Subgroups — C₂×D₆₁⋊#368 #30624128

Order 15616.

662 conjugacy classes of subgroups.

The subgroup lattice has 290 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	2	yes	maximal
2	C₂×#16840691	C₂	1	2	yes	maximal, complemented
3	D₆₁⋊#368	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16840690	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×D₆₁⋊M₁₆	—	2	2	no	—
7	C₂×#14519848	C₂²	1	4	yes	—
8	D₆₁⋊#108	—	2	4	no	—
9	D₆₁⋊#108	C₂²	1	8	yes	—
10	C₂×#14519847	—	2	2	no	—
11	#16840691	C₂²	1	8	yes	complemented
12	#16840690	C₂²	1	8	yes	complemented
13	C₂²×C₈×D₆₁	C₂²	1	1	yes	characteristic
14	C₂²×#11758185	C₄	1	1	yes	characteristic
15	C₂×C₁₆×D₆₁	C₄	1	1	yes	characteristic
16	C₂×C₁₆×D₆₁	C₄	1	4	yes	—
17	C₂×C₁₆×D₆₁	C₄	1	1	yes	characteristic
18	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×D₆₁	C₂²	1	4	yes	—
20	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₆₁	C₄	1	1	yes	characteristic
22	C₂×#11758213	—	2	4	no	—
23	D₆₁⋊M₁₆	—	2	8	no	—
24	D₆₁⋊M₁₆	—	4	4	no	—
25	C₂×#11758217	—	4	2	no	—
26	C₂×D₆₁⋊C₈	—	4	2	no	—
27	#14519848	—	2	4	no	—
28	#14519848	C₂³	1	8	yes	—
29	#14519847	—	2	8	no	—
30	C₂²×C₄×D₆₁	D₄	1	1	yes	characteristic
31	C₂²×#92418	Q₈	1	1	yes	characteristic
32	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
35	C₂×C₈×D₆₁	D₄	1	2	yes	—
36	C₂×C₈×D₆₁	Q₈	1	2	yes	—
37	C₂×C₈×D₆₁	Q₈	1	1	yes	characteristic
38	C₂×C₈×D₆₁	C₂³	1	2	yes	—
39	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
40	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
41	C₂×#11758185	C₂×C₄	1	4	yes	—
42	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
43	C₁₆×D₆₁	C₂×C₄	1	2	yes	—
44	C₁₆×D₆₁	C₂×C₄	1	4	yes	—
45	C₁₆×D₆₁	C₂×C₄	1	4	yes	—
46	C₁₆×D₆₁	C₂×C₄	1	2	yes	—
47	C₁₆×D₆₁	C₂³	1	2	yes	—
48	C₁₆×D₆₁	C₂³	1	2	yes	—
49	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	—
52	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
53	C₂³×D₆₁	Q₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×Dic₆₁	D₈	1	1	yes	characteristic
56	C₂×C₄×D₆₁	C₄⋊C₄	1	2	yes	—
57	C₂×C₄×D₆₁	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×D₆₁	D₈	1	1	yes	characteristic
59	C₂×C₄×D₆₁	D₈	1	2	yes	—
60	C₂×C₄×D₆₁	Q₁₆	1	2	yes	—
61	C₂×C₄×D₆₁	Q₁₆	1	1	yes	characteristic
62	C₂×C₄×D₆₁	C₂×D₄	1	2	yes	—
63	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
64	#11758217	—	4	8	no	—
65	C₂×#92420	—	4	4	no	—
66	#11758213	—	2	8	no	—
67	#11758213	—	4	4	no	—
68	D₆₁⋊C₈	—	4	8	no	—
69	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
71	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
73	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
74	C₂×#92418	C₄⋊C₄	1	2	yes	—
75	C₂×#92418	C₂×Q₈	1	2	yes	—
76	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
77	C₈×D₆₁	C₄⋊C₄	1	2	yes	—
78	C₈×D₆₁	C₄⋊C₄	1	2	yes	—
79	C₈×D₆₁	C₄⋊C₄	1	2	yes	—
80	C₈×D₆₁	C₂²×C₄	1	2	yes	—
81	C₈×D₆₁	C₂²×C₄	1	1	yes	characteristic
82	C₈×D₆₁	C₂×D₄	1	2	yes	—
83	C₈×D₆₁	C₂×D₄	1	1	yes	characteristic
84	C₈×D₆₁	C₂×Q₈	1	2	yes	—
85	C₈×D₆₁	C₂×Q₈	1	1	yes	characteristic
86	C₈×D₆₁	C₂⁴	1	1	yes	characteristic
87	C₁₆×C₆₁	C₂²×C₄	1	2	yes	—
88	C₁₆×C₆₁	C₂²×C₄	1	2	yes	—
89	#11758185	C₂²×C₄	1	2	yes	—
90	#11758185	C₂²×C₄	1	2	yes	—
91	C₂³×C₆₁	#107	1	1	yes	characteristic
92	C₂²×D₆₁	—	2	4	no	—
93	C₂²×D₆₁	#107	1	2	yes	—
94	C₂²×D₆₁	#107	1	1	yes	characteristic
95	C₂²×D₆₁	C₂×Q₁₆	1	1	yes	characteristic
96	C₂²×D₆₁	C₂×Q₁₆	1	2	yes	—
97	C₂×C₄×C₆₁	#107	1	2	yes	—
98	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×Dic₆₁	#107	1	2	yes	—
102	C₂×Dic₆₁	#107	1	1	yes	characteristic
103	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
104	C₂×Dic₆₁	C₂×D₈	1	2	yes	—
105	C₄×D₆₁	#107	1	2	yes	—
106	C₄×D₆₁	#107	1	2	yes	—
107	C₄×D₆₁	#107	1	2	yes	—
108	C₄×D₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₄×D₆₁	C₂×C₄⋊C₄	1	2	yes	—
110	C₄×D₆₁	C₂×D₈	1	2	yes	—
111	C₄×D₆₁	C₂×D₈	1	1	yes	characteristic
112	C₄×D₆₁	C₂×Q₁₆	1	2	yes	—
113	C₄×D₆₁	C₂×Q₁₆	1	1	yes	characteristic
114	C₄×D₆₁	C₂²×D₄	1	1	yes	characteristic
115	#92420	—	4	8	no	—
116	C₈×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
117	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
119	#92418	C₂×C₄⋊C₄	1	2	yes	—
120	#92418	C₂×C₄⋊C₄	1	1	yes	characteristic
121	#92418	C₂²×Q₈	1	1	yes	characteristic
122	C₂²×#368	—	61	1	no	maximal, Sylow
123	C₂²×C₆₁	—	2	2	no	—
124	C₂²×C₆₁	C₂×#107	1	2	yes	—
125	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
126	C₂×D₆₁	—	2	2	no	—
127	C₂×D₆₁	—	2	4	no	—
128	C₂×D₆₁	—	2	4	no	—
129	C₂×D₆₁	#368	1	4	yes	complemented
130	C₂×D₆₁	C₂×#107	1	1	yes	characteristic
131	C₂×D₆₁	C₂×#107	1	2	yes	—
132	C₂×D₆₁	C₂²×Q₁₆	1	1	yes	characteristic
133	C₄×C₆₁	C₂×#107	1	1	yes	characteristic
134	C₄×C₆₁	C₂×#107	1	2	yes	—
135	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
136	Dic₆₁	C₂×#107	1	1	yes	characteristic
137	Dic₆₁	C₂×#107	1	2	yes	—
138	Dic₆₁	C₂²×D₈	1	1	yes	characteristic
139	C₂³×C₁₆	—	61	1	no	—
140	C₂²×#108	—	61	2	no	—
141	C₂×#368	—	61	2	no	—
142	C₂×#368	—	61	8	no	—
143	C₂×#368	—	61	2	no	—
144	C₂×C₆₁	—	2	2	no	—
145	C₂×C₆₁	C₂×#368	1	2	yes	complemented
146	C₂×C₆₁	C₂²×#107	1	1	yes	characteristic
147	D₆₁	—	2	1	no	—
148	D₆₁	—	2	1	no	—
149	D₆₁	C₂×#368	1	4	yes	complemented
150	C₂²×M₁₆	—	122	2	no	—
151	C₂³×C₈	—	61	1	no	—
152	C₂²×C₁₆	—	61	1	no	—
153	C₂²×C₁₆	—	61	1	no	—
154	C₂²×C₁₆	—	61	4	no	—
155	C₂²×C₁₆	—	61	1	no	—
156	C₂²×C₁₆	—	61	1	no	—
157	C₂²×C₁₆	—	61	4	no	—
158	C₂²×C₁₆	—	61	1	no	—
159	C₂²×C₁₆	#6923	1	1	yes	characteristic
160	C₂×#108	—	61	8	no	—
161	C₂×#108	—	61	4	no	—
162	C₂×#108	—	122	2	no	—
163	C₂×#108	—	122	4	no	—
164	#368	—	61	8	no	—
165	#368	—	61	8	no	—
166	C₆₁	C₂²×#368	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₄	—	61	1	no	—
168	C₂×M₁₆	—	122	8	no	—
169	C₂×M₁₆	—	122	4	no	—
170	C₂×M₁₆	—	244	2	no	—
171	C₂×M₁₆	—	244	4	no	—
172	C₂²×C₈	—	61	1	no	—
173	C₂²×C₈	—	61	1	no	—
174	C₂²×C₈	—	61	2	no	—
175	C₂²×C₈	—	61	1	no	—
176	C₂²×C₈	—	61	2	no	—
177	C₂²×C₈	—	61	2	no	—
178	C₂²×C₈	—	61	1	no	—
179	C₂²×C₈	—	61	1	no	—
180	C₂²×C₈	—	61	2	no	—
181	C₂²×C₈	—	244	2	no	—
182	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
183	C₂×C₁₆	—	61	1	no	—
184	C₂×C₁₆	—	61	4	no	—
185	C₂×C₁₆	—	61	1	no	—
186	C₂×C₁₆	—	61	2	no	—
187	C₂×C₁₆	—	61	2	no	—
188	C₂×C₁₆	—	61	2	no	—
189	C₂×C₁₆	—	61	4	no	—
190	C₂×C₁₆	—	61	4	no	—
191	C₂×C₁₆	—	61	2	no	—
192	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
193	C₂×C₁₆	C₂×#6923	1	4	yes	—
194	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
195	#108	—	61	8	no	—
196	#108	—	122	8	no	—
197	#108	—	122	4	no	—
198	C₂⁴	—	61	1	no	—
199	C₂²×C₄	—	61	1	no	—
200	C₂²×C₄	—	61	2	no	—
201	C₂²×C₄	—	61	2	no	—
202	C₂²×C₄	—	61	1	no	—
203	C₂²×C₄	—	61	1	no	—
204	C₂²×C₄	—	61	1	no	—
205	C₂²×C₄	—	61	2	no	—
206	C₂²×C₄	—	61	1	no	—
207	C₂²×C₄	—	61	2	no	—
208	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
209	M₁₆	—	122	8	no	—
210	M₁₆	—	244	4	no	—
211	M₁₆	—	244	8	no	—
212	C₂×C₈	—	61	1	no	—
213	C₂×C₈	—	61	2	no	—
214	C₂×C₈	—	61	1	no	—
215	C₂×C₈	—	61	2	no	—
216	C₂×C₈	—	61	2	no	—
217	C₂×C₈	—	61	1	no	—
218	C₂×C₈	—	61	2	no	—
219	C₂×C₈	—	61	2	no	—
220	C₂×C₈	—	61	2	no	—
221	C₂×C₈	—	61	2	no	—
222	C₂×C₈	—	61	1	no	—
223	C₂×C₈	—	61	1	no	—
224	C₂×C₈	—	61	2	no	—
225	C₂×C₈	—	61	1	no	—
226	C₂×C₈	—	244	8	no	—
227	C₂×C₈	—	244	4	no	—
228	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
229	C₂×C₈	C₄⋊#6923	1	2	yes	—
230	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
231	C₂×C₈	C₂²×#6923	1	2	yes	—
232	C₁₆	—	61	2	no	—
233	C₁₆	—	61	2	no	—
234	C₁₆	C₂²×#6923	1	2	yes	—
235	C₁₆	C₂²×#6923	1	2	yes	—
236	C₂³	—	61	1	no	—
237	C₂³	—	61	2	no	—
238	C₂³	—	61	2	no	—
239	C₂³	—	61	1	no	—
240	C₂³	—	122	4	no	—
241	C₂³	#14519846	1	1	yes	characteristic
242	C₂×C₄	—	61	2	no	—
243	C₂×C₄	—	61	1	no	—
244	C₂×C₄	—	61	1	no	—
245	C₂×C₄	—	61	2	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	—	61	1	no	—
251	C₂×C₄	—	61	1	no	—
252	C₂×C₄	—	61	2	no	—
253	C₂×C₄	—	61	2	no	—
254	C₂×C₄	—	61	1	no	—
255	C₂×C₄	—	61	1	no	—
256	C₂×C₄	#14519846	1	2	yes	—
257	C₂×C₄	#14519846	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄⋊#6923	1	2	yes	—
260	C₈	—	61	2	no	—
261	C₈	—	61	1	no	—
262	C₈	—	61	1	no	—
263	C₈	—	244	8	no	—
264	C₈	C₂×C₄⋊#6923	1	2	yes	—
265	C₈	C₂×C₄⋊#6923	1	1	yes	characteristic
266	C₈	C₂³×#6923	1	1	yes	characteristic
267	C₂²	—	2	2	no	—
268	C₂²	—	61	4	no	—
269	C₂²	—	61	2	no	—
270	C₂²	—	61	1	no	—
271	C₂²	—	61	1	no	—
272	C₂²	—	122	4	no	—
273	C₂²	—	122	4	no	—
274	C₂²	—	122	2	no	—
275	C₂²	C₂×#14519846	1	2	yes	—
276	C₂²	C₂×#14519846	1	1	yes	characteristic
277	C₄	—	61	1	no	—
278	C₄	—	61	1	no	—
279	C₄	—	61	2	no	—
280	C₄	C₂×#14519846	1	1	yes	characteristic
281	C₄	C₂×#14519846	1	2	yes	—
282	C₄	C₂²×C₄⋊#6923	1	1	yes	characteristic
283	C₂	—	2	2	no	—
284	C₂	—	61	4	no	—
285	C₂	—	122	1	no	—
286	C₂	—	122	1	no	—
287	C₂	D₆₁⋊#368	1	2	yes	direct product factor
288	C₂	C₂²×#14519846	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14519846
commutator	C₈×C₆₁	C₂³×C₄
Frattini	C₈	C₂³×#6923
Fitting	C₂²×C₁₆×C₆₁	C₄
socle	C₂²×C₆₁	C₂×#107
prime radical	C₂³×D₆₁	Q₁₆
cosocle	C₈×D₆₁	C₂⁴
hypercenter	C₂²×C₁₆	#6923
hypocenter	C₆₁	C₂²×#368
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
2. C₂×#16840691	C₂	semidirect
3. D₆₁⋊#368	C₂	direct (N × H)
4. C₂×#16840690	C₂	semidirect
11. #16840691	C₂²	semidirect
12. #16840690	C₂²	semidirect
129. C₂×D₆₁	#368	semidirect
145. C₂×C₆₁	C₂×#368	semidirect
149. D₆₁	C₂×#368	semidirect
166. C₆₁	C₂²×#368	semidirect
287. C₂	D₆₁⋊#368	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×#16840691	2
D₆₁⋊#368	2
C₂×#16840690	2
C₂²×C₁₆×D₆₁	2
C₂²×#368	61

Subgroups — C2×D61⋊#368 #30624128

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₆₁⋊#368 #30624128