Subgroups — #29569475

Order 15616.

722 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839815	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#92418	C₂	1	1	yes	maximal, characteristic
3	#21443654	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×#92418	—	2	1	no	—
6	#16839815	—	2	2	no	—
7	#16839815	C₄	1	4	yes	—
8	C₂×C₄⋊#92418	—	2	1	no	—
9	C₂×C₄⋊#92418	—	2	2	no	—
10	C₂×C₄⋊#92418	C₄	1	4	yes	—
11	C₂×C₄⋊#92418	C₂²	1	2	yes	—
12	C₂×C₄⋊#92418	C₂²	1	2	yes	—
13	C₂×C₂²⋊#92418	—	2	1	no	—
14	C₂×C₄×#92418	—	2	1	no	—
15	C₂²×M₁₆×C₆₁	C₄	1	2	yes	—
16	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
17	C₄×M₁₆×C₆₁	—	2	2	no	—
18	C₄×M₁₆×C₆₁	—	2	2	no	—
19	C₂×#97×C₆₁	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
21	C₂²×#92418	—	2	4	no	—
22	C₂²×#92418	—	2	2	no	—
23	C₂²×#92418	—	2	2	no	—
24	C₂²×#92418	—	4	4	no	—
25	C₄⋊#92418	—	2	4	no	—
26	C₄⋊#92418	—	2	2	no	—
27	C₄⋊#92418	—	2	2	no	—
28	C₄⋊#92418	C₂×C₄	1	4	yes	—
29	C₄⋊#92418	C₂×C₄	1	4	yes	—
30	C₂²⋊#92418	—	2	4	no	—
31	C₄×#92418	—	2	4	no	—
32	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₆₁	Q₈	1	1	yes	characteristic
35	C₂×M₁₆×C₆₁	—	2	4	no	—
36	C₂×M₁₆×C₆₁	—	2	4	no	—
37	C₂×M₁₆×C₆₁	C₈	1	8	yes	complemented
38	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₁	D₄	1	2	yes	—
44	C₂×C₄²×C₆₁	D₄	1	2	yes	—
45	C₂×C₄²×C₆₁	Q₈	1	1	yes	characteristic
46	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
47	#97×C₆₁	—	2	4	no	—
48	C₄×C₈×C₆₁	—	2	2	no	—
49	C₄×C₈×C₆₁	—	2	2	no	—
50	C₂⁴×C₆₁	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	—	2	2	no	—
52	C₂²×C₄×C₆₁	—	2	2	no	—
53	C₂²×C₄×C₆₁	—	4	2	no	—
54	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
55	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
64	C₂²×C₄×C₆₁	M₁₆	1	4	yes	—
65	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
68	M₁₆×C₆₁	—	2	16	no	—
69	C₂×C₈×C₆₁	—	2	4	no	—
70	C₂×C₈×C₆₁	—	2	4	no	—
71	C₂×C₈×C₆₁	C₂×C₈	1	8	yes	complemented
72	C₄²×C₆₁	C₄²	1	2	yes	—
73	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
77	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
78	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
82	C₂×#92418	—	2	4	no	—
83	C₂×#92418	—	2	4	no	—
84	C₂×#92418	—	4	8	no	—
85	C₂×#92418	—	4	2	no	—
86	C₂×#92418	—	4	4	no	—
87	C₂×#92418	—	4	2	no	—
88	C₂³×C₆₁	—	2	2	no	—
89	C₂³×C₆₁	—	2	2	no	—
90	C₂³×C₆₁	#95	1	2	yes	—
91	C₂³×C₆₁	C₄⋊C₈	1	4	yes	—
92	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₆₁	—	2	2	no	—
94	C₂×C₄×C₆₁	—	2	2	no	—
95	C₂×C₄×C₆₁	—	2	2	no	—
96	C₂×C₄×C₆₁	—	2	4	no	—
97	C₂×C₄×C₆₁	—	2	2	no	—
98	C₂×C₄×C₆₁	—	2	2	no	—
99	C₂×C₄×C₆₁	—	4	2	no	—
100	C₂×C₄×C₆₁	—	4	2	no	—
101	C₂×C₄×C₆₁	#95	1	2	yes	—
102	C₂×C₄×C₆₁	#95	1	2	yes	—
103	C₂×C₄×C₆₁	#95	1	2	yes	—
104	C₂×C₄×C₆₁	C₄×C₈	1	4	yes	—
105	C₂×C₄×C₆₁	#97	1	4	yes	—
106	C₂×C₄×C₆₁	C₂²⋊C₈	1	8	yes	—
107	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
108	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
115	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
116	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×C₄×C₆₁	#137	1	2	yes	—
118	C₈×C₆₁	—	2	8	no	—
119	#92418	—	4	4	no	—
120	#92418	—	4	4	no	—
121	#7873	—	61	1	no	maximal, Sylow
122	C₂²×C₆₁	—	2	8	no	—
123	C₂²×C₆₁	—	2	2	no	—
124	C₂²×C₆₁	—	2	2	no	—
125	C₂²×C₆₁	#336	1	4	yes	—
126	C₂²×C₆₁	C₂×#95	1	1	yes	characteristic
127	C₂²×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
128	C₂²×C₆₁	#428	1	2	yes	—
129	C₂²×C₆₁	#430	1	2	yes	—
130	C₄×C₆₁	—	2	2	no	—
131	C₄×C₆₁	—	2	2	no	—
132	C₄×C₆₁	—	2	1	no	—
133	C₄×C₆₁	—	2	1	no	—
134	C₄×C₆₁	#336	1	4	yes	—
135	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
136	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
137	C₄×C₆₁	C₂×C₂²⋊C₈	1	2	yes	—
138	C₄×C₆₁	#417	1	2	yes	—
139	C₄×C₆₁	#418	1	1	yes	characteristic
140	C₄×C₆₁	#419	1	1	yes	characteristic
141	C₂×#431	—	61	1	no	—
142	C₂²×C₄⋊C₈	—	61	1	no	—
143	C₂×C₄×M₁₆	D₆₁	1	1	yes	characteristic
144	#1277	—	61	4	no	—
145	C₂×C₆₁	—	2	4	no	—
146	C₂×C₆₁	#1565	1	4	yes	—
147	C₂×C₆₁	C₂×#336	1	1	yes	characteristic
148	C₂×C₆₁	#1736	1	1	yes	characteristic
149	C₂×C₆₁	#1738	1	1	yes	characteristic
150	C₂²×M₁₆	Dic₆₁	1	2	yes	—
151	C₂³×C₈	—	122	1	no	—
152	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic
153	#431	—	61	4	no	—
154	#431	—	122	2	no	—
155	C₂×C₄⋊C₈	—	61	4	no	—
156	C₂×C₄⋊C₈	—	61	2	no	—
157	C₂×C₄⋊C₈	—	61	2	no	—
158	C₂×C₄⋊C₈	—	122	1	no	—
159	C₂×C₄⋊C₈	—	122	2	no	—
160	C₂×C₂²⋊C₈	—	122	1	no	—
161	C₄×M₁₆	—	2	2	no	—
162	C₄×M₁₆	—	2	2	no	—
163	C₂×#97	Dic₆₁	1	2	yes	—
164	C₂×C₄×C₈	—	122	1	no	—
165	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
166	C₆₁	#7873	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₄	Dic₁₂₂	1	1	yes	characteristic
168	C₂³×C₄	D₂₄₄	1	1	yes	characteristic
169	C₂³×C₄	C₂×Dic₆₁	1	1	yes	characteristic
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	—	2	4	no	—
172	C₂×M₁₆	#92418	1	8	yes	complemented
173	C₂²×C₈	—	122	4	no	—
174	C₂²×C₈	—	122	2	no	—
175	C₂²×C₈	—	122	2	no	—
176	C₂²×C₈	—	244	4	no	—
177	C₂²×C₈	C₂×Dic₆₁	1	4	yes	—
178	C₂×C₄²	Dic₁₂₂	1	1	yes	characteristic
179	C₂×C₄²	C₄×D₆₁	1	2	yes	—
180	C₂×C₄²	C₄×D₆₁	1	2	yes	—
181	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
182	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
183	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
184	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
185	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
186	C₄⋊C₈	—	61	4	no	—
187	C₄⋊C₈	—	61	4	no	—
188	C₄⋊C₈	—	122	4	no	—
189	C₄⋊C₈	—	122	2	no	—
190	C₄⋊C₈	—	122	2	no	—
191	C₂²⋊C₈	—	122	4	no	—
192	#97	—	2	4	no	—
193	C₄×C₈	—	2	2	no	—
194	C₄×C₈	—	2	2	no	—
195	C₄×C₈	—	122	4	no	—
196	C₂⁴	C₄⋊Dic₆₁	1	1	yes	characteristic
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	C₂×#92418	1	4	yes	—
201	C₂²×C₄	#11758194	1	4	yes	—
202	C₂²×C₄	C₄×Dic₆₁	1	2	yes	—
203	C₂²×C₄	#11758196	1	2	yes	—
204	C₂²×C₄	#11758196	1	2	yes	—
205	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
206	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
207	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
208	C₂²×C₄	#11758198	1	2	yes	—
209	C₂²×C₄	#11758198	1	2	yes	—
210	C₂²×C₄	C₂²⋊Dic₆₁	1	2	yes	—
211	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
213	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
214	M₁₆	—	2	16	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	122	4	no	—
218	C₂×C₈	—	122	4	no	—
219	C₂×C₈	—	244	8	no	—
220	C₂×C₈	—	244	2	no	—
221	C₂×C₈	—	244	2	no	—
222	C₂×C₈	—	244	4	no	—
223	C₂×C₈	C₂×#92418	1	8	yes	complemented
224	C₄²	C₄×Dic₆₁	1	2	yes	—
225	C₄²	#11758196	1	2	yes	—
226	C₄²	#11758196	1	2	yes	—
227	C₄²	#11758198	1	2	yes	—
228	C₄²	#11758198	1	2	yes	—
229	C₄²	C₂²⋊Dic₆₁	1	2	yes	—
230	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
231	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
232	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
233	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	#14519684	1	2	yes	—
237	C₂³	C₄⋊#92418	1	4	yes	—
238	C₂³	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	#14519684	1	2	yes	—
248	C₂×C₄	#14519684	1	2	yes	—
249	C₂×C₄	#14519684	1	2	yes	—
250	C₂×C₄	C₄×#92418	1	4	yes	—
251	C₂×C₄	#14519688	1	4	yes	—
252	C₂×C₄	C₂²⋊#92418	1	8	yes	—
253	C₂×C₄	C₄⋊#92418	1	4	yes	—
254	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
255	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
256	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
258	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
259	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
260	C₂×C₄	C₂²×#92418	1	2	yes	—
261	C₂×C₄	C₂×#11758194	1	2	yes	—
262	C₂×C₄	#14519805	1	2	yes	—
263	C₂×C₄	#14519810	1	2	yes	—
264	C₈	—	2	8	no	—
265	C₈	—	244	4	no	—
266	C₈	—	244	4	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	8	no	—
270	C₂²	#16839517	1	4	yes	—
271	C₂²	C₂×#14519684	1	1	yes	characteristic
272	C₂²	C₂×C₄⋊#92418	1	2	yes	—
273	C₂²	#16839804	1	2	yes	—
274	C₂²	#16839811	1	2	yes	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	#16839517	1	4	yes	—
280	C₄	C₂×C₄×#92418	1	1	yes	characteristic
281	C₄	C₂×#14519688	1	1	yes	characteristic
282	C₄	C₂×C₂²⋊#92418	1	2	yes	—
283	C₄	#16839764	1	2	yes	—
284	C₄	#16839768	1	1	yes	characteristic
285	C₄	#16839772	1	1	yes	characteristic
286	C₂	—	2	4	no	—
287	C₂	#21444571	1	4	yes	—
288	C₂	C₂×#16839517	1	1	yes	characteristic
289	C₂	#21445539	1	1	yes	characteristic
290	C₂	#21445546	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₄₄
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₄×M₁₆×C₆₁	C₂
socle	C₂³×C₆₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₁	C₄⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×C₄×M₁₆	D₆₁
hypocenter	C₆₁	#7873
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #21443654	C₂	semidirect
37. C₂×M₁₆×C₆₁	C₈	semidirect
71. C₂×C₈×C₆₁	C₂×C₈	semidirect
166. C₆₁	#7873	semidirect
172. C₂×M₁₆	#92418	semidirect
223. C₂×C₈	C₂×#92418	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839815	2
C₂²×C₄⋊#92418	2
#21443654	2
C₂×C₄×M₁₆×C₆₁	2
#7873	61