Subgroups — #29570125

Order 15616.

730 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839786	C₂	1	1	yes	maximal, characteristic
2	#21448226	C₂	1	1	yes	maximal, characteristic
3	#21443703	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11758194	—	2	1	no	—
6	C₂×#14519736	—	2	1	no	—
7	#16839786	—	2	2	no	—
8	#16839786	C₄	1	4	yes	complemented
9	C₂×C₄⋊#92418	—	2	1	no	—
10	C₂×C₄⋊#92418	C₂²	1	2	yes	—
11	#16839632	C₂²	1	2	yes	—
12	#16839628	—	2	1	no	—
13	#16839622	—	2	1	no	—
14	#16839622	C₄	1	2	yes	—
15	C₄×C₄⋊Dic₆₁	—	2	1	no	—
16	C₄×C₄⋊Dic₆₁	C₄	1	2	yes	—
17	C₂³×C₈×C₆₁	—	2	1	no	—
18	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
20	C₂×C₄⋊C₈×C₆₁	—	2	2	no	—
21	C₂×C₄⋊C₈×C₆₁	C₄	1	4	yes	complemented
22	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
23	C₂×#11758194	—	2	4	no	—
24	C₂×#11758194	—	4	2	no	—
25	C₂×#11758194	—	4	2	no	—
26	C₂²×#92418	—	2	2	no	—
27	#14519736	—	4	2	no	—
28	#14519736	—	4	2	no	—
29	C₂×C₄⋊Dic₆₁	—	2	2	no	—
30	C₂×C₄⋊Dic₆₁	—	2	2	no	—
31	C₂×C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊Dic₆₁	—	2	2	no	—
33	C₂×C₄×Dic₆₁	—	2	2	no	—
34	C₄⋊#92418	—	2	4	no	—
35	C₄⋊#92418	—	2	2	no	—
36	C₄⋊#92418	C₂×C₄	1	4	yes	—
37	#14519684	—	2	2	no	—
38	#14519684	—	2	2	no	—
39	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₁	—	2	4	no	—
43	C₂²×C₈×C₆₁	—	2	2	no	—
44	C₂²×C₈×C₆₁	—	4	4	no	—
45	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₁	D₄	1	2	yes	—
51	C₂×C₄²×C₆₁	Q₈	1	2	yes	—
52	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₆₁	—	2	4	no	—
54	C₄⋊C₈×C₆₁	—	2	2	no	—
55	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
56	C₂⁴×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	—	2	2	no	—
58	C₂²×C₄×C₆₁	—	2	2	no	—
59	C₂²×C₄×C₆₁	—	4	2	no	—
60	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
61	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₆₁	D₈	1	2	yes	—
70	C₂²×C₄×C₆₁	SD₁₆	1	2	yes	—
71	C₂²×C₄×C₆₁	SD₁₆	1	2	yes	—
72	C₂²×C₄×C₆₁	Q₁₆	1	2	yes	—
73	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×Dic₆₁	—	2	4	no	—
77	C₂×C₈×C₆₁	—	2	4	no	—
78	C₂×C₈×C₆₁	—	4	8	no	—
79	C₂×C₈×C₆₁	—	4	4	no	—
80	C₄²×C₆₁	C₄²	1	2	yes	—
81	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
84	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
85	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
86	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
90	C₂²⋊Dic₆₁	—	4	4	no	—
91	C₄⋊Dic₆₁	—	4	2	no	—
92	C₄⋊Dic₆₁	—	4	2	no	—
93	C₄⋊Dic₆₁	—	4	2	no	—
94	C₄⋊Dic₆₁	—	4	2	no	—
95	C₄⋊Dic₆₁	—	4	4	no	—
96	C₄×Dic₆₁	—	4	4	no	—
97	#11758194	—	4	8	no	—
98	C₂×#92418	—	2	4	no	—
99	C₂×#92418	—	4	2	no	—
100	C₂×#92418	—	4	2	no	—
101	C₂³×C₆₁	—	2	2	no	—
102	C₂³×C₆₁	—	2	2	no	—
103	C₂³×C₆₁	#95	1	2	yes	—
104	C₂³×C₆₁	#102	1	2	yes	—
105	C₂³×C₆₁	#103	1	2	yes	—
106	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₆₁	—	2	2	no	—
108	C₂×C₄×C₆₁	—	2	2	no	—
109	C₂×C₄×C₆₁	—	2	2	no	—
110	C₂×C₄×C₆₁	—	2	2	no	—
111	C₂×C₄×C₆₁	—	2	2	no	—
112	C₂×C₄×C₆₁	—	2	4	no	—
113	C₂×C₄×C₆₁	—	4	2	no	—
114	C₂×C₄×C₆₁	—	4	2	no	—
115	C₂×C₄×C₆₁	#95	1	2	yes	—
116	C₂×C₄×C₆₁	#95	1	2	yes	—
117	C₂×C₄×C₆₁	#95	1	2	yes	—
118	C₂×C₄×C₆₁	#102	1	2	yes	—
119	C₂×C₄×C₆₁	#102	1	4	yes	—
120	C₂×C₄×C₆₁	#103	1	2	yes	—
121	C₂×C₄×C₆₁	#103	1	4	yes	—
122	C₂×C₄×C₆₁	#106	1	4	yes	—
123	C₂×C₄×C₆₁	#107	1	4	yes	—
124	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
134	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
135	C₂×C₄×C₆₁	#137	1	2	yes	—
136	C₂×Dic₆₁	—	4	4	no	—
137	C₂×Dic₆₁	—	4	4	no	—
138	C₂×Dic₆₁	—	4	4	no	—
139	C₈×C₆₁	—	4	4	no	—
140	#92418	—	4	4	no	—
141	#8045	—	61	1	no	maximal, Sylow
142	C₂²×C₆₁	—	2	4	no	—
143	C₂²×C₆₁	—	2	4	no	—
144	C₂²×C₆₁	—	2	2	no	—
145	C₂²×C₆₁	—	2	2	no	—
146	C₂²×C₆₁	#340	1	4	yes	—
147	C₂²×C₆₁	C₂×#95	1	1	yes	characteristic
148	C₂²×C₆₁	#413	1	2	yes	—
149	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
150	C₂²×C₆₁	C₂×#103	1	1	yes	characteristic
151	C₂²×C₆₁	#417	1	2	yes	—
152	C₄×C₆₁	—	2	2	no	—
153	C₄×C₆₁	—	2	2	no	—
154	C₄×C₆₁	—	2	1	no	—
155	C₄×C₆₁	—	2	1	no	—
156	C₄×C₆₁	#340	1	4	yes	—
157	C₄×C₆₁	C₂×#102	1	1	yes	characteristic
158	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
159	C₄×C₆₁	#418	1	1	yes	characteristic
160	C₄×C₆₁	#419	1	1	yes	characteristic
161	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
162	C₄×C₆₁	C₂×#107	1	1	yes	characteristic
163	C₄×C₆₁	#428	1	2	yes	—
164	Dic₆₁	—	8	4	no	—
165	C₂×#423	—	61	1	no	—
166	C₂²×C₄⋊C₈	D₆₁	1	1	yes	characteristic
167	#2285	—	61	1	no	—
168	#1296	—	61	4	no	—
169	C₂×C₆₁	—	2	4	no	—
170	C₂×C₆₁	#1547	1	2	yes	—
171	C₂×C₆₁	#1550	1	2	yes	—
172	C₂×C₆₁	C₂×#340	1	1	yes	characteristic
173	C₂×C₆₁	#1736	1	1	yes	characteristic
174	C₂×C₆₁	#1742	1	1	yes	characteristic
175	C₂²×M₁₆	—	122	1	no	—
176	C₂³×C₈	—	2	1	no	—
177	C₂×#117	—	122	1	no	—
178	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic
179	#423	—	61	4	no	—
180	#423	—	122	2	no	—
181	C₂×C₄⋊C₈	—	2	1	no	—
182	C₂×C₄⋊C₈	—	2	2	no	—
183	C₂×C₄⋊C₈	—	61	2	no	—
184	C₂×C₄⋊C₈	—	122	1	no	—
185	C₂×C₄⋊C₈	Dic₆₁	1	4	yes	complemented
186	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
187	#384	—	61	2	no	—
188	#382	—	122	1	no	—
189	#380	—	61	2	no	—
190	#380	—	122	1	no	—
191	C₄×C₄⋊C₄	—	61	2	no	—
192	C₄×C₄⋊C₄	—	122	1	no	—
193	C₆₁	#8045	1	1	yes	characteristic, complemented, Sylow
194	C₂³×C₄	C₄×D₆₁	1	1	yes	characteristic
195	C₂³×C₄	D₂₄₄	1	1	yes	characteristic
196	C₂³×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
197	C₂×M₁₆	—	122	4	no	—
198	C₂×M₁₆	—	244	2	no	—
199	C₂×M₁₆	—	244	2	no	—
200	C₂²×C₈	—	2	4	no	—
201	C₂²×C₈	—	2	2	no	—
202	C₂²×C₈	—	4	4	no	—
203	C₂²×C₈	—	122	2	no	—
204	#117	—	244	2	no	—
205	#117	—	244	2	no	—
206	C₂×C₄⋊C₄	—	61	4	no	—
207	C₂×C₄⋊C₄	—	122	2	no	—
208	C₂×C₄⋊C₄	—	122	2	no	—
209	C₂×C₂²⋊C₄	—	122	2	no	—
210	C₂×C₄²	—	122	2	no	—
211	C₂×C₄²	Dic₁₂₂	1	2	yes	—
212	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
213	C₂×C₄²	C₄×D₆₁	1	2	yes	—
214	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
215	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
216	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
217	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
218	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
219	C₄⋊C₈	—	2	4	no	—
220	C₄⋊C₈	—	2	2	no	—
221	C₄⋊C₈	—	61	4	no	—
222	C₄⋊C₈	—	122	4	no	—
223	C₄⋊C₈	—	122	2	no	—
224	C₄⋊C₈	C₂×Dic₆₁	1	4	yes	—
225	#95	—	122	2	no	—
226	#95	—	122	2	no	—
227	C₂⁴	#11758198	1	1	yes	characteristic
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	122	4	no	—
232	C₂²×C₄	#11758190	1	2	yes	—
233	C₂²×C₄	#11758190	1	2	yes	—
234	C₂²×C₄	D₄₈₈	1	2	yes	—
235	C₂²×C₄	Dic₂₄₄	1	2	yes	—
236	C₂²×C₄	C₄×Dic₆₁	1	2	yes	—
237	C₂²×C₄	#11758196	1	2	yes	—
238	C₂²×C₄	#11758196	1	2	yes	—
239	C₂²×C₄	C₄⋊Dic₆₁	1	2	yes	—
240	C₂²×C₄	#11758198	1	1	yes	characteristic
241	C₂²×C₄	#11758198	1	1	yes	characteristic
242	C₂²×C₄	#11758198	1	1	yes	characteristic
243	C₂²×C₄	#11758198	1	2	yes	—
244	C₂²×C₄	C₂²⋊Dic₆₁	1	2	yes	—
245	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
248	M₁₆	—	244	8	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	4	8	no	—
251	C₂×C₈	—	4	4	no	—
252	C₂×C₈	—	122	4	no	—
253	C₂×C₈	—	244	2	no	—
254	C₂×C₈	—	244	2	no	—
255	C₄⋊C₄	—	244	4	no	—
256	C₄⋊C₄	—	244	2	no	—
257	C₄⋊C₄	—	244	2	no	—
258	C₄⋊C₄	—	244	2	no	—
259	C₄⋊C₄	—	244	2	no	—
260	C₂²⋊C₄	—	244	4	no	—
261	C₄²	—	244	4	no	—
262	C₄²	C₄×Dic₆₁	1	2	yes	—
263	C₄²	#11758196	1	2	yes	—
264	C₄²	#11758196	1	2	yes	—
265	C₄²	C₄⋊Dic₆₁	1	2	yes	—
266	C₄²	#11758198	1	2	yes	—
267	C₄²	C₂²⋊Dic₆₁	1	2	yes	—
268	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
269	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
270	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
271	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
272	C₂³	—	2	2	no	—
273	C₂³	—	2	2	no	—
274	C₂³	#14519684	1	2	yes	—
275	C₂³	#14519699	1	2	yes	—
276	C₂³	#14519702	1	2	yes	—
277	C₂³	C₂×#11758198	1	1	yes	characteristic
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	244	4	no	—
287	C₂×C₄	—	244	4	no	—
288	C₂×C₄	—	244	4	no	—
289	C₂×C₄	#14519684	1	2	yes	—
290	C₂×C₄	#14519684	1	2	yes	—
291	C₂×C₄	#14519684	1	2	yes	—
292	C₂×C₄	#14519699	1	4	yes	—
293	C₂×C₄	#14519699	1	2	yes	—
294	C₂×C₄	#14519702	1	4	yes	—
295	C₂×C₄	#14519702	1	2	yes	—
296	C₂×C₄	#14519709	1	4	yes	—
297	C₂×C₄	#14519711	1	4	yes	—
298	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
299	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
300	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
301	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
302	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
304	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
305	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
306	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
307	C₂×C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
308	C₂×C₄	#14519803	1	2	yes	—
309	C₂×C₄	#14519808	1	2	yes	—
310	C₈	—	4	4	no	—
311	C₈	—	244	4	no	—
312	C₂²	—	2	4	no	—
313	C₂²	—	2	4	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	#16839525	1	4	yes	—
317	C₂²	C₂×#14519684	1	1	yes	characteristic
318	C₂²	#16839744	1	2	yes	—
319	C₂²	C₂×#14519699	1	1	yes	characteristic
320	C₂²	C₂×#14519702	1	1	yes	characteristic
321	C₂²	#16839761	1	2	yes	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	488	4	no	—
327	C₄	#16839525	1	4	yes	—
328	C₄	C₂×#14519699	1	1	yes	characteristic
329	C₄	C₂×#14519702	1	1	yes	characteristic
330	C₄	#16839767	1	1	yes	characteristic
331	C₄	#16839771	1	1	yes	characteristic
332	C₄	C₂×#14519709	1	1	yes	characteristic
333	C₄	C₂×#14519711	1	1	yes	characteristic
334	C₄	#16839802	1	2	yes	—
335	C₂	—	2	4	no	—
336	C₂	#21444494	1	2	yes	—
337	C₂	#21444507	1	2	yes	—
338	C₂	C₂×#16839525	1	1	yes	characteristic
339	C₂	#21445538	1	1	yes	characteristic
340	C₂	#21445557	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11758198
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂²×C₄⋊C₈×C₆₁	C₂
socle	C₂³×C₆₁	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₆₁	C₂²⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂²×C₄⋊C₈	D₆₁
hypocenter	C₆₁	#8045
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. #21443703	C₂	semidirect
8. #16839786	C₄	semidirect
21. C₂×C₄⋊C₈×C₆₁	C₄	semidirect
185. C₂×C₄⋊C₈	Dic₆₁	semidirect
193. C₆₁	#8045	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839786	2
#21448226	2
#21443703	2
C₂²×C₄⋊C₈×C₆₁	2
#8045	61