Subgroups — C₂×#21462666 #30625716

Order 15616.

2890 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	#21462666	C₂	1	16	yes	maximal, direct product factor
2	C₂×D₄×#92420	C₂	1	8	yes	maximal, complemented
3	C₂×#16840778	C₂	1	2	yes	maximal, complemented
4	C₂²×C₂²⋊#92420	C₂	1	4	yes	maximal, complemented
5	C₂²×D₄×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×#92420	C₂²	1	8	yes	complemented
7	D₄×#92420	C₂²	1	64	yes	complemented
8	#16840778	C₂²	1	16	yes	complemented
9	C₂×C₄⋊#92420	C₂²	1	4	yes	complemented
10	C₂×C₂²⋊#92420	C₂²	1	4	yes	—
11	C₂×C₂²⋊#92420	C₂²	1	32	yes	complemented
12	C₂×C₂²⋊#92420	C₂²	1	8	yes	complemented
13	C₂×C₄×#92420	C₂²	1	4	yes	complemented
14	C₂⁴×Dic₆₁	C₂²	1	2	yes	—
15	C₂×D₄×Dic₆₁	C₄	1	8	yes	—
16	C₂×D₄×Dic₆₁	C₄	1	4	yes	—
17	C₂×D₄×Dic₆₁	C₂²	1	8	yes	—
18	C₂×D₄×Dic₆₁	C₂²	1	4	yes	—
19	C₂²×C₄⋊Dic₆₁	C₄	1	1	yes	characteristic
20	C₂²×C₂²⋊Dic₆₁	C₄	1	2	yes	—
21	C₂²×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
22	C₂³×D₄×C₆₁	C₄	1	1	yes	characteristic
23	C₂²×#92420	—	4	16	no	—
24	C₂²×#92420	C₂³	1	8	yes	—
25	C₂²×#92420	C₂³	1	32	yes	complemented
26	C₄⋊#92420	C₂³	1	16	yes	complemented
27	C₂²⋊#92420	C₂³	1	16	yes	—
28	C₂²⋊#92420	C₂³	1	32	yes	complemented
29	C₄×#92420	C₂³	1	16	yes	complemented
30	C₂³×Dic₆₁	—	4	8	no	—
31	C₂³×Dic₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂³×Dic₆₁	C₂×C₄	1	2	yes	—
33	C₂³×Dic₆₁	C₂×C₄	1	8	yes	—
34	C₂³×Dic₆₁	C₂×C₄	1	4	yes	—
35	C₂³×Dic₆₁	C₂³	1	1	yes	characteristic
36	C₂³×Dic₆₁	C₂³	1	4	yes	—
37	C₂³×Dic₆₁	C₂³	1	8	yes	—
38	D₄×Dic₆₁	C₂×C₄	1	16	yes	—
39	D₄×Dic₆₁	C₂×C₄	1	32	yes	—
40	D₄×Dic₆₁	C₂³	1	16	yes	—
41	C₂×C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
42	C₂×C₄⋊Dic₆₁	C₂×C₄	1	8	yes	—
43	C₂×C₂²⋊Dic₆₁	C₂×C₄	1	8	yes	—
44	C₂×C₂²⋊Dic₆₁	C₂×C₄	1	16	yes	—
45	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
46	C₂×C₄×Dic₆₁	C₂×C₄	1	4	yes	—
47	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
48	C₂×C₄×Dic₆₁	C₂³	1	4	yes	—
49	C₂⁵×C₆₁	C₂×C₄	1	2	yes	—
50	C₂²×D₄×C₆₁	C₈	1	16	yes	complemented
51	C₂²×D₄×C₆₁	C₂×C₄	1	4	yes	—
52	C₂²×D₄×C₆₁	C₂×C₄	1	8	yes	—
53	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
54	C₂⁴×C₆₁	—	4	4	no	—
55	C₂⁴×C₆₁	—	4	4	no	—
56	C₂⁴×C₆₁	C₂×C₈	1	16	yes	complemented
57	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₂⁴×C₆₁	C₂²×C₄	1	4	yes	—
59	C₂⁴×C₆₁	C₂²×C₄	1	8	yes	—
60	C₂×D₄×C₆₁	C₂×C₈	1	32	yes	complemented
61	C₂×D₄×C₆₁	C₂×C₈	1	64	yes	complemented
62	C₂×D₄×C₆₁	C₂²×C₄	1	16	yes	—
63	C₂²×C₄×C₆₁	C₂×C₈	1	8	yes	complemented
64	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
66	C₂²×Dic₆₁	—	2	4	no	—
67	C₂²×Dic₆₁	—	2	16	no	—
68	C₂²×Dic₆₁	—	4	16	no	—
69	C₂²×Dic₆₁	—	4	16	no	—
70	C₂²×Dic₆₁	—	4	8	no	—
71	C₂²×Dic₆₁	—	4	4	no	—
72	C₂²×Dic₆₁	—	4	2	no	—
73	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
74	C₂²×Dic₆₁	C₂²×C₄	1	2	yes	—
75	C₂²×Dic₆₁	C₂²×C₄	1	4	yes	—
76	C₂²×Dic₆₁	C₂²×C₄	1	8	yes	—
77	C₂²×Dic₆₁	C₂²×C₄	1	2	yes	—
78	C₂²×Dic₆₁	C₂²×C₄	1	8	yes	—
79	C₂²×Dic₆₁	C₂²×C₄	1	16	yes	—
80	C₂²×Dic₆₁	C₂²×C₄	1	4	yes	—
81	C₂²×Dic₆₁	C₂⁴	1	2	yes	—
82	C₂²×Dic₆₁	C₂⁴	1	8	yes	—
83	C₂²×Dic₆₁	C₂⁴	1	1	yes	characteristic
84	C₂×#92420	—	2	16	no	—
85	C₂×#92420	—	4	32	no	—
86	C₂×#92420	C₂⁴	1	16	yes	—
87	C₂²⋊Dic₆₁	C₂²×C₄	1	32	yes	—
88	C₄⋊Dic₆₁	C₂²×C₄	1	16	yes	—
89	C₄×Dic₆₁	C₂²×C₄	1	8	yes	—
90	C₄×Dic₆₁	C₂²×C₄	1	4	yes	—
91	C₄×Dic₆₁	C₂⁴	1	4	yes	—
92	C₂³×C₆₁	—	2	2	no	—
93	C₂³×C₆₁	—	2	8	no	—
94	C₂³×C₆₁	—	2	2	no	—
95	C₂³×C₆₁	—	2	8	no	—
96	C₂³×C₆₁	—	4	8	no	—
97	C₂³×C₆₁	—	4	32	no	—
98	C₂³×C₆₁	—	4	8	no	—
99	C₂³×C₆₁	C₂²×C₈	1	4	yes	—
100	C₂³×C₆₁	C₂²×C₈	1	16	yes	—
101	C₂³×C₆₁	C₂²×C₈	1	32	yes	complemented
102	C₂³×C₆₁	C₂³×C₄	1	8	yes	—
103	C₂³×C₆₁	C₂³×C₄	1	2	yes	—
104	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
105	D₄×C₆₁	C₂²×C₈	1	64	yes	complemented
106	C₂×C₄×C₆₁	C₂²×C₈	1	8	yes	—
107	C₂×C₄×C₆₁	C₂²×C₈	1	16	yes	complemented
108	C₂×C₄×C₆₁	C₂³×C₄	1	4	yes	—
109	C₂×Dic₆₁	—	2	2	no	—
110	C₂×Dic₆₁	—	2	8	no	—
111	C₂×Dic₆₁	—	2	4	no	—
112	C₂×Dic₆₁	—	2	4	no	—
113	C₂×Dic₆₁	—	2	16	no	—
114	C₂×Dic₆₁	—	2	16	no	—
115	C₂×Dic₆₁	—	2	4	no	—
116	C₂×Dic₆₁	—	4	16	no	—
117	C₂×Dic₆₁	—	4	8	no	—
118	C₂×Dic₆₁	—	4	4	no	—
119	C₂×Dic₆₁	C₂³×C₄	1	2	yes	—
120	C₂×Dic₆₁	C₂³×C₄	1	4	yes	—
121	C₂×Dic₆₁	C₂³×C₄	1	8	yes	—
122	C₂×Dic₆₁	C₂³×C₄	1	1	yes	characteristic
123	C₂×Dic₆₁	H(5,𝔽₂)	1	4	yes	—
124	C₂×Dic₆₁	C₂⁵	1	1	yes	characteristic
125	#92420	—	2	16	no	—
126	C₂×#2972	—	61	1	no	maximal, Sylow
127	C₂²×C₆₁	—	2	2	no	—
128	C₂²×C₆₁	—	2	8	no	—
129	C₂²×C₆₁	—	2	8	no	—
130	C₂²×C₆₁	—	2	32	no	—
131	C₂²×C₆₁	—	2	2	no	—
132	C₂²×C₆₁	—	2	8	no	—
133	C₂²×C₆₁	—	4	32	no	—
134	C₂²×C₆₁	#518	1	2	yes	—
135	C₂²×C₆₁	C₂³×C₈	1	4	yes	—
136	C₂²×C₆₁	C₂³×C₈	1	16	yes	—
137	C₂²×C₆₁	C₂³×C₈	1	2	yes	—
138	C₂²×C₆₁	#568	1	2	yes	—
139	C₂²×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
140	C₄×C₆₁	C₂³×C₈	1	8	yes	—
141	Dic₆₁	—	2	2	no	—
142	Dic₆₁	—	2	8	no	—
143	Dic₆₁	—	2	4	no	—
144	Dic₆₁	#518	1	2	yes	—
145	Dic₆₁	C₂×H(5,𝔽₂)	1	2	yes	—
146	C₂²×C₄×D₄	—	61	1	no	—
147	#2972	—	61	16	no	—
148	C₂×C₈×D₄	—	61	8	no	—
149	C₂×#431	—	61	2	no	—
150	C₂²×C₂²⋊C₈	—	61	4	no	—
151	C₂×C₆₁	—	2	4	no	—
152	C₂×C₆₁	—	2	16	no	—
153	C₂×C₆₁	#2972	1	4	yes	complemented
154	C₂×C₆₁	C₂×#518	1	1	yes	characteristic
155	C₂×C₆₁	C₂⁴×C₈	1	1	yes	characteristic
156	C₂×C₆₁	C₂×#568	1	1	yes	characteristic
157	C₂³×D₄	#6923	1	1	yes	characteristic
158	C₂⁴×C₄	—	61	2	no	—
159	C₂³×C₈	—	61	8	no	—
160	C₂×C₄×D₄	—	61	8	no	—
161	C₂×C₄×D₄	—	61	4	no	—
162	C₂×C₄×D₄	—	61	4	no	—
163	C₂×C₄×D₄	—	61	8	no	—
164	C₂²×C₄⋊C₄	—	61	1	no	—
165	C₂²×C₂²⋊C₄	—	61	2	no	—
166	C₂²×C₄²	—	61	1	no	—
167	C₈×D₄	—	61	64	no	—
168	#431	—	61	16	no	—
169	C₂×C₄⋊C₈	—	61	4	no	—
170	C₂×C₂²⋊C₈	—	61	4	no	—
171	C₂×C₂²⋊C₈	—	61	8	no	—
172	C₂×C₂²⋊C₈	—	61	32	no	—
173	C₂×C₄×C₈	—	61	4	no	—
174	C₆₁	C₂×#2972	1	1	yes	characteristic, complemented, Sylow
175	C₂⁵	C₂×#6923	1	2	yes	—
176	C₂²×D₄	#92420	1	16	yes	complemented
177	C₂²×D₄	C₂×#6923	1	4	yes	—
178	C₂²×D₄	C₂×#6923	1	8	yes	—
179	C₂³×C₄	—	61	4	no	—
180	C₂³×C₄	—	61	1	no	—
181	C₂³×C₄	—	61	1	no	—
182	C₂³×C₄	—	61	2	no	—
183	C₂³×C₄	—	61	8	no	—
184	C₂³×C₄	—	61	4	no	—
185	C₂³×C₄	—	61	8	no	—
186	C₂³×C₄	—	244	8	no	—
187	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
188	C₂²×C₈	—	61	8	no	—
189	C₂²×C₈	—	61	32	no	—
190	C₂²×C₈	—	244	16	no	—
191	C₄×D₄	—	61	16	no	—
192	C₄×D₄	—	61	16	no	—
193	C₄×D₄	—	61	32	no	—
194	C₂×C₄⋊C₄	—	61	4	no	—
195	C₂×C₄⋊C₄	—	61	8	no	—
196	C₂×C₂²⋊C₄	—	61	8	no	—
197	C₂×C₂²⋊C₄	—	61	16	no	—
198	C₂×C₄²	—	61	2	no	—
199	C₂×C₄²	—	61	4	no	—
200	C₂×C₄²	—	61	2	no	—
201	C₂×C₄²	—	61	4	no	—
202	C₄⋊C₈	—	61	16	no	—
203	C₂²⋊C₈	—	61	16	no	—
204	C₂²⋊C₈	—	61	32	no	—
205	C₄×C₈	—	61	16	no	—
206	C₂⁴	—	4	4	no	—
207	C₂⁴	—	4	4	no	—
208	C₂⁴	C₂×#92420	1	16	yes	complemented
209	C₂⁴	C₂²×#6923	1	4	yes	—
210	C₂⁴	C₂²×#6923	1	1	yes	characteristic
211	C₂⁴	C₂²×#6923	1	8	yes	—
212	C₂×D₄	C₂×#92420	1	32	yes	complemented
213	C₂×D₄	C₂×#92420	1	64	yes	complemented
214	C₂×D₄	C₂²×#6923	1	16	yes	—
215	C₂²×C₄	—	61	1	no	—
216	C₂²×C₄	—	61	2	no	—
217	C₂²×C₄	—	61	8	no	—
218	C₂²×C₄	—	61	2	no	—
219	C₂²×C₄	—	61	8	no	—
220	C₂²×C₄	—	61	2	no	—
221	C₂²×C₄	—	61	1	no	—
222	C₂²×C₄	—	61	4	no	—
223	C₂²×C₄	—	61	4	no	—
224	C₂²×C₄	—	61	8	no	—
225	C₂²×C₄	—	61	16	no	—
226	C₂²×C₄	—	122	16	no	—
227	C₂²×C₄	—	122	4	no	—
228	C₂²×C₄	—	244	16	no	—
229	C₂²×C₄	—	244	16	no	—
230	C₂²×C₄	—	244	4	no	—
231	C₂²×C₄	—	244	8	no	—
232	C₂²×C₄	—	244	2	no	—
233	C₂²×C₄	C₂×#92420	1	8	yes	complemented
234	C₂²×C₄	C₂²×#6923	1	2	yes	—
235	C₂²×C₄	C₂²×#6923	1	4	yes	—
236	C₂×C₈	—	61	16	no	—
237	C₂×C₈	—	122	16	no	—
238	C₂×C₈	—	244	32	no	—
239	C₄⋊C₄	—	61	16	no	—
240	C₂²⋊C₄	—	61	32	no	—
241	C₄²	—	61	4	no	—
242	C₄²	—	61	4	no	—
243	C₄²	—	61	8	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	8	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	8	no	—
248	C₂³	—	4	8	no	—
249	C₂³	—	4	32	no	—
250	C₂³	—	4	8	no	—
251	C₂³	C₂²×#92420	1	4	yes	—
252	C₂³	C₂²×#92420	1	16	yes	—
253	C₂³	C₂²×#92420	1	32	yes	complemented
254	C₂³	C₂³×#6923	1	8	yes	—
255	C₂³	C₂³×#6923	1	2	yes	—
256	C₂³	C₂³×#6923	1	1	yes	characteristic
257	D₄	C₂²×#92420	1	64	yes	complemented
258	C₂×C₄	—	61	4	no	—
259	C₂×C₄	—	61	1	no	—
260	C₂×C₄	—	61	1	no	—
261	C₂×C₄	—	61	4	no	—
262	C₂×C₄	—	61	2	no	—
263	C₂×C₄	—	61	8	no	—
264	C₂×C₄	—	122	16	no	—
265	C₂×C₄	—	122	16	no	—
266	C₂×C₄	—	122	4	no	—
267	C₂×C₄	—	122	4	no	—
268	C₂×C₄	—	122	8	no	—
269	C₂×C₄	—	122	2	no	—
270	C₂×C₄	—	122	4	no	—
271	C₂×C₄	—	244	4	no	—
272	C₂×C₄	—	244	8	no	—
273	C₂×C₄	—	244	16	no	—
274	C₂×C₄	C₂²×#92420	1	8	yes	—
275	C₂×C₄	C₂²×#92420	1	16	yes	complemented
276	C₂×C₄	C₂³×#6923	1	4	yes	—
277	C₈	—	122	16	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	8	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	8	no	—
282	C₂²	—	2	8	no	—
283	C₂²	—	2	32	no	—
284	C₂²	—	4	32	no	—
285	C₂²	#16840829	1	2	yes	—
286	C₂²	C₂³×#92420	1	4	yes	—
287	C₂²	C₂³×#92420	1	16	yes	—
288	C₂²	C₂³×#92420	1	2	yes	—
289	C₂²	#16840840	1	2	yes	—
290	C₂²	C₂⁴×#6923	1	1	yes	characteristic
291	C₄	—	61	2	no	—
292	C₄	—	61	2	no	—
293	C₄	—	122	8	no	—
294	C₄	—	122	2	no	—
295	C₄	—	122	4	no	—
296	C₄	C₂³×#92420	1	8	yes	—
297	C₂	—	2	4	no	—
298	C₂	—	2	16	no	—
299	C₂	#21462666	1	4	yes	direct product factor
300	C₂	C₂×#16840829	1	1	yes	characteristic
301	C₂	C₂⁴×#92420	1	1	yes	characteristic
302	C₂	C₂×#16840840	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×#6923
commutator	C₂×C₆₁	C₂⁴×C₈
Frattini	C₂²	C₂⁴×#6923
Fitting, prime radical	C₂³×D₄×C₆₁	C₄
socle	C₂³×C₆₁	C₂³×C₄
cosocle	C₂×Dic₆₁	C₂⁵
hypercenter	C₂³×D₄	#6923
hypocenter	C₆₁	C₂×#2972
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂³×D₄

Complements

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

N	H ≅ G/N	type
1. #21462666	C₂	direct (N × H)
2. C₂×D₄×#92420	C₂	semidirect
3. C₂×#16840778	C₂	semidirect
4. C₂²×C₂²⋊#92420	C₂	semidirect
6. C₂³×#92420	C₂²	semidirect
7. D₄×#92420	C₂²	semidirect
8. #16840778	C₂²	semidirect
9. C₂×C₄⋊#92420	C₂²	semidirect
11. C₂×C₂²⋊#92420	C₂²	semidirect
12. C₂×C₂²⋊#92420	C₂²	semidirect
13. C₂×C₄×#92420	C₂²	semidirect
25. C₂²×#92420	C₂³	semidirect
26. C₄⋊#92420	C₂³	semidirect
28. C₂²⋊#92420	C₂³	semidirect
29. C₄×#92420	C₂³	semidirect
50. C₂²×D₄×C₆₁	C₈	semidirect
56. C₂⁴×C₆₁	C₂×C₈	semidirect
60. C₂×D₄×C₆₁	C₂×C₈	semidirect
61. C₂×D₄×C₆₁	C₂×C₈	semidirect
63. C₂²×C₄×C₆₁	C₂×C₈	semidirect
101. C₂³×C₆₁	C₂²×C₈	semidirect
105. D₄×C₆₁	C₂²×C₈	semidirect
107. C₂×C₄×C₆₁	C₂²×C₈	semidirect
153. C₂×C₆₁	#2972	semidirect
174. C₆₁	C₂×#2972	semidirect
176. C₂²×D₄	#92420	semidirect
208. C₂⁴	C₂×#92420	semidirect
212. C₂×D₄	C₂×#92420	semidirect
213. C₂×D₄	C₂×#92420	semidirect
233. C₂²×C₄	C₂×#92420	semidirect
253. C₂³	C₂²×#92420	semidirect
257. D₄	C₂²×#92420	semidirect
275. C₂×C₄	C₂²×#92420	semidirect
299. C₂	#21462666	direct (N × H)

Maximal subgroups

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

subgroup	index
#21462666	2
C₂×D₄×#92420	2
C₂×#16840778	2
C₂²×C₂²⋊#92420	2
C₂²×D₄×Dic₆₁	2
C₂×#2972	61

Subgroups — C2×#21462666 #30625716

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21462666 #30625716