Subgroups — #29572954

Order 15616.

378 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445815	C₂	1	2	yes	maximal
2	#21444578	C₂	1	1	yes	maximal, characteristic
3	#21444575	C₂	1	1	yes	maximal, characteristic
4	#21443694	C₂	1	2	yes	maximal
5	C₂×C₆₁×#446	C₂	1	1	yes	maximal, characteristic
6	#16839786	C₂²	1	2	yes	—
7	#16839786	C₂²	1	2	yes	—
8	C₂×#14519696	—	2	2	no	—
9	#16839531	—	2	2	no	—
10	#16839515	—	2	1	no	—
11	#16839515	—	2	1	no	—
12	#16839513	—	2	1	no	—
13	#16839513	—	2	1	no	—
14	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#446	—	2	1	no	—
16	C₆₁×#446	—	2	1	no	—
17	C₆₁×#446	—	2	1	no	—
18	C₆₁×#446	—	2	1	no	—
19	C₂×C₄⋊C₈×C₆₁	C₄	1	2	yes	—
20	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×#97×C₆₁	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×#11758194	—	2	2	no	—
24	C₂×#11758194	—	2	2	no	—
25	C₄⋊#92418	—	2	2	no	—
26	C₄⋊#92418	—	2	2	no	—
27	C₄⋊#92418	—	2	2	no	—
28	C₄⋊#92418	—	2	2	no	—
29	#14519696	—	4	4	no	—
30	C₂²×Q₈×C₆₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₄×Q₈×C₆₁	—	2	1	no	—
35	C₄×Q₈×C₆₁	—	2	1	no	—
36	C₄×Q₈×C₆₁	—	2	1	no	—
37	C₄×Q₈×C₆₁	—	2	1	no	—
38	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₆₁	Q₈	1	2	yes	—
40	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₆₁	—	2	2	no	—
43	C₄⋊C₈×C₆₁	—	2	2	no	—
44	C₄⋊C₈×C₆₁	—	2	1	no	—
45	C₄⋊C₈×C₆₁	—	2	1	no	—
46	#97×C₆₁	—	2	2	no	—
47	#97×C₆₁	—	2	2	no	—
48	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
49	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
50	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
52	C₂×Q₈×C₆₁	—	2	1	no	—
53	C₂×Q₈×C₆₁	—	2	1	no	—
54	C₂×Q₈×C₆₁	—	2	2	no	—
55	C₂×Q₈×C₆₁	—	2	2	no	—
56	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₆₁	—	2	2	no	—
62	C₂×C₈×C₆₁	—	2	2	no	—
63	C₂×C₈×C₆₁	—	4	1	no	—
64	C₂×C₈×C₆₁	—	4	2	no	—
65	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₄⋊C₄×C₆₁	—	2	1	no	—
70	C₄⋊C₄×C₆₁	—	2	1	no	—
71	C₄⋊C₄×C₆₁	—	2	2	no	—
72	C₄⋊C₄×C₆₁	—	2	2	no	—
73	C₄²×C₆₁	—	2	2	no	—
74	C₄²×C₆₁	—	2	2	no	—
75	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₆₁	#41	1	1	yes	characteristic
78	C₄²×C₆₁	#41	1	1	yes	characteristic
79	#11758194	—	4	2	no	—
80	#11758194	—	4	2	no	—
81	#11758194	—	4	2	no	—
82	#11758194	—	4	2	no	—
83	C₂×#92418	—	4	2	no	—
84	C₂×#92418	—	4	2	no	—
85	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
86	Q₈×C₆₁	—	4	2	no	—
87	Q₈×C₆₁	—	4	1	no	—
88	Q₈×C₆₁	—	4	1	no	—
89	C₂×C₄×C₆₁	—	2	1	no	—
90	C₂×C₄×C₆₁	—	2	1	no	—
91	C₂×C₄×C₆₁	—	2	2	no	—
92	C₂×C₄×C₆₁	—	2	2	no	—
93	C₂×C₄×C₆₁	—	2	4	no	—
94	C₂×C₄×C₆₁	—	4	1	no	—
95	C₂×C₄×C₆₁	—	4	2	no	—
96	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
98	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
99	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
103	C₂×C₄×C₆₁	#137	1	2	yes	—
104	C₈×C₆₁	—	2	1	no	—
105	C₈×C₆₁	—	2	1	no	—
106	C₈×C₆₁	—	2	1	no	—
107	C₈×C₆₁	—	4	2	no	—
108	C₈×C₆₁	#101	1	2	yes	complemented
109	#92418	—	8	2	no	—
110	#92418	—	8	2	no	—
111	#8619	—	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	#428	1	2	yes	—
116	C₄×C₆₁	—	2	1	no	—
117	C₄×C₆₁	—	2	1	no	—
118	C₄×C₆₁	—	2	1	no	—
119	C₄×C₆₁	—	4	2	no	—
120	C₄×C₆₁	—	4	2	no	—
121	C₄×C₆₁	C₂×#101	1	1	yes	characteristic
122	C₄×C₆₁	#413	1	1	yes	characteristic
123	C₄×C₆₁	#468	1	2	yes	—
124	C₄×C₆₁	#469	1	1	yes	characteristic
125	C₄×C₆₁	#470	1	1	yes	characteristic
126	C₂×#446	D₆₁	1	1	yes	characteristic
127	#1801	—	61	2	no	—
128	#1567	—	61	1	no	—
129	#1566	—	61	1	no	—
130	#1292	—	61	2	no	—
131	C₂×C₆₁	—	2	2	no	—
132	C₂×C₆₁	#1801	1	1	yes	characteristic
133	C₂×C₆₁	#1809	1	1	yes	characteristic
134	C₂×C₆₁	#1813	1	1	yes	characteristic
135	C₂×C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
136	#446	—	2	1	no	—
137	#446	—	2	1	no	—
138	#446	—	2	1	no	—
139	#446	—	2	1	no	—
140	#423	—	61	2	no	—
141	#423	—	61	2	no	—
142	C₂×C₄⋊C₈	Dic₆₁	1	2	yes	—
143	C₂×C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
144	C₂×#101	—	122	2	no	—
145	C₂×#97	Dic₆₁	1	2	yes	—
146	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
147	#343	—	122	2	no	—
148	#335	—	122	1	no	—
149	#335	—	122	1	no	—
150	#334	—	122	1	no	—
151	#334	—	122	1	no	—
152	C₆₁	#8619	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₈	D₂₄₄	1	1	yes	characteristic
154	C₂×M₁₆	—	122	2	no	—
155	C₂×M₁₆	—	122	2	no	—
156	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
157	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
158	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
159	C₄×Q₈	—	2	1	no	—
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	—	2	1	no	—
162	C₄×Q₈	—	2	1	no	—
163	C₂×C₄⋊C₄	Dic₁₂₂	1	2	yes	—
164	C₂×C₄⋊C₄	D₂₄₄	1	1	yes	characteristic
165	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
166	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	2	1	no	—
170	C₄⋊C₈	—	2	1	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	#101	—	244	4	no	—
176	#97	—	2	2	no	—
177	#97	—	2	2	no	—
178	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
179	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	1	no	—
184	C₂×Q₈	—	2	1	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂²×C₄	C₄⋊Dic₆₁	1	2	yes	—
187	C₂²×C₄	C₄⋊Dic₆₁	1	2	yes	—
188	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
190	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
191	M₁₆	—	244	2	no	—
192	M₁₆	—	244	2	no	—
193	M₁₆	—	244	2	no	—
194	M₁₆	—	244	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	244	2	no	—
200	C₂×C₈	—	244	2	no	—
201	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	#11758222	1	1	yes	characteristic
212	C₄²	#11758222	1	1	yes	characteristic
213	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
214	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
215	C₂³	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
216	Q₈	—	4	2	no	—
217	Q₈	—	4	1	no	—
218	Q₈	—	4	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
227	C₂×C₄	#14519736	1	1	yes	characteristic
228	C₂×C₄	#14519751	1	1	yes	characteristic
229	C₂×C₄	#14519751	1	1	yes	characteristic
230	C₂×C₄	#14519756	1	1	yes	characteristic
231	C₂×C₄	#14519756	1	1	yes	characteristic
232	C₂×C₄	#14519805	1	2	yes	—
233	C₂×C₄	#14519810	1	2	yes	—
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	4	2	no	—
238	C₈	—	488	2	no	—
239	C₈	—	488	2	no	—
240	C₈	#14519696	1	2	yes	complemented
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	#16839622	1	1	yes	characteristic
244	C₂²	#16839804	1	2	yes	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	2	no	—
250	C₄	C₂×#14519696	1	1	yes	characteristic
251	C₄	#16839746	1	1	yes	characteristic
252	C₄	#16840008	1	2	yes	—
253	C₄	#16840015	1	1	yes	characteristic
254	C₄	#16840020	1	1	yes	characteristic
255	C₂	—	2	2	no	—
256	C₂	#21445815	1	1	yes	characteristic
257	C₂	#21445852	1	1	yes	characteristic
258	C₂	#21445869	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16839622
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#446	C₂
socle	C₂²×C₆₁	#380
prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#446	D₆₁
hypocenter	C₆₁	#8619
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
108. C₈×C₆₁	#101	semidirect
152. C₆₁	#8619	semidirect
240. C₈	#14519696	semidirect

Maximal subgroups

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

subgroup	index
#21445815	2
#21444578	2
#21444575	2
#21443694	2
C₂×C₆₁×#446	2
#8619	61