Subgroups — C₂×#21461366 #30611999

Order 15616.

830 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840778	C₂	1	2	yes	maximal
2	#21461366	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16840625	C₂	1	2	yes	maximal, complemented
4	C₂×#16840623	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	#16840778	—	2	4	no	—
7	#16840778	C₂²	1	8	yes	—
8	C₂×C₄⋊#92420	C₂²	1	4	yes	—
9	C₂×C₂²⋊#92420	—	2	2	no	—
10	C₂×C₄×#92420	—	2	2	no	—
11	#16840625	C₂²	1	8	yes	complemented
12	#16840623	C₂²	1	8	yes	complemented
13	C₂³×#92418	C₄	1	1	yes	characteristic
14	C₂²×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₆₁	C₄	1	4	yes	—
18	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₆₁	C₂²	1	4	yes	—
21	C₂³×C₈×C₆₁	C₄	1	1	yes	characteristic
22	C₂²×#92420	—	2	4	no	—
23	C₄⋊#92420	—	2	4	no	—
24	C₄⋊#92420	C₂³	1	8	yes	—
25	C₂²⋊#92420	—	2	8	no	—
26	C₄×#92420	—	2	8	no	—
27	C₂³×Dic₆₁	D₄	1	1	yes	characteristic
28	C₂³×Dic₆₁	Q₈	1	1	yes	characteristic
29	C₂²×#92418	—	2	4	no	—
30	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
31	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
32	C₂²×#92418	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
34	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
36	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
37	C₂×C₄×Dic₆₁	Q₈	1	2	yes	—
38	C₂×C₄×Dic₆₁	Q₈	1	1	yes	characteristic
39	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
40	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
41	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
42	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
43	C₈×Dic₆₁	C₂×C₄	1	4	yes	—
44	C₈×Dic₆₁	C₂×C₄	1	4	yes	—
45	C₈×Dic₆₁	C₂³	1	2	yes	—
46	C₈×Dic₆₁	C₂³	1	2	yes	—
47	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₁	C₈	1	8	yes	complemented
49	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
52	C₂⁴×C₆₁	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
56	C₂²×C₄×C₆₁	M₁₆	1	4	yes	—
57	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
59	C₂²×Dic₆₁	—	2	4	no	—
60	C₂²×Dic₆₁	—	2	4	no	—
61	C₂²×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
63	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
64	C₂²×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×Dic₆₁	C₂×D₄	1	2	yes	—
66	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
68	C₂²×Dic₆₁	C₂×Q₈	1	2	yes	—
69	C₂×#92420	—	2	8	no	—
70	C₂×#92420	—	4	8	no	—
71	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
72	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
73	C₂×C₈×C₆₁	C₂×C₈	1	16	yes	complemented
74	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
76	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
77	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
78	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
79	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
80	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
81	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
82	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
83	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
84	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
85	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
86	C₂×#92418	—	2	8	no	—
87	C₂×#92418	—	2	2	no	—
88	C₂×#92418	—	2	2	no	—
89	C₂×#92418	C₂²×C₄	1	2	yes	—
90	C₂×#92418	C₂²×C₄	1	2	yes	—
91	C₂³×C₆₁	—	2	2	no	—
92	C₂³×C₆₁	—	2	2	no	—
93	C₂³×C₆₁	C₄⋊C₈	1	4	yes	—
94	C₂³×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₄⋊C₈	1	8	yes	—
97	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
98	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
101	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
102	C₂×C₄×C₆₁	C₂×M₁₆	1	4	yes	—
103	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
104	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
105	C₂×Dic₆₁	—	2	2	no	—
106	C₂×Dic₆₁	—	2	2	no	—
107	C₂×Dic₆₁	—	2	4	no	—
108	C₂×Dic₆₁	—	2	4	no	—
109	C₂×Dic₆₁	—	2	4	no	—
110	C₂×Dic₆₁	—	2	4	no	—
111	C₂×Dic₆₁	#108	1	4	yes	—
112	C₂×Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×Dic₆₁	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×Dic₆₁	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×Dic₆₁	#135	1	4	yes	—
117	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
118	C₂×Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
119	#92420	—	4	8	no	—
120	C₈×C₆₁	C₂²×C₈	1	4	yes	—
121	C₈×C₆₁	C₂²×C₈	1	4	yes	—
122	#92418	—	2	2	no	—
123	#92418	—	2	2	no	—
124	C₂×#1564	—	61	1	no	maximal, Sylow
125	C₂²×C₆₁	—	2	8	no	—
126	C₂²×C₆₁	—	2	2	no	—
127	C₂²×C₆₁	—	2	2	no	—
128	C₂²×C₆₁	C₂×C₄⋊C₈	1	4	yes	—
129	C₂²×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
130	C₂²×C₆₁	#427	1	2	yes	—
131	C₂²×C₆₁	C₂×#108	1	2	yes	—
132	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
134	C₄×C₆₁	C₂×C₄⋊C₈	1	4	yes	—
135	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
136	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
137	Dic₆₁	—	2	2	no	—
138	Dic₆₁	—	2	1	no	—
139	Dic₆₁	—	2	1	no	—
140	Dic₆₁	#427	1	2	yes	—
141	Dic₆₁	C₂×#108	1	4	yes	—
142	Dic₆₁	C₂×#135	1	2	yes	—
143	C₂×#431	—	61	2	no	—
144	C₂²×C₄×C₈	—	61	1	no	—
145	#1564	—	61	8	no	—
146	C₂×#335	—	61	2	no	—
147	C₂×#334	—	61	2	no	—
148	C₂×C₆₁	—	2	4	no	—
149	C₂×C₆₁	#1564	1	4	yes	complemented
150	C₂×C₆₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
151	C₂×C₆₁	C₂×#427	1	1	yes	characteristic
152	C₂×C₆₁	C₂²×#108	1	1	yes	characteristic
153	C₂³×C₈	—	61	1	no	—
154	C₂³×C₈	#6923	1	1	yes	characteristic
155	C₂²×C₄²	—	61	1	no	—
156	#431	—	61	8	no	—
157	#431	—	122	4	no	—
158	C₂×C₄⋊C₈	—	61	4	no	—
159	C₂×C₂²⋊C₈	—	122	2	no	—
160	C₂×C₄×C₈	—	61	1	no	—
161	C₂×C₄×C₈	—	61	1	no	—
162	C₂×C₄×C₈	—	61	1	no	—
163	C₂×C₄×C₈	—	61	1	no	—
164	C₂×C₄×C₈	—	61	4	no	—
165	C₂×C₄×C₈	—	61	4	no	—
166	C₂×C₄×C₈	—	122	2	no	—
167	#335	—	61	8	no	—
168	#334	—	61	8	no	—
169	C₆₁	C₂×#1564	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₄	—	61	1	no	—
171	C₂³×C₄	—	61	1	no	—
172	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
173	C₂²×C₈	—	61	1	no	—
174	C₂²×C₈	—	61	1	no	—
175	C₂²×C₈	—	61	4	no	—
176	C₂²×C₈	—	122	4	no	—
177	C₂²×C₈	—	122	4	no	—
178	C₂²×C₈	#92420	1	8	yes	complemented
179	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
180	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
181	C₂²×C₈	C₂×#6923	1	4	yes	—
182	C₂×C₄²	—	61	2	no	—
183	C₂×C₄²	—	61	2	no	—
184	C₂×C₄²	—	61	1	no	—
185	C₂×C₄²	—	61	1	no	—
186	C₂×C₄²	—	61	1	no	—
187	C₂×C₄²	—	61	1	no	—
188	C₂×C₄²	—	61	2	no	—
189	C₂×C₄²	—	61	2	no	—
190	C₄⋊C₈	—	61	8	no	—
191	C₄⋊C₈	—	122	4	no	—
192	C₂²⋊C₈	—	122	8	no	—
193	C₄×C₈	—	61	2	no	—
194	C₄×C₈	—	61	2	no	—
195	C₄×C₈	—	61	2	no	—
196	C₄×C₈	—	61	2	no	—
197	C₄×C₈	—	61	4	no	—
198	C₄×C₈	—	61	4	no	—
199	C₄×C₈	—	122	8	no	—
200	C₂⁴	C₄⋊#6923	1	1	yes	characteristic
201	C₂²×C₄	—	61	1	no	—
202	C₂²×C₄	—	61	1	no	—
203	C₂²×C₄	—	61	2	no	—
204	C₂²×C₄	—	61	2	no	—
205	C₂²×C₄	—	61	2	no	—
206	C₂²×C₄	—	61	1	no	—
207	C₂²×C₄	—	61	2	no	—
208	C₂²×C₄	—	61	1	no	—
209	C₂²×C₄	—	122	4	no	—
210	C₂²×C₄	—	122	4	no	—
211	C₂²×C₄	#11758213	1	4	yes	—
212	C₂²×C₄	C₄⋊#6923	1	2	yes	—
213	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
214	C₂²×C₄	C₂×#92420	1	4	yes	—
215	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
216	C₂²×C₄	C₂²×#6923	1	2	yes	—
217	C₂×C₈	—	61	2	no	—
218	C₂×C₈	—	61	2	no	—
219	C₂×C₈	—	122	8	no	—
220	C₂×C₈	—	122	2	no	—
221	C₂×C₈	—	122	2	no	—
222	C₂×C₈	—	122	8	no	—
223	C₂×C₈	—	244	8	no	—
224	C₂×C₈	C₂×#92420	1	4	yes	—
225	C₂×C₈	C₂×#92420	1	4	yes	—
226	C₂×C₈	C₂×#92420	1	16	yes	complemented
227	C₂×C₈	C₂²×#6923	1	2	yes	—
228	C₂×C₈	C₂²×#6923	1	2	yes	—
229	C₄²	—	61	2	no	—
230	C₄²	—	61	1	no	—
231	C₄²	—	61	2	no	—
232	C₄²	—	61	2	no	—
233	C₄²	—	61	2	no	—
234	C₄²	—	61	1	no	—
235	C₄²	—	61	2	no	—
236	C₄²	—	61	1	no	—
237	C₄²	—	61	1	no	—
238	C₄²	—	61	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	C₄⋊#92420	1	4	yes	—
242	C₂³	C₂×C₄⋊#6923	1	2	yes	—
243	C₂³	C₂×C₄⋊#6923	1	1	yes	characteristic
244	C₂×C₄	—	61	4	no	—
245	C₂×C₄	—	61	4	no	—
246	C₂×C₄	—	61	1	no	—
247	C₂×C₄	—	61	2	no	—
248	C₂×C₄	—	61	1	no	—
249	C₂×C₄	—	61	2	no	—
250	C₂×C₄	—	61	1	no	—
251	C₂×C₄	—	61	1	no	—
252	C₂×C₄	—	122	4	no	—
253	C₂×C₄	—	122	4	no	—
254	C₂×C₄	—	122	4	no	—
255	C₂×C₄	—	122	4	no	—
256	C₂×C₄	—	122	2	no	—
257	C₂×C₄	—	122	2	no	—
258	C₂×C₄	C₄⋊#92420	1	8	yes	—
259	C₂×C₄	C₄⋊#92420	1	4	yes	—
260	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
261	C₂×C₄	C₂×C₄⋊#6923	1	2	yes	—
262	C₂×C₄	C₂²×#92420	1	2	yes	—
263	C₂×C₄	C₂²×#92420	1	4	yes	—
264	C₂×C₄	C₂×#11758213	1	4	yes	—
265	C₂×C₄	C₂×#11758213	1	2	yes	—
266	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
267	C₈	—	122	2	no	—
268	C₈	—	122	2	no	—
269	C₈	—	244	8	no	—
270	C₈	C₂²×#92420	1	4	yes	—
271	C₈	C₂²×#92420	1	4	yes	—
272	C₂²	—	2	8	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	C₂×C₄⋊#92420	1	4	yes	—
276	C₂²	C₂×C₄⋊#92420	1	2	yes	—
277	C₂²	#16840772	1	2	yes	—
278	C₂²	D₆₁⋊#108	1	2	yes	—
279	C₂²	C₂²×C₄⋊#6923	1	1	yes	characteristic
280	C₄	—	61	2	no	—
281	C₄	—	61	2	no	—
282	C₄	—	61	4	no	—
283	C₄	—	122	2	no	—
284	C₄	—	122	1	no	—
285	C₄	—	122	1	no	—
286	C₄	C₂×C₄⋊#92420	1	2	yes	—
287	C₄	C₂×C₄⋊#92420	1	4	yes	—
288	C₄	C₂³×#92420	1	1	yes	characteristic
289	C₄	C₂²×#11758213	1	1	yes	characteristic
290	C₂	—	2	4	no	—
291	C₂	#21461366	1	4	yes	direct product factor
292	C₂	C₂²×C₄⋊#92420	1	1	yes	characteristic
293	C₂	C₂×#16840772	1	1	yes	characteristic
294	C₂	C₂×D₆₁⋊#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄⋊#6923
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₂³×C₈×C₆₁	C₄
socle	C₂³×C₆₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₁	C₄⋊C₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	C₂³×C₈	#6923
hypocenter	C₆₁	C₂×#1564
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. #21461366	C₂	direct (N × H)
3. C₂×#16840625	C₂	semidirect
4. C₂×#16840623	C₂	semidirect
11. #16840625	C₂²	semidirect
12. #16840623	C₂²	semidirect
48. C₂²×C₈×C₆₁	C₈	semidirect
73. C₂×C₈×C₆₁	C₂×C₈	semidirect
149. C₂×C₆₁	#1564	semidirect
169. C₆₁	C₂×#1564	semidirect
178. C₂²×C₈	#92420	semidirect
226. C₂×C₈	C₂×#92420	semidirect
291. C₂	#21461366	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×#16840778	2
#21461366	2
C₂×#16840625	2
C₂×#16840623	2
C₂²×C₈×Dic₆₁	2
C₂×#1564	61

Subgroups — C2×#21461366 #30611999

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21461366 #30611999