Subgroups — C₂×#21462166 #30624115

Order 15616.

658 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₆₁⋊#108	C₂	1	1	yes	maximal, characteristic
2	C₂×#16840772	C₂	1	1	yes	maximal, characteristic
3	#21462166	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16840687	C₂	1	2	yes	maximal, complemented
5	C₂×#16840686	C₂	1	2	yes	maximal, complemented
6	C₂²×#14519716	C₂	1	1	yes	maximal, characteristic
7	C₂×D₆₁⋊M₁₆	—	2	1	no	—
8	C₂×#14519856	—	2	1	no	—
9	C₂×#14519848	—	2	1	no	—
10	D₆₁⋊#108	—	2	2	no	—
11	D₆₁⋊#108	C₂²	1	4	yes	—
12	C₂×#14519847	C₂²	1	2	yes	—
13	#16840772	—	2	2	no	—
14	#16840772	C₂²	1	4	yes	—
15	C₂×#14519846	—	2	1	no	—
16	C₂×#14519845	C₂²	1	2	yes	—
17	#16840687	C₂²	1	8	yes	complemented
18	#16840686	C₂²	1	8	yes	complemented
19	C₂²×C₈×D₆₁	C₂²	1	1	yes	characteristic
20	C₂×#14519716	C₄	1	1	yes	characteristic, complemented
21	C₂×#14519716	C₄	1	1	yes	characteristic, complemented
22	C₂×#14519716	C₄	1	4	yes	complemented
23	C₂×#14519716	C₂²	1	1	yes	characteristic
24	C₂×#14519716	C₂²	1	1	yes	characteristic
25	C₂×#14519716	C₂²	1	4	yes	—
26	C₂²×#11758185	C₄	1	1	yes	characteristic, complemented
27	C₂²×C₁₆×C₆₁	C₄	1	1	yes	characteristic, complemented
28	C₂×#11758213	—	2	2	no	—
29	D₆₁⋊M₁₆	—	2	4	no	—
30	D₆₁⋊M₁₆	—	4	2	no	—
31	C₂×#11758217	—	4	1	no	—
32	C₂×D₆₁⋊C₈	—	4	1	no	—
33	#14519856	—	2	4	no	—
34	#14519856	—	4	2	no	—
35	C₂×C₄⋊#6923	—	2	2	no	—
36	C₂×C₂²⋊#6923	—	4	1	no	—
37	C₂×C₄×#6923	—	4	1	no	—
38	#14519848	—	2	4	no	—
39	#14519847	—	2	2	no	—
40	#14519847	C₂³	1	4	yes	—
41	#14519846	—	2	4	no	—
42	#14519845	—	2	2	no	—
43	#14519845	C₂³	1	4	yes	—
44	C₂²×C₄×D₆₁	D₄	1	1	yes	characteristic
45	C₂²×#92418	Q₈	1	1	yes	characteristic
46	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
49	C₂×C₈×D₆₁	D₄	1	2	yes	—
50	C₂×C₈×D₆₁	Q₈	1	2	yes	—
51	C₂×C₈×D₆₁	Q₈	1	1	yes	characteristic
52	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
53	C₂×C₈×D₆₁	C₂³	1	2	yes	—
54	#14519716	C₂×C₄	1	2	yes	—
55	#14519716	C₂×C₄	1	2	yes	—
56	#14519716	C₂×C₄	1	4	yes	complemented
57	#14519716	C₂×C₄	1	4	yes	complemented
58	#14519716	C₂³	1	2	yes	—
59	#14519716	C₂³	1	2	yes	—
60	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
61	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
62	C₂×#11758185	C₂×C₄	1	4	yes	complemented
63	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
64	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
65	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	complemented
67	C₂³×D₆₁	SD₁₆	1	1	yes	characteristic
68	C₂²×#6923	—	4	2	no	—
69	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×Dic₆₁	SD₁₆	1	1	yes	characteristic
71	C₂×C₄×D₆₁	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×D₆₁	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×D₆₁	SD₁₆	1	1	yes	characteristic
74	C₂×C₄×D₆₁	SD₁₆	1	2	yes	—
75	C₂×C₄×D₆₁	SD₁₆	1	1	yes	characteristic
76	C₂×C₄×D₆₁	SD₁₆	1	2	yes	—
77	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×D₆₁	C₂×D₄	1	2	yes	—
79	C₂²⋊#6923	—	4	4	no	—
80	#11758217	—	4	4	no	—
81	C₂×#92420	—	4	2	no	—
82	C₄⋊#6923	—	2	4	no	—
83	C₄⋊#6923	—	4	2	no	—
84	C₄×#6923	—	4	4	no	—
85	#11758213	—	2	4	no	—
86	#11758213	—	4	2	no	—
87	D₆₁⋊C₈	—	4	4	no	—
88	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
90	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
92	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
93	C₂×#92418	C₄⋊C₄	1	2	yes	—
94	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
95	C₂×#92418	C₂×Q₈	1	2	yes	—
96	C₈×D₆₁	C₄⋊C₄	1	2	yes	—
97	C₈×D₆₁	C₄⋊C₄	1	2	yes	—
98	C₈×D₆₁	C₄⋊C₄	1	2	yes	—
99	C₈×D₆₁	C₂²×C₄	1	1	yes	characteristic
100	C₈×D₆₁	C₂²×C₄	1	2	yes	—
101	C₈×D₆₁	C₂×D₄	1	1	yes	characteristic
102	C₈×D₆₁	C₂×D₄	1	2	yes	—
103	C₈×D₆₁	C₂×Q₈	1	1	yes	characteristic
104	C₈×D₆₁	C₂×Q₈	1	2	yes	—
105	C₈×D₆₁	C₂⁴	1	1	yes	characteristic
106	C₁₆×C₆₁	C₂²×C₄	1	2	yes	—
107	C₁₆×C₆₁	C₂²×C₄	1	2	yes	—
108	#11758185	C₂²×C₄	1	2	yes	—
109	#11758185	C₂²×C₄	1	2	yes	—
110	C₂³×C₆₁	#106	1	1	yes	characteristic
111	C₂²×D₆₁	—	4	2	no	—
112	C₂²×D₆₁	#106	1	1	yes	characteristic
113	C₂²×D₆₁	#106	1	2	yes	—
114	C₂²×D₆₁	C₂×SD₁₆	1	1	yes	characteristic
115	C₂²×D₆₁	C₂×SD₁₆	1	2	yes	—
116	C₂×#6923	—	4	4	no	—
117	C₂×#6923	—	8	4	no	—
118	C₂×C₄×C₆₁	#106	1	2	yes	—
119	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
120	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×Dic₆₁	#106	1	2	yes	—
123	C₂×Dic₆₁	#106	1	1	yes	characteristic
124	C₂×Dic₆₁	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×Dic₆₁	C₂×SD₁₆	1	2	yes	—
126	C₄×D₆₁	#106	1	2	yes	—
127	C₄×D₆₁	#106	1	2	yes	—
128	C₄×D₆₁	#106	1	2	yes	—
129	C₄×D₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₄×D₆₁	C₂×C₄⋊C₄	1	2	yes	—
131	C₄×D₆₁	C₂×SD₁₆	1	2	yes	—
132	C₄×D₆₁	C₂×SD₁₆	1	1	yes	characteristic
133	C₄×D₆₁	C₂×SD₁₆	1	2	yes	—
134	C₄×D₆₁	C₂×SD₁₆	1	1	yes	characteristic
135	C₄×D₆₁	C₂²×D₄	1	1	yes	characteristic
136	#92420	—	4	4	no	—
137	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₈×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
139	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
140	#92418	C₂×C₄⋊C₄	1	1	yes	characteristic
141	#92418	C₂×C₄⋊C₄	1	2	yes	—
142	#92418	C₂²×Q₈	1	1	yes	characteristic
143	C₂×#2155	—	61	1	no	maximal, Sylow
144	C₂²×C₆₁	—	2	2	no	—
145	C₂²×C₆₁	C₂×#106	1	2	yes	—
146	C₂²×C₆₁	C₂×#106	1	1	yes	characteristic
147	C₂×D₆₁	—	2	2	no	—
148	C₂×D₆₁	—	2	2	no	—
149	C₂×D₆₁	—	4	2	no	—
150	C₂×D₆₁	—	4	2	no	—
151	C₂×D₆₁	C₂×#106	1	2	yes	—
152	C₂×D₆₁	C₂×#106	1	1	yes	characteristic
153	C₂×D₆₁	C₂²×SD₁₆	1	1	yes	characteristic
154	#6923	—	8	4	no	—
155	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
156	C₄×C₆₁	C₂×#106	1	2	yes	—
157	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
158	Dic₆₁	C₂×#106	1	2	yes	—
159	Dic₆₁	C₂×#106	1	1	yes	characteristic
160	Dic₆₁	C₂²×SD₁₆	1	1	yes	characteristic
161	C₂²×M₃₂	—	61	1	no	—
162	C₂²×#108	—	61	1	no	—
163	C₂×#427	—	61	1	no	—
164	#2155	—	61	8	no	—
165	C₂×#365	—	61	2	no	—
166	C₂×#365	—	61	2	no	—
167	C₂×C₆₁	—	2	2	no	—
168	C₂×C₆₁	#2155	1	2	yes	complemented
169	C₂×C₆₁	C₂²×#106	1	1	yes	characteristic
170	D₆₁	—	2	1	no	—
171	D₆₁	—	2	1	no	—
172	D₆₁	—	2	2	no	—
173	C₂²×M₁₆	—	122	1	no	—
174	C₂³×C₈	—	61	1	no	—
175	C₂×#117	—	122	1	no	—
176	C₂×M₃₂	—	61	1	no	—
177	C₂×M₃₂	—	61	1	no	—
178	C₂×M₃₂	—	61	1	no	—
179	C₂×M₃₂	—	61	1	no	—
180	C₂×M₃₂	—	61	4	no	—
181	C₂×M₃₂	—	61	4	no	—
182	C₂²×C₁₆	—	61	1	no	—
183	C₂²×C₁₆	#6923	1	1	yes	characteristic, complemented
184	C₂×#108	—	61	2	no	—
185	C₂×#108	—	61	4	no	—
186	C₂×#108	—	122	2	no	—
187	C₂×#108	—	122	1	no	—
188	#427	—	61	4	no	—
189	#427	—	122	2	no	—
190	C₂×#107	—	122	1	no	—
191	C₂×#106	—	61	2	no	—
192	#365	—	61	8	no	—
193	#365	—	61	8	no	—
194	C₆₁	C₂×#2155	1	1	yes	characteristic, complemented, Sylow
195	C₂³×C₄	—	61	1	no	—
196	C₂×M₁₆	—	122	4	no	—
197	C₂×M₁₆	—	122	2	no	—
198	C₂×M₁₆	—	244	1	no	—
199	C₂×M₁₆	—	244	2	no	—
200	C₂²×C₈	—	61	1	no	—
201	C₂²×C₈	—	61	1	no	—
202	C₂²×C₈	—	61	2	no	—
203	C₂²×C₈	—	61	1	no	—
204	C₂²×C₈	—	61	2	no	—
205	C₂²×C₈	—	61	1	no	—
206	C₂²×C₈	—	61	2	no	—
207	C₂²×C₈	—	61	1	no	—
208	C₂²×C₈	—	61	2	no	—
209	C₂²×C₈	—	244	1	no	—
210	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
211	#117	—	122	4	no	—
212	#117	—	244	2	no	—
213	C₂×C₄⋊C₄	—	122	2	no	—
214	C₂×C₂²⋊C₄	—	244	1	no	—
215	C₂×C₄²	—	244	1	no	—
216	M₃₂	—	61	2	no	—
217	M₃₂	—	61	2	no	—
218	M₃₂	—	61	2	no	—
219	M₃₂	—	61	2	no	—
220	M₃₂	—	61	4	no	—
221	M₃₂	—	61	4	no	—
222	C₂×C₁₆	—	61	1	no	—
223	C₂×C₁₆	—	61	1	no	—
224	C₂×C₁₆	—	61	4	no	—
225	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
226	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
227	C₂×C₁₆	C₂×#6923	1	4	yes	complemented
228	#108	—	61	4	no	—
229	#108	—	122	2	no	—
230	#108	—	122	4	no	—
231	#107	—	122	4	no	—
232	#106	—	61	4	no	—
233	#106	—	122	2	no	—
234	C₂⁴	—	61	1	no	—
235	C₂²×C₄	—	61	1	no	—
236	C₂²×C₄	—	61	2	no	—
237	C₂²×C₄	—	61	2	no	—
238	C₂²×C₄	—	61	1	no	—
239	C₂²×C₄	—	61	1	no	—
240	C₂²×C₄	—	61	1	no	—
241	C₂²×C₄	—	61	2	no	—
242	C₂²×C₄	—	61	1	no	—
243	C₂²×C₄	—	61	2	no	—
244	C₂²×C₄	—	244	2	no	—
245	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
246	M₁₆	—	122	4	no	—
247	M₁₆	—	244	2	no	—
248	M₁₆	—	244	4	no	—
249	C₂×C₈	—	61	1	no	—
250	C₂×C₈	—	61	2	no	—
251	C₂×C₈	—	61	1	no	—
252	C₂×C₈	—	61	2	no	—
253	C₂×C₈	—	61	1	no	—
254	C₂×C₈	—	61	2	no	—
255	C₂×C₈	—	61	2	no	—
256	C₂×C₈	—	61	2	no	—
257	C₂×C₈	—	61	2	no	—
258	C₂×C₈	—	61	1	no	—
259	C₂×C₈	—	61	2	no	—
260	C₂×C₈	—	61	1	no	—
261	C₂×C₈	—	61	1	no	—
262	C₂×C₈	—	61	2	no	—
263	C₂×C₈	—	244	4	no	—
264	C₂×C₈	—	244	2	no	—
265	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
266	C₂×C₈	C₄⋊#6923	1	2	yes	—
267	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
268	C₂×C₈	C₂²×#6923	1	2	yes	—
269	C₄⋊C₄	—	122	4	no	—
270	C₄⋊C₄	—	244	2	no	—
271	C₂²⋊C₄	—	244	4	no	—
272	C₄²	—	244	4	no	—
273	C₁₆	—	61	2	no	—
274	C₁₆	—	61	2	no	—
275	C₁₆	C₂²×#6923	1	2	yes	—
276	C₁₆	C₂²×#6923	1	2	yes	—
277	C₂³	—	61	1	no	—
278	C₂³	—	61	1	no	—
279	C₂³	—	61	2	no	—
280	C₂³	—	61	2	no	—
281	C₂³	—	244	2	no	—
282	C₂³	#14519845	1	1	yes	characteristic
283	C₂×C₄	—	61	2	no	—
284	C₂×C₄	—	61	1	no	—
285	C₂×C₄	—	61	1	no	—
286	C₂×C₄	—	61	2	no	—
287	C₂×C₄	—	61	2	no	—
288	C₂×C₄	—	61	2	no	—
289	C₂×C₄	—	61	2	no	—
290	C₂×C₄	—	61	2	no	—
291	C₂×C₄	—	61	1	no	—
292	C₂×C₄	—	61	2	no	—
293	C₂×C₄	—	61	1	no	—
294	C₂×C₄	—	61	1	no	—
295	C₂×C₄	—	61	2	no	—
296	C₂×C₄	—	61	1	no	—
297	C₂×C₄	—	488	4	no	—
298	C₂×C₄	—	244	4	no	—
299	C₂×C₄	#14519845	1	2	yes	—
300	C₂×C₄	#14519845	1	1	yes	characteristic
301	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
302	C₂×C₄	C₂×C₄⋊#6923	1	2	yes	—
303	C₈	—	61	1	no	—
304	C₈	—	61	2	no	—
305	C₈	—	61	1	no	—
306	C₈	—	244	4	no	—
307	C₈	C₂×C₄⋊#6923	1	1	yes	characteristic
308	C₈	C₂×C₄⋊#6923	1	2	yes	—
309	C₈	C₂³×#6923	1	1	yes	characteristic
310	C₂²	—	2	2	no	—
311	C₂²	—	61	1	no	—
312	C₂²	—	61	2	no	—
313	C₂²	—	61	1	no	—
314	C₂²	—	122	2	no	—
315	C₂²	—	122	2	no	—
316	C₂²	—	244	2	no	—
317	C₂²	—	244	2	no	—
318	C₂²	C₂×#14519845	1	2	yes	—
319	C₂²	C₂×#14519845	1	1	yes	characteristic
320	C₄	—	61	1	no	—
321	C₄	—	61	1	no	—
322	C₄	—	61	2	no	—
323	C₄	—	488	4	no	—
324	C₄	C₂×#14519845	1	1	yes	characteristic
325	C₄	C₂×#14519845	1	2	yes	—
326	C₄	C₂²×C₄⋊#6923	1	1	yes	characteristic
327	C₂	—	2	2	no	—
328	C₂	—	122	1	no	—
329	C₂	—	122	1	no	—
330	C₂	—	122	2	no	—
331	C₂	#21462166	1	2	yes	direct product factor
332	C₂	C₂²×#14519845	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14519845
commutator	C₈×C₆₁	C₂³×C₄
Frattini	C₈	C₂³×#6923
Fitting	C₂²×C₁₆×C₆₁	C₄
socle	C₂²×C₆₁	C₂×#106
prime radical	C₂³×D₆₁	SD₁₆
cosocle	C₈×D₆₁	C₂⁴
hypercenter	C₂²×C₁₆	#6923
hypocenter	C₆₁	C₂×#2155
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #21462166	C₂	direct (N × H)
4. C₂×#16840687	C₂	semidirect
5. C₂×#16840686	C₂	semidirect
17. #16840687	C₂²	semidirect
18. #16840686	C₂²	semidirect
20. C₂×#14519716	C₄	semidirect
21. C₂×#14519716	C₄	semidirect
22. C₂×#14519716	C₄	semidirect
26. C₂²×#11758185	C₄	semidirect
27. C₂²×C₁₆×C₆₁	C₄	semidirect
56. #14519716	C₂×C₄	semidirect
57. #14519716	C₂×C₄	semidirect
62. C₂×#11758185	C₂×C₄	semidirect
66. C₂×C₁₆×C₆₁	C₂×C₄	semidirect
168. C₂×C₆₁	#2155	semidirect
183. C₂²×C₁₆	#6923	semidirect
194. C₆₁	C₂×#2155	semidirect
227. C₂×C₁₆	C₂×#6923	semidirect
331. C₂	#21462166	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×D₆₁⋊#108	2
C₂×#16840772	2
#21462166	2
C₂×#16840687	2
C₂×#16840686	2
C₂²×#14519716	2
C₂×#2155	61

Subgroups — C2×#21462166 #30624115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21462166 #30624115