Subgroups — C₂×#21443776 #29589727

Order 15616.

422 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839841	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839538	C₂	1	1	yes	maximal, characteristic, complemented
3	#21443776	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₆₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#11758189	—	2	1	no	—
6	C₂×C₄×D₂₄₄	C₄	1	1	yes	characteristic
7	C₂²×#11758185	—	2	1	no	—
8	#16839841	—	2	1	no	—
9	#16839841	—	2	1	no	—
10	#16839841	C₄	1	2	yes	—
11	#16839841	C₂²	1	2	yes	—
12	C₂×C₄⋊#92418	C₄	1	1	yes	characteristic
13	C₂×#14519689	—	2	1	no	—
14	#16839538	—	2	1	no	—
15	#16839538	C₂²	1	2	yes	complemented
16	C₂²×C₁₆×C₆₁	—	2	1	no	—
17	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×C₄⋊C₁₆	—	2	1	no	—
19	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	complemented
20	C₂²×D₂₄₄	C₈	1	1	yes	characteristic
21	C₂²×C₄×D₆₁	—	2	1	no	—
22	C₂×#11758189	—	2	2	no	—
23	C₂×#11758189	—	2	2	no	—
24	C₂×#11758189	—	4	2	no	—
25	C₂×#11758189	—	4	2	no	—
26	C₂²×#92418	—	2	1	no	—
27	C₄×D₂₄₄	—	2	1	no	—
28	C₄×D₂₄₄	—	2	1	no	—
29	C₄×D₂₄₄	C₈	1	2	yes	—
30	C₄×D₂₄₄	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊Dic₆₁	C₈	1	1	yes	characteristic
32	C₂×#11758198	—	2	1	no	—
33	C₂×#11758185	—	2	4	no	—
34	C₂×#11758185	—	2	2	no	—
35	C₄⋊#92418	—	2	1	no	—
36	C₄⋊#92418	C₂×C₄	1	2	yes	—
37	#14519689	—	2	2	no	—
38	#14519689	—	2	2	no	—
39	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₁₆×C₆₁	—	2	2	no	—
43	C₂×C₁₆×C₆₁	—	4	2	no	—
44	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
48	C₂³×D₆₁	—	2	1	no	—
49	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
52	C₂²×Dic₆₁	—	2	1	no	—
53	C₂×D₂₄₄	—	2	1	no	—
54	C₂×D₂₄₄	—	4	2	no	—
55	C₂×D₂₄₄	C₂×C₈	1	2	yes	—
56	C₂×C₄×D₆₁	—	2	2	no	—
57	C₂×C₄×D₆₁	—	2	2	no	—
58	C₂×C₄×D₆₁	—	4	2	no	—
59	C₂×C₄×D₆₁	—	4	2	no	—
60	C₂×C₈×C₆₁	—	2	2	no	—
61	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₆₁	D₈	1	2	yes	—
64	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
65	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
69	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
70	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
71	#11758198	—	2	2	no	—
72	#11758198	—	2	2	no	—
73	C₄⋊Dic₆₁	—	2	1	no	—
74	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
75	C₂×#92418	—	2	2	no	—
76	C₂×#92418	—	4	2	no	—
77	#11758189	—	4	4	no	—
78	#11758189	—	4	2	no	—
79	#11758189	—	4	2	no	—
80	C₁₆×C₆₁	—	4	2	no	—
81	#11758185	—	2	4	no	—
82	C₂³×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
83	C₂²×D₆₁	—	2	2	no	—
84	C₂²×D₆₁	—	4	2	no	—
85	C₂²×D₆₁	—	4	2	no	—
86	C₂²×D₆₁	—	4	2	no	—
87	C₂×C₄×C₆₁	—	2	2	no	—
88	C₂×C₄×C₆₁	—	2	2	no	—
89	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
90	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
91	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
92	C₂×C₄×C₆₁	#102	1	2	yes	—
93	C₂×C₄×C₆₁	#104	1	2	yes	—
94	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
97	C₂×Dic₆₁	—	2	2	no	—
98	C₂×Dic₆₁	—	4	2	no	—
99	D₂₄₄	—	4	2	no	—
100	D₂₄₄	—	4	2	no	—
101	C₄×D₆₁	—	4	4	no	—
102	C₄×D₆₁	—	4	2	no	—
103	C₄×D₆₁	—	4	2	no	—
104	C₈×C₆₁	—	2	1	no	—
105	C₈×C₆₁	—	2	1	no	—
106	C₈×C₆₁	#102	1	2	yes	—
107	C₈×C₆₁	C₂×D₈	1	1	yes	characteristic
108	C₈×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
109	#92418	—	4	2	no	—
110	C₂×#1330	—	61	1	no	maximal, Sylow
111	C₂²×C₆₁	D₄⋊C₈	1	2	yes	—
112	C₂²×C₆₁	#349	1	2	yes	—
113	C₂²×C₆₁	#350	1	2	yes	—
114	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
115	C₂×D₆₁	—	4	2	no	—
116	C₂×D₆₁	—	4	2	no	—
117	C₂×D₆₁	—	4	2	no	—
118	C₂×D₆₁	—	8	4	no	—
119	C₄×C₆₁	—	2	1	no	—
120	C₄×C₆₁	—	2	1	no	—
121	C₄×C₆₁	—	2	1	no	—
122	C₄×C₆₁	—	2	1	no	—
123	C₄×C₆₁	D₄⋊C₈	1	2	yes	—
124	C₄×C₆₁	C₂×#102	1	1	yes	characteristic
125	C₄×C₆₁	C₂×#104	1	1	yes	characteristic
126	Dic₆₁	—	4	2	no	—
127	C₂×#436	—	61	1	no	—
128	C₂×C₄⋊C₁₆	D₆₁	1	1	yes	characteristic, complemented
129	C₂×#346	—	61	1	no	—
130	#1330	—	61	4	no	—
131	C₂×C₆₁	#1330	1	4	yes	complemented
132	C₂×C₆₁	C₂×D₄⋊C₈	1	1	yes	characteristic
133	C₂×C₆₁	C₂×#349	1	1	yes	characteristic
134	C₂×C₆₁	C₂×#350	1	1	yes	characteristic
135	D₆₁	—	8	2	no	—
136	C₂²×M₁₆	—	122	1	no	—
137	C₂×C₄×D₄	—	61	1	no	—
138	C₂²×C₁₆	—	2	1	no	—
139	C₂²×C₁₆	—	122	1	no	—
140	#436	—	61	2	no	—
141	#436	—	61	2	no	—
142	#436	—	122	1	no	—
143	#436	—	122	1	no	—
144	C₂×C₄⋊C₈	—	61	1	no	—
145	C₂×C₂²⋊C₈	—	122	1	no	—
146	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
147	C₄⋊C₁₆	—	2	1	no	—
148	C₄⋊C₁₆	C₂×D₆₁	1	2	yes	complemented
149	#346	—	61	2	no	—
150	#346	—	122	1	no	—
151	C₆₁	C₂×#1330	1	1	yes	characteristic, complemented, Sylow
152	C₂²×D₄	—	61	1	no	—
153	C₂³×C₄	—	122	1	no	—
154	C₂×M₁₆	—	122	2	no	—
155	C₂×M₁₆	—	122	2	no	—
156	C₂×M₁₆	—	244	2	no	—
157	C₂×M₁₆	—	244	2	no	—
158	C₂²×C₈	—	122	1	no	—
159	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
160	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
161	C₄×D₄	—	61	2	no	—
162	C₄×D₄	—	61	2	no	—
163	C₄×D₄	—	122	1	no	—
164	C₄×D₄	—	122	1	no	—
165	C₂×C₄⋊C₄	—	61	1	no	—
166	C₂×C₂²⋊C₄	—	122	1	no	—
167	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
168	C₂×C₁₆	—	2	2	no	—
169	C₂×C₁₆	—	4	2	no	—
170	C₂×C₁₆	—	122	4	no	—
171	C₂×C₁₆	—	122	2	no	—
172	C₄⋊C₈	—	61	2	no	—
173	C₄⋊C₈	—	122	1	no	—
174	C₂²⋊C₈	—	122	2	no	—
175	C₂²⋊C₈	—	122	2	no	—
176	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
177	C₄×C₈	D₂₄₄	1	1	yes	characteristic
178	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
179	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
180	C₂⁴	—	122	1	no	—
181	C₂×D₄	—	61	2	no	—
182	C₂×D₄	—	122	1	no	—
183	C₂×D₄	—	244	2	no	—
184	C₂²×C₄	—	122	1	no	—
185	C₂²×C₄	—	122	2	no	—
186	C₂²×C₄	—	122	2	no	—
187	C₂²×C₄	—	244	2	no	—
188	C₂²×C₄	—	244	2	no	—
189	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
190	C₂²×C₄	#11758189	1	1	yes	characteristic
191	C₂²×C₄	#11758198	1	1	yes	characteristic
192	M₁₆	—	244	4	no	—
193	M₁₆	—	244	2	no	—
194	M₁₆	—	244	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	122	2	no	—
197	C₂×C₈	—	244	2	no	—
198	C₂×C₈	#11758198	1	1	yes	characteristic
199	C₂×C₈	#11758198	1	1	yes	characteristic
200	C₂×C₈	#11758199	1	2	yes	—
201	C₂×C₈	#11758201	1	2	yes	—
202	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
203	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
204	C₄⋊C₄	—	61	2	no	—
205	C₄⋊C₄	—	122	1	no	—
206	C₂²⋊C₄	—	122	2	no	—
207	C₂²⋊C₄	—	122	2	no	—
208	C₄²	C₈×D₆₁	1	1	yes	characteristic
209	C₄²	#11758189	1	1	yes	characteristic
210	C₄²	#11758198	1	1	yes	characteristic
211	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
212	C₁₆	—	4	2	no	—
213	C₁₆	—	122	4	no	—
214	C₂³	—	122	2	no	—
215	C₂³	—	244	2	no	—
216	C₂³	—	244	2	no	—
217	C₂³	—	244	2	no	—
218	C₂³	#14519689	1	1	yes	characteristic
219	D₄	—	244	2	no	—
220	D₄	—	244	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	122	2	no	—
224	C₂×C₄	—	244	2	no	—
225	C₂×C₄	—	244	4	no	—
226	C₂×C₄	—	244	2	no	—
227	C₂×C₄	—	244	2	no	—
228	C₂×C₄	#14519689	1	1	yes	characteristic
229	C₂×C₄	#14519689	1	1	yes	characteristic
230	C₂×C₄	#14519689	1	1	yes	characteristic
231	C₂×C₄	#14519697	1	2	yes	—
232	C₂×C₄	#14519703	1	2	yes	—
233	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
234	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
235	C₂×C₄	C₂×#11758189	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	244	2	no	—
239	C₈	#14519697	1	2	yes	—
240	C₈	C₂×#11758199	1	1	yes	characteristic
241	C₈	C₂×#11758201	1	1	yes	characteristic
242	C₂²	—	488	4	no	—
243	C₂²	—	244	2	no	—
244	C₂²	—	244	2	no	—
245	C₂²	—	244	2	no	—
246	C₂²	#16839489	1	2	yes	—
247	C₂²	#16839543	1	2	yes	—
248	C₂²	#16839545	1	2	yes	—
249	C₂²	C₂×#14519689	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	244	2	no	—
255	C₄	#16839489	1	2	yes	—
256	C₄	C₂×#14519697	1	1	yes	characteristic
257	C₄	C₂×#14519703	1	1	yes	characteristic
258	C₂	—	488	2	no	—
259	C₂	#21443776	1	4	yes	direct product factor
260	C₂	C₂×#16839489	1	1	yes	characteristic
261	C₂	C₂×#16839543	1	1	yes	characteristic
262	C₂	C₂×#16839545	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11758198
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×C₄⋊C₁₆	C₂
socle	C₂³×C₆₁	C₂²⋊C₈
prime radical	C₂²×D₂₄₄	C₈
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₆₁
hypocenter	C₆₁	C₂×#1330
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16839538	C₂	semidirect
3. #21443776	C₂	direct (N × H)
4. C₂×C₆₁×C₄⋊C₁₆	C₂	semidirect
15. #16839538	C₂²	semidirect
19. C₆₁×C₄⋊C₁₆	C₂²	semidirect
128. C₂×C₄⋊C₁₆	D₆₁	semidirect
131. C₂×C₆₁	#1330	semidirect
148. C₄⋊C₁₆	C₂×D₆₁	semidirect
151. C₆₁	C₂×#1330	semidirect
259. C₂	#21443776	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16839841	2
C₂×#16839538	2
#21443776	2
C₂×C₆₁×C₄⋊C₁₆	2
C₂×#1330	61

Subgroups — C2×#21443776 #29589727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21443776 #29589727