Subgroups — #30601672

Order 15616.

606 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	#21461940	C₂	1	2	yes	maximal, complemented
2	C₂×#16840625	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄⋊#92420	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×#92420	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92420	C₂²	1	2	yes	—
7	C₂×C₂²⋊#92420	—	2	4	no	—
8	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
9	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
10	#16840625	—	2	2	no	—
11	D₄⋊#92420	—	2	4	no	—
12	C₈×#92420	—	2	2	no	—
13	C₂×D₄×Dic₆₁	C₂²	1	2	yes	—
14	D₈×Dic₆₁	—	2	2	no	—
15	D₈×Dic₆₁	—	2	2	no	—
16	C₂×#14519711	C₄	1	1	yes	characteristic
17	C₂×#14519698	C₄	1	2	yes	—
18	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
19	C₂²×D₈×C₆₁	C₄	1	1	yes	characteristic
20	C₂²×#92420	—	2	1	no	—
21	C₂²×#92420	—	2	2	no	—
22	C₂²×#92420	D₄	1	2	yes	—
23	C₄⋊#92420	—	2	2	no	—
24	C₄⋊#92420	—	2	2	no	—
25	C₂²⋊#92420	—	4	8	no	—
26	C₄×#92420	—	2	1	no	—
27	C₄×#92420	—	2	1	no	—
28	C₄×#92420	—	2	1	no	—
29	C₄×#92420	D₄	1	2	yes	—
30	C₂³×Dic₆₁	—	2	2	no	—
31	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
32	D₄×Dic₆₁	—	2	2	no	—
33	D₄×Dic₆₁	—	2	2	no	—
34	D₄×Dic₆₁	—	2	2	no	—
35	D₄×Dic₆₁	—	2	2	no	—
36	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊Dic₆₁	—	2	2	no	—
38	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
39	#14519711	—	2	2	no	—
40	#14519698	—	2	4	no	—
41	C₈×Dic₆₁	D₄	1	2	yes	—
42	C₈×Dic₆₁	D₄	1	2	yes	—
43	C₂²×D₄×C₆₁	C₂×C₄	1	2	yes	—
44	C₂×D₈×C₆₁	—	2	2	no	—
45	C₂×D₈×C₆₁	—	2	2	no	—
46	C₂×D₈×C₆₁	C₈	1	4	yes	complemented
47	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
48	C₂⁴×C₆₁	—	2	2	no	—
49	C₂×D₄×C₆₁	—	2	2	no	—
50	C₂×D₄×C₆₁	—	2	2	no	—
51	C₂×D₄×C₆₁	—	2	2	no	—
52	C₂×D₄×C₆₁	—	2	2	no	—
53	C₂×D₄×C₆₁	—	2	2	no	—
54	C₂×D₄×C₆₁	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₆₁	—	2	2	no	—
57	C₂²×Dic₆₁	—	4	2	no	—
58	C₂²×Dic₆₁	—	4	2	no	—
59	C₂²×Dic₆₁	—	4	4	no	—
60	C₂²×Dic₆₁	—	4	4	no	—
61	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₆₁	#41	1	1	yes	characteristic
63	C₂×#92420	—	2	4	no	—
64	C₂×#92420	—	2	2	no	—
65	C₂×#92420	—	2	2	no	—
66	C₂×#92420	—	2	2	no	—
67	C₂×#92420	—	4	2	no	—
68	C₂×#92420	—	4	4	no	—
69	C₂×#92420	D₈	1	4	yes	complemented
70	D₈×C₆₁	—	2	8	no	—
71	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
73	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
74	C₂²⋊Dic₆₁	—	4	2	no	—
75	C₂²⋊Dic₆₁	—	4	2	no	—
76	C₄⋊Dic₆₁	—	2	2	no	—
77	C₄⋊Dic₆₁	—	2	2	no	—
78	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
79	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
80	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
81	C₄×Dic₆₁	#41	1	1	yes	characteristic
82	C₂×#92418	—	2	1	no	—
83	C₂×#92418	—	2	1	no	—
84	C₂×#92418	C₂²⋊C₄	1	2	yes	—
85	C₂³×C₆₁	—	2	4	no	—
86	C₂³×C₆₁	—	4	2	no	—
87	C₂³×C₆₁	—	4	2	no	—
88	C₂³×C₆₁	—	4	4	no	—
89	C₂³×C₆₁	—	4	4	no	—
90	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
91	D₄×C₆₁	—	2	4	no	—
92	D₄×C₆₁	—	2	4	no	—
93	D₄×C₆₁	—	4	2	no	—
94	D₄×C₆₁	—	4	2	no	—
95	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
100	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
101	C₂×Dic₆₁	—	2	2	no	—
102	C₂×Dic₆₁	—	4	2	no	—
103	C₂×Dic₆₁	—	4	4	no	—
104	C₂×Dic₆₁	—	8	4	no	—
105	C₂×Dic₆₁	—	8	2	no	—
106	C₂×Dic₆₁	—	8	2	no	—
107	C₂×Dic₆₁	#121	1	1	yes	characteristic
108	C₂×Dic₆₁	#121	1	1	yes	characteristic
109	C₂×Dic₆₁	#124	1	1	yes	characteristic
110	C₂×Dic₆₁	#127	1	1	yes	characteristic
111	C₂×Dic₆₁	C₂×D₈	1	2	yes	—
112	C₂×Dic₆₁	#135	1	2	yes	—
113	#92420	—	2	4	no	—
114	#92420	—	2	4	no	—
115	#92420	—	8	2	no	—
116	C₈×C₆₁	C₂²⋊C₈	1	4	yes	complemented
117	#92418	—	2	2	no	—
118	#9657	—	61	1	no	maximal, Sylow
119	C₂²×C₆₁	—	4	8	no	—
120	C₂²×C₆₁	—	4	4	no	—
121	C₂²×C₆₁	—	8	4	no	—
122	C₂²×C₆₁	—	8	2	no	—
123	C₂²×C₆₁	—	8	2	no	—
124	C₂²×C₆₁	#390	1	1	yes	characteristic
125	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
126	C₂²×C₆₁	#436	1	1	yes	characteristic
127	C₂²×C₆₁	C₄×D₈	1	2	yes	—
128	C₂²×C₆₁	#443	1	2	yes	—
129	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₄×C₆₁	#408	1	1	yes	characteristic
131	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
132	C₄×C₆₁	#436	1	1	yes	characteristic
133	Dic₆₁	—	2	1	no	—
134	Dic₆₁	—	2	1	no	—
135	Dic₆₁	—	8	2	no	—
136	Dic₆₁	#466	1	2	yes	—
137	Dic₆₁	#493	1	1	yes	characteristic
138	Dic₆₁	#495	1	1	yes	characteristic
139	C₂×C₄×D₈	—	61	1	no	—
140	#1954	—	61	2	no	—
141	C₂×#335	—	61	1	no	—
142	C₂×D₄⋊C₈	—	61	2	no	—
143	C₂×C₈²	—	61	1	no	—
144	C₂×C₆₁	—	8	4	no	—
145	C₂×C₆₁	C₈×D₈	1	2	yes	—
146	C₂×C₆₁	#1589	1	2	yes	—
147	C₂×C₆₁	#1954	1	1	yes	characteristic
148	C₂×C₆₁	#1970	1	1	yes	characteristic
149	C₂×C₆₁	#1974	1	1	yes	characteristic
150	C₂²×D₈	#6923	1	1	yes	characteristic
151	C₂×C₄×D₄	—	61	2	no	—
152	C₄×D₈	—	122	2	no	—
153	C₄×D₈	—	122	2	no	—
154	C₂×#107	—	61	1	no	—
155	C₂×C₄⋊C₈	—	61	2	no	—
156	C₂×#102	—	61	2	no	—
157	C₂×C₂²⋊C₈	—	122	4	no	—
158	C₂×C₄×C₈	—	61	1	no	—
159	C₂×C₄×C₈	—	61	1	no	—
160	C₂×C₄×C₈	—	61	1	no	—
161	#335	—	122	2	no	—
162	D₄⋊C₈	—	122	4	no	—
163	C₈²	—	122	2	no	—
164	C₆₁	#9657	1	1	yes	characteristic, complemented, Sylow
165	C₂²×D₄	C₂×#6923	1	2	yes	—
166	C₂³×C₄	—	122	2	no	—
167	C₂×D₈	—	2	2	no	—
168	C₂×D₈	—	2	2	no	—
169	C₂×D₈	#92420	1	4	yes	complemented
170	C₂²×C₈	—	61	2	no	—
171	C₂²×C₈	—	61	1	no	—
172	C₂²×C₈	—	122	1	no	—
173	C₂²×C₈	—	122	2	no	—
174	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
175	C₄×D₄	—	122	2	no	—
176	C₄×D₄	—	122	2	no	—
177	C₄×D₄	—	122	2	no	—
178	C₄×D₄	—	122	2	no	—
179	C₂×C₄⋊C₄	—	61	2	no	—
180	C₂×C₂²⋊C₄	—	122	2	no	—
181	C₂×C₄²	—	61	1	no	—
182	#107	—	122	2	no	—
183	C₄⋊C₈	—	122	2	no	—
184	C₄⋊C₈	—	122	2	no	—
185	#102	—	122	4	no	—
186	C₂²⋊C₈	—	244	8	no	—
187	C₄×C₈	—	61	2	no	—
188	C₄×C₈	—	61	2	no	—
189	C₄×C₈	—	61	2	no	—
190	C₄×C₈	—	122	1	no	—
191	C₄×C₈	—	122	1	no	—
192	C₄×C₈	—	122	1	no	—
193	C₂⁴	—	2	2	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	C₂×#92420	1	4	yes	—
200	C₂²×C₄	—	61	1	no	—
201	C₂²×C₄	—	61	1	no	—
202	C₂²×C₄	—	122	2	no	—
203	C₂²×C₄	—	244	4	no	—
204	C₂²×C₄	—	244	2	no	—
205	C₂²×C₄	—	244	2	no	—
206	C₂²×C₄	—	244	4	no	—
207	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
208	D₈	—	2	8	no	—
209	C₂×C₈	—	61	4	no	—
210	C₂×C₈	—	61	2	no	—
211	C₂×C₈	—	122	2	no	—
212	C₂×C₈	—	122	2	no	—
213	C₂×C₈	—	122	2	no	—
214	C₂×C₈	—	122	4	no	—
215	C₂×C₈	—	122	1	no	—
216	C₂×C₈	—	122	1	no	—
217	C₂×C₈	—	244	4	no	—
218	C₂×C₈	—	244	2	no	—
219	C₂×C₈	C₂×#92420	1	2	yes	—
220	C₂×C₈	#11758217	1	2	yes	—
221	C₂×C₈	C₂²⋊#6923	1	2	yes	—
222	C₄⋊C₄	—	122	2	no	—
223	C₄⋊C₄	—	122	2	no	—
224	C₂²⋊C₄	—	244	2	no	—
225	C₂²⋊C₄	—	244	2	no	—
226	C₄²	—	61	1	no	—
227	C₄²	—	61	1	no	—
228	C₄²	—	61	1	no	—
229	C₄²	—	61	1	no	—
230	C₂³	—	2	4	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	4	2	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	4	4	no	—
235	C₂³	D₄×#6923	1	1	yes	characteristic
236	D₄	—	2	4	no	—
237	D₄	—	2	4	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	C₂×C₄	—	61	1	no	—
241	C₂×C₄	—	61	1	no	—
242	C₂×C₄	—	61	2	no	—
243	C₂×C₄	—	61	2	no	—
244	C₂×C₄	—	61	1	no	—
245	C₂×C₄	—	61	1	no	—
246	C₂×C₄	—	122	2	no	—
247	C₂×C₄	—	488	4	no	—
248	C₂×C₄	—	488	2	no	—
249	C₂×C₄	—	488	2	no	—
250	C₂×C₄	—	244	2	no	—
251	C₂×C₄	—	244	4	no	—
252	C₂×C₄	C₂×C₂²⋊#6923	1	1	yes	characteristic
253	C₂×C₄	D₄×#6923	1	1	yes	characteristic
254	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
255	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
256	C₂×C₄	#14519864	1	1	yes	characteristic
257	C₂×C₄	#14519864	1	1	yes	characteristic
258	C₈	—	122	4	no	—
259	C₈	—	122	4	no	—
260	C₈	—	122	2	no	—
261	C₈	—	488	2	no	—
262	C₈	C₂²⋊#92420	1	4	yes	complemented
263	C₂²	—	4	8	no	—
264	C₂²	—	4	4	no	—
265	C₂²	—	8	4	no	—
266	C₂²	—	8	2	no	—
267	C₂²	—	8	2	no	—
268	C₂²	#16840714	1	1	yes	characteristic
269	C₂²	D₄×#92420	1	1	yes	characteristic
270	C₂²	#16840797	1	1	yes	characteristic
271	C₂²	D₈×#6923	1	2	yes	—
272	C₂²	#16840804	1	2	yes	—
273	C₄	—	61	2	no	—
274	C₄	—	61	1	no	—
275	C₄	—	61	1	no	—
276	C₄	—	122	1	no	—
277	C₄	—	122	1	no	—
278	C₄	—	488	2	no	—
279	C₄	C₂×C₂²⋊#92420	1	1	yes	characteristic
280	C₄	#16840736	1	1	yes	characteristic
281	C₄	D₄×#92420	1	1	yes	characteristic
282	C₄	#16840797	1	1	yes	characteristic
283	C₂	—	8	4	no	—
284	C₂	D₈×#92420	1	2	yes	—
285	C₂	#21461452	1	2	yes	—
286	C₂	#21461940	1	1	yes	characteristic
287	C₂	#21461966	1	1	yes	characteristic
288	C₂	#21461972	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#6923
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting, prime radical	C₂²×D₈×C₆₁	C₄
socle	C₂³×C₆₁	C₄×D₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂²×D₈	#6923
hypocenter	C₆₁	#9657
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×D₈

Complements

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

N	H ≅ G/N	type
1. #21461940	C₂	semidirect
2. C₂×#16840625	C₂	semidirect
3. C₂×D₄⋊#92420	C₂	semidirect
4. C₂×C₈×#92420	C₂	semidirect
46. C₂×D₈×C₆₁	C₈	semidirect
69. C₂×#92420	D₈	semidirect
116. C₈×C₆₁	C₂²⋊C₈	semidirect
164. C₆₁	#9657	semidirect
169. C₂×D₈	#92420	semidirect
262. C₈	C₂²⋊#92420	semidirect

Maximal subgroups

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

subgroup	index
#21461940	2
C₂×#16840625	2
C₂×D₄⋊#92420	2
C₂×C₈×#92420	2
C₂×D₈×Dic₆₁	2
#9657	61