Subgroups — C₂×#21445210 #29662366

Order 15616.

702 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839969	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
3	#21445210	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16839515	C₂	1	1	yes	maximal, characteristic
5	C₂×#16839508	C₂	1	2	yes	maximal
6	C₂×C₆₁×#500	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11758202	—	2	2	no	—
8	C₂×C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
9	#16839969	—	2	4	no	—
10	#16839969	C₂²	1	8	yes	—
11	C₈×Dic₁₂₂	—	2	2	no	—
12	C₈×Dic₁₂₂	C₂²	1	4	yes	—
13	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#92418	C₂²	1	2	yes	—
15	C₂×#14519706	—	2	1	no	—
16	C₂×#14519701	—	2	2	no	—
17	C₂×C₈×Dic₆₁	—	2	1	no	—
18	#16839515	C₂²	1	4	yes	—
19	#16839508	C₂²	1	8	yes	—
20	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
21	C₆₁×#500	—	2	2	no	—
22	C₆₁×#500	C₂²	1	4	yes	—
23	C₂×#107×C₆₁	—	2	1	no	—
24	C₂×#107×C₆₁	—	2	1	no	—
25	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₂₂	D₄	1	1	yes	characteristic
27	C₂×#11758202	—	2	8	no	—
28	C₂×#11758202	—	4	8	no	—
29	C₂²×#92418	—	2	1	no	—
30	C₂²×#92418	—	2	2	no	—
31	C₄×Dic₁₂₂	—	2	1	no	—
32	C₄×Dic₁₂₂	—	2	1	no	—
33	C₄×Dic₁₂₂	D₄	1	2	yes	—
34	C₄×Dic₁₂₂	C₂³	1	2	yes	—
35	C₂×C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
36	C₂×#11758196	—	2	1	no	—
37	C₂×C₄×Dic₆₁	—	2	1	no	—
38	C₄⋊#92418	—	2	2	no	—
39	C₄⋊#92418	—	2	1	no	—
40	C₄⋊#92418	C₂³	1	4	yes	—
41	C₄⋊#92418	C₂³	1	2	yes	—
42	#14519706	—	2	4	no	—
43	#14519701	—	2	8	no	—
44	C₈×Dic₆₁	—	2	4	no	—
45	C₂²×Q₈×C₆₁	—	2	2	no	—
46	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
48	C₄⋊Q₈×C₆₁	—	2	2	no	—
49	C₄⋊Q₈×C₆₁	—	2	2	no	—
50	C₄⋊Q₈×C₆₁	—	2	2	no	—
51	C₄⋊Q₈×C₆₁	C₂³	1	4	yes	—
52	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
53	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
54	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
55	#107×C₆₁	—	2	4	no	—
56	#107×C₆₁	—	2	4	no	—
57	C₄×C₈×C₆₁	D₄	1	2	yes	—
58	C₄×C₈×C₆₁	C₂³	1	2	yes	—
59	C₂×Q₈×C₆₁	—	2	4	no	—
60	C₂×Q₈×C₆₁	—	4	2	no	—
61	C₂×Q₈×C₆₁	—	4	4	no	—
62	C₂×Q₈×C₆₁	—	4	4	no	—
63	C₂²×C₄×C₆₁	—	4	2	no	—
64	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
67	C₂²×Dic₆₁	—	2	1	no	—
68	C₂²×Dic₆₁	—	2	1	no	—
69	C₂×Dic₁₂₂	—	2	1	no	—
70	C₂×Dic₁₂₂	—	2	2	no	—
71	C₂×Dic₁₂₂	D₈	1	4	yes	—
72	C₂×Dic₁₂₂	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₆₁	—	2	2	no	—
74	C₂×C₈×C₆₁	Q₁₆	1	4	yes	complemented
75	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
77	C₄⋊C₄×C₆₁	—	2	4	no	—
78	C₄⋊C₄×C₆₁	—	2	4	no	—
79	C₄⋊C₄×C₆₁	—	4	2	no	—
80	C₄⋊C₄×C₆₁	—	4	2	no	—
81	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₆₁	#41	1	1	yes	characteristic
84	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
85	#11758202	—	4	16	no	—
86	C₄⋊Dic₆₁	—	2	1	no	—
87	C₄⋊Dic₆₁	C₂×D₄	1	2	yes	—
88	#11758196	—	2	2	no	—
89	#11758196	—	2	2	no	—
90	C₄×Dic₆₁	—	2	2	no	—
91	C₄×Dic₆₁	—	2	2	no	—
92	C₂×#92418	—	2	2	no	—
93	C₂×#92418	—	2	4	no	—
94	C₂×#92418	—	4	4	no	—
95	C₂×#92418	—	4	2	no	—
96	C₂³×C₆₁	#121	1	1	yes	characteristic
97	Q₈×C₆₁	—	4	4	no	—
98	Q₈×C₆₁	—	4	4	no	—
99	Q₈×C₆₁	—	8	4	no	—
100	C₂×C₄×C₆₁	—	2	2	no	—
101	C₂×C₄×C₆₁	—	4	4	no	—
102	C₂×C₄×C₆₁	—	8	4	no	—
103	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
104	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
105	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
108	C₂×C₄×C₆₁	#135	1	2	yes	—
109	C₂×C₄×C₆₁	#137	1	2	yes	—
110	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
113	C₂×Dic₆₁	—	2	2	no	—
114	C₂×Dic₆₁	—	2	4	no	—
115	C₂×Dic₆₁	—	2	2	no	—
116	C₂×Dic₆₁	—	4	2	no	—
117	Dic₁₂₂	—	2	2	no	—
118	Dic₁₂₂	—	2	4	no	—
119	Dic₁₂₂	C₂×D₈	1	4	yes	—
120	C₈×C₆₁	—	2	2	no	—
121	C₈×C₆₁	C₂×Q₁₆	1	4	yes	complemented
122	#92418	—	4	4	no	—
123	#92418	—	4	2	no	—
124	C₂×#1674	—	61	1	no	maximal, Sylow
125	C₂²×C₆₁	#462	1	2	yes	—
126	C₂²×C₆₁	#466	1	2	yes	—
127	C₂²×C₆₁	#473	1	2	yes	—
128	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
129	C₄×C₆₁	—	2	1	no	—
130	C₄×C₆₁	—	2	1	no	—
131	C₄×C₆₁	—	8	4	no	—
132	C₄×C₆₁	#462	1	2	yes	—
133	C₄×C₆₁	#466	1	2	yes	—
134	C₄×C₆₁	C₂²×D₈	1	1	yes	characteristic
135	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
136	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
137	C₄×C₆₁	C₂×#137	1	1	yes	characteristic
138	Dic₆₁	—	2	4	no	—
139	Dic₆₁	—	4	2	no	—
140	C₂×#500	D₆₁	1	1	yes	characteristic
141	C₂×#462	—	61	2	no	—
142	C₂×C₈×Q₈	—	61	1	no	—
143	#1674	—	61	8	no	—
144	C₂×#335	—	61	1	no	—
145	C₂×#332	—	61	2	no	—
146	C₂×C₆₁	#1674	1	4	yes	complemented
147	C₂×C₆₁	C₂×#462	1	1	yes	characteristic
148	C₂×C₆₁	C₂×#466	1	1	yes	characteristic
149	C₂×C₆₁	C₂×#473	1	1	yes	characteristic
150	C₂²×Q₁₆	—	122	2	no	—
151	C₂×C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
152	C₂×C₄×Q₈	—	61	1	no	—
153	#500	—	2	2	no	—
154	#500	C₂×D₆₁	1	4	yes	—
155	#462	—	61	8	no	—
156	#462	—	122	4	no	—
157	C₈×Q₈	—	61	4	no	—
158	C₈×Q₈	—	122	2	no	—
159	C₂×#107	—	2	1	no	—
160	C₂×#107	—	2	1	no	—
161	C₂×C₄⋊C₈	—	61	1	no	—
162	C₂×C₄⋊C₈	—	61	2	no	—
163	C₂×C₄⋊C₈	—	122	1	no	—
164	C₂×#103	—	122	2	no	—
165	C₂×C₄×C₈	—	122	1	no	—
166	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
167	#335	—	61	4	no	—
168	#332	—	61	8	no	—
169	C₆₁	C₂×#1674	1	1	yes	characteristic, complemented, Sylow
170	C₂²×Q₈	—	2	2	no	—
171	C₂²×Q₈	—	61	1	no	—
172	C₂×Q₁₆	—	122	8	no	—
173	C₂×Q₁₆	—	244	8	no	—
174	C₂²×C₈	—	122	2	no	—
175	C₂²×C₈	—	122	1	no	—
176	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
177	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
178	C₄⋊Q₈	—	2	2	no	—
179	C₄⋊Q₈	—	2	2	no	—
180	C₄⋊Q₈	—	2	2	no	—
181	C₄⋊Q₈	C₂²×D₆₁	1	4	yes	—
182	C₄×Q₈	—	61	2	no	—
183	C₄×Q₈	—	61	2	no	—
184	C₄×Q₈	—	122	1	no	—
185	C₄×Q₈	—	122	1	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	61	1	no	—
189	C₂×C₄⋊C₄	—	122	1	no	—
190	C₂×C₄²	—	122	1	no	—
191	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
192	#107	—	2	4	no	—
193	#107	—	2	4	no	—
194	C₄⋊C₈	—	61	2	no	—
195	C₄⋊C₈	—	61	4	no	—
196	C₄⋊C₈	—	122	4	no	—
197	C₄⋊C₈	—	122	2	no	—
198	C₄⋊C₈	—	122	1	no	—
199	#103	—	122	8	no	—
200	C₄×C₈	—	122	4	no	—
201	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
202	C₄×C₈	C₂²×D₆₁	1	2	yes	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	—	4	2	no	—
205	C₂×Q₈	—	4	4	no	—
206	C₂×Q₈	—	4	4	no	—
207	C₂×Q₈	—	61	2	no	—
208	C₂×Q₈	—	61	4	no	—
209	C₂×Q₈	—	122	1	no	—
210	C₂×Q₈	—	122	2	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	—	122	1	no	—
213	C₂²×C₄	—	122	1	no	—
214	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
215	C₂²×C₄	#11758224	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
217	Q₁₆	—	244	16	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	122	4	no	—
220	C₂×C₈	—	122	2	no	—
221	C₂×C₈	—	244	2	no	—
222	C₂×C₈	—	244	4	no	—
223	C₂×C₈	#11758202	1	4	yes	complemented
224	C₂×C₈	C₂×C₂²⋊D₆₁	1	2	yes	—
225	C₂×C₈	C₂×C₂²⋊D₆₁	1	2	yes	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	61	2	no	—
231	C₄⋊C₄	—	122	2	no	—
232	C₄⋊C₄	—	122	1	no	—
233	C₄⋊C₄	—	122	2	no	—
234	C₄²	—	122	2	no	—
235	C₄²	—	122	2	no	—
236	C₄²	D₄×D₆₁	1	1	yes	characteristic
237	C₄²	#11758224	1	1	yes	characteristic
238	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
239	C₄²	C₂³×D₆₁	1	1	yes	characteristic
240	C₂³	#14519748	1	1	yes	characteristic
241	Q₈	—	4	4	no	—
242	Q₈	—	4	4	no	—
243	Q₈	—	8	4	no	—
244	Q₈	—	61	4	no	—
245	Q₈	—	122	4	no	—
246	Q₈	—	122	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	8	4	no	—
250	C₂×C₄	—	122	4	no	—
251	C₂×C₄	—	122	2	no	—
252	C₂×C₄	—	122	2	no	—
253	C₂×C₄	—	244	2	no	—
254	C₂×C₄	#14519748	1	1	yes	characteristic
255	C₂×C₄	#14519748	1	1	yes	characteristic
256	C₂×C₄	#14519748	1	1	yes	characteristic
257	C₂×C₄	D₈×D₆₁	1	2	yes	—
258	C₂×C₄	C₂×#11758202	1	2	yes	—
259	C₂×C₄	#14519796	1	2	yes	—
260	C₂×C₄	#14519808	1	2	yes	—
261	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
263	C₂×C₄	C₂×#11758224	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	244	4	no	—
266	C₈	—	244	2	no	—
267	C₈	C₂×#11758202	1	4	yes	complemented
268	C₂²	#16839969	1	2	yes	—
269	C₂²	#16839995	1	2	yes	—
270	C₂²	#16840034	1	2	yes	—
271	C₂²	C₂×#14519748	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	8	4	no	—
275	C₄	—	122	4	no	—
276	C₄	—	244	2	no	—
277	C₄	#16839969	1	2	yes	—
278	C₄	#16839995	1	2	yes	—
279	C₄	C₂×D₈×D₆₁	1	1	yes	characteristic
280	C₄	C₂²×#11758202	1	1	yes	characteristic
281	C₄	C₂×#14519796	1	1	yes	characteristic
282	C₄	C₂×#14519808	1	1	yes	characteristic
283	C₂	#21445210	1	4	yes	direct product factor
284	C₂	C₂×#16839969	1	1	yes	characteristic
285	C₂	C₂×#16839995	1	1	yes	characteristic
286	C₂	C₂×#16840034	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519748
commutator, cosocle	C₄²×C₆₁	C₂⁴
Frattini	C₄²	C₂³×D₆₁
Fitting	C₂×C₆₁×#500	C₂
socle, prime radical	C₂³×C₆₁	#121
hypercenter	C₂×#500	D₆₁
hypocenter	C₆₁	C₂×#1674
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #21445210	C₂	direct (N × H)
74. C₂×C₈×C₆₁	Q₁₆	semidirect
121. C₈×C₆₁	C₂×Q₁₆	semidirect
146. C₂×C₆₁	#1674	semidirect
169. C₆₁	C₂×#1674	semidirect
223. C₂×C₈	#11758202	semidirect
267. C₈	C₂×#11758202	semidirect
283. C₂	#21445210	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×#16839969	2
C₂×C₈×Dic₁₂₂	2
#21445210	2
C₂×#16839515	2
C₂×#16839508	2
C₂×C₆₁×#500	2
C₂×#1674	61

Subgroups — C2×#21445210 #29662366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21445210 #29662366