Subgroups — C₂×#21460649 #30596087

Order 15616.

830 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#92420	C₂	1	2	yes	maximal
2	#21460649	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×#92420	—	2	2	no	—
5	C₂×C₄⋊#92420	—	2	2	no	—
6	C₂×C₄⋊#92420	—	2	4	no	—
7	C₂×C₄⋊#92420	C₄	1	8	yes	—
8	C₂×C₄⋊#92420	C₂²	1	4	yes	—
9	C₂³×#92418	C₄	1	1	yes	characteristic
10	C₂²×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₆₁	—	2	4	no	—
12	C₂×C₈×Dic₆₁	C₄	1	2	yes	—
13	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
14	C₂³×C₈×C₆₁	C₄	1	1	yes	characteristic
15	C₂²×#92420	—	2	8	no	—
16	C₂²×#92420	—	2	4	no	—
17	C₂²×#92420	—	4	4	no	—
18	C₂²×#92420	—	4	4	no	—
19	C₄⋊#92420	—	2	8	no	—
20	C₄⋊#92420	—	2	4	no	—
21	C₄⋊#92420	C₂×C₄	1	8	yes	—
22	C₂³×Dic₆₁	D₄	1	1	yes	characteristic
23	C₂³×Dic₆₁	Q₈	1	1	yes	characteristic
24	C₂²×#92418	—	2	2	no	—
25	C₂²×#92418	—	2	2	no	—
26	C₂²×#92418	C₈	1	4	yes	complemented
27	C₂²×#92418	C₂×C₄	1	2	yes	—
28	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₆₁	C₂×C₄	1	4	yes	—
30	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
31	C₂×C₄×Dic₆₁	D₄	1	4	yes	—
32	C₂×C₄×Dic₆₁	Q₈	1	1	yes	characteristic
33	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
34	C₈×Dic₆₁	—	2	4	no	—
35	C₈×Dic₆₁	—	2	4	no	—
36	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₁	—	2	2	no	—
38	C₂²×C₈×C₆₁	—	2	2	no	—
39	C₂²×C₈×C₆₁	C₈	1	4	yes	complemented
40	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
41	C₂⁴×C₆₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
43	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
48	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
49	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×Dic₆₁	—	2	2	no	—
51	C₂²×Dic₆₁	—	2	2	no	—
52	C₂²×Dic₆₁	C₂²⋊C₄	1	4	yes	—
53	C₂²×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×Dic₆₁	C₄⋊C₄	1	4	yes	—
55	C₂²×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×Dic₆₁	D₈	1	2	yes	—
57	C₂²×Dic₆₁	SD₁₆	1	4	yes	—
58	C₂²×Dic₆₁	Q₁₆	1	2	yes	—
59	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
61	C₂×#92420	—	2	8	no	—
62	C₂×#92420	—	4	16	no	—
63	C₂×#92420	—	4	4	no	—
64	C₂×#92420	—	4	4	no	—
65	C₂×C₈×C₆₁	—	2	8	no	—
66	C₂×C₈×C₆₁	—	2	2	no	—
67	C₂×C₈×C₆₁	—	2	2	no	—
68	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	complemented
69	C₄×Dic₆₁	C₄²	1	2	yes	—
70	C₄×Dic₆₁	C₂²⋊C₄	1	2	yes	—
71	C₄×Dic₆₁	C₂²⋊C₄	1	4	yes	—
72	C₄×Dic₆₁	C₄⋊C₄	1	4	yes	—
73	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
74	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
75	C₂×#92418	—	2	8	no	—
76	C₂×#92418	—	2	2	no	—
77	C₂×#92418	—	2	2	no	—
78	C₂×#92418	C₂×C₈	1	4	yes	complemented
79	C₂³×C₆₁	#95	1	2	yes	—
80	C₂³×C₆₁	C₄⋊C₈	1	2	yes	—
81	C₂³×C₆₁	C₄⋊C₈	1	2	yes	—
82	C₂³×C₆₁	#106	1	2	yes	—
83	C₂³×C₆₁	#107	1	2	yes	—
84	C₂³×C₆₁	#108	1	2	yes	—
85	C₂³×C₆₁	#108	1	2	yes	—
86	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₆₁	#95	1	2	yes	—
88	C₂×C₄×C₆₁	C₄×C₈	1	4	yes	—
89	C₂×C₄×C₆₁	#97	1	4	yes	—
90	C₂×C₄×C₆₁	C₂²⋊C₈	1	4	yes	—
91	C₂×C₄×C₆₁	C₂²⋊C₈	1	4	yes	—
92	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
93	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
94	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
100	C₂×Dic₆₁	—	2	8	no	—
101	C₂×Dic₆₁	—	2	1	no	—
102	C₂×Dic₆₁	—	2	1	no	—
103	C₂×Dic₆₁	—	2	1	no	—
104	C₂×Dic₆₁	—	2	1	no	—
105	C₂×Dic₆₁	#95	1	2	yes	—
106	C₂×Dic₆₁	#95	1	2	yes	—
107	C₂×Dic₆₁	#102	1	8	yes	—
108	C₂×Dic₆₁	#103	1	8	yes	—
109	C₂×Dic₆₁	#106	1	2	yes	—
110	C₂×Dic₆₁	#107	1	2	yes	—
111	C₂×Dic₆₁	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂×Dic₆₁	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
114	C₂×Dic₆₁	C₂×SD₁₆	1	2	yes	—
115	C₂×Dic₆₁	C₂×Q₁₆	1	1	yes	characteristic
116	#92420	—	4	8	no	—
117	C₈×C₆₁	—	2	4	no	—
118	#92418	—	2	4	no	—
119	C₂×#1277	—	61	1	no	maximal, Sylow
120	C₂²×C₆₁	#334	1	4	yes	—
121	C₂²×C₆₁	#334	1	4	yes	—
122	C₂²×C₆₁	#335	1	4	yes	—
123	C₂²×C₆₁	#335	1	4	yes	—
124	C₂²×C₆₁	#336	1	4	yes	—
125	C₂²×C₆₁	#340	1	4	yes	—
126	C₂²×C₆₁	#341	1	4	yes	—
127	C₂²×C₆₁	C₂×#95	1	1	yes	characteristic
128	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
129	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
130	C₂²×C₆₁	C₂×#106	1	1	yes	characteristic
131	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
132	C₂²×C₆₁	C₂×#108	1	1	yes	characteristic
133	C₂²×C₆₁	C₂×#108	1	1	yes	characteristic
134	C₄×C₆₁	#336	1	4	yes	—
135	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
136	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
137	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
138	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	Dic₆₁	—	2	2	no	—
140	Dic₆₁	—	2	2	no	—
141	Dic₆₁	#340	1	4	yes	—
142	Dic₆₁	C₂×#102	1	2	yes	—
143	Dic₆₁	C₂×#103	1	2	yes	—
144	C₂²×C₄⋊C₈	—	61	2	no	—
145	C₂²×C₄×C₈	—	61	1	no	—
146	#1277	—	61	4	no	—
147	C₂×C₆₁	#1277	1	8	yes	complemented
148	C₂×C₆₁	C₂×#334	1	1	yes	characteristic
149	C₂×C₆₁	C₂×#334	1	1	yes	characteristic
150	C₂×C₆₁	C₂×#335	1	1	yes	characteristic
151	C₂×C₆₁	C₂×#335	1	1	yes	characteristic
152	C₂×C₆₁	C₂×#336	1	1	yes	characteristic
153	C₂×C₆₁	C₂×#340	1	1	yes	characteristic
154	C₂×C₆₁	C₂×#341	1	1	yes	characteristic
155	C₂³×C₈	—	61	1	no	—
156	C₂³×C₈	—	122	2	no	—
157	C₂³×C₈	#6923	1	1	yes	characteristic
158	C₂²×C₄²	—	61	1	no	—
159	C₂×C₄⋊C₈	—	61	8	no	—
160	C₂×C₄⋊C₈	—	61	4	no	—
161	C₂×C₄⋊C₈	—	122	2	no	—
162	C₂×C₄⋊C₈	—	122	4	no	—
163	C₂×C₄×C₈	—	61	2	no	—
164	C₂×C₄×C₈	—	61	2	no	—
165	C₂×C₄×C₈	—	122	4	no	—
166	C₆₁	C₂×#1277	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₄	—	61	1	no	—
168	C₂³×C₄	—	61	1	no	—
169	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	61	4	no	—
173	C₂²×C₈	—	61	2	no	—
174	C₂²×C₈	—	122	2	no	—
175	C₂²×C₈	—	122	2	no	—
176	C₂²×C₈	—	122	8	no	—
177	C₂²×C₈	—	122	4	no	—
178	C₂²×C₈	—	244	4	no	—
179	C₂²×C₈	—	244	4	no	—
180	C₂²×C₈	#92420	1	4	yes	complemented
181	C₂²×C₈	C₂×#6923	1	2	yes	—
182	C₂×C₄²	—	61	4	no	—
183	C₂×C₄²	—	61	1	no	—
184	C₂×C₄²	—	61	1	no	—
185	C₂×C₄²	—	61	4	no	—
186	C₂×C₄²	—	61	1	no	—
187	C₂×C₄²	—	61	1	no	—
188	C₄⋊C₈	—	61	8	no	—
189	C₄⋊C₈	—	122	8	no	—
190	C₄⋊C₈	—	122	4	no	—
191	C₄×C₈	—	122	4	no	—
192	C₄×C₈	—	122	4	no	—
193	C₂⁴	C₄⋊#6923	1	1	yes	characteristic
194	C₂²×C₄	—	61	4	no	—
195	C₂²×C₄	—	61	4	no	—
196	C₂²×C₄	—	61	2	no	—
197	C₂²×C₄	—	61	2	no	—
198	C₂²×C₄	—	61	1	no	—
199	C₂²×C₄	—	61	1	no	—
200	C₂²×C₄	—	61	4	no	—
201	C₂²×C₄	—	61	1	no	—
202	C₂²×C₄	—	61	1	no	—
203	C₂²×C₄	—	122	2	no	—
204	C₂²×C₄	—	122	2	no	—
205	C₂²×C₄	D₆₁⋊C₈	1	2	yes	—
206	C₂²×C₄	#11758213	1	2	yes	—
207	C₂²×C₄	C₄×#6923	1	2	yes	—
208	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
209	C₂²×C₄	C₂×#92420	1	2	yes	—
210	C₂²×C₄	#11758217	1	2	yes	—
211	C₂²×C₄	C₂²⋊#6923	1	2	yes	—
212	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
213	C₂×C₈	—	2	8	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	61	4	no	—
217	C₂×C₈	—	122	8	no	—
218	C₂×C₈	—	122	2	no	—
219	C₂×C₈	—	122	2	no	—
220	C₂×C₈	—	122	8	no	—
221	C₂×C₈	—	244	4	no	—
222	C₂×C₈	—	244	4	no	—
223	C₂×C₈	—	244	16	no	—
224	C₂×C₈	C₂×#92420	1	4	yes	complemented
225	C₄²	—	61	4	no	—
226	C₄²	—	61	4	no	—
227	C₄²	—	61	2	no	—
228	C₄²	—	61	2	no	—
229	C₄²	—	61	2	no	—
230	C₄²	—	61	2	no	—
231	C₂³	(C₂×#6923)⋊C₄	1	2	yes	—
232	C₂³	C₄⋊#92420	1	2	yes	—
233	C₂³	#14519844	1	2	yes	—
234	C₂³	#14519845	1	2	yes	—
235	C₂³	#14519846	1	2	yes	—
236	C₂³	#14519847	1	2	yes	—
237	C₂³	#14519848	1	2	yes	—
238	C₂³	C₂×C₄⋊#6923	1	1	yes	characteristic
239	C₂×C₄	—	61	2	no	—
240	C₂×C₄	—	61	2	no	—
241	C₂×C₄	—	61	2	no	—
242	C₂×C₄	—	61	1	no	—
243	C₂×C₄	—	61	1	no	—
244	C₂×C₄	—	61	8	no	—
245	C₂×C₄	—	61	8	no	—
246	C₂×C₄	—	61	2	no	—
247	C₂×C₄	—	61	2	no	—
248	C₂×C₄	—	61	2	no	—
249	C₂×C₄	—	61	2	no	—
250	C₂×C₄	—	122	1	no	—
251	C₂×C₄	—	122	1	no	—
252	C₂×C₄	—	122	1	no	—
253	C₂×C₄	—	122	1	no	—
254	C₂×C₄	—	122	8	no	—
255	C₂×C₄	(C₂×#6923)⋊C₄	1	2	yes	—
256	C₂×C₄	C₄×#92420	1	4	yes	—
257	C₂×C₄	#14519832	1	4	yes	—
258	C₂×C₄	C₂²⋊#92420	1	4	yes	—
259	C₂×C₄	#14519834	1	4	yes	—
260	C₂×C₄	C₄⋊#92420	1	2	yes	—
261	C₂×C₄	#14519844	1	2	yes	—
262	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
263	C₂×C₄	C₂×C₂²⋊#6923	1	1	yes	characteristic
264	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
265	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
266	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
267	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
268	C₈	—	2	4	no	—
269	C₈	—	122	4	no	—
270	C₈	—	244	8	no	—
271	C₂²	#16840623	1	4	yes	—
272	C₂²	#16840624	1	4	yes	—
273	C₂²	#16840625	1	4	yes	—
274	C₂²	#16840626	1	4	yes	—
275	C₂²	#16840628	1	4	yes	—
276	C₂²	#16840636	1	4	yes	—
277	C₂²	#16840638	1	4	yes	—
278	C₂²	C₂×(C₂×#6923)⋊C₄	1	1	yes	characteristic
279	C₂²	C₂×C₄⋊#92420	1	1	yes	characteristic
280	C₂²	C₂×#14519844	1	1	yes	characteristic
281	C₂²	C₂×#14519845	1	1	yes	characteristic
282	C₂²	C₂×#14519846	1	1	yes	characteristic
283	C₂²	C₂×#14519847	1	1	yes	characteristic
284	C₂²	C₂×#14519848	1	1	yes	characteristic
285	C₄	—	61	4	no	—
286	C₄	—	61	2	no	—
287	C₄	—	61	2	no	—
288	C₄	—	122	2	no	—
289	C₄	—	122	2	no	—
290	C₄	#16840628	1	4	yes	—
291	C₄	C₂×C₄×#92420	1	1	yes	characteristic
292	C₄	C₂×#14519832	1	1	yes	characteristic
293	C₄	C₂×C₂²⋊#92420	1	1	yes	characteristic
294	C₄	C₂×#14519834	1	1	yes	characteristic
295	C₂	#21460649	1	8	yes	direct product factor
296	C₂	C₂×#16840623	1	1	yes	characteristic
297	C₂	C₂×#16840624	1	1	yes	characteristic
298	C₂	C₂×#16840625	1	1	yes	characteristic
299	C₂	C₂×#16840626	1	1	yes	characteristic
300	C₂	C₂×#16840628	1	1	yes	characteristic
301	C₂	C₂×#16840636	1	1	yes	characteristic
302	C₂	C₂×#16840638	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₄⋊#6923
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂³×C₈×C₆₁	C₄
socle, prime radical	C₂⁴×C₆₁	C₄⋊C₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂³×C₈	#6923
hypocenter	C₆₁	C₂×#1277
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. #21460649	C₂	direct (N × H)
26. C₂²×#92418	C₈	semidirect
39. C₂²×C₈×C₆₁	C₈	semidirect
68. C₂×C₈×C₆₁	C₂×C₈	semidirect
78. C₂×#92418	C₂×C₈	semidirect
147. C₂×C₆₁	#1277	semidirect
166. C₆₁	C₂×#1277	semidirect
180. C₂²×C₈	#92420	semidirect
224. C₂×C₈	C₂×#92420	semidirect
295. C₂	#21460649	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×C₄⋊#92420	2
#21460649	2
C₂²×C₈×Dic₆₁	2
C₂×#1277	61

Subgroups — C2×#21460649 #30596087

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21460649 #30596087