Subgroups — #29580012

Order 15616.

512 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14519777	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#92418	C₂	1	1	yes	maximal, characteristic
3	#21443658	C₂	1	4	yes	maximal
4	C₆₁×#1767	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519777	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₂₂	—	2	1	no	—
7	C₂×C₄⋊#92418	—	2	1	no	—
8	C₂×C₄×#92418	C₂²	1	2	yes	—
9	#16839668	—	2	1	no	—
10	#16839660	—	2	1	no	—
11	#16839660	C₄	1	2	yes	—
12	#16839632	C₄	1	1	yes	characteristic
13	C₄×C₄⋊Dic₆₁	—	2	1	no	—
14	C₄×C₄⋊Dic₆₁	C₂²	1	2	yes	—
15	#16839517	—	2	1	no	—
16	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
17	C₂×#97×C₆₁	—	2	1	no	—
18	C₄³×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×#336	—	2	1	no	—
20	C₂²×Dic₁₂₂	—	2	1	no	—
21	C₂²×#92418	—	2	2	no	—
22	#14519777	—	2	4	no	—
23	C₄×Dic₁₂₂	—	4	4	no	—
24	C₂×C₄⋊Dic₆₁	—	2	1	no	—
25	C₂×C₄⋊Dic₆₁	—	2	1	no	—
26	C₂×C₄⋊Dic₆₁	—	2	1	no	—
27	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
29	C₂×#11758196	—	2	2	no	—
30	C₂×C₄×Dic₆₁	—	2	2	no	—
31	C₄⋊#92418	—	2	4	no	—
32	C₄×#92418	—	2	4	no	—
33	#14519684	—	2	2	no	—
34	#14519684	—	2	2	no	—
35	C₂²×C₈×C₆₁	—	2	2	no	—
36	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₆₁	—	2	2	no	—
44	C₄⋊C₈×C₆₁	Q₈	1	4	yes	complemented
45	#97×C₆₁	—	4	2	no	—
46	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₆₁	—	2	2	no	—
54	C₂²×Dic₆₁	—	2	2	no	—
55	C₂×Dic₁₂₂	—	2	4	no	—
56	C₂×Dic₁₂₂	—	4	2	no	—
57	C₂×Dic₁₂₂	—	4	2	no	—
58	C₂×C₈×C₆₁	—	2	4	no	—
59	C₂×C₈×C₆₁	—	4	2	no	—
60	C₂×C₈×C₆₁	—	4	2	no	—
61	C₄²×C₆₁	—	2	2	no	—
62	C₄²×C₆₁	—	2	2	no	—
63	C₄²×C₆₁	—	2	2	no	—
64	C₄²×C₆₁	—	2	2	no	—
65	C₄²×C₆₁	—	4	1	no	—
66	C₄²×C₆₁	SD₁₆	1	2	yes	—
67	C₄²×C₆₁	Q₁₆	1	2	yes	—
68	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
69	C₄²×C₆₁	#41	1	2	yes	—
70	C₄⋊Dic₆₁	—	2	4	no	—
71	C₄⋊Dic₆₁	—	2	2	no	—
72	C₄⋊Dic₆₁	—	2	4	no	—
73	C₄⋊Dic₆₁	—	4	2	no	—
74	C₄⋊Dic₆₁	—	4	2	no	—
75	C₄⋊Dic₆₁	M₁₆	1	4	yes	—
76	#11758196	—	4	4	no	—
77	C₄×Dic₆₁	—	4	4	no	—
78	C₂×#92418	—	2	4	no	—
79	C₂×#92418	—	4	2	no	—
80	C₂×#92418	—	4	2	no	—
81	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₁	—	2	1	no	—
83	C₂×C₄×C₆₁	—	2	1	no	—
84	C₂×C₄×C₆₁	—	2	1	no	—
85	C₂×C₄×C₆₁	—	2	1	no	—
86	C₂×C₄×C₆₁	—	2	1	no	—
87	C₂×C₄×C₆₁	—	2	1	no	—
88	C₂×C₄×C₆₁	—	2	1	no	—
89	C₂×C₄×C₆₁	—	2	1	no	—
90	C₂×C₄×C₆₁	—	2	1	no	—
91	C₂×C₄×C₆₁	—	2	1	no	—
92	C₂×C₄×C₆₁	—	2	1	no	—
93	C₂×C₄×C₆₁	#103	1	2	yes	—
94	C₂×C₄×C₆₁	#103	1	2	yes	—
95	C₂×C₄×C₆₁	#104	1	2	yes	—
96	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
97	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
98	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
101	C₂×C₄×C₆₁	#131	1	2	yes	—
102	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
104	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
105	C₂×Dic₆₁	—	2	4	no	—
106	C₂×Dic₆₁	—	4	2	no	—
107	C₂×Dic₆₁	—	4	2	no	—
108	C₂×Dic₆₁	—	4	2	no	—
109	C₂×Dic₆₁	—	4	2	no	—
110	C₂×Dic₆₁	—	4	2	no	—
111	Dic₁₂₂	—	4	8	no	—
112	C₈×C₆₁	—	4	4	no	—
113	#92418	—	4	4	no	—
114	#9868	—	61	1	no	maximal, Sylow
115	C₂²×C₆₁	#391	1	1	yes	characteristic
116	C₂²×C₆₁	#407	1	1	yes	characteristic
117	C₂²×C₆₁	#408	1	1	yes	characteristic
118	C₂²×C₆₁	C₂×#103	1	1	yes	characteristic
119	C₂²×C₆₁	#418	1	1	yes	characteristic
120	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
121	C₂²×C₆₁	#421	1	1	yes	characteristic
122	C₄×C₆₁	—	2	1	no	—
123	C₄×C₆₁	—	2	1	no	—
124	C₄×C₆₁	—	2	2	no	—
125	C₄×C₆₁	—	2	2	no	—
126	C₄×C₆₁	—	2	2	no	—
127	C₄×C₆₁	—	4	1	no	—
128	C₄×C₆₁	#446	1	4	yes	—
129	C₄×C₆₁	#476	1	2	yes	—
130	C₄×C₆₁	#477	1	2	yes	—
131	Dic₆₁	—	4	4	no	—
132	Dic₆₁	—	8	2	no	—
133	C₄×C₄⋊Q₈	—	61	1	no	—
134	#1767	D₆₁	1	1	yes	characteristic
135	C₄×C₄⋊C₈	—	61	1	no	—
136	#1279	—	61	4	no	—
137	C₂×C₆₁	#1487	1	1	yes	characteristic
138	C₂×C₆₁	#1490	1	1	yes	characteristic
139	C₂×C₆₁	#1493	1	1	yes	characteristic
140	C₂×C₆₁	#1496	1	1	yes	characteristic
141	C₂×C₆₁	#1985	1	1	yes	characteristic
142	C₂×C₆₁	#1986	1	1	yes	characteristic
143	C₂×C₆₁	#1987	1	1	yes	characteristic
144	C₂×C₄⋊Q₈	—	61	1	no	—
145	C₂×C₄×Q₈	—	122	1	no	—
146	C₂×C₄⋊C₈	—	122	1	no	—
147	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
148	C₂×#97	—	2	1	no	—
149	C₂×C₄×C₈	—	61	2	no	—
150	#391	—	122	1	no	—
151	#389	—	61	2	no	—
152	#389	—	122	1	no	—
153	#384	—	61	1	no	—
154	C₄×C₄⋊C₄	—	61	2	no	—
155	C₄×C₄⋊C₄	—	122	1	no	—
156	C₄³	C₂×D₆₁	1	1	yes	characteristic
157	#336	—	2	1	no	—
158	#336	—	122	1	no	—
159	C₆₁	#9868	1	1	yes	characteristic, complemented, Sylow
160	C₂²×Q₈	—	122	1	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	122	2	no	—
163	C₄⋊Q₈	—	122	4	no	—
164	C₄×Q₈	—	244	4	no	—
165	C₂×C₄⋊C₄	—	61	2	no	—
166	C₂×C₄⋊C₄	—	61	2	no	—
167	C₂×C₄⋊C₄	—	122	1	no	—
168	C₂×C₄⋊C₄	—	122	1	no	—
169	C₂×C₄⋊C₄	—	122	2	no	—
170	C₂×C₄⋊C₄	—	122	1	no	—
171	C₂×C₄²	—	122	2	no	—
172	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
173	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
174	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
175	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
176	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
177	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
178	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	122	4	no	—
181	C₄⋊C₈	Dic₁₂₂	1	4	yes	complemented
182	#97	—	4	2	no	—
183	C₄×C₈	—	122	4	no	—
184	#95	—	122	2	no	—
185	#95	—	122	2	no	—
186	C₂×Q₈	—	122	4	no	—
187	C₂×Q₈	—	244	2	no	—
188	C₂×Q₈	—	244	2	no	—
189	C₂²×C₄	—	122	2	no	—
190	C₂²×C₄	—	122	2	no	—
191	C₂²×C₄	#11758198	1	1	yes	characteristic
192	C₂²×C₄	#11758198	1	1	yes	characteristic
193	C₂²×C₄	#11758198	1	1	yes	characteristic
194	C₂²×C₄	#11758198	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	122	4	no	—
202	C₂×C₈	—	244	2	no	—
203	C₂×C₈	—	244	2	no	—
204	C₄⋊C₄	—	61	4	no	—
205	C₄⋊C₄	—	122	4	no	—
206	C₄⋊C₄	—	122	2	no	—
207	C₄⋊C₄	—	122	4	no	—
208	C₄⋊C₄	—	244	4	no	—
209	C₄⋊C₄	—	244	2	no	—
210	C₄⋊C₄	—	244	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	4	1	no	—
216	C₄²	—	244	4	no	—
217	C₄²	#11758200	1	2	yes	—
218	C₄²	#11758202	1	2	yes	—
219	C₄²	C₂×Dic₁₂₂	1	2	yes	—
220	C₄²	#11758222	1	2	yes	—
221	C₂³	C₂×#11758198	1	1	yes	characteristic
222	Q₈	—	244	8	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	122	4	no	—
235	C₂×C₄	—	244	2	no	—
236	C₂×C₄	—	244	2	no	—
237	C₂×C₄	—	244	2	no	—
238	C₂×C₄	—	244	2	no	—
239	C₂×C₄	—	244	2	no	—
240	C₂×C₄	#14519700	1	2	yes	—
241	C₂×C₄	#14519700	1	2	yes	—
242	C₂×C₄	#14519703	1	2	yes	—
243	C₂×C₄	C₄×Dic₁₂₂	1	2	yes	—
244	C₂×C₄	#14519756	1	2	yes	—
245	C₂×C₄	#14519765	1	1	yes	characteristic
246	C₂×C₄	#14519777	1	1	yes	characteristic
247	C₂×C₄	M₁₆×D₆₁	1	2	yes	—
248	C₂×C₄	#14519785	1	2	yes	—
249	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
250	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
251	C₂×C₄	#14519802	1	2	yes	—
252	C₈	—	4	4	no	—
253	C₈	—	244	4	no	—
254	C₂²	#16839668	1	1	yes	characteristic
255	C₂²	#16839725	1	1	yes	characteristic
256	C₂²	#16839730	1	1	yes	characteristic
257	C₂²	C₂×#14519700	1	1	yes	characteristic
258	C₂²	#16839766	1	1	yes	characteristic
259	C₂²	C₂×#14519703	1	1	yes	characteristic
260	C₂²	#16839778	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	488	2	no	—
268	C₄	—	244	4	no	—
269	C₄	#16839890	1	4	yes	—
270	C₄	#16840052	1	2	yes	—
271	C₄	#16840057	1	2	yes	—
272	C₂	#21444247	1	1	yes	characteristic
273	C₂	#21444261	1	1	yes	characteristic
274	C₂	#21444273	1	1	yes	characteristic
275	C₂	#21444287	1	1	yes	characteristic
276	C₂	#21446759	1	1	yes	characteristic
277	C₂	#21446763	1	1	yes	characteristic
278	C₂	#21446767	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₁
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1767	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1767	D₆₁
hypocenter	C₆₁	#9868
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
44. C₄⋊C₈×C₆₁	Q₈	semidirect
159. C₆₁	#9868	semidirect
181. C₄⋊C₈	Dic₁₂₂	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14519777	2
C₄×C₄⋊#92418	2
#21443658	2
C₆₁×#1767	2
#9868	61