Subgroups — C₂×#21460659 #30596276

Order 15616.

678 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	C₂×#16840778	C₂	1	2	yes	maximal
2	#21460659	C₂	1	4	yes	maximal, direct product factor
3	C₂²×#14519687	C₂	1	1	yes	maximal, characteristic
4	#16840778	—	2	2	no	—
5	#16840778	—	2	2	no	—
6	#16840778	C₄	1	4	yes	—
7	#16840778	C₂²	1	4	yes	—
8	C₂×C₄⋊#92420	—	2	2	no	—
9	C₂×C₂²⋊#92420	C₄	1	4	yes	—
10	C₂×C₄×#92420	—	2	2	no	—
11	C₂³×#92418	C₄	1	1	yes	characteristic
12	C₂²×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
13	C₂×#14519687	—	2	2	no	—
14	C₂×#14519687	—	2	2	no	—
15	C₂×#14519687	C₄	1	2	yes	—
16	C₂×#14519687	C₂²	1	2	yes	—
17	C₂³×C₈×C₆₁	C₄	1	1	yes	characteristic
18	C₂²×#92420	—	2	4	no	—
19	C₄⋊#92420	—	2	4	no	—
20	C₄⋊#92420	—	2	4	no	—
21	C₂²⋊#92420	—	2	4	no	—
22	C₂²⋊#92420	C₂×C₄	1	8	yes	—
23	C₄×#92420	—	2	4	no	—
24	C₄×#92420	—	2	4	no	—
25	C₂³×Dic₆₁	C₂×C₄	1	2	yes	—
26	C₂²×#92418	—	2	1	no	—
27	C₂²×#92418	—	2	1	no	—
28	C₂²×#92418	—	2	2	no	—
29	C₂²×#92418	C₈	1	4	yes	complemented
30	C₂²×#92418	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
33	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
34	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
35	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
36	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
37	C₂×C₄×Dic₆₁	Q₈	1	1	yes	characteristic
38	C₂×C₄×Dic₆₁	Q₈	1	1	yes	characteristic
39	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
40	#14519687	—	2	4	no	—
41	#14519687	—	2	4	no	—
42	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₁	—	2	1	no	—
44	C₂²×C₈×C₆₁	—	2	1	no	—
45	C₂²×C₈×C₆₁	—	2	2	no	—
46	C₂²×C₈×C₆₁	C₈	1	4	yes	complemented
47	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
48	C₂⁴×C₆₁	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
57	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
58	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
59	C₂²×Dic₆₁	—	4	4	no	—
60	C₂²×Dic₆₁	C₄²	1	2	yes	—
61	C₂²×Dic₆₁	C₂²⋊C₄	1	2	yes	—
62	C₂²×Dic₆₁	C₂²⋊C₄	1	2	yes	—
63	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
64	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
65	C₂²×Dic₆₁	C₂²×C₄	1	2	yes	—
66	C₂×#92420	—	2	8	no	—
67	C₂×#92420	—	4	8	no	—
68	C₂×C₈×C₆₁	—	2	4	no	—
69	C₂×C₈×C₆₁	—	2	4	no	—
70	C₂×C₈×C₆₁	—	2	2	no	—
71	C₂×C₈×C₆₁	—	2	2	no	—
72	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	complemented
73	C₄×Dic₆₁	C₂²⋊C₄	1	2	yes	—
74	C₄×Dic₆₁	C₂²⋊C₄	1	2	yes	—
75	C₄×Dic₆₁	C₂²⋊C₄	1	2	yes	—
76	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
77	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
78	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
79	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
80	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
81	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
82	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
83	C₂×#92418	—	2	4	no	—
84	C₂×#92418	—	2	4	no	—
85	C₂×#92418	—	2	2	no	—
86	C₂×#92418	—	2	2	no	—
87	C₂×#92418	C₂×C₈	1	4	yes	complemented
88	C₂³×C₆₁	—	2	4	no	—
89	C₂³×C₆₁	#95	1	1	yes	characteristic
90	C₂³×C₆₁	#95	1	1	yes	characteristic
91	C₂³×C₆₁	C₄×C₈	1	2	yes	—
92	C₂³×C₆₁	#97	1	2	yes	—
93	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
95	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
97	C₂×C₄×C₆₁	#97	1	2	yes	—
98	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
99	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
100	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
101	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
102	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
103	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
104	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
105	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
106	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
112	C₂×Dic₆₁	—	2	4	no	—
113	C₂×Dic₆₁	—	2	4	no	—
114	C₂×Dic₆₁	—	4	2	no	—
115	C₂×Dic₆₁	—	4	2	no	—
116	C₂×Dic₆₁	—	4	2	no	—
117	C₂×Dic₆₁	—	4	2	no	—
118	C₂×Dic₆₁	#95	1	2	yes	—
119	C₂×Dic₆₁	#95	1	2	yes	—
120	C₂×Dic₆₁	C₂×C₂²⋊C₄	1	2	yes	—
121	C₂×Dic₆₁	C₂×C₄⋊C₄	1	2	yes	—
122	#92420	—	4	8	no	—
123	C₈×C₆₁	—	2	4	no	—
124	#92418	—	2	4	no	—
125	C₂×#1281	—	61	1	no	maximal, Sylow
126	C₂²×C₆₁	—	2	4	no	—
127	C₂²×C₆₁	—	2	4	no	—
128	C₂²×C₆₁	—	2	2	no	—
129	C₂²×C₆₁	—	2	2	no	—
130	C₂²×C₆₁	#336	1	2	yes	—
131	C₂²×C₆₁	#336	1	2	yes	—
132	C₂²×C₆₁	#344	1	4	yes	—
133	C₂²×C₆₁	C₂×#95	1	1	yes	characteristic
134	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
135	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
136	C₄×C₆₁	#336	1	2	yes	—
137	C₄×C₆₁	#336	1	2	yes	—
138	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
140	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
141	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
142	Dic₆₁	—	2	2	no	—
143	Dic₆₁	—	2	2	no	—
144	Dic₆₁	—	2	2	no	—
145	Dic₆₁	—	2	2	no	—
146	C₂×#431	—	61	2	no	—
147	C₂²×#97	—	61	1	no	—
148	#1281	—	61	4	no	—
149	C₂×C₆₁	—	2	2	no	—
150	C₂×C₆₁	—	2	1	no	—
151	C₂×C₆₁	—	2	1	no	—
152	C₂×C₆₁	#1281	1	4	yes	complemented
153	C₂×C₆₁	C₂×#336	1	1	yes	characteristic
154	C₂×C₆₁	C₂×#344	1	2	yes	—
155	C₂³×C₈	—	61	1	no	—
156	C₂³×C₈	#6923	1	1	yes	characteristic
157	C₂²×C₄²	—	61	1	no	—
158	#431	—	61	4	no	—
159	#431	—	61	4	no	—
160	#431	—	122	2	no	—
161	#431	—	122	2	no	—
162	C₂×C₄⋊C₈	—	122	2	no	—
163	C₂×C₂²⋊C₈	—	61	4	no	—
164	C₂×#97	—	61	2	no	—
165	C₂×#97	—	61	2	no	—
166	C₂×#97	—	122	2	no	—
167	C₂×#97	—	122	2	no	—
168	C₂×C₄×C₈	—	122	2	no	—
169	C₆₁	C₂×#1281	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₄	—	61	2	no	—
171	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	2	1	no	—
174	C₂²×C₈	—	2	1	no	—
175	C₂²×C₈	—	61	4	no	—
176	C₂²×C₈	—	61	2	no	—
177	C₂²×C₈	—	122	2	no	—
178	C₂²×C₈	—	122	1	no	—
179	C₂²×C₈	—	122	1	no	—
180	C₂²×C₈	—	122	4	no	—
181	C₂²×C₈	#92420	1	4	yes	complemented
182	C₂²×C₈	C₂×#6923	1	2	yes	—
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	C₂×C₄²	—	61	1	no	—
191	C₂×C₄²	—	61	1	no	—
192	C₄⋊C₈	—	122	4	no	—
193	C₄⋊C₈	—	122	4	no	—
194	C₂²⋊C₈	—	61	8	no	—
195	C₂²⋊C₈	—	122	4	no	—
196	#97	—	122	4	no	—
197	#97	—	122	4	no	—
198	C₄×C₈	—	122	4	no	—
199	C₄×C₈	—	122	4	no	—
200	C₂⁴	C₄×#6923	1	1	yes	characteristic
201	C₂²×C₄	—	61	2	no	—
202	C₂²×C₄	—	61	2	no	—
203	C₂²×C₄	—	61	2	no	—
204	C₂²×C₄	—	61	2	no	—
205	C₂²×C₄	—	61	2	no	—
206	C₂²×C₄	—	61	2	no	—
207	C₂²×C₄	—	244	4	no	—
208	C₂²×C₄	D₆₁⋊C₈	1	2	yes	—
209	C₂²×C₄	#11758213	1	2	yes	—
210	C₂²×C₄	C₄×#6923	1	1	yes	characteristic
211	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
212	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
213	C₂²×C₄	C₂×#92420	1	2	yes	—
214	C₂²×C₄	#11758217	1	2	yes	—
215	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
217	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	61	4	no	—
223	C₂×C₈	—	122	4	no	—
224	C₂×C₈	—	122	4	no	—
225	C₂×C₈	—	122	2	no	—
226	C₂×C₈	—	122	2	no	—
227	C₂×C₈	—	122	8	no	—
228	C₂×C₈	—	244	8	no	—
229	C₂×C₈	C₂×#92420	1	4	yes	complemented
230	C₄²	—	61	2	no	—
231	C₄²	—	61	2	no	—
232	C₄²	—	61	2	no	—
233	C₄²	—	61	2	no	—
234	C₄²	—	61	2	no	—
235	C₄²	—	61	2	no	—
236	C₄²	—	61	1	no	—
237	C₄²	—	61	1	no	—
238	C₄²	—	61	1	no	—
239	C₄²	—	61	1	no	—
240	C₂³	—	2	4	no	—
241	C₂³	(C₂×#6923)⋊C₄	1	1	yes	characteristic
242	C₂³	(C₂×#6923)⋊C₄	1	1	yes	characteristic
243	C₂³	C₄×#92420	1	2	yes	—
244	C₂³	#14519832	1	2	yes	—
245	C₂³	C₂×C₄×#6923	1	1	yes	characteristic
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	4	no	—
251	C₂×C₄	—	122	4	no	—
252	C₂×C₄	—	244	2	no	—
253	C₂×C₄	—	244	2	no	—
254	C₂×C₄	—	244	2	no	—
255	C₂×C₄	—	244	2	no	—
256	C₂×C₄	(C₂×#6923)⋊C₄	1	1	yes	characteristic
257	C₂×C₄	(C₂×#6923)⋊C₄	1	1	yes	characteristic
258	C₂×C₄	C₄×#92420	1	2	yes	—
259	C₂×C₄	#14519832	1	2	yes	—
260	C₂×C₄	C₂²⋊#92420	1	2	yes	—
261	C₂×C₄	C₂²⋊#92420	1	2	yes	—
262	C₂×C₄	#14519834	1	2	yes	—
263	C₂×C₄	#14519834	1	2	yes	—
264	C₂×C₄	C₄⋊#92420	1	2	yes	—
265	C₂×C₄	C₄⋊#92420	1	2	yes	—
266	C₂×C₄	#14519844	1	2	yes	—
267	C₂×C₄	#14519844	1	2	yes	—
268	C₂×C₄	C₂×C₂²⋊#6923	1	1	yes	characteristic
269	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
270	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
271	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
272	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
273	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
274	C₈	—	2	4	no	—
275	C₈	—	122	4	no	—
276	C₈	—	244	8	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	4	no	—
281	C₂²	#16840628	1	2	yes	—
282	C₂²	#16840628	1	2	yes	—
283	C₂²	#16840644	1	4	yes	—
284	C₂²	C₂×(C₂×#6923)⋊C₄	1	1	yes	characteristic
285	C₂²	C₂×C₄×#92420	1	1	yes	characteristic
286	C₂²	C₂×#14519832	1	1	yes	characteristic
287	C₄	—	122	2	no	—
288	C₄	—	122	2	no	—
289	C₄	—	122	2	no	—
290	C₄	—	122	2	no	—
291	C₄	#16840628	1	2	yes	—
292	C₄	#16840628	1	2	yes	—
293	C₄	C₂×C₂²⋊#92420	1	1	yes	characteristic
294	C₄	C₂×#14519834	1	1	yes	characteristic
295	C₄	C₂×C₄⋊#92420	1	1	yes	characteristic
296	C₄	C₂×#14519844	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	#21460659	1	4	yes	direct product factor
301	C₂	C₂×#16840628	1	1	yes	characteristic
302	C₂	C₂×#16840644	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊#6923
commutator	C₂²×C₆₁	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂³×C₈×C₆₁	C₄
socle	C₂³×C₆₁	#95
prime radical	C₂⁴×C₆₁	C₄²
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂³×C₈	#6923
hypocenter	C₆₁	C₂×#1281
LCS	G ▷ C₂²×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. #21460659	C₂	direct (N × H)
29. C₂²×#92418	C₈	semidirect
46. C₂²×C₈×C₆₁	C₈	semidirect
72. C₂×C₈×C₆₁	C₂×C₈	semidirect
87. C₂×#92418	C₂×C₈	semidirect
152. C₂×C₆₁	#1281	semidirect
169. C₆₁	C₂×#1281	semidirect
181. C₂²×C₈	#92420	semidirect
229. C₂×C₈	C₂×#92420	semidirect
300. C₂	#21460659	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
#21460659	2
C₂²×#14519687	2
C₂×#1281	61

Subgroups — C2×#21460659 #30596276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21460659 #30596276