Subgroups — #29650579

Order 15616.

734 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454531	C₂	1	2	yes	maximal, complemented
2	#21444733	C₂	1	4	yes	maximal, complemented
3	#21444643	C₂	1	4	yes	maximal, complemented
4	#21444575	C₂	1	1	yes	maximal, characteristic, complemented
5	#21444270	C₂	1	2	yes	maximal, complemented
6	C₂×#16839483	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#3169	C₂	1	1	yes	maximal, characteristic
8	C₂×#14519784	—	2	2	no	—
9	#16839840	—	2	2	no	—
10	#16839840	C₂²	1	4	yes	—
11	D₄×#92418	—	2	2	no	—
12	D₄×#92418	C₂²	1	4	yes	—
13	#16839786	C₂²	1	2	yes	—
14	#16839729	—	2	2	no	—
15	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
16	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
17	#16839513	C₂²	1	4	yes	—
18	D₄⋊#92418	C₂²	1	8	yes	—
19	#16839483	C₂²	1	2	yes	—
20	#16839483	C₂²	1	2	yes	—
21	C₆₁×#532	C₂²	1	2	yes	—
22	C₆₁×#500	C₄	1	1	yes	characteristic
23	C₆₁×#493	C₄	1	1	yes	characteristic
24	C₆₁×#486	C₄	1	2	yes	—
25	C₆₁×#466	C₄	1	4	yes	—
26	C₄×D₈×C₆₁	C₂²	1	4	yes	—
27	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
28	#14519784	—	4	8	no	—
29	C₂×#11758194	—	2	2	no	—
30	C₂×#11758194	—	2	4	no	—
31	C₂²×#92418	—	2	1	no	—
32	C₂²×#92418	—	2	4	no	—
33	C₂²×#92418	D₄	1	2	yes	—
34	C₄⋊#92418	—	2	2	no	—
35	C₄⋊#92418	C₂³	1	4	yes	—
36	C₂²⋊#92418	—	2	4	no	—
37	C₂²⋊#92418	—	2	4	no	—
38	C₄×#92418	—	2	1	no	—
39	C₄×#92418	D₄	1	2	yes	—
40	C₄×#92418	C₂³	1	2	yes	—
41	C₄×#92418	C₂³	1	1	yes	characteristic
42	C₄×#92418	C₂³	1	1	yes	characteristic
43	C₂×#41×C₆₁	—	2	2	no	—
44	C₂×D₈×C₆₁	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
46	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
47	#127×C₆₁	C₂×C₄	1	2	yes	—
48	#124×C₆₁	—	2	2	no	—
49	#121×C₆₁	—	2	2	no	—
50	#121×C₆₁	C₂×C₄	1	4	yes	—
51	C₄×D₄×C₆₁	—	2	2	no	—
52	C₄×D₄×C₆₁	C₂³	1	4	yes	—
53	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
54	#107×C₆₁	C₂×C₄	1	4	yes	—
55	#102×C₆₁	C₂×C₄	1	8	yes	—
56	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₆₁	C₂³	1	2	yes	—
59	#41×C₆₁	—	4	4	no	—
60	#41×C₆₁	—	4	4	no	—
61	C₂×Q₈×C₆₁	—	2	2	no	—
62	C₂×D₄×C₆₁	—	2	4	no	—
63	C₂×D₄×C₆₁	—	2	2	no	—
64	C₂×D₄×C₆₁	—	2	2	no	—
65	C₂×D₄×C₆₁	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₆₁	—	2	4	no	—
67	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
70	D₈×C₆₁	—	2	8	no	—
71	C₂×C₈×C₆₁	—	2	4	no	—
72	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
74	C₄⋊C₄×C₆₁	—	2	2	no	—
75	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
76	C₂²⋊C₄×C₆₁	—	2	4	no	—
77	C₂²⋊C₄×C₆₁	—	2	4	no	—
78	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₆₁	#41	1	1	yes	characteristic
81	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
82	#11758194	—	4	8	no	—
83	#11758194	—	4	4	no	—
84	C₂×#92418	—	2	4	no	—
85	C₂×#92418	—	2	4	no	—
86	C₂×#92418	—	2	2	no	—
87	C₂×#92418	—	4	8	no	—
88	C₂×#92418	—	4	2	no	—
89	C₂×#92418	C₂×D₄	1	2	yes	—
90	C₂×#92418	C₂×D₄	1	2	yes	—
91	C₂³×C₆₁	—	2	4	no	—
92	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
93	Q₈×C₆₁	—	4	4	no	—
94	D₄×C₆₁	—	2	8	no	—
95	D₄×C₆₁	—	4	8	no	—
96	D₄×C₆₁	—	4	4	no	—
97	D₄×C₆₁	—	4	4	no	—
98	C₂×C₄×C₆₁	—	2	2	no	—
99	C₂×C₄×C₆₁	—	2	2	no	—
100	C₂×C₄×C₆₁	—	2	4	no	—
101	C₂×C₄×C₆₁	—	4	4	no	—
102	C₂×C₄×C₆₁	—	4	4	no	—
103	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
107	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
110	C₈×C₆₁	—	2	2	no	—
111	C₈×C₆₁	#131	1	4	yes	complemented
112	#92418	—	2	2	no	—
113	#92418	—	2	2	no	—
114	#92418	—	4	2	no	—
115	#92418	—	4	4	no	—
116	#17962	—	61	1	no	maximal, Sylow
117	C₂²×C₆₁	—	2	1	no	—
118	C₂²×C₆₁	—	2	1	no	—
119	C₂²×C₆₁	—	4	4	no	—
120	C₂²×C₆₁	—	4	4	no	—
121	C₂²×C₆₁	—	4	4	no	—
122	C₂²×C₆₁	#442	1	2	yes	—
123	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
124	C₄×C₆₁	—	2	1	no	—
125	C₄×C₆₁	—	2	1	no	—
126	C₄×C₆₁	—	4	4	no	—
127	C₄×C₆₁	#442	1	2	yes	—
128	C₄×C₆₁	#444	1	2	yes	—
129	C₄×C₆₁	C₂×#131	1	2	yes	—
130	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
131	#3169	D₆₁	1	1	yes	characteristic
132	#2931	—	61	2	no	—
133	#1594	—	61	4	no	—
134	#1580	—	61	4	no	—
135	#1566	—	61	1	no	—
136	#1492	—	61	2	no	—
137	C₂×#322	—	61	1	no	—
138	C₂×C₆₁	—	2	2	no	—
139	C₂×C₆₁	—	8	4	no	—
140	C₂×C₆₁	#2931	1	1	yes	characteristic
141	C₂×C₆₁	C₂×#442	1	1	yes	characteristic
142	C₂×C₆₁	C₂×#444	1	1	yes	characteristic
143	C₂×#131	—	122	2	no	—
144	#532	C₂×D₆₁	1	2	yes	—
145	#500	Dic₆₁	1	1	yes	characteristic
146	#493	Dic₆₁	1	1	yes	characteristic
147	#486	Dic₆₁	1	2	yes	—
148	#466	Dic₆₁	1	4	yes	—
149	C₄×D₈	C₂×D₆₁	1	4	yes	—
150	#436	—	61	4	no	—
151	#436	—	122	2	no	—
152	C₈×D₄	—	61	4	no	—
153	C₈×D₄	—	122	2	no	—
154	#423	—	61	2	no	—
155	#408	—	122	2	no	—
156	C₂×C₄×C₈	—	61	1	no	—
157	C₂×C₄×C₈	—	61	1	no	—
158	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
159	#334	—	61	4	no	—
160	D₄⋊C₈	—	61	8	no	—
161	#322	—	61	2	no	—
162	#322	—	61	2	no	—
163	C₆₁	#17962	1	1	yes	characteristic, complemented, Sylow
164	C₂×#41	—	2	2	no	—
165	C₂×D₈	C₂×Dic₆₁	1	4	yes	—
166	#131	—	244	8	no	—
167	C₂×M₁₆	—	122	4	no	—
168	C₂×M₁₆	—	122	2	no	—
169	C₂²×C₈	—	61	2	no	—
170	C₂²×C₈	—	122	4	no	—
171	C₂²×C₈	—	122	1	no	—
172	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
173	C₄⋊Q₈	C₂×Dic₆₁	1	2	yes	—
174	#127	C₂×Dic₆₁	1	2	yes	—
175	#124	—	2	2	no	—
176	#121	—	2	2	no	—
177	#121	C₂×Dic₆₁	1	4	yes	—
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	C₂²×D₆₁	1	4	yes	—
180	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
181	#107	C₂×Dic₆₁	1	4	yes	—
182	C₄⋊C₈	—	61	4	no	—
183	C₄⋊C₈	—	122	2	no	—
184	#102	C₂×Dic₆₁	1	8	yes	—
185	C₂²⋊C₈	—	122	4	no	—
186	C₂²⋊C₈	—	122	4	no	—
187	C₄×C₈	—	61	2	no	—
188	C₄×C₈	—	61	1	no	—
189	C₄×C₈	—	61	1	no	—
190	C₄×C₈	—	61	2	no	—
191	C₄×C₈	—	122	1	no	—
192	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
193	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₆₁	1	2	yes	—
195	#41	—	4	4	no	—
196	#41	—	4	4	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	C₂²×Dic₆₁	1	4	yes	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
204	C₂²×C₄	#11758224	1	1	yes	characteristic
205	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
206	D₈	—	2	8	no	—
207	M₁₆	—	244	8	no	—
208	M₁₆	—	244	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	61	2	no	—
211	C₂×C₈	—	61	2	no	—
212	C₂×C₈	—	122	4	no	—
213	C₂×C₈	—	122	4	no	—
214	C₂×C₈	—	122	2	no	—
215	C₂×C₈	—	244	8	no	—
216	C₂×C₈	—	244	2	no	—
217	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
218	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	C₂²×Dic₆₁	1	4	yes	—
221	C₂²⋊C₄	—	2	4	no	—
222	C₂²⋊C₄	—	2	4	no	—
223	C₄²	D₄×D₆₁	1	1	yes	characteristic
224	C₄²	#11758224	1	1	yes	characteristic
225	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
226	C₄²	C₂³×D₆₁	1	1	yes	characteristic
227	C₂³	—	2	4	no	—
228	C₂³	D₄×Dic₆₁	1	1	yes	characteristic
229	Q₈	—	4	4	no	—
230	D₄	—	2	8	no	—
231	D₄	—	4	8	no	—
232	D₄	—	4	4	no	—
233	D₄	—	4	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
240	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
241	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
242	C₂×C₄	#14519801	1	4	yes	—
243	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
244	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
245	C₂×C₄	C₂×#11758224	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	—	122	2	no	—
248	C₈	—	122	2	no	—
249	C₈	—	244	2	no	—
250	C₈	—	244	4	no	—
251	C₈	#14519784	1	4	yes	complemented
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	4	4	no	—
257	C₂²	#16839874	1	2	yes	—
258	C₂²	C₂×D₄×Dic₆₁	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	4	4	no	—
262	C₄	#16839874	1	2	yes	—
263	C₄	#16839884	1	2	yes	—
264	C₄	C₂×#14519784	1	2	yes	—
265	C₄	C₂×#14519801	1	2	yes	—
266	C₂	—	2	2	no	—
267	C₂	—	8	4	no	—
268	C₂	#21454531	1	1	yes	characteristic
269	C₂	C₂×#16839874	1	1	yes	characteristic
270	C₂	C₂×#16839884	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₄²	C₂³×D₆₁
Fitting, prime radical	C₆₁×#3169	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	#3169	D₆₁
hypocenter	C₆₁	#17962
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #21454531	C₂	semidirect
2. #21444733	C₂	semidirect
3. #21444643	C₂	semidirect
4. #21444575	C₂	semidirect
5. #21444270	C₂	semidirect
6. C₂×#16839483	C₂	semidirect
111. C₈×C₆₁	#131	semidirect
163. C₆₁	#17962	semidirect
251. C₈	#14519784	semidirect

Maximal subgroups

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

subgroup	index
#21454531	2
#21444733	2
#21444643	2
#21444575	2
#21444270	2
C₂×#16839483	2
C₆₁×#3169	2
#17962	61