Subgroups — #29594897

Order 15616.

630 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14519717	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14519711	C₂	1	1	yes	maximal, characteristic
3	#21443910	C₂	1	4	yes	maximal, complemented
4	C₂×C₆₁×#427	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₆₁	—	2	1	no	—
6	C₂×#14519717	—	2	2	no	—
7	C₂×#14519717	—	2	2	no	—
8	C₂×#14519717	C₄	1	4	yes	complemented
9	C₂²×#11758185	C₂²	1	2	yes	—
10	C₂×#14519711	—	2	1	no	—
11	C₂×#14519711	—	2	2	no	—
12	C₂×#14519711	C₄	1	4	yes	complemented
13	C₂×#14519711	C₂²	1	2	yes	—
14	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×#117×C₆₁	—	2	1	no	—
16	C₆₁×#427	—	2	2	no	—
17	C₆₁×#427	C₄	1	4	yes	complemented
18	C₂×#107×C₆₁	C₂²	1	2	yes	—
19	C₂×#106×C₆₁	—	2	1	no	—
20	C₂³×Dic₆₁	—	4	1	no	—
21	C₂×C₄⋊Dic₆₁	—	2	4	no	—
22	C₂×C₄⋊Dic₆₁	—	2	2	no	—
23	C₂×C₄⋊Dic₆₁	—	4	1	no	—
24	C₂×C₄⋊Dic₆₁	—	4	2	no	—
25	#14519717	—	2	8	no	—
26	C₂×#11758185	—	2	2	no	—
27	C₂×#11758185	—	2	2	no	—
28	C₂×#11758185	C₂×C₄	1	4	yes	complemented
29	#14519711	—	2	4	no	—
30	#14519711	—	2	2	no	—
31	#14519711	C₂×C₄	1	4	yes	—
32	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₁	D₄	1	2	yes	—
38	C₂²×C₈×C₆₁	D₄	1	2	yes	—
39	C₂²×C₈×C₆₁	Q₈	1	2	yes	—
40	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
41	#117×C₆₁	—	2	4	no	—
42	#117×C₆₁	—	4	2	no	—
43	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
44	C₂×C₂²⋊C₄×C₆₁	—	4	1	no	—
45	C₂×C₄²×C₆₁	—	4	1	no	—
46	#107×C₆₁	—	2	2	no	—
47	#107×C₆₁	C₂×C₄	1	4	yes	complemented
48	#106×C₆₁	—	2	4	no	—
49	C₂⁴×C₆₁	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	—	4	2	no	—
51	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	SD₁₆	1	2	yes	—
56	C₂²×C₄×C₆₁	SD₁₆	1	2	yes	—
57	C₂²×C₄×C₆₁	Q₁₆	1	2	yes	—
58	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₆₁	—	4	4	no	—
60	C₂²×Dic₆₁	—	4	2	no	—
61	C₂²×Dic₆₁	—	8	4	no	—
62	C₂×C₈×C₆₁	C₄²	1	4	yes	—
63	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₆₁	C₂²⋊C₄	1	4	yes	—
66	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
67	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
68	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
69	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
75	C₄⋊C₄×C₆₁	—	2	4	no	—
76	C₄⋊C₄×C₆₁	—	4	2	no	—
77	C₂²⋊C₄×C₆₁	—	4	4	no	—
78	C₄²×C₆₁	—	4	4	no	—
79	C₄⋊Dic₆₁	—	2	4	no	—
80	C₄⋊Dic₆₁	—	4	4	no	—
81	C₄⋊Dic₆₁	—	4	2	no	—
82	#11758185	—	2	4	no	—
83	C₂³×C₆₁	—	2	2	no	—
84	C₂³×C₆₁	—	2	2	no	—
85	C₂³×C₆₁	#102	1	2	yes	—
86	C₂³×C₆₁	#102	1	2	yes	—
87	C₂³×C₆₁	#107	1	2	yes	—
88	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₆₁	—	4	4	no	—
90	C₂×C₄×C₆₁	—	8	2	no	—
91	C₂×C₄×C₆₁	—	8	2	no	—
92	C₂×C₄×C₆₁	#95	1	4	yes	—
93	C₂×C₄×C₆₁	#102	1	2	yes	—
94	C₂×C₄×C₆₁	#102	1	2	yes	—
95	C₂×C₄×C₆₁	#103	1	4	yes	—
96	C₂×C₄×C₆₁	#103	1	4	yes	—
97	C₂×C₄×C₆₁	#106	1	4	yes	—
98	C₂×C₄×C₆₁	#107	1	2	yes	—
99	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
105	C₂×Dic₆₁	—	4	4	no	—
106	C₂×Dic₆₁	—	8	8	no	—
107	C₂×Dic₆₁	—	8	4	no	—
108	C₈×C₆₁	#95	1	4	yes	—
109	C₈×C₆₁	C₂×C₄²	1	1	yes	characteristic
110	C₈×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
113	#12557	—	61	1	no	maximal, Sylow
114	C₂²×C₆₁	—	2	4	no	—
115	C₂²×C₆₁	—	2	2	no	—
116	C₂²×C₆₁	—	2	2	no	—
117	C₂²×C₆₁	—	2	4	no	—
118	C₂²×C₆₁	#340	1	4	yes	—
119	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
120	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
121	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
122	C₂²×C₆₁	#509	1	2	yes	—
123	C₂²×C₆₁	#510	1	2	yes	—
124	C₄×C₆₁	—	8	4	no	—
125	C₄×C₆₁	#340	1	4	yes	—
126	C₄×C₆₁	C₂×#95	1	1	yes	characteristic
127	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
128	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
129	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
130	Dic₆₁	—	8	4	no	—
131	C₂²×M₃₂	—	61	1	no	—
132	C₂×#427	D₆₁	1	1	yes	characteristic
133	C₂²×#107	—	61	1	no	—
134	#1380	—	61	4	no	—
135	C₂×C₆₁	—	2	4	no	—
136	C₂×C₆₁	C₂×#340	1	1	yes	characteristic
137	C₂×C₆₁	#2139	1	2	yes	—
138	C₂×C₆₁	#2140	1	1	yes	characteristic
139	C₂×C₆₁	#2141	1	1	yes	characteristic
140	C₂×C₆₁	#2159	1	2	yes	—
141	C₂³×C₈	C₂×D₆₁	1	1	yes	characteristic
142	C₂×#117	—	2	1	no	—
143	C₂²×C₄⋊C₄	—	122	1	no	—
144	C₂×M₃₂	—	61	4	no	—
145	C₂×M₃₂	—	122	2	no	—
146	C₂×M₃₂	—	122	2	no	—
147	C₂²×C₁₆	—	61	2	no	—
148	#427	—	2	2	no	—
149	#427	Dic₆₁	1	4	yes	complemented
150	C₂×#107	—	61	4	no	—
151	C₂×#107	—	61	2	no	—
152	C₂×#107	—	122	2	no	—
153	C₂×#107	—	122	1	no	—
154	C₂×#107	C₂×D₆₁	1	2	yes	—
155	C₂×#106	—	2	1	no	—
156	C₆₁	#12557	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₄	—	244	1	no	—
158	C₂³×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
159	C₂²×C₈	Dic₁₂₂	1	2	yes	—
160	C₂²×C₈	C₄×D₆₁	1	2	yes	—
161	C₂²×C₈	C₄×D₆₁	1	2	yes	—
162	C₂²×C₈	D₂₄₄	1	2	yes	—
163	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
164	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
165	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
166	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
167	#117	—	2	4	no	—
168	#117	—	4	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	122	4	no	—
171	C₂×C₄⋊C₄	—	122	2	no	—
172	C₂×C₄⋊C₄	—	244	1	no	—
173	C₂×C₄⋊C₄	—	244	2	no	—
174	C₂×C₂²⋊C₄	—	4	1	no	—
175	C₂×C₄²	—	4	1	no	—
176	M₃₂	—	122	8	no	—
177	C₂×C₁₆	—	61	4	no	—
178	C₂×C₁₆	—	122	2	no	—
179	C₂×C₁₆	—	122	2	no	—
180	#107	—	2	2	no	—
181	#107	—	61	4	no	—
182	#107	—	122	4	no	—
183	#107	—	122	2	no	—
184	#107	C₂×Dic₆₁	1	4	yes	complemented
185	#106	—	2	4	no	—
186	C₂⁴	#11758199	1	1	yes	characteristic
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	488	4	no	—
189	C₂²×C₄	—	244	4	no	—
190	C₂²×C₄	—	244	2	no	—
191	C₂²×C₄	#11758196	1	2	yes	—
192	C₂²×C₄	#11758198	1	2	yes	—
193	C₂²×C₄	#11758199	1	1	yes	characteristic
194	C₂²×C₄	#11758200	1	2	yes	—
195	C₂²×C₄	#11758201	1	2	yes	—
196	C₂²×C₄	#11758202	1	2	yes	—
197	C₂²×C₄	C₂²⋊Dic₆₁	1	2	yes	—
198	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
199	C₂×C₈	C₄×Dic₆₁	1	4	yes	—
200	C₂×C₈	#11758196	1	2	yes	—
201	C₂×C₈	#11758196	1	4	yes	—
202	C₂×C₈	C₄⋊Dic₆₁	1	4	yes	—
203	C₂×C₈	#11758198	1	2	yes	—
204	C₂×C₈	#11758198	1	4	yes	—
205	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
206	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
207	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
208	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
209	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
210	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
211	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	122	4	no	—
215	C₄⋊C₄	—	244	4	no	—
216	C₄⋊C₄	—	244	2	no	—
217	C₂²⋊C₄	—	4	4	no	—
218	C₄²	—	4	4	no	—
219	C₁₆	—	122	4	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	#14519697	1	2	yes	—
223	C₂³	#14519698	1	2	yes	—
224	C₂³	#14519710	1	2	yes	—
225	C₂³	C₂×#11758199	1	1	yes	characteristic
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	8	2	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	488	8	no	—
230	C₂×C₄	—	488	4	no	—
231	C₂×C₄	—	244	4	no	—
232	C₂×C₄	#14519684	1	4	yes	—
233	C₂×C₄	#14519697	1	2	yes	—
234	C₂×C₄	#14519698	1	2	yes	—
235	C₂×C₄	#14519700	1	4	yes	—
236	C₂×C₄	#14519701	1	4	yes	—
237	C₂×C₄	#14519708	1	4	yes	—
238	C₂×C₄	#14519710	1	2	yes	—
239	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
240	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
241	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
242	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
243	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
244	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
245	C₈	#14519684	1	4	yes	—
246	C₈	C₂×C₄×Dic₆₁	1	1	yes	characteristic
247	C₈	C₂×#11758198	1	1	yes	characteristic
248	C₈	C₂×#11758196	1	1	yes	characteristic
249	C₈	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
250	C₂²	—	2	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	#16839524	1	4	yes	—
255	C₂²	C₂×#14519697	1	1	yes	characteristic
256	C₂²	C₂×#14519698	1	1	yes	characteristic
257	C₂²	C₂×#14519710	1	1	yes	characteristic
258	C₂²	#16840197	1	2	yes	—
259	C₂²	#16840202	1	2	yes	—
260	C₄	—	8	4	no	—
261	C₄	—	488	4	no	—
262	C₄	#16839524	1	4	yes	—
263	C₄	C₂×#14519684	1	1	yes	characteristic
264	C₄	C₂×#14519700	1	1	yes	characteristic
265	C₄	C₂×#14519701	1	1	yes	characteristic
266	C₄	C₂×#14519708	1	1	yes	characteristic
267	C₂	—	2	4	no	—
268	C₂	C₂×#16839524	1	1	yes	characteristic
269	C₂	#21447531	1	2	yes	—
270	C₂	#21447534	1	1	yes	characteristic
271	C₂	#21447538	1	1	yes	characteristic
272	C₂	#21447608	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11758199
commutator	C₈×C₆₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×#427	C₂
socle	C₂³×C₆₁	C₂×D₈
prime radical	C₂⁴×C₆₁	D₈
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×#427	D₆₁
hypocenter	C₆₁	#12557
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
3. #21443910	C₂	semidirect
8. C₂×#14519717	C₄	semidirect
12. C₂×#14519711	C₄	semidirect
17. C₆₁×#427	C₄	semidirect
28. C₂×#11758185	C₂×C₄	semidirect
47. #107×C₆₁	C₂×C₄	semidirect
149. #427	Dic₆₁	semidirect
156. C₆₁	#12557	semidirect
184. #107	C₂×Dic₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14519717	2
C₂²×#14519711	2
#21443910	2
C₂×C₆₁×#427	2
#12557	61