Subgroups — C₂×#21460643 #30596004

Order 15616.

758 conjugacy classes of subgroups.

The subgroup lattice has 320 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	#21460643	C₂	1	4	yes	maximal, direct product factor
3	C₂×#16839608	C₂	1	1	yes	maximal, characteristic
4	#16840778	—	2	2	no	—
5	#16840778	—	2	2	no	—
6	#16840778	C₄	1	4	yes	complemented
7	#16840778	C₂²	1	4	yes	—
8	C₂×C₄⋊#92420	—	2	2	no	—
9	C₂×C₂²⋊#92420	—	2	2	no	—
10	C₂×C₄×#92420	C₄	1	4	yes	complemented
11	C₂²×C₄×Dic₆₁	C₄	1	1	yes	characteristic
12	C₂²×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
13	#16839608	—	2	2	no	—
14	#16839608	C₄	1	2	yes	—
15	#16839608	C₂²	1	2	yes	—
16	C₂×#14519684	—	2	2	no	—
17	C₂²×C₄²×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	—	2	4	no	—
23	C₄×#92420	—	2	4	no	—
24	C₄×#92420	C₂×C₄	1	8	yes	complemented
25	C₂³×Dic₆₁	—	2	1	no	—
26	C₂³×Dic₆₁	D₄	1	2	yes	—
27	C₂×C₄×Dic₆₁	—	2	1	no	—
28	C₂×C₄×Dic₆₁	—	2	2	no	—
29	C₂×C₄×Dic₆₁	C₈	1	4	yes	complemented
30	C₂×C₄×Dic₆₁	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	C₂×C₄	1	2	yes	—
34	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
35	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
36	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
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	#14519684	—	2	4	no	—
41	#14519684	—	2	4	no	—
42	C₂³×C₄×C₆₁	—	2	1	no	—
43	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₁	—	2	1	no	—
45	C₂×C₄²×C₆₁	—	2	2	no	—
46	C₂×C₄²×C₆₁	C₈	1	4	yes	complemented
47	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
48	C₂⁴×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	—	2	4	no	—
50	C₂²×C₄×C₆₁	—	2	2	no	—
51	C₂²×C₄×C₆₁	—	4	4	no	—
52	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
60	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
61	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₆₁	—	2	4	no	—
63	C₂²×Dic₆₁	—	2	2	no	—
64	C₂²×Dic₆₁	—	4	4	no	—
65	C₂²×Dic₆₁	—	4	4	no	—
66	C₂²×Dic₆₁	C₂²⋊C₄	1	2	yes	—
67	C₂²×Dic₆₁	C₂²⋊C₄	1	2	yes	—
68	C₂²×Dic₆₁	C₂²⋊C₄	1	2	yes	—
69	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
70	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
71	C₂²×Dic₆₁	C₂×D₄	1	2	yes	—
72	C₂×#92420	—	2	8	no	—
73	C₂×#92420	—	4	8	no	—
74	C₄²×C₆₁	—	2	4	no	—
75	C₄²×C₆₁	—	2	2	no	—
76	C₄²×C₆₁	C₂×C₈	1	4	yes	complemented
77	C₄×Dic₆₁	—	2	4	no	—
78	C₄×Dic₆₁	—	2	2	no	—
79	C₄×Dic₆₁	C₄²	1	2	yes	—
80	C₄×Dic₆₁	C₂²⋊C₄	1	2	yes	—
81	C₄×Dic₆₁	C₂²⋊C₄	1	2	yes	—
82	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
83	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
84	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
85	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
86	C₄×Dic₆₁	C₂×C₈	1	4	yes	complemented
87	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
88	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
89	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
90	C₂³×C₆₁	—	2	4	no	—
91	C₂³×C₆₁	#95	1	1	yes	characteristic
92	C₂³×C₆₁	#95	1	1	yes	characteristic
93	C₂³×C₆₁	C₂²⋊C₈	1	2	yes	—
94	C₂³×C₆₁	C₂²⋊C₈	1	2	yes	—
95	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₆₁	—	2	4	no	—
97	C₂×C₄×C₆₁	—	4	4	no	—
98	C₂×C₄×C₆₁	—	4	8	no	—
99	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
100	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
102	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
103	C₂×C₄×C₆₁	#97	1	2	yes	—
104	C₂×C₄×C₆₁	#97	1	2	yes	—
105	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
106	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
107	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
108	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
109	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
110	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
111	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
116	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
117	C₂×Dic₆₁	—	2	4	no	—
118	C₂×Dic₆₁	—	2	4	no	—
119	C₂×Dic₆₁	—	2	4	no	—
120	C₂×Dic₆₁	—	4	2	no	—
121	C₂×Dic₆₁	—	4	2	no	—
122	C₂×Dic₆₁	—	4	2	no	—
123	C₂×Dic₆₁	—	4	2	no	—
124	C₂×Dic₆₁	—	4	4	no	—
125	C₂×Dic₆₁	—	4	8	no	—
126	C₂×Dic₆₁	#95	1	2	yes	—
127	C₂×Dic₆₁	#95	1	2	yes	—
128	C₂×Dic₆₁	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂×Dic₆₁	C₂×C₄⋊C₄	1	2	yes	—
130	#92420	—	4	8	no	—
131	C₂×#1274	—	61	1	no	maximal, Sylow
132	C₂²×C₆₁	—	2	2	no	—
133	C₂²×C₆₁	—	2	2	no	—
134	C₂²×C₆₁	—	2	4	no	—
135	C₂²×C₆₁	—	2	4	no	—
136	C₂²×C₆₁	#336	1	2	yes	—
137	C₂²×C₆₁	#336	1	2	yes	—
138	C₂²×C₆₁	H(3,ℤ/4)	1	4	yes	—
139	C₂²×C₆₁	C₂×#95	1	1	yes	characteristic
140	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
141	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
142	C₄×C₆₁	—	4	4	no	—
143	C₄×C₆₁	#336	1	2	yes	—
144	C₄×C₆₁	#336	1	2	yes	—
145	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
146	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
147	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
148	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
149	Dic₆₁	—	2	2	no	—
150	Dic₆₁	—	2	2	no	—
151	Dic₆₁	—	2	2	no	—
152	Dic₆₁	—	2	2	no	—
153	Dic₆₁	—	4	4	no	—
154	C₂×#431	—	61	2	no	—
155	C₂×#376	—	61	1	no	—
156	#1274	—	61	4	no	—
157	C₂×C₆₁	—	2	1	no	—
158	C₂×C₆₁	—	2	1	no	—
159	C₂×C₆₁	—	2	2	no	—
160	C₂×C₆₁	#1274	1	4	yes	complemented
161	C₂×C₆₁	C₂×#336	1	1	yes	characteristic
162	C₂×C₆₁	C₂×H(3,ℤ/4)	1	2	yes	—
163	C₂²×C₄²	—	61	1	no	—
164	C₂²×C₄²	—	61	1	no	—
165	C₂²×C₄²	#6923	1	1	yes	characteristic
166	#431	—	61	4	no	—
167	#431	—	61	4	no	—
168	#431	—	122	2	no	—
169	#431	—	122	2	no	—
170	C₂×C₄⋊C₈	—	122	2	no	—
171	C₂×C₂²⋊C₈	—	122	2	no	—
172	C₂×C₄×C₈	—	61	4	no	—
173	#376	—	61	2	no	—
174	#376	—	61	2	no	—
175	#376	—	122	2	no	—
176	C₂×#95	—	122	2	no	—
177	C₆₁	C₂×#1274	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₄	—	2	1	no	—
179	C₂³×C₄	—	61	2	no	—
180	C₂³×C₄	—	122	1	no	—
181	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
182	C₂²×C₈	—	122	4	no	—
183	C₂×C₄²	—	2	2	no	—
184	C₂×C₄²	—	2	1	no	—
185	C₂×C₄²	—	61	2	no	—
186	C₂×C₄²	—	61	2	no	—
187	C₂×C₄²	—	61	1	no	—
188	C₂×C₄²	—	61	1	no	—
189	C₂×C₄²	—	61	1	no	—
190	C₂×C₄²	—	61	1	no	—
191	C₂×C₄²	—	61	2	no	—
192	C₂×C₄²	—	61	1	no	—
193	C₂×C₄²	—	61	1	no	—
194	C₂×C₄²	—	61	4	no	—
195	C₂×C₄²	—	61	2	no	—
196	C₂×C₄²	—	122	2	no	—
197	C₂×C₄²	—	122	1	no	—
198	C₂×C₄²	#92420	1	4	yes	complemented
199	C₂×C₄²	C₂×#6923	1	2	yes	—
200	C₄⋊C₈	—	122	4	no	—
201	C₄⋊C₈	—	122	4	no	—
202	C₂²⋊C₈	—	122	4	no	—
203	C₂²⋊C₈	—	122	4	no	—
204	C₄×C₈	—	61	8	no	—
205	C₄×C₈	—	122	4	no	—
206	#95	—	122	4	no	—
207	#95	—	122	4	no	—
208	C₂⁴	C₂²⋊#6923	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	—	61	2	no	—
213	C₂²×C₄	—	61	2	no	—
214	C₂²×C₄	—	61	2	no	—
215	C₂²×C₄	—	61	2	no	—
216	C₂²×C₄	—	61	2	no	—
217	C₂²×C₄	—	61	2	no	—
218	C₂²×C₄	—	122	4	no	—
219	C₂²×C₄	—	122	2	no	—
220	C₂²×C₄	—	244	4	no	—
221	C₂²×C₄	—	244	4	no	—
222	C₂²×C₄	D₆₁⋊C₈	1	2	yes	—
223	C₂²×C₄	#11758213	1	2	yes	—
224	C₂²×C₄	C₄×#6923	1	1	yes	characteristic
225	C₂²×C₄	C₄×#6923	1	1	yes	characteristic
226	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
227	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
228	C₂²×C₄	C₂×#92420	1	2	yes	—
229	C₂²×C₄	#11758217	1	2	yes	—
230	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
231	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
232	C₂×C₈	—	122	8	no	—
233	C₂×C₈	—	244	8	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	61	2	no	—
237	C₄²	—	61	2	no	—
238	C₄²	—	61	2	no	—
239	C₄²	—	61	2	no	—
240	C₄²	—	61	2	no	—
241	C₄²	—	61	2	no	—
242	C₄²	—	61	1	no	—
243	C₄²	—	61	1	no	—
244	C₄²	—	61	1	no	—
245	C₄²	—	61	1	no	—
246	C₄²	—	61	4	no	—
247	C₄²	—	122	4	no	—
248	C₄²	—	122	2	no	—
249	C₄²	C₂×#92420	1	4	yes	complemented
250	C₂³	—	2	4	no	—
251	C₂³	(C₂×#6923)⋊C₄	1	1	yes	characteristic
252	C₂³	(C₂×#6923)⋊C₄	1	1	yes	characteristic
253	C₂³	C₂²⋊#92420	1	2	yes	—
254	C₂³	#14519834	1	2	yes	—
255	C₂³	C₂×C₂²⋊#6923	1	1	yes	characteristic
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	61	2	no	—
260	C₂×C₄	—	61	2	no	—
261	C₂×C₄	—	61	2	no	—
262	C₂×C₄	—	61	2	no	—
263	C₂×C₄	—	122	4	no	—
264	C₂×C₄	—	122	4	no	—
265	C₂×C₄	—	122	4	no	—
266	C₂×C₄	—	244	2	no	—
267	C₂×C₄	—	244	2	no	—
268	C₂×C₄	—	244	2	no	—
269	C₂×C₄	—	244	2	no	—
270	C₂×C₄	—	244	8	no	—
271	C₂×C₄	—	244	4	no	—
272	C₂×C₄	(C₂×#6923)⋊C₄	1	1	yes	characteristic
273	C₂×C₄	(C₂×#6923)⋊C₄	1	1	yes	characteristic
274	C₂×C₄	C₄×#92420	1	2	yes	—
275	C₂×C₄	C₄×#92420	1	2	yes	—
276	C₂×C₄	#14519832	1	2	yes	—
277	C₂×C₄	#14519832	1	2	yes	—
278	C₂×C₄	C₂²⋊#92420	1	2	yes	—
279	C₂×C₄	#14519834	1	2	yes	—
280	C₂×C₄	C₄⋊#92420	1	2	yes	—
281	C₂×C₄	C₄⋊#92420	1	2	yes	—
282	C₂×C₄	#14519844	1	2	yes	—
283	C₂×C₄	#14519844	1	2	yes	—
284	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
285	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
286	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
287	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
288	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
289	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
290	C₈	—	244	8	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	#16840628	1	2	yes	—
296	C₂²	#16840628	1	2	yes	—
297	C₂²	#16840629	1	4	yes	—
298	C₂²	C₂×(C₂×#6923)⋊C₄	1	1	yes	characteristic
299	C₂²	C₂×C₂²⋊#92420	1	1	yes	characteristic
300	C₂²	C₂×#14519834	1	1	yes	characteristic
301	C₄	—	4	4	no	—
302	C₄	—	122	2	no	—
303	C₄	—	122	2	no	—
304	C₄	—	122	2	no	—
305	C₄	—	122	2	no	—
306	C₄	—	244	4	no	—
307	C₄	#16840628	1	2	yes	—
308	C₄	#16840628	1	2	yes	—
309	C₄	C₂×C₄×#92420	1	1	yes	characteristic
310	C₄	C₂×#14519832	1	1	yes	characteristic
311	C₄	C₂×C₄⋊#92420	1	1	yes	characteristic
312	C₄	C₂×#14519844	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	1	no	—
315	C₂	—	2	2	no	—
316	C₂	#21460643	1	4	yes	direct product factor
317	C₂	C₂×#16840628	1	1	yes	characteristic
318	C₂	C₂×#16840629	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₄×#6923
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂²×C₄²×C₆₁	C₄
socle	C₂³×C₆₁	#95
prime radical	C₂⁴×C₆₁	C₂²⋊C₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂²×C₄²	#6923
hypocenter	C₆₁	C₂×#1274
LCS	G ▷ C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. #21460643	C₂	direct (N × H)
6. #16840778	C₄	semidirect
10. C₂×C₄×#92420	C₄	semidirect
24. C₄×#92420	C₂×C₄	semidirect
29. C₂×C₄×Dic₆₁	C₈	semidirect
46. C₂×C₄²×C₆₁	C₈	semidirect
76. C₄²×C₆₁	C₂×C₈	semidirect
86. C₄×Dic₆₁	C₂×C₈	semidirect
160. C₂×C₆₁	#1274	semidirect
177. C₆₁	C₂×#1274	semidirect
198. C₂×C₄²	#92420	semidirect
249. C₄²	C₂×#92420	semidirect
316. C₂	#21460643	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
#21460643	2
C₂×#16839608	2
C₂×#1274	61

Subgroups — C2×#21460643 #30596004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21460643 #30596004