Subgroups — #29573163

Order 15616.

374 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839892	C₂	1	1	yes	maximal, characteristic
2	#21445991	C₂	1	1	yes	maximal, characteristic
3	#21444249	C₂	1	2	yes	maximal
4	#21443715	C₂	1	2	yes	maximal
5	C₆₁×#1564	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
7	#16839892	—	2	1	no	—
8	#16839892	—	2	1	no	—
9	#16839892	C₄	1	2	yes	—
10	#16839892	C₄	1	2	yes	—
11	C₂×#14519713	—	2	1	no	—
12	C₂×#14519713	—	2	1	no	—
13	#16839786	C₂²	1	2	yes	—
14	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
15	C₂×#14519706	—	2	1	no	—
16	C₂×#14519687	—	2	1	no	—
17	#16839494	—	2	4	no	—
18	C₆₁×#431	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₆₁×#335	—	2	1	no	—
21	C₆₁×#334	—	2	1	no	—
22	C₂²×Dic₁₂₂	Q₈	1	1	yes	characteristic
23	C₂×#11758194	—	2	2	no	—
24	C₂²×#92418	—	2	1	no	—
25	C₄×Dic₁₂₂	—	2	1	no	—
26	C₄×Dic₁₂₂	—	2	1	no	—
27	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
28	C₄×Dic₁₂₂	D₄	1	2	yes	—
29	C₂×C₄⋊Dic₆₁	Q₈	1	1	yes	characteristic
30	C₂×#11758196	—	2	1	no	—
31	C₂×C₄×Dic₆₁	—	2	1	no	—
32	#14519713	—	4	2	no	—
33	#14519713	—	4	2	no	—
34	C₄⋊#92418	—	2	1	no	—
35	C₄⋊#92418	—	2	2	no	—
36	C₄⋊#92418	—	2	2	no	—
37	C₄⋊#92418	C₂×C₄	1	2	yes	—
38	#14519706	—	2	2	no	—
39	#14519706	—	2	2	no	—
40	#14519687	—	2	2	no	—
41	#14519687	—	2	2	no	—
42	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₆₁	—	2	2	no	—
46	C₂²⋊C₈×C₆₁	—	2	2	no	—
47	C₄×C₈×C₆₁	—	2	2	no	—
48	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
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	1	yes	characteristic
55	C₂²×Dic₆₁	—	2	1	no	—
56	C₂²×Dic₆₁	—	2	1	no	—
57	C₂×Dic₁₂₂	—	2	1	no	—
58	C₂×Dic₁₂₂	—	4	2	no	—
59	C₂×Dic₁₂₂	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₆₁	—	2	1	no	—
61	C₂×C₈×C₆₁	—	2	1	no	—
62	C₂×C₈×C₆₁	—	4	2	no	—
63	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
68	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
69	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₆₁	#41	1	1	yes	characteristic
73	C₄⋊Dic₆₁	—	2	1	no	—
74	C₄⋊Dic₆₁	C₄⋊C₄	1	2	yes	—
75	#11758196	—	2	2	no	—
76	#11758196	—	2	2	no	—
77	C₄×Dic₆₁	—	2	2	no	—
78	C₄×Dic₆₁	—	2	2	no	—
79	#11758194	—	4	2	no	—
80	#11758194	—	4	2	no	—
81	C₂×#92418	—	2	2	no	—
82	C₂×#92418	—	4	1	no	—
83	C₂×#92418	—	4	1	no	—
84	C₂×#92418	—	4	2	no	—
85	C₂³×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₆₁	—	2	2	no	—
87	C₂×C₄×C₆₁	—	4	1	no	—
88	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
91	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
92	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
93	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
98	C₂×Dic₆₁	—	2	2	no	—
99	C₂×Dic₆₁	—	2	2	no	—
100	C₂×Dic₆₁	—	4	2	no	—
101	C₂×Dic₆₁	—	4	2	no	—
102	Dic₁₂₂	—	2	4	no	—
103	Dic₁₂₂	—	4	2	no	—
104	C₈×C₆₁	—	2	1	no	—
105	C₈×C₆₁	—	2	1	no	—
106	C₈×C₆₁	—	8	2	no	—
107	C₈×C₆₁	#103	1	2	yes	—
108	C₈×C₆₁	#103	1	2	yes	—
109	#92418	—	4	2	no	—
110	#92418	—	8	2	no	—
111	#8663	—	61	1	no	maximal, Sylow
112	C₂²×C₆₁	—	2	1	no	—
113	C₂²×C₆₁	—	2	1	no	—
114	C₂²×C₆₁	#380	1	1	yes	characteristic
115	C₂²×C₆₁	#475	1	1	yes	characteristic
116	C₂²×C₆₁	#477	1	1	yes	characteristic
117	C₄×C₆₁	—	2	1	no	—
118	C₄×C₆₁	—	2	1	no	—
119	C₄×C₆₁	—	2	1	no	—
120	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
121	C₄×C₆₁	#417	1	1	yes	characteristic
122	C₄×C₆₁	#430	1	2	yes	—
123	C₄×C₆₁	#460	1	1	yes	characteristic
124	C₄×C₆₁	#462	1	1	yes	characteristic
125	Dic₆₁	—	4	2	no	—
126	Dic₆₁	—	4	2	no	—
127	C₂×#446	—	61	1	no	—
128	#1848	—	61	1	no	—
129	#1564	D₆₁	1	1	yes	characteristic
130	#1487	—	61	2	no	—
131	#1301	—	61	2	no	—
132	C₂×C₆₁	—	2	1	no	—
133	C₂×C₆₁	—	2	1	no	—
134	C₂×C₆₁	#1803	1	1	yes	characteristic
135	C₂×C₆₁	#1811	1	1	yes	characteristic
136	C₂×C₆₁	#1814	1	1	yes	characteristic
137	C₂×C₄×Q₈	—	61	1	no	—
138	#446	—	61	2	no	—
139	#446	—	61	2	no	—
140	#446	—	122	1	no	—
141	#446	—	122	1	no	—
142	#431	C₂×D₆₁	1	2	yes	—
143	C₂×#108	—	122	1	no	—
144	C₂×#108	—	122	1	no	—
145	#423	—	61	2	no	—
146	C₂×C₄⋊C₈	—	61	1	no	—
147	C₂×C₄⋊C₈	—	122	1	no	—
148	C₂×#97	—	122	1	no	—
149	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
150	#335	—	2	1	no	—
151	#334	—	2	1	no	—
152	Q₈⋊C₈	—	122	4	no	—
153	C₆₁	#8663	1	1	yes	characteristic, complemented, Sylow
154	C₂²×Q₈	—	61	1	no	—
155	C₂×M₁₆	—	122	2	no	—
156	C₂²×C₈	—	122	1	no	—
157	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
158	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
159	C₄×Q₈	—	61	2	no	—
160	C₄×Q₈	—	61	2	no	—
161	C₄×Q₈	—	122	1	no	—
162	C₄×Q₈	—	122	1	no	—
163	C₂×C₄⋊C₄	—	61	1	no	—
164	C₂×C₄⋊C₄	—	122	1	no	—
165	C₂×C₄²	—	122	1	no	—
166	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
167	#108	—	244	2	no	—
168	#108	—	244	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	61	2	no	—
171	C₄⋊C₈	—	122	2	no	—
172	C₄⋊C₈	—	122	2	no	—
173	C₄⋊C₈	—	122	2	no	—
174	C₄⋊C₈	—	122	2	no	—
175	C₄⋊C₈	—	122	1	no	—
176	C₂²⋊C₈	—	2	2	no	—
177	#97	—	122	2	no	—
178	#97	—	122	2	no	—
179	C₄×C₈	—	2	2	no	—
180	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
181	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
183	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
184	C₂×Q₈	—	61	2	no	—
185	C₂×Q₈	—	122	1	no	—
186	C₂×Q₈	—	244	2	no	—
187	C₂²×C₄	—	122	1	no	—
188	C₂²×C₄	—	122	1	no	—
189	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
190	C₂²×C₄	#11758224	1	1	yes	characteristic
191	C₂²×C₄	Q₈×D₆₁	1	1	yes	characteristic
192	M₁₆	—	244	2	no	—
193	M₁₆	—	244	2	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	122	2	no	—
198	C₂×C₈	—	244	2	no	—
199	C₂×C₈	—	244	1	no	—
200	C₂×C₈	—	244	1	no	—
201	C₂×C₈	#11758190	1	2	yes	—
202	C₂×C₈	Dic₂₄₄	1	2	yes	—
203	C₂×C₈	#11758198	1	1	yes	characteristic
204	C₂×C₈	#11758198	1	1	yes	characteristic
205	C₂×C₈	#11758198	1	1	yes	characteristic
206	C₂×C₈	#11758198	1	1	yes	characteristic
207	C₄⋊C₄	—	61	2	no	—
208	C₄⋊C₄	—	122	2	no	—
209	C₄⋊C₄	—	122	2	no	—
210	C₄⋊C₄	—	122	1	no	—
211	C₄²	—	122	2	no	—
212	C₄²	—	122	2	no	—
213	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
214	C₄²	D₄×D₆₁	1	1	yes	characteristic
215	C₄²	#11758226	1	1	yes	characteristic
216	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
217	C₂³	#14519757	1	1	yes	characteristic
218	Q₈	—	122	4	no	—
219	Q₈	—	244	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	122	2	no	—
223	C₂×C₄	—	122	2	no	—
224	C₂×C₄	—	244	2	no	—
225	C₂×C₄	—	244	2	no	—
226	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
227	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
228	C₂×C₄	#14519735	1	1	yes	characteristic
229	C₂×C₄	#14519748	1	1	yes	characteristic
230	C₂×C₄	#14519752	1	1	yes	characteristic
231	C₂×C₄	#14519753	1	1	yes	characteristic
232	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
233	C₂×C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
234	C₂×C₄	#14519805	1	1	yes	characteristic
235	C₂×C₄	#14519810	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	8	2	no	—
239	C₈	—	488	2	no	—
240	C₈	—	244	2	no	—
241	C₈	#14519702	1	2	yes	—
242	C₈	#14519702	1	2	yes	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	#16839619	1	1	yes	characteristic
246	C₂²	#16840048	1	1	yes	characteristic
247	C₂²	#16840058	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	244	2	no	—
252	C₄	—	244	2	no	—
253	C₄	C₂×#14519702	1	1	yes	characteristic
254	C₄	#16839764	1	1	yes	characteristic
255	C₄	#16839809	1	2	yes	—
256	C₄	#16839963	1	1	yes	characteristic
257	C₄	#16839973	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	#21445825	1	1	yes	characteristic
261	C₂	#21445858	1	1	yes	characteristic
262	C₂	#21445870	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14519752
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1564	C₂
socle	C₂²×C₆₁	#380
prime radical	C₂³×C₆₁	C₂²⋊Q₈
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1564	D₆₁
hypocenter	C₆₁	#8663
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
153. C₆₁	#8663	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839892	2
#21445991	2
#21444249	2
#21443715	2
C₆₁×#1564	2
#8663	61