Subgroups — #29569570

Order 15616.

798 conjugacy classes of subgroups.

The subgroup lattice has 332 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	C₂²×C₄×#92418	C₂	1	1	yes	maximal, characteristic
3	#21443660	C₂	1	4	yes	maximal, complemented
4	C₆₁×#2285	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11758194	—	2	1	no	—
6	C₂³×#92418	—	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	C₂×C₄×#92418	—	2	2	no	—
12	C₂×C₄×#92418	—	2	1	no	—
13	C₂×C₄×#92418	C₄	1	4	yes	complemented
14	C₂×C₄×#92418	C₂²	1	2	yes	—
15	C₂×#117×C₆₁	—	2	1	no	—
16	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#384	C₄	1	2	yes	—
18	C₆₁×#382	—	2	1	no	—
19	C₆₁×#380	—	2	1	no	—
20	C₆₁×#380	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₆₁	—	2	1	no	—
22	C₄×C₄⋊C₄×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	8	no	—
27	C₂²×#92418	—	2	4	no	—
28	C₂²×#92418	—	2	2	no	—
29	C₂²×#92418	—	2	2	no	—
30	C₄⋊#92418	—	2	4	no	—
31	C₄⋊#92418	—	2	2	no	—
32	C₄⋊#92418	C₂×C₄	1	4	yes	—
33	C₄×#92418	—	2	4	no	—
34	C₄×#92418	—	2	2	no	—
35	C₄×#92418	C₂×C₄	1	4	yes	—
36	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
39	#117×C₆₁	—	4	2	no	—
40	#117×C₆₁	—	4	2	no	—
41	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
44	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
45	C₂×C₄²×C₆₁	—	2	2	no	—
46	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₁	D₄	1	2	yes	—
50	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₆₁	Q₈	1	2	yes	—
53	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
54	#95×C₆₁	—	2	2	no	—
55	#95×C₆₁	—	2	2	no	—
56	C₂⁴×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	—	2	4	no	—
58	C₂²×C₄×C₆₁	—	2	4	no	—
59	C₂²×C₄×C₆₁	—	2	4	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	2	yes	—
63	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
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	C₄⋊C₄×C₆₁	—	2	8	no	—
77	C₄⋊C₄×C₆₁	—	4	4	no	—
78	C₄⋊C₄×C₆₁	—	4	4	no	—
79	C₂²⋊C₄×C₆₁	—	4	4	no	—
80	C₄²×C₆₁	—	4	2	no	—
81	C₄²×C₆₁	—	4	2	no	—
82	C₄²×C₆₁	C₄²	1	2	yes	—
83	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
86	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
87	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
88	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
92	#11758194	—	4	8	no	—
93	C₂×#92418	—	2	8	no	—
94	C₂×#92418	—	2	16	no	—
95	C₂×#92418	—	2	4	no	—
96	C₂×#92418	—	2	4	no	—
97	C₂×#92418	—	4	2	no	—
98	C₂×#92418	—	4	2	no	—
99	C₂³×C₆₁	—	2	2	no	—
100	C₂³×C₆₁	—	2	2	no	—
101	C₂³×C₆₁	#95	1	2	yes	—
102	C₂³×C₆₁	#102	1	2	yes	—
103	C₂³×C₆₁	#103	1	2	yes	—
104	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₁	—	2	2	no	—
106	C₂×C₄×C₆₁	—	2	2	no	—
107	C₂×C₄×C₆₁	—	2	4	no	—
108	C₂×C₄×C₆₁	—	2	4	no	—
109	C₂×C₄×C₆₁	—	2	4	no	—
110	C₂×C₄×C₆₁	—	2	4	no	—
111	C₂×C₄×C₆₁	—	4	4	no	—
112	C₂×C₄×C₆₁	—	4	4	no	—
113	C₂×C₄×C₆₁	—	4	4	no	—
114	C₂×C₄×C₆₁	#95	1	2	yes	—
115	C₂×C₄×C₆₁	#95	1	2	yes	—
116	C₂×C₄×C₆₁	#95	1	2	yes	—
117	C₂×C₄×C₆₁	#102	1	2	yes	—
118	C₂×C₄×C₆₁	#102	1	4	yes	—
119	C₂×C₄×C₆₁	#103	1	2	yes	—
120	C₂×C₄×C₆₁	#103	1	4	yes	—
121	C₂×C₄×C₆₁	#104	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	#131	1	4	yes	—
131	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
135	#92418	—	2	8	no	—
136	#92418	—	4	4	no	—
137	#7897	—	61	1	no	maximal, Sylow
138	C₂²×C₆₁	—	2	2	no	—
139	C₂²×C₆₁	—	2	2	no	—
140	C₂²×C₆₁	—	2	4	no	—
141	C₂²×C₆₁	—	2	4	no	—
142	C₂²×C₆₁	#340	1	4	yes	—
143	C₂²×C₆₁	C₂×#95	1	1	yes	characteristic
144	C₂²×C₆₁	#408	1	2	yes	—
145	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
146	C₂²×C₆₁	C₂×#103	1	1	yes	characteristic
147	C₂²×C₆₁	C₂×#104	1	2	yes	—
148	C₄×C₆₁	—	2	2	no	—
149	C₄×C₆₁	—	2	1	no	—
150	C₄×C₆₁	—	2	1	no	—
151	C₄×C₆₁	—	8	4	no	—
152	C₄×C₆₁	#339	1	4	yes	—
153	C₄×C₆₁	#340	1	4	yes	—
154	C₄×C₆₁	#405	1	2	yes	—
155	C₄×C₆₁	C₂×#102	1	1	yes	characteristic
156	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
157	C₄×C₆₁	#424	1	2	yes	—
158	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
159	C₄×C₆₁	C₂×#107	1	1	yes	characteristic
160	C₂×#423	—	61	1	no	—
161	C₂²×C₄×C₈	—	61	1	no	—
162	#2285	D₆₁	1	1	yes	characteristic
163	#1279	—	61	4	no	—
164	C₂×C₆₁	—	2	4	no	—
165	C₂×C₆₁	#1492	1	2	yes	—
166	C₂×C₆₁	#1493	1	2	yes	—
167	C₂×C₆₁	#1729	1	1	yes	characteristic
168	C₂×C₆₁	C₂×#339	1	1	yes	characteristic
169	C₂×C₆₁	C₂×#340	1	1	yes	characteristic
170	C₂²×M₁₆	—	122	1	no	—
171	C₂³×C₈	—	122	1	no	—
172	C₂×#117	—	2	1	no	—
173	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic
174	#423	—	61	4	no	—
175	#423	—	122	2	no	—
176	C₂×C₄⋊C₈	—	61	2	no	—
177	C₂×C₄⋊C₈	—	122	1	no	—
178	C₂×C₄×C₈	—	61	4	no	—
179	C₂×C₄×C₈	—	61	2	no	—
180	C₂×C₄×C₈	—	122	2	no	—
181	C₂×C₄×C₈	—	122	1	no	—
182	#384	Dic₆₁	1	2	yes	—
183	#382	—	2	1	no	—
184	#380	—	2	1	no	—
185	#380	Dic₆₁	1	2	yes	—
186	C₄×C₄⋊C₄	—	2	1	no	—
187	C₄×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
188	C₆₁	#7897	1	1	yes	characteristic, complemented, Sylow
189	C₂³×C₄	C₂×Dic₆₁	1	1	yes	characteristic
190	C₂³×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
191	C₂³×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
192	C₂×M₁₆	—	122	4	no	—
193	C₂×M₁₆	—	244	2	no	—
194	C₂×M₁₆	—	244	2	no	—
195	C₂²×C₈	—	122	8	no	—
196	C₂²×C₈	—	122	4	no	—
197	C₂²×C₈	—	122	2	no	—
198	C₂²×C₈	—	122	2	no	—
199	#117	—	4	2	no	—
200	#117	—	4	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	C₂×Dic₆₁	1	4	yes	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	Dic₁₂₂	1	2	yes	—
207	C₂×C₄²	C₄×D₆₁	1	2	yes	—
208	C₂×C₄²	C₄×D₆₁	1	2	yes	—
209	C₂×C₄²	D₂₄₄	1	2	yes	—
210	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
212	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
213	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
214	C₄⋊C₈	—	61	4	no	—
215	C₄⋊C₈	—	122	4	no	—
216	C₄⋊C₈	—	122	2	no	—
217	C₄×C₈	—	61	4	no	—
218	C₄×C₈	—	122	4	no	—
219	C₄×C₈	—	122	2	no	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	C₂⁴	C₂²⋊Dic₆₁	1	1	yes	characteristic
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	C₄×Dic₆₁	1	2	yes	—
227	C₂²×C₄	#11758196	1	2	yes	—
228	C₂²×C₄	#11758196	1	2	yes	—
229	C₂²×C₄	C₄⋊Dic₆₁	1	2	yes	—
230	C₂²×C₄	#11758198	1	2	yes	—
231	C₂²×C₄	#11758198	1	2	yes	—
232	C₂²×C₄	#11758199	1	2	yes	—
233	C₂²×C₄	#11758200	1	2	yes	—
234	C₂²×C₄	#11758201	1	2	yes	—
235	C₂²×C₄	#11758202	1	2	yes	—
236	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
237	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
238	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
239	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
242	M₁₆	—	244	8	no	—
243	C₂×C₈	—	122	8	no	—
244	C₂×C₈	—	122	16	no	—
245	C₂×C₈	—	122	4	no	—
246	C₂×C₈	—	122	4	no	—
247	C₂×C₈	—	244	2	no	—
248	C₂×C₈	—	244	2	no	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₂²⋊C₄	—	4	4	no	—
253	C₄²	—	4	2	no	—
254	C₄²	—	4	2	no	—
255	C₄²	C₄×Dic₆₁	1	2	yes	—
256	C₄²	#11758196	1	2	yes	—
257	C₄²	#11758196	1	2	yes	—
258	C₄²	C₄⋊Dic₆₁	1	2	yes	—
259	C₄²	#11758198	1	2	yes	—
260	C₄²	#11758198	1	2	yes	—
261	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
262	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
263	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
264	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	#14519684	1	2	yes	—
268	C₂³	#14519698	1	2	yes	—
269	C₂³	#14519701	1	2	yes	—
270	C₂³	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	#14519684	1	2	yes	—
281	C₂×C₄	#14519684	1	2	yes	—
282	C₂×C₄	#14519684	1	2	yes	—
283	C₂×C₄	#14519697	1	4	yes	—
284	C₂×C₄	#14519698	1	2	yes	—
285	C₂×C₄	#14519700	1	4	yes	—
286	C₂×C₄	#14519701	1	2	yes	—
287	C₂×C₄	#14519704	1	4	yes	—
288	C₂×C₄	#14519708	1	4	yes	—
289	C₂×C₄	#14519710	1	4	yes	—
290	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
291	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
292	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
293	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
294	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
295	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
296	C₂×C₄	#14519784	1	4	yes	—
297	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
298	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
299	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
300	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
301	C₈	—	122	8	no	—
302	C₈	—	244	4	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	4	no	—
307	C₂²	#16839524	1	4	yes	—
308	C₂²	C₂×#14519684	1	1	yes	characteristic
309	C₂²	#16839729	1	2	yes	—
310	C₂²	C₂×#14519698	1	1	yes	characteristic
311	C₂²	C₂×#14519701	1	1	yes	characteristic
312	C₂²	C₂×#14519704	1	2	yes	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	8	4	no	—
317	C₄	#16839521	1	4	yes	—
318	C₄	#16839524	1	4	yes	—
319	C₄	#16839718	1	2	yes	—
320	C₄	C₂×#14519697	1	1	yes	characteristic
321	C₄	C₂×#14519700	1	1	yes	characteristic
322	C₄	#16839789	1	2	yes	—
323	C₄	C₂×#14519708	1	1	yes	characteristic
324	C₄	C₂×#14519710	1	1	yes	characteristic
325	C₂	—	2	4	no	—
326	C₂	#21444270	1	2	yes	—
327	C₂	#21444274	1	2	yes	—
328	C₂	#21445516	1	1	yes	characteristic
329	C₂	C₂×#16839521	1	1	yes	characteristic
330	C₂	C₂×#16839524	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#2285	C₂
socle	C₂³×C₆₁	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₆₁	C₂²⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#2285	D₆₁
hypocenter	C₆₁	#7897
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
3. #21443660	C₂	semidirect
8. #16839786	C₄	semidirect
13. C₂×C₄×#92418	C₄	semidirect
188. C₆₁	#7897	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
C₂²×C₄×#92418	2
#21443660	2
C₆₁×#2285	2
#7897	61