Subgroups — #29709983

Order 15616.

664 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454382	C₂	1	1	yes	maximal, characteristic
2	#21447393	C₂	1	8	yes	maximal, complemented
3	C₂×#16839533	C₂	1	2	yes	maximal
4	#21445530	C₂	1	2	yes	maximal
5	C₂×#16839519	C₂	1	1	yes	maximal, characteristic
6	C₂×C₆₁×#430	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11758194	C₂²	1	1	yes	characteristic
8	C₂×#14519736	C₂²	1	2	yes	—
9	C₂×#14519736	C₂²	1	1	yes	characteristic
10	#16839790	—	2	2	no	—
11	#16839788	—	2	2	no	—
12	#16839785	C₂²	1	4	yes	—
13	#16839533	C₂²	1	8	yes	—
14	#16839523	C₂²	1	8	yes	—
15	#16839519	C₂²	1	4	yes	—
16	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
17	C₂²×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#430	—	2	2	no	—
19	C₆₁×#430	C₂²	1	4	yes	—
20	C₂×#108×C₆₁	—	2	2	no	—
21	C₂×#11758194	—	2	1	no	—
22	C₂×#11758194	—	2	2	no	—
23	C₂×#11758194	—	2	2	no	—
24	C₂×#11758194	C₂³	1	2	yes	—
25	C₂²×#92418	—	2	1	no	—
26	#14519736	—	2	1	no	—
27	#14519736	—	2	1	no	—
28	#14519736	—	2	2	no	—
29	#14519736	—	2	2	no	—
30	#14519736	—	2	2	no	—
31	#14519736	Q₈	1	2	yes	—
32	#14519736	Q₈	1	2	yes	—
33	#14519736	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₆₁	—	2	1	no	—
35	C₂×C₄⋊Dic₆₁	—	2	2	no	—
36	C₂×C₂²⋊Dic₆₁	—	2	1	no	—
37	C₂×C₂²⋊Dic₆₁	—	2	2	no	—
38	C₂×C₄×Dic₆₁	—	2	2	no	—
39	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
40	#14519706	—	2	8	no	—
41	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₆₁	—	2	2	no	—
43	C₂×M₁₆×C₆₁	—	2	2	no	—
44	C₂×M₁₆×C₆₁	—	2	2	no	—
45	C₂×M₁₆×C₆₁	—	2	2	no	—
46	C₂×M₁₆×C₆₁	—	2	2	no	—
47	C₂×M₁₆×C₆₁	—	2	2	no	—
48	C₂×M₁₆×C₆₁	—	2	2	no	—
49	C₂×M₁₆×C₆₁	D₄	1	2	yes	—
50	C₂×M₁₆×C₆₁	C₂³	1	4	yes	—
51	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
52	C₂²×C₈×C₆₁	—	2	2	no	—
53	C₂²×C₈×C₆₁	D₄	1	2	yes	—
54	#108×C₆₁	—	2	8	no	—
55	C₂⁴×C₆₁	#41	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	#41	1	2	yes	—
62	C₂²×C₄×C₆₁	#41	1	2	yes	—
63	C₂²×C₄×C₆₁	#41	1	2	yes	—
64	C₂²×C₄×C₆₁	#41	1	2	yes	—
65	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
66	C₂²×Dic₆₁	—	2	2	no	—
67	C₂²×Dic₆₁	—	4	2	no	—
68	M₁₆×C₆₁	—	2	4	no	—
69	M₁₆×C₆₁	—	2	4	no	—
70	M₁₆×C₆₁	—	2	4	no	—
71	M₁₆×C₆₁	—	2	4	no	—
72	M₁₆×C₆₁	—	4	2	no	—
73	M₁₆×C₆₁	—	4	2	no	—
74	C₂×C₈×C₆₁	—	2	2	no	—
75	C₂×C₈×C₆₁	—	2	2	no	—
76	C₂×C₈×C₆₁	—	2	4	no	—
77	C₂×C₈×C₆₁	—	4	2	no	—
78	C₂×C₈×C₆₁	—	4	2	no	—
79	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
80	C₂²⋊Dic₆₁	—	2	2	no	—
81	C₂²⋊Dic₆₁	—	2	2	no	—
82	C₂²⋊Dic₆₁	—	2	4	no	—
83	C₂²⋊Dic₆₁	—	4	2	no	—
84	C₄⋊Dic₆₁	—	2	2	no	—
85	C₄⋊Dic₆₁	—	2	2	no	—
86	C₄⋊Dic₆₁	—	2	4	no	—
87	C₄⋊Dic₆₁	—	4	2	no	—
88	C₄×Dic₆₁	—	2	2	no	—
89	C₄×Dic₆₁	—	2	4	no	—
90	C₄×Dic₆₁	—	4	2	no	—
91	C₄×Dic₆₁	C₂×Q₈	1	4	yes	—
92	#11758194	—	2	4	no	—
93	#11758194	—	4	1	no	—
94	#11758194	—	4	1	no	—
95	C₂×#92418	—	2	2	no	—
96	C₂×#92418	—	4	2	no	—
97	C₂³×C₆₁	—	8	1	no	—
98	C₂³×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂³×C₆₁	#126	1	2	yes	—
101	C₂³×C₆₁	#126	1	2	yes	—
102	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₆₁	#123	1	2	yes	—
106	C₂×C₄×C₆₁	#123	1	1	yes	characteristic
107	C₂×C₄×C₆₁	#123	1	1	yes	characteristic
108	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
109	C₂×C₄×C₆₁	#124	1	2	yes	—
110	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
111	C₂×C₄×C₆₁	#125	1	2	yes	—
112	C₂×C₄×C₆₁	#125	1	2	yes	—
113	C₂×C₄×C₆₁	#125	1	2	yes	—
114	C₂×C₄×C₆₁	#125	1	2	yes	—
115	C₂×C₄×C₆₁	#126	1	2	yes	—
116	C₂×C₄×C₆₁	#126	1	2	yes	—
117	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
120	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
121	C₂×Dic₆₁	—	2	4	no	—
122	C₂×Dic₆₁	—	4	4	no	—
123	C₂×Dic₆₁	—	8	2	no	—
124	C₂×Dic₆₁	—	8	4	no	—
125	C₈×C₆₁	—	2	4	no	—
126	C₈×C₆₁	—	4	4	no	—
127	#92418	—	4	2	no	—
128	#23751	—	61	1	no	maximal, Sylow
129	C₂²×C₆₁	—	4	2	no	—
130	C₂²×C₆₁	—	4	1	no	—
131	C₂²×C₆₁	—	4	1	no	—
132	C₂²×C₆₁	—	4	1	no	—
133	C₂²×C₆₁	—	4	2	no	—
134	C₂²×C₆₁	#400	1	2	yes	—
135	C₂²×C₆₁	#400	1	1	yes	characteristic
136	C₂²×C₆₁	#400	1	1	yes	characteristic
137	C₂²×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
138	C₂²×C₆₁	C₂×#126	1	2	yes	—
139	C₄×C₆₁	#400	1	2	yes	—
140	C₄×C₆₁	#400	1	1	yes	characteristic
141	C₄×C₆₁	#400	1	1	yes	characteristic
142	C₄×C₆₁	C₂×#123	1	1	yes	characteristic
143	C₄×C₆₁	C₂×#124	1	1	yes	characteristic
144	C₄×C₆₁	C₂×#125	1	2	yes	—
145	Dic₆₁	—	4	4	no	—
146	Dic₆₁	—	8	4	no	—
147	C₂×#430	D₆₁	1	1	yes	characteristic
148	#2905	—	61	1	no	—
149	#2102	—	61	8	no	—
150	C₂×#344	—	61	2	no	—
151	#1733	—	61	2	no	—
152	C₂×H(3,ℤ/4)	—	61	1	no	—
153	C₂×C₆₁	—	2	1	no	—
154	C₂×C₆₁	—	2	2	no	—
155	C₂×C₆₁	—	2	1	no	—
156	C₂×C₆₁	—	2	2	no	—
157	C₂×C₆₁	—	2	1	no	—
158	C₂×C₆₁	C₂×#400	1	1	yes	characteristic
159	C₂²×M₁₆	—	61	1	no	—
160	C₂²×M₁₆	C₂×D₆₁	1	2	yes	—
161	C₂²×M₁₆	C₂×D₆₁	1	1	yes	characteristic
162	C₂×#117	—	61	1	no	—
163	C₂×#117	—	61	2	no	—
164	#430	—	2	2	no	—
165	#430	C₂×D₆₁	1	4	yes	—
166	C₂×#108	—	2	2	no	—
167	#424	—	122	2	no	—
168	#424	—	122	2	no	—
169	#423	—	61	4	no	—
170	#344	—	61	8	no	—
171	#339	—	61	8	no	—
172	H(3,ℤ/4)	—	61	4	no	—
173	C₆₁	#23751	1	1	yes	characteristic, complemented, Sylow
174	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	61	2	no	—
183	C₂×M₁₆	—	122	1	no	—
184	C₂×M₁₆	—	122	2	no	—
185	C₂×M₁₆	—	122	2	no	—
186	C₂×M₁₆	C₂²⋊D₆₁	1	2	yes	—
187	C₂×M₁₆	C₂²×D₆₁	1	4	yes	—
188	C₂×M₁₆	C₂²×D₆₁	1	2	yes	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	122	1	no	—
191	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
192	#117	—	61	2	no	—
193	#117	—	61	2	no	—
194	#117	—	61	4	no	—
195	#117	—	122	1	no	—
196	#117	—	122	1	no	—
197	#117	—	122	2	no	—
198	#117	—	122	2	no	—
199	#117	—	122	2	no	—
200	C₂×C₄⋊C₄	—	122	1	no	—
201	C₂×C₄⋊C₄	—	122	2	no	—
202	C₂×C₂²⋊C₄	—	122	1	no	—
203	C₂×C₂²⋊C₄	—	122	2	no	—
204	C₂×C₄²	—	61	2	no	—
205	C₂×C₄²	—	122	2	no	—
206	#108	—	2	8	no	—
207	C₄⋊C₈	—	122	8	no	—
208	C₂⁴	#11758226	1	1	yes	characteristic
209	C₂²×C₄	—	122	2	no	—
210	C₂²×C₄	—	244	2	no	—
211	C₂²×C₄	#11758222	1	2	yes	—
212	C₂²×C₄	#11758222	1	2	yes	—
213	C₂²×C₄	#11758224	1	2	yes	—
214	C₂²×C₄	#11758224	1	2	yes	—
215	C₂²×C₄	Q₈×D₆₁	1	1	yes	characteristic
216	C₂²×C₄	Q₈×D₆₁	1	1	yes	characteristic
217	C₂²×C₄	#11758226	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
220	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	122	4	no	—
228	M₁₆	—	244	1	no	—
229	M₁₆	—	244	1	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	122	2	no	—
236	C₂×C₈	—	244	2	no	—
237	C₂×C₈	C₂×C₂²⋊D₆₁	1	4	yes	—
238	C₄⋊C₄	—	122	2	no	—
239	C₄⋊C₄	—	122	2	no	—
240	C₄⋊C₄	—	122	4	no	—
241	C₄⋊C₄	—	244	2	no	—
242	C₂²⋊C₄	—	122	2	no	—
243	C₂²⋊C₄	—	122	2	no	—
244	C₂²⋊C₄	—	122	4	no	—
245	C₂²⋊C₄	—	244	2	no	—
246	C₄²	—	61	4	no	—
247	C₄²	—	122	2	no	—
248	C₄²	—	122	4	no	—
249	C₄²	—	244	2	no	—
250	C₂³	—	8	1	no	—
251	C₂³	#14519753	1	1	yes	characteristic
252	C₂³	#14519753	1	1	yes	characteristic
253	C₂³	#14519772	1	2	yes	—
254	C₂³	#14519772	1	2	yes	—
255	C₂³	C₂×#11758226	1	1	yes	characteristic
256	C₂×C₄	—	122	4	no	—
257	C₂×C₄	—	488	4	no	—
258	C₂×C₄	—	488	2	no	—
259	C₂×C₄	—	244	4	no	—
260	C₂×C₄	#14519753	1	1	yes	characteristic
261	C₂×C₄	#14519753	1	1	yes	characteristic
262	C₂×C₄	#14519760	1	1	yes	characteristic
263	C₂×C₄	#14519760	1	1	yes	characteristic
264	C₂×C₄	#14519760	1	2	yes	—
265	C₂×C₄	#14519764	1	1	yes	characteristic
266	C₂×C₄	#14519764	1	2	yes	—
267	C₂×C₄	#14519764	1	1	yes	characteristic
268	C₂×C₄	#14519767	1	2	yes	—
269	C₂×C₄	#14519767	1	2	yes	—
270	C₂×C₄	#14519767	1	2	yes	—
271	C₂×C₄	#14519767	1	2	yes	—
272	C₂×C₄	#14519772	1	2	yes	—
273	C₂×C₄	#14519772	1	2	yes	—
274	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
275	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
276	C₂×C₄	C₂×#11758224	1	2	yes	—
277	C₂×C₄	C₂×#11758222	1	2	yes	—
278	C₈	—	2	4	no	—
279	C₈	—	4	4	no	—
280	C₈	—	244	2	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	2	no	—
286	C₂²	#16839700	1	1	yes	characteristic
287	C₂²	#16839700	1	1	yes	characteristic
288	C₂²	#16839700	1	2	yes	—
289	C₂²	C₂×#14519753	1	1	yes	characteristic
290	C₂²	C₂×#14519772	1	2	yes	—
291	C₄	—	488	4	no	—
292	C₄	—	244	4	no	—
293	C₄	#16839700	1	1	yes	characteristic
294	C₄	#16839700	1	1	yes	characteristic
295	C₄	#16839700	1	2	yes	—
296	C₄	C₂×#14519760	1	1	yes	characteristic
297	C₄	C₂×#14519764	1	1	yes	characteristic
298	C₄	C₂×#14519767	1	2	yes	—
299	C₂	—	2	2	no	—
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	—	2	2	no	—
303	C₂	—	2	1	no	—
304	C₂	C₂×#16839700	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14519764
commutator, cosocle	C₂²×C₄×C₆₁	C₂⁴
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₂×C₆₁×#430	C₂
socle	C₂×C₆₁	C₂×#400
prime radical	C₂⁴×C₆₁	#41
hypercenter	C₂×#430	D₆₁
hypocenter	C₆₁	#23751
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#430

Complements

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

N	H ≅ G/N	type
2. #21447393	C₂	semidirect
173. C₆₁	#23751	semidirect

Maximal subgroups

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

subgroup	index
#21454382	2
#21447393	2
C₂×#16839533	2
#21445530	2
C₂×#16839519	2
C₂×C₆₁×#430	2
#23751	61