—
190	C₂×C₄	—	122	1	no	—
191	C₂×C₄	—	122	1	no	—
192	C₂×C₄	—	122	2	no	—
193	C₂×C₄	—	122	2	no	—
194	C₂×C₄	—	122	2	no	—
195	C₂×C₄	—	122	2	no	—
196	C₂×C₄	#14519689	1	1	yes	characteristic
197	C₂×C₄	#14519689	1	1	yes	characteristic
198	C₂×C₄	C₁₆×D₆₁	1	1	yes	characteristic
199	C₂×C₄	#14519716	1	1	yes	characteristic
200	C₂×C₄	#14519716	1	1	yes	characteristic
201	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
202	C₈	—	122	2	no	—
203	C₈	#14519689	1	1	yes	characteristic
204	C₈	#14519689	1	1	yes	characteristic
205	C₈	C₂×#11758198	1	1	yes	characteristic
206	C₈	C₂×#11758189	1	1	yes	characteristic
207	C₂²	—	2	2	no	—
208	C₂²	—	122	2	no	—
209	C₂²	—	244	2	no	—
210	C₂²	—	244	2	no	—
211	C₂²	—	244	2	no	—
212	C₂²	#16839541	1	1	yes	characteristic
213	C₂²	#16839541	1	1	yes	characteristic
214	C₂²	C₂×C₁₆×D₆₁	1	1	yes	characteristic
215	C₄	—	122	2	no	—
216	C₄	#16839541	1	1	yes	characteristic
217	C₄	#16839541	1	1	yes	characteristic
218	C₄	C₂×#14519689	1	1	yes	characteristic
219	C₄	C₂×#14519716	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	—	244	2	no	—
223	C₂	#21443958	1	2	yes	direct product factor
224	C₂	C₂×#16839541	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#11758198
commutator	C₂²×C₆₁	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₆₁
Fitting	C₂×C₆₁×M₆₄	C₂
socle	C₂²×C₆₁	C₂²⋊C₁₆
prime radical	C₂³×D₆₁	C₁₆
cosocle	C₂×C₁₆×C₆₁	C₂³
hypercenter	C₂×M₆₄	D₆₁
hypocenter	C₆₁	C₂×#1400
LCS	G ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
2. C₂×#16839596	C₂	semidirect
3. #21443958	C₂	direct (N × H)
4. C₂×C₆₁×M₆₄	C₂	semidirect
14. #16839596	C₂²	semidirect
18. C₆₁×M₆₄	C₂²	semidirect
110. C₂×M₆₄	D₆₁	semidirect
114. C₂×C₆₁	#1400	semidirect
129. M₆₄	C₂×D₆₁	semidirect
132. C₆₁	C₂×#1400	semidirect
223. C₂	#21443958	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14519716	2
C₂×#16839596	2
#21443958	2
C₂×C₆₁×M₆₄	2
C₂×#1400	61

Subgroups — C2×#21443958 #29600213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21443958 #29600213