Subgroups — #29580015

Order 15616.

512 conjugacy classes of subgroups.

The subgroup lattice has 284 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	#21445672	C₂	1	1	yes	maximal, characteristic
3	#21443659	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₆₁	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	C₂²	1	2	yes	—
8	C₂×#14519688	—	2	1	no	—
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	—	2	1	no	—
17	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
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	2	no	—
24	C₄×Dic₁₂₂	—	4	2	no	—
25	C₂×C₄⋊Dic₆₁	—	2	1	no	—
26	C₂×C₄⋊Dic₆₁	—	2	1	no	—
27	C₂×C₄⋊Dic₆₁	—	2	1	no	—
28	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
30	C₂×#11758196	—	2	2	no	—
31	C₂×C₄×Dic₆₁	—	2	2	no	—
32	C₄⋊#92418	—	2	2	no	—
33	C₄⋊#92418	Q₈	1	4	yes	complemented
34	#14519688	—	4	2	no	—
35	#14519684	—	2	2	no	—
36	#14519684	—	2	2	no	—
37	C₂²×C₈×C₆₁	—	2	2	no	—
38	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	C₂×C₄	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	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₆₁	—	2	4	no	—
46	C₄×C₈×C₆₁	—	2	4	no	—
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	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×Dic₆₁	—	2	2	no	—
55	C₂²×Dic₆₁	—	2	2	no	—
56	C₂×Dic₁₂₂	—	2	4	no	—
57	C₂×Dic₁₂₂	—	4	2	no	—
58	C₂×Dic₁₂₂	—	4	2	no	—
59	C₂×C₈×C₆₁	—	2	4	no	—
60	C₂×C₈×C₆₁	—	4	2	no	—
61	C₂×C₈×C₆₁	—	4	2	no	—
62	C₄²×C₆₁	—	2	2	no	—
63	C₄²×C₆₁	—	2	2	no	—
64	C₄²×C₆₁	—	2	2	no	—
65	C₄²×C₆₁	—	2	2	no	—
66	C₄²×C₆₁	—	4	1	no	—
67	C₄²×C₆₁	D₈	1	2	yes	—
68	C₄²×C₆₁	SD₁₆	1	2	yes	—
69	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
70	C₄²×C₆₁	#41	1	2	yes	—
71	C₄⋊Dic₆₁	—	2	4	no	—
72	C₄⋊Dic₆₁	—	2	2	no	—
73	C₄⋊Dic₆₁	—	2	4	no	—
74	C₄⋊Dic₆₁	—	4	2	no	—
75	C₄⋊Dic₆₁	—	4	2	no	—
76	C₄⋊Dic₆₁	M₁₆	1	4	yes	—
77	#11758196	—	4	4	no	—
78	C₄×Dic₆₁	—	4	2	no	—
79	C₄×Dic₆₁	—	4	2	no	—
80	C₂×#92418	—	2	4	no	—
81	C₂×#92418	—	4	2	no	—
82	C₂×#92418	—	4	2	no	—
83	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
94	C₂×C₄×C₆₁	—	2	1	no	—
95	C₂×C₄×C₆₁	#102	1	2	yes	—
96	C₂×C₄×C₆₁	#102	1	2	yes	—
97	C₂×C₄×C₆₁	#104	1	2	yes	—
98	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
99	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
100	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
103	C₂×C₄×C₆₁	#131	1	2	yes	—
104	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₆₁	#137	1	2	yes	—
107	C₂×Dic₆₁	—	2	4	no	—
108	C₂×Dic₆₁	—	4	2	no	—
109	C₂×Dic₆₁	—	4	2	no	—
110	C₂×Dic₆₁	—	4	2	no	—
111	C₂×Dic₆₁	—	4	2	no	—
112	C₂×Dic₆₁	—	4	2	no	—
113	Dic₁₂₂	—	4	8	no	—
114	C₈×C₆₁	—	4	4	no	—
115	#92418	—	4	4	no	—
116	#9869	—	61	1	no	maximal, Sylow
117	C₂²×C₆₁	#391	1	1	yes	characteristic
118	C₂²×C₆₁	#407	1	1	yes	characteristic
119	C₂²×C₆₁	#408	1	1	yes	characteristic
120	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
121	C₂²×C₆₁	#419	1	1	yes	characteristic
122	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
123	C₂²×C₆₁	#421	1	1	yes	characteristic
124	C₄×C₆₁	—	2	1	no	—
125	C₄×C₆₁	—	2	1	no	—
126	C₄×C₆₁	—	2	2	no	—
127	C₄×C₆₁	—	2	2	no	—
128	C₄×C₆₁	—	2	2	no	—
129	C₄×C₆₁	—	4	1	no	—
130	C₄×C₆₁	#446	1	4	yes	—
131	C₄×C₆₁	#474	1	2	yes	—
132	C₄×C₆₁	#475	1	2	yes	—
133	Dic₆₁	—	4	4	no	—
134	Dic₆₁	—	8	2	no	—
135	C₄×C₄⋊Q₈	—	61	1	no	—
136	#1767	—	61	1	no	—
137	C₄×C₄⋊C₈	D₆₁	1	1	yes	characteristic
138	#1279	—	61	4	no	—
139	C₂×C₆₁	#1486	1	1	yes	characteristic
140	C₂×C₆₁	#1491	1	1	yes	characteristic
141	C₂×C₆₁	#1492	1	1	yes	characteristic
142	C₂×C₆₁	#1497	1	1	yes	characteristic
143	C₂×C₆₁	#1985	1	1	yes	characteristic
144	C₂×C₆₁	#1986	1	1	yes	characteristic
145	C₂×C₆₁	#1988	1	1	yes	characteristic
146	C₂×C₄⋊Q₈	—	61	1	no	—
147	C₂×C₄×Q₈	—	122	1	no	—
148	C₂×C₄⋊C₈	—	2	1	no	—
149	C₂×C₄⋊C₈	—	61	2	no	—
150	C₂×#97	—	122	1	no	—
151	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
152	#391	—	122	1	no	—
153	#389	—	61	2	no	—
154	#389	—	122	1	no	—
155	#384	—	61	1	no	—
156	C₄×C₄⋊C₄	—	61	2	no	—
157	C₄×C₄⋊C₄	—	122	1	no	—
158	C₄³	C₂×D₆₁	1	1	yes	characteristic
159	#336	—	2	1	no	—
160	#336	—	122	1	no	—
161	C₆₁	#9869	1	1	yes	characteristic, complemented, Sylow
162	C₂²×Q₈	—	122	1	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	122	2	no	—
165	C₄⋊Q₈	—	122	4	no	—
166	C₄×Q₈	—	244	2	no	—
167	C₄×Q₈	—	244	2	no	—
168	C₂×C₄⋊C₄	—	61	2	no	—
169	C₂×C₄⋊C₄	—	61	2	no	—
170	C₂×C₄⋊C₄	—	122	1	no	—
171	C₂×C₄⋊C₄	—	122	1	no	—
172	C₂×C₄⋊C₄	—	122	2	no	—
173	C₂×C₄⋊C₄	—	122	1	no	—
174	C₂×C₄²	—	122	2	no	—
175	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
176	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
177	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
178	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
179	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
180	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
181	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
182	C₄⋊C₈	—	2	4	no	—
183	C₄⋊C₈	—	61	4	no	—
184	C₄⋊C₈	—	122	2	no	—
185	#97	—	244	2	no	—
186	C₄×C₈	—	2	4	no	—
187	#95	—	122	2	no	—
188	#95	—	122	2	no	—
189	C₂×Q₈	—	122	4	no	—
190	C₂×Q₈	—	244	2	no	—
191	C₂×Q₈	—	244	2	no	—
192	C₂²×C₄	—	122	2	no	—
193	C₂²×C₄	—	122	2	no	—
194	C₂²×C₄	#11758198	1	1	yes	characteristic
195	C₂²×C₄	#11758198	1	1	yes	characteristic
196	C₂²×C₄	#11758198	1	1	yes	characteristic
197	C₂²×C₄	#11758198	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	122	4	no	—
205	C₂×C₈	—	244	2	no	—
206	C₂×C₈	—	244	2	no	—
207	C₄⋊C₄	—	61	4	no	—
208	C₄⋊C₄	—	122	4	no	—
209	C₄⋊C₄	—	122	2	no	—
210	C₄⋊C₄	—	122	4	no	—
211	C₄⋊C₄	—	244	4	no	—
212	C₄⋊C₄	—	244	2	no	—
213	C₄⋊C₄	—	244	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	4	1	no	—
219	C₄²	—	244	2	no	—
220	C₄²	—	244	2	no	—
221	C₄²	#11758190	1	2	yes	—
222	C₄²	D₄₈₈	1	2	yes	—
223	C₄²	#11758224	1	2	yes	—
224	C₄²	Q₈×D₆₁	1	2	yes	—
225	C₂³	C₂×#11758198	1	1	yes	characteristic
226	Q₈	—	244	8	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	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	122	4	no	—
239	C₂×C₄	—	244	2	no	—
240	C₂×C₄	—	244	2	no	—
241	C₂×C₄	—	244	2	no	—
242	C₂×C₄	—	244	2	no	—
243	C₂×C₄	—	244	2	no	—
244	C₂×C₄	#14519699	1	2	yes	—
245	C₂×C₄	#14519699	1	2	yes	—
246	C₂×C₄	#14519705	1	2	yes	—
247	C₂×C₄	#14519742	1	2	yes	—
248	C₂×C₄	#14519757	1	2	yes	—
249	C₂×C₄	#14519764	1	1	yes	characteristic
250	C₂×C₄	#14519775	1	1	yes	characteristic
251	C₂×C₄	M₁₆×D₆₁	1	2	yes	—
252	C₂×C₄	#14519783	1	2	yes	—
253	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
254	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
255	C₂×C₄	#14519810	1	2	yes	—
256	C₈	—	4	4	no	—
257	C₈	—	244	4	no	—
258	C₂²	#16839667	1	1	yes	characteristic
259	C₂²	#16839725	1	1	yes	characteristic
260	C₂²	#16839728	1	1	yes	characteristic
261	C₂²	C₂×#14519699	1	1	yes	characteristic
262	C₂²	#16839772	1	1	yes	characteristic
263	C₂²	C₂×#14519705	1	1	yes	characteristic
264	C₂²	#16839780	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	4	1	no	—
271	C₄	—	488	2	no	—
272	C₄	—	244	4	no	—
273	C₄	#16839891	1	4	yes	—
274	C₄	#16840043	1	2	yes	—
275	C₄	#16840048	1	2	yes	—
276	C₂	#21444245	1	1	yes	characteristic
277	C₂	#21444267	1	1	yes	characteristic
278	C₂	#21444271	1	1	yes	characteristic
279	C₂	#21444293	1	1	yes	characteristic
280	C₂	#21446758	1	1	yes	characteristic
281	C₂	#21446762	1	1	yes	characteristic
282	C₂	#21446770	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₄₄
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₄×C₄⋊C₈×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₄×C₄⋊C₈	D₆₁
hypocenter	C₆₁	#9869
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
33. C₄⋊#92418	Q₈	semidirect
161. C₆₁	#9869	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
#21445672	2
#21443659	2
C₄×C₄⋊C₈×C₆₁	2
#9869	61