Subgroups — #29588299

Order 15616.

574 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839533	C₂	1	2	yes	maximal
2	#21445545	C₂	1	2	yes	maximal
3	C₂×#16839527	C₂	1	1	yes	maximal, characteristic
4	#21445515	C₂	1	1	yes	maximal, characteristic
5	C₂²×#108×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758194	C₂²	1	2	yes	—
7	C₂²×#11758194	C₂²	1	1	yes	characteristic
8	C₂×#14519736	C₂²	1	1	yes	characteristic
9	#16839533	—	2	4	no	—
10	#16839527	C₄	1	4	yes	complemented
11	#16839526	—	2	4	no	—
12	#16839516	—	2	2	no	—
13	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
14	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×#108×C₆₁	—	2	4	no	—
16	C₂×#108×C₆₁	—	2	2	no	—
17	C₂×#11758194	—	2	2	no	—
18	C₂×#11758194	—	2	1	no	—
19	C₂×#11758194	—	2	2	no	—
20	C₂×#11758194	—	2	2	no	—
21	C₂×#11758194	—	2	2	no	—
22	C₂×#11758194	—	2	2	no	—
23	C₂×#11758194	—	2	2	no	—
24	C₂×#11758194	—	2	2	no	—
25	C₂×#11758194	C₂×C₄	1	4	yes	—
26	C₂×#11758194	D₄	1	2	yes	—
27	C₂²×#92418	—	2	1	no	—
28	C₂²×#92418	—	2	2	no	—
29	#14519736	—	2	2	no	—
30	#14519736	—	2	2	no	—
31	C₂×C₄⋊Dic₆₁	—	2	1	no	—
32	C₂×C₂²⋊Dic₆₁	Q₈	1	2	yes	—
33	C₂×C₄×Dic₆₁	—	2	1	no	—
34	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₆₁	—	2	2	no	—
36	C₂×M₁₆×C₆₁	—	2	2	no	—
37	C₂×M₁₆×C₆₁	—	2	2	no	—
38	C₂×M₁₆×C₆₁	—	2	2	no	—
39	C₂×M₁₆×C₆₁	—	2	2	no	—
40	C₂×M₁₆×C₆₁	D₄	1	4	yes	—
41	C₂²×C₈×C₆₁	—	2	2	no	—
42	C₂²×C₈×C₆₁	—	2	2	no	—
43	C₂²×C₈×C₆₁	—	2	2	no	—
44	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₆₁	D₄	1	2	yes	—
46	C₂²×C₈×C₆₁	Q₈	1	2	yes	—
47	#108×C₆₁	—	2	8	no	—
48	C₂⁴×C₆₁	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	#41	1	2	yes	—
55	C₂²×C₄×C₆₁	#41	1	2	yes	—
56	C₂²×C₄×C₆₁	#41	1	2	yes	—
57	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
59	C₂²×Dic₆₁	—	2	2	no	—
60	M₁₆×C₆₁	—	2	4	no	—
61	M₁₆×C₆₁	—	2	4	no	—
62	M₁₆×C₆₁	—	4	2	no	—
63	M₁₆×C₆₁	—	4	2	no	—
64	C₂×C₈×C₆₁	—	2	4	no	—
65	C₂×C₈×C₆₁	—	2	2	no	—
66	C₂×C₈×C₆₁	—	2	4	no	—
67	C₂×C₈×C₆₁	—	2	2	no	—
68	C₂×C₈×C₆₁	—	2	4	no	—
69	C₂×C₈×C₆₁	—	4	2	no	—
70	C₂×C₈×C₆₁	—	4	2	no	—
71	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
72	C₂²⋊Dic₆₁	—	2	4	no	—
73	C₄⋊Dic₆₁	—	4	2	no	—
74	C₄×Dic₆₁	—	4	1	no	—
75	C₄×Dic₆₁	—	4	1	no	—
76	#11758194	—	2	4	no	—
77	#11758194	—	2	4	no	—
78	#11758194	—	2	4	no	—
79	#11758194	—	4	2	no	—
80	#11758194	—	4	2	no	—
81	#11758194	—	4	2	no	—
82	C₂×#92418	—	2	4	no	—
83	C₂×#92418	—	2	2	no	—
84	C₂×#92418	—	4	2	no	—
85	C₂×#92418	—	4	2	no	—
86	C₂×#92418	—	4	1	no	—
87	C₂×#92418	—	4	1	no	—
88	C₂³×C₆₁	—	4	2	no	—
89	C₂³×C₆₁	C₄×Q₈	1	1	yes	characteristic
90	C₂³×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂³×C₆₁	C₂²⋊Q₈	1	2	yes	—
92	C₂³×C₆₁	#125	1	2	yes	—
93	C₂³×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₆₁	#117	1	2	yes	—
96	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
99	C₂×C₄×C₆₁	#121	1	2	yes	—
100	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₆₁	#123	1	1	yes	characteristic
103	C₂×C₄×C₆₁	#123	1	2	yes	—
104	C₂×C₄×C₆₁	#123	1	2	yes	—
105	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
106	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
107	C₂×C₄×C₆₁	#124	1	2	yes	—
108	C₂×C₄×C₆₁	#125	1	2	yes	—
109	C₂×C₄×C₆₁	#125	1	1	yes	characteristic
110	C₂×C₄×C₆₁	#126	1	2	yes	—
111	C₂×C₄×C₆₁	#126	1	2	yes	—
112	C₂×Dic₆₁	—	4	4	no	—
113	C₂×Dic₆₁	—	4	2	no	—
114	C₈×C₆₁	—	2	4	no	—
115	C₈×C₆₁	—	4	4	no	—
116	#92418	—	4	4	no	—
117	#92418	—	4	2	no	—
118	#11272	—	61	1	no	maximal, Sylow
119	C₂²×C₆₁	—	2	2	no	—
120	C₂²×C₆₁	—	2	2	no	—
121	C₂²×C₆₁	—	2	2	no	—
122	C₂²×C₆₁	—	4	2	no	—
123	C₂²×C₆₁	—	4	2	no	—
124	C₂²×C₆₁	#387	1	2	yes	—
125	C₂²×C₆₁	#389	1	1	yes	characteristic
126	C₂²×C₆₁	#393	1	1	yes	characteristic
127	C₂²×C₆₁	#398	1	2	yes	—
128	C₂²×C₆₁	#400	1	1	yes	characteristic
129	C₄×C₆₁	#382	1	1	yes	characteristic
130	C₄×C₆₁	#385	1	1	yes	characteristic
131	C₄×C₆₁	#388	1	2	yes	—
132	C₄×C₆₁	#394	1	1	yes	characteristic
133	C₄×C₆₁	#397	1	1	yes	characteristic
134	C₄×C₆₁	#400	1	2	yes	—
135	Dic₆₁	—	8	2	no	—
136	C₂²×#108	D₆₁	1	1	yes	characteristic
137	C₂×#344	—	61	2	no	—
138	#1738	—	61	2	no	—
139	C₂×#341	—	61	1	no	—
140	#1729	—	61	1	no	—
141	C₂×C₆₁	—	2	1	no	—
142	C₂×C₆₁	—	2	2	no	—
143	C₂×C₆₁	—	2	2	no	—
144	C₂×C₆₁	—	2	1	no	—
145	C₂×C₆₁	#1442	1	1	yes	characteristic
146	C₂×C₆₁	#1953	1	2	yes	—
147	C₂²×M₁₆	—	61	1	no	—
148	C₂²×M₁₆	—	61	2	no	—
149	C₂²×M₁₆	C₂×D₆₁	1	2	yes	—
150	C₂³×C₈	C₂×D₆₁	1	1	yes	characteristic
151	C₂×#117	—	61	1	no	—
152	C₂×#108	—	2	4	no	—
153	C₂×#108	—	2	2	no	—
154	#344	—	122	4	no	—
155	#341	—	61	4	no	—
156	#341	—	122	4	no	—
157	#336	—	122	2	no	—
158	C₆₁	#11272	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	61	4	no	—
166	C₂×M₁₆	—	61	2	no	—
167	C₂×M₁₆	—	122	1	no	—
168	C₂×M₁₆	—	122	2	no	—
169	C₂×M₁₆	—	122	2	no	—
170	C₂×M₁₆	—	122	2	no	—
171	C₂×M₁₆	—	122	2	no	—
172	C₂×M₁₆	—	122	2	no	—
173	C₂×M₁₆	—	122	2	no	—
174	C₂×M₁₆	—	122	2	no	—
175	C₂×M₁₆	C₂²⋊D₆₁	1	4	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	122	2	no	—
180	C₂²×C₈	—	122	1	no	—
181	C₂²×C₈	Dic₁₂₂	1	2	yes	—
182	C₂²×C₈	C₄×D₆₁	1	2	yes	—
183	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
184	#117	—	122	2	no	—
185	#117	—	122	2	no	—
186	C₂×C₄⋊C₄	—	122	1	no	—
187	C₂×C₂²⋊C₄	—	61	2	no	—
188	C₂×C₄²	—	122	1	no	—
189	#108	—	2	8	no	—
190	C₂⁴	Q₈×D₆₁	1	1	yes	characteristic
191	C₂²×C₄	—	122	2	no	—
192	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
194	C₂²×C₄	#11758222	1	2	yes	—
195	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
196	C₂²×C₄	#11758224	1	2	yes	—
197	C₂²×C₄	#11758224	1	1	yes	characteristic
198	C₂²×C₄	#11758226	1	2	yes	—
199	C₂²×C₄	#11758226	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
202	M₁₆	—	2	4	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	4	2	no	—
205	M₁₆	—	4	2	no	—
206	M₁₆	—	122	4	no	—
207	M₁₆	—	122	4	no	—
208	M₁₆	—	122	4	no	—
209	M₁₆	—	244	2	no	—
210	M₁₆	—	244	2	no	—
211	M₁₆	—	244	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	122	4	no	—
220	C₂×C₈	—	122	2	no	—
221	C₂×C₈	—	244	2	no	—
222	C₂×C₈	—	244	1	no	—
223	C₂×C₈	—	244	2	no	—
224	C₂×C₈	—	244	1	no	—
225	C₂×C₈	#11758196	1	4	yes	—
226	C₄⋊C₄	—	244	2	no	—
227	C₂²⋊C₄	—	122	4	no	—
228	C₄²	—	244	1	no	—
229	C₄²	—	244	1	no	—
230	C₂³	—	4	2	no	—
231	C₂³	#14519742	1	1	yes	characteristic
232	C₂³	#14519753	1	1	yes	characteristic
233	C₂³	#14519753	1	2	yes	—
234	C₂³	#14519767	1	2	yes	—
235	C₂³	#14519776	1	1	yes	characteristic
236	C₂×C₄	—	244	4	no	—
237	C₂×C₄	—	244	2	no	—
238	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
239	C₂×C₄	#14519735	1	2	yes	—
240	C₂×C₄	C₄×C₂²⋊D₆₁	1	2	yes	—
241	C₂×C₄	#14519739	1	1	yes	characteristic
242	C₂×C₄	#14519746	1	1	yes	characteristic
243	C₂×C₄	#14519748	1	1	yes	characteristic
244	C₂×C₄	#14519749	1	2	yes	—
245	C₂×C₄	#14519756	1	2	yes	—
246	C₂×C₄	#14519758	1	2	yes	—
247	C₂×C₄	#14519760	1	1	yes	characteristic
248	C₂×C₄	#14519762	1	2	yes	—
249	C₂×C₄	#14519764	1	1	yes	characteristic
250	C₂×C₄	#14519766	1	1	yes	characteristic
251	C₂×C₄	#14519766	1	2	yes	—
252	C₂×C₄	#14519768	1	1	yes	characteristic
253	C₂×C₄	#14519768	1	2	yes	—
254	C₂×C₄	#14519772	1	2	yes	—
255	C₂×C₄	#14519772	1	2	yes	—
256	C₈	—	2	4	no	—
257	C₈	—	4	4	no	—
258	C₈	—	244	2	no	—
259	C₈	—	244	4	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	#16839643	1	2	yes	—
266	C₂²	#16839658	1	1	yes	characteristic
267	C₂²	#16839673	1	1	yes	characteristic
268	C₂²	#16839695	1	2	yes	—
269	C₂²	#16839700	1	1	yes	characteristic
270	C₄	—	488	2	no	—
271	C₄	#16839627	1	1	yes	characteristic
272	C₄	#16839635	1	1	yes	characteristic
273	C₄	#16839652	1	2	yes	—
274	C₄	#16839676	1	1	yes	characteristic
275	C₄	#16839688	1	1	yes	characteristic
276	C₄	#16839701	1	2	yes	—
277	C₂	—	2	1	no	—
278	C₂	—	2	2	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	2	no	—
281	C₂	#21444078	1	1	yes	characteristic
282	C₂	#21446607	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14519748
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₁
Fitting	C₂²×#108×C₆₁	C₂
socle	C₂²×C₆₁	#389
prime radical	C₂⁴×C₆₁	C₂×Q₈
cosocle	C₂³×C₄×C₆₁	C₂³
hypercenter	C₂²×#108	D₆₁
hypocenter	C₆₁	#11272
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
10. #16839527	C₄	semidirect
158. C₆₁	#11272	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839533	2
#21445545	2
C₂×#16839527	2
#21445515	2
C₂²×#108×C₆₁	2
#11272	61