Subgroups — #29569501

Order 15616.

618 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839815	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839786	C₂	1	1	yes	maximal, characteristic
3	#21443654	C₂	1	4	yes	maximal, complemented
4	C₂×C₆₁×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11758194	—	2	1	no	—
6	#16839815	—	2	2	no	—
7	#16839815	C₄	1	4	yes	—
8	#16839786	—	2	2	no	—
9	#16839786	C₄	1	4	yes	—
10	C₂×C₄⋊#92418	—	2	1	no	—
11	C₂×C₄⋊#92418	C₂²	1	2	yes	—
12	C₂×C₄⋊#92418	C₂²	1	2	yes	—
13	C₂×C₂²⋊#92418	—	2	1	no	—
14	C₂×C₄×#92418	—	2	1	no	—
15	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#431	—	2	2	no	—
17	C₆₁×#431	—	2	2	no	—
18	C₂×C₄⋊C₈×C₆₁	C₄	1	2	yes	—
19	C₂×C₂²⋊C₈×C₆₁	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
21	C₂×#11758194	—	2	4	no	—
22	C₂×#11758194	—	4	4	no	—
23	C₂²×#92418	—	2	2	no	—
24	C₂²×#92418	—	2	2	no	—
25	C₄⋊#92418	—	2	4	no	—
26	C₄⋊#92418	—	2	2	no	—
27	C₄⋊#92418	—	2	2	no	—
28	C₄⋊#92418	C₂×C₄	1	4	yes	—
29	C₄⋊#92418	C₂×C₄	1	4	yes	—
30	C₂²⋊#92418	—	2	4	no	—
31	C₄×#92418	—	2	4	no	—
32	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₆₁	Q₈	1	1	yes	characteristic
35	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₁	D₄	1	2	yes	—
41	C₂×C₄²×C₆₁	D₄	1	2	yes	—
42	C₂×C₄²×C₆₁	Q₈	1	1	yes	characteristic
43	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₆₁	—	2	4	no	—
45	C₂²⋊C₈×C₆₁	—	2	4	no	—
46	C₄×C₈×C₆₁	—	2	2	no	—
47	C₄×C₈×C₆₁	—	2	2	no	—
48	C₂⁴×C₆₁	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	—	2	4	no	—
50	C₂²×C₄×C₆₁	—	2	4	no	—
51	C₂²×C₄×C₆₁	—	2	2	no	—
52	C₂²×C₄×C₆₁	—	2	2	no	—
53	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
54	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₈×C₆₁	—	2	4	no	—
66	C₂×C₈×C₆₁	—	2	8	no	—
67	C₄²×C₆₁	C₄²	1	2	yes	—
68	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
72	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
73	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
77	#11758194	—	4	8	no	—
78	C₂×#92418	—	2	4	no	—
79	C₂×#92418	—	2	4	no	—
80	C₂×#92418	—	4	2	no	—
81	C₂×#92418	—	4	4	no	—
82	C₂×#92418	—	4	2	no	—
83	C₂³×C₆₁	—	2	2	no	—
84	C₂³×C₆₁	—	2	2	no	—
85	C₂³×C₆₁	—	2	2	no	—
86	C₂³×C₆₁	#95	1	2	yes	—
87	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₆₁	—	2	2	no	—
89	C₂×C₄×C₆₁	—	2	2	no	—
90	C₂×C₄×C₆₁	—	2	2	no	—
91	C₂×C₄×C₆₁	—	2	4	no	—
92	C₂×C₄×C₆₁	—	2	4	no	—
93	C₂×C₄×C₆₁	—	2	4	no	—
94	C₂×C₄×C₆₁	—	2	4	no	—
95	C₂×C₄×C₆₁	—	2	4	no	—
96	C₂×C₄×C₆₁	—	2	2	no	—
97	C₂×C₄×C₆₁	—	2	2	no	—
98	C₂×C₄×C₆₁	#95	1	2	yes	—
99	C₂×C₄×C₆₁	#95	1	2	yes	—
100	C₂×C₄×C₆₁	#95	1	2	yes	—
101	C₂×C₄×C₆₁	#108	1	4	yes	—
102	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₆₁	#131	1	4	yes	—
109	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
110	C₂×C₄×C₆₁	#137	1	2	yes	—
111	C₈×C₆₁	—	4	4	no	—
112	#92418	—	4	4	no	—
113	#92418	—	4	4	no	—
114	#7880	—	61	1	no	maximal, Sylow
115	C₂²×C₆₁	—	2	2	no	—
116	C₂²×C₆₁	—	2	2	no	—
117	C₂²×C₆₁	—	2	2	no	—
118	C₂²×C₆₁	—	4	4	no	—
119	C₂²×C₆₁	C₂×#95	1	1	yes	characteristic
120	C₂²×C₆₁	#424	1	2	yes	—
121	C₂²×C₆₁	#428	1	2	yes	—
122	C₂²×C₆₁	C₂×#108	1	2	yes	—
123	C₄×C₆₁	—	2	2	no	—
124	C₄×C₆₁	—	2	2	no	—
125	C₄×C₆₁	—	2	1	no	—
126	C₄×C₆₁	—	2	1	no	—
127	C₄×C₆₁	#341	1	4	yes	—
128	C₄×C₆₁	#405	1	2	yes	—
129	C₄×C₆₁	#408	1	2	yes	—
130	C₄×C₆₁	#417	1	2	yes	—
131	C₄×C₆₁	#418	1	1	yes	characteristic
132	C₄×C₆₁	#419	1	1	yes	characteristic
133	C₂×#431	—	61	1	no	—
134	C₂×#431	D₆₁	1	1	yes	characteristic
135	C₂×#423	—	61	1	no	—
136	#1277	—	61	4	no	—
137	C₂×C₆₁	—	4	2	no	—
138	C₂×C₆₁	#1574	1	4	yes	—
139	C₂×C₆₁	#1729	1	1	yes	characteristic
140	C₂×C₆₁	#1736	1	1	yes	characteristic
141	C₂×C₆₁	C₂×#341	1	1	yes	characteristic
142	C₂²×M₁₆	—	122	1	no	—
143	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic
144	#431	—	2	2	no	—
145	#431	—	2	2	no	—
146	#431	—	61	4	no	—
147	#431	—	122	2	no	—
148	#423	—	61	4	no	—
149	#423	—	122	2	no	—
150	C₂×C₄⋊C₈	—	61	2	no	—
151	C₂×C₄⋊C₈	—	61	2	no	—
152	C₂×C₄⋊C₈	—	122	1	no	—
153	C₂×C₄⋊C₈	Dic₆₁	1	2	yes	—
154	C₂×C₂²⋊C₈	—	122	1	no	—
155	C₂×C₂²⋊C₈	Dic₆₁	1	2	yes	—
156	C₂×C₄×C₈	—	122	1	no	—
157	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
158	C₆₁	#7880	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₄	Dic₁₂₂	1	1	yes	characteristic
160	C₂³×C₄	D₂₄₄	1	1	yes	characteristic
161	C₂³×C₄	C₂×Dic₆₁	1	1	yes	characteristic
162	C₂×M₁₆	—	122	4	no	—
163	C₂×M₁₆	—	244	4	no	—
164	C₂²×C₈	—	122	2	no	—
165	C₂²×C₈	—	122	2	no	—
166	C₂²×C₈	C₂×Dic₆₁	1	4	yes	—
167	C₂×C₄²	Dic₁₂₂	1	1	yes	characteristic
168	C₂×C₄²	C₄×D₆₁	1	2	yes	—
169	C₂×C₄²	C₄×D₆₁	1	2	yes	—
170	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
171	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
172	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
173	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
174	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
175	C₄⋊C₈	—	2	4	no	—
176	C₄⋊C₈	—	61	4	no	—
177	C₄⋊C₈	—	61	4	no	—
178	C₄⋊C₈	—	122	4	no	—
179	C₄⋊C₈	—	122	2	no	—
180	C₄⋊C₈	—	122	2	no	—
181	C₂²⋊C₈	—	2	4	no	—
182	C₂²⋊C₈	—	122	4	no	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	122	4	no	—
186	C₂⁴	C₄⋊Dic₆₁	1	1	yes	characteristic
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	C₄×Dic₆₁	1	2	yes	—
192	C₂²×C₄	#11758196	1	2	yes	—
193	C₂²×C₄	#11758196	1	2	yes	—
194	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
195	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
196	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
197	C₂²×C₄	#11758198	1	2	yes	—
198	C₂²×C₄	#11758198	1	2	yes	—
199	C₂²×C₄	C₂²⋊Dic₆₁	1	2	yes	—
200	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
202	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
203	M₁₆	—	244	8	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	8	no	—
206	C₂×C₈	—	122	4	no	—
207	C₂×C₈	—	122	4	no	—
208	C₂×C₈	—	244	2	no	—
209	C₂×C₈	—	244	2	no	—
210	C₂×C₈	—	244	4	no	—
211	C₄²	C₄×Dic₆₁	1	2	yes	—
212	C₄²	#11758196	1	2	yes	—
213	C₄²	#11758196	1	2	yes	—
214	C₄²	#11758198	1	2	yes	—
215	C₄²	#11758198	1	2	yes	—
216	C₄²	C₂²⋊Dic₆₁	1	2	yes	—
217	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
218	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
219	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
220	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	#14519684	1	2	yes	—
225	C₂³	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	#14519684	1	2	yes	—
237	C₂×C₄	#14519684	1	2	yes	—
238	C₂×C₄	#14519684	1	2	yes	—
239	C₂×C₄	#14519713	1	4	yes	—
240	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
241	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
242	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
243	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
244	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
245	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
246	C₂×C₄	#14519784	1	4	yes	—
247	C₂×C₄	#14519805	1	2	yes	—
248	C₂×C₄	#14519810	1	2	yes	—
249	C₈	—	4	4	no	—
250	C₈	—	244	4	no	—
251	C₈	—	244	4	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	4	4	no	—
256	C₂²	C₂×#14519684	1	1	yes	characteristic
257	C₂²	#16839789	1	2	yes	—
258	C₂²	#16839804	1	2	yes	—
259	C₂²	C₂×#14519713	1	2	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	#16839527	1	4	yes	—
265	C₄	#16839718	1	2	yes	—
266	C₄	#16839729	1	2	yes	—
267	C₄	#16839764	1	2	yes	—
268	C₄	#16839768	1	1	yes	characteristic
269	C₄	#16839772	1	1	yes	characteristic
270	C₂	—	4	2	no	—
271	C₂	#21444608	1	4	yes	—
272	C₂	#21445516	1	1	yes	characteristic
273	C₂	#21445539	1	1	yes	characteristic
274	C₂	C₂×#16839527	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#431	C₂
socle	C₂³×C₆₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₁	C₄⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#431	D₆₁
hypocenter	C₆₁	#7880
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #21443654	C₂	semidirect
158. C₆₁	#7880	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839815	2
C₂×#16839786	2
#21443654	2
C₂×C₆₁×#431	2
#7880	61