Subgroups — #29584231

Order 15616.

486 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840142	C₂	1	1	yes	maximal, characteristic
2	#21446771	C₂	1	2	yes	maximal
3	C₂×#16839509	C₂	1	2	yes	maximal
4	#21443654	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1939	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₄₄	—	2	1	no	—
7	C₂²×Dic₂₄₄	—	2	1	no	—
8	C₂×#14519777	C₂²	1	2	yes	—
9	#16840142	—	2	4	no	—
10	C₂×C₄⋊#92418	C₂²	1	2	yes	—
11	#16839524	—	2	2	no	—
12	#16839524	—	2	2	no	—
13	#16839509	—	2	4	no	—
14	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
15	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#336	—	2	1	no	—
19	C₂²×Dic₁₂₂	—	2	2	no	—
20	C₂×Dic₂₄₄	—	2	4	no	—
21	C₂×Dic₂₄₄	—	2	4	no	—
22	C₂×Dic₂₄₄	—	4	2	no	—
23	C₂×Dic₂₄₄	—	4	2	no	—
24	C₂×Dic₂₄₄	—	4	2	no	—
25	C₂×Dic₂₄₄	—	4	2	no	—
26	C₂²×#92418	—	2	2	no	—
27	#14519777	—	2	2	no	—
28	#14519777	—	2	2	no	—
29	#14519777	—	2	2	no	—
30	#14519777	D₄	1	4	yes	—
31	C₂×C₄⋊Dic₆₁	—	2	2	no	—
32	C₂×C₄⋊Dic₆₁	—	2	2	no	—
33	C₄⋊#92418	—	2	2	no	—
34	C₄⋊#92418	—	2	2	no	—
35	C₂²×C₈×C₆₁	—	2	1	no	—
36	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₁	—	2	1	no	—
41	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₆₁	—	2	2	no	—
43	C₄⋊C₈×C₆₁	—	4	2	no	—
44	C₄×C₈×C₆₁	—	2	2	no	—
45	#95×C₆₁	—	2	1	no	—
46	C₂²×C₄×C₆₁	—	2	1	no	—
47	C₂²×C₄×C₆₁	—	2	1	no	—
48	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
51	C₂²×Dic₆₁	—	4	2	no	—
52	C₂×Dic₁₂₂	—	2	4	no	—
53	C₂×Dic₁₂₂	—	4	2	no	—
54	C₂×Dic₁₂₂	—	4	2	no	—
55	C₂×Dic₁₂₂	—	4	2	no	—
56	C₂×Dic₁₂₂	—	4	2	no	—
57	C₂×Dic₁₂₂	—	4	2	no	—
58	C₂×C₈×C₆₁	—	2	1	no	—
59	C₂×C₈×C₆₁	—	2	1	no	—
60	C₂×C₈×C₆₁	—	2	1	no	—
61	C₂×C₈×C₆₁	—	2	1	no	—
62	C₂×C₈×C₆₁	—	4	1	no	—
63	C₂×C₈×C₆₁	—	4	1	no	—
64	C₂×C₈×C₆₁	—	4	1	no	—
65	C₂×C₈×C₆₁	D₈	1	2	yes	—
66	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
67	C₄⋊C₄×C₆₁	—	2	1	no	—
68	C₄⋊C₄×C₆₁	—	2	2	no	—
69	C₄⋊C₄×C₆₁	Q₁₆	1	2	yes	complemented
70	C₄²×C₆₁	—	4	1	no	—
71	C₄²×C₆₁	—	4	1	no	—
72	C₄²×C₆₁	C₂×D₄	1	2	yes	—
73	C₄²×C₆₁	#41	1	2	yes	—
74	C₄⋊Dic₆₁	—	2	4	no	—
75	C₄⋊Dic₆₁	—	2	4	no	—
76	C₄⋊Dic₆₁	—	4	2	no	—
77	C₄⋊Dic₆₁	—	4	2	no	—
78	C₂×#92418	—	4	2	no	—
79	C₂×#92418	—	4	2	no	—
80	C₂×#92418	—	4	2	no	—
81	Dic₂₄₄	—	4	8	no	—
82	Dic₂₄₄	—	4	8	no	—
83	C₂³×C₆₁	#121	1	1	yes	characteristic
84	C₂×C₄×C₆₁	—	2	1	no	—
85	C₂×C₄×C₆₁	—	2	1	no	—
86	C₂×C₄×C₆₁	—	2	2	no	—
87	C₂×C₄×C₆₁	—	4	1	no	—
88	C₂×C₄×C₆₁	—	4	1	no	—
89	C₂×C₄×C₆₁	—	4	1	no	—
90	C₂×C₄×C₆₁	—	4	1	no	—
91	C₂×C₄×C₆₁	—	4	1	no	—
92	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
93	C₂×C₄×C₆₁	#121	1	2	yes	—
94	C₂×C₄×C₆₁	#123	1	2	yes	—
95	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
99	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
100	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂×C₄×C₆₁	#137	1	2	yes	—
103	C₂×Dic₆₁	—	4	4	no	—
104	C₂×Dic₆₁	—	8	2	no	—
105	C₂×Dic₆₁	—	8	2	no	—
106	Dic₁₂₂	—	4	4	no	—
107	Dic₁₂₂	—	4	4	no	—
108	Dic₁₂₂	—	8	2	no	—
109	Dic₁₂₂	—	8	2	no	—
110	C₈×C₆₁	—	2	2	no	—
111	C₈×C₆₁	—	2	2	no	—
112	C₈×C₆₁	—	8	1	no	—
113	#92418	—	8	2	no	—
114	#10538	—	61	1	no	maximal, Sylow
115	C₂²×C₆₁	#394	1	1	yes	characteristic
116	C₂²×C₆₁	#459	1	1	yes	characteristic
117	C₂²×C₆₁	#460	1	1	yes	characteristic
118	C₂²×C₆₁	#467	1	1	yes	characteristic
119	C₂²×C₆₁	#470	1	1	yes	characteristic
120	C₂²×C₆₁	#473	1	1	yes	characteristic
121	C₂²×C₆₁	#473	1	1	yes	characteristic
122	C₄×C₆₁	—	2	2	no	—
123	C₄×C₆₁	—	4	2	no	—
124	C₄×C₆₁	—	8	1	no	—
125	C₄×C₆₁	#447	1	1	yes	characteristic
126	C₄×C₆₁	#450	1	1	yes	characteristic
127	C₄×C₆₁	#462	1	2	yes	—
128	C₄×C₆₁	#464	1	2	yes	—
129	C₄×C₆₁	#486	1	1	yes	characteristic
130	C₄×C₆₁	#490	1	1	yes	characteristic
131	Dic₆₁	—	8	4	no	—
132	C₂×#494	—	61	1	no	—
133	#1988	—	61	2	no	—
134	#1939	D₆₁	1	1	yes	characteristic
135	C₂×#332	—	61	2	no	—
136	#1277	—	61	1	no	—
137	C₂×C₆₁	#1669	1	1	yes	characteristic
138	C₂×C₆₁	#1670	1	1	yes	characteristic
139	C₂×C₆₁	#1689	1	1	yes	characteristic
140	C₂×C₆₁	#1690	1	1	yes	characteristic
141	C₂×C₆₁	#2041	1	1	yes	characteristic
142	C₂×C₆₁	#2047	1	1	yes	characteristic
143	C₂×C₆₁	#2056	1	1	yes	characteristic
144	C₂²×Q₁₆	—	122	1	no	—
145	C₂²×Q₁₆	—	122	1	no	—
146	C₂×C₄⋊Q₈	—	61	2	no	—
147	#494	—	122	4	no	—
148	C₂×C₄⋊C₈	—	2	1	no	—
149	C₂×C₄⋊C₈	—	61	2	no	—
150	C₂×C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
152	C₄×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
153	#340	—	122	2	no	—
154	#340	—	122	2	no	—
155	#336	—	2	1	no	—
156	#332	—	122	4	no	—
157	C₆₁	#10538	1	1	yes	characteristic, complemented, Sylow
158	C₂²×Q₈	—	122	2	no	—
159	C₂×Q₁₆	—	122	4	no	—
160	C₂×Q₁₆	—	122	4	no	—
161	C₂×Q₁₆	—	244	2	no	—
162	C₂×Q₁₆	—	244	2	no	—
163	C₂×Q₁₆	—	244	2	no	—
164	C₂×Q₁₆	—	244	2	no	—
165	C₂²×C₈	—	2	1	no	—
166	C₂²×C₈	—	122	2	no	—
167	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
168	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
169	C₄⋊Q₈	—	61	4	no	—
170	C₄⋊Q₈	—	122	2	no	—
171	C₄⋊Q₈	—	122	2	no	—
172	C₄⋊Q₈	—	122	2	no	—
173	C₂×C₄⋊C₄	—	122	2	no	—
174	C₂×C₄⋊C₄	—	122	2	no	—
175	C₂×C₄⋊C₄	D₂₄₄	1	1	yes	characteristic
176	C₂×C₄⋊C₄	D₂₄₄	1	1	yes	characteristic
177	C₂×C₄²	—	2	1	no	—
178	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	4	2	no	—
181	C₄⋊C₈	—	122	2	no	—
182	C₄⋊C₈	—	122	2	no	—
183	C₄×C₈	—	2	2	no	—
184	#95	—	2	1	no	—
185	C₂×Q₈	—	122	4	no	—
186	C₂×Q₈	—	244	2	no	—
187	C₂×Q₈	—	244	2	no	—
188	C₂×Q₈	—	244	2	no	—
189	C₂×Q₈	—	244	2	no	—
190	C₂×Q₈	—	244	2	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	244	2	no	—
194	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
195	C₂²×C₄	#11758222	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
197	Q₁₆	—	244	8	no	—
198	Q₁₆	—	244	8	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	244	2	no	—
207	C₂×C₈	—	244	2	no	—
208	C₂×C₈	—	244	2	no	—
209	C₂×C₈	#11758199	1	2	yes	—
210	C₂×C₈	#11758201	1	2	yes	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	122	4	no	—
214	C₄⋊C₄	—	122	4	no	—
215	C₄⋊C₄	—	244	2	no	—
216	C₄⋊C₄	—	244	2	no	—
217	C₄⋊C₄	Dic₂₄₄	1	2	yes	complemented
218	C₄²	—	4	1	no	—
219	C₄²	—	4	1	no	—
220	C₄²	D₄×D₆₁	1	2	yes	—
221	C₄²	#11758226	1	2	yes	—
222	C₂³	#14519751	1	1	yes	characteristic
223	Q₈	—	488	2	no	—
224	Q₈	—	488	2	no	—
225	Q₈	—	244	4	no	—
226	Q₈	—	244	4	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	488	2	no	—
236	C₂×C₄	—	488	2	no	—
237	C₂×C₄	—	244	4	no	—
238	C₂×C₄	#14519745	1	1	yes	characteristic
239	C₂×C₄	#14519752	1	2	yes	—
240	C₂×C₄	#14519762	1	2	yes	—
241	C₂×C₄	#14519766	1	1	yes	characteristic
242	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
243	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
244	C₂×C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
245	C₂×C₄	#14519799	1	1	yes	characteristic
246	C₂×C₄	#14519802	1	1	yes	characteristic
247	C₂×C₄	#14519802	1	1	yes	characteristic
248	C₂×C₄	#14519810	1	2	yes	—
249	C₈	—	2	2	no	—
250	C₈	—	2	2	no	—
251	C₈	—	8	1	no	—
252	C₈	—	488	2	no	—
253	C₂²	#16839679	1	1	yes	characteristic
254	C₂²	#16839957	1	1	yes	characteristic
255	C₂²	#16839962	1	1	yes	characteristic
256	C₂²	#16840003	1	1	yes	characteristic
257	C₂²	#16840020	1	1	yes	characteristic
258	C₂²	#16840037	1	1	yes	characteristic
259	C₂²	#16840037	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	8	1	no	—
263	C₄	—	488	4	no	—
264	C₄	#16839895	1	1	yes	characteristic
265	C₄	#16839912	1	1	yes	characteristic
266	C₄	#16839973	1	2	yes	—
267	C₄	#16839985	1	2	yes	—
268	C₄	#16840110	1	1	yes	characteristic
269	C₄	#16840128	1	1	yes	characteristic
270	C₂	#21445184	1	1	yes	characteristic
271	C₂	#21445189	1	1	yes	characteristic
272	C₂	#21445308	1	1	yes	characteristic
273	C₂	#21445313	1	1	yes	characteristic
274	C₂	#21447053	1	1	yes	characteristic
275	C₂	#21447087	1	1	yes	characteristic
276	C₂	#21447142	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519751
commutator, cosocle	C₂×C₄²×C₆₁	C₂³
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1939	C₂
socle, prime radical	C₂³×C₆₁	#121
hypercenter	#1939	D₆₁
hypocenter	C₆₁	#10538
LCS	G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1939

Complements

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

N	H ≅ G/N	type
69. C₄⋊C₄×C₆₁	Q₁₆	semidirect
157. C₆₁	#10538	semidirect
217. C₄⋊C₄	Dic₂₄₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840142	2
#21446771	2
C₂×#16839509	2
#21443654	2
C₆₁×#1939	2
#10538	61