Subgroups — #29648997

Order 15616.

778 conjugacy classes of subgroups.

The subgroup lattice has 276 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	2	yes	maximal
2	#21444571	C₂	1	8	yes	maximal, complemented
3	#21444567	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
5	#16839815	—	2	2	no	—
6	#16839815	—	2	2	no	—
7	#16839815	C₂²	1	4	yes	—
8	#16839815	C₂²	1	4	yes	—
9	C₂×C₄⋊#92418	C₂²	1	4	yes	—
10	C₂×C₂²⋊#92418	—	2	2	no	—
11	C₂×C₄×#92418	—	2	2	no	—
12	#16839515	C₂²	1	8	yes	complemented
13	#16839513	C₂²	1	8	yes	complemented
14	C₂²×M₁₆×C₆₁	C₄	1	1	yes	characteristic
15	C₂²×M₁₆×C₆₁	C₄	1	1	yes	characteristic
16	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
17	C₄×M₁₆×C₆₁	C₄	1	4	yes	—
18	C₄×M₁₆×C₆₁	C₂²	1	4	yes	—
19	C₂×#97×C₆₁	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
21	C₂²×#92418	—	2	4	no	—
22	C₄⋊#92418	—	2	4	no	—
23	C₄⋊#92418	C₂³	1	8	yes	—
24	C₂²⋊#92418	—	2	4	no	—
25	C₂²⋊#92418	—	2	4	no	—
26	C₄×#92418	—	2	4	no	—
27	C₄×#92418	—	2	4	no	—
28	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₆₁	Q₈	1	1	yes	characteristic
31	C₂×M₁₆×C₆₁	—	2	4	no	—
32	C₂×M₁₆×C₆₁	C₈	1	8	yes	complemented
33	C₂×M₁₆×C₆₁	C₂×C₄	1	4	yes	—
34	C₂×M₁₆×C₆₁	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₆₁	D₄	1	2	yes	—
40	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₁	Q₈	1	1	yes	characteristic
42	C₂×C₄²×C₆₁	Q₈	1	2	yes	—
43	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
44	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
45	#97×C₆₁	C₂×C₄	1	8	yes	—
46	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
47	C₄×C₈×C₆₁	C₂³	1	4	yes	—
48	C₂⁴×C₆₁	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	—	4	2	no	—
50	C₂²×C₄×C₆₁	—	4	2	no	—
51	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
58	C₂²×C₄×C₆₁	M₁₆	1	4	yes	—
59	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
65	M₁₆×C₆₁	—	2	8	no	—
66	M₁₆×C₆₁	C₂×C₈	1	16	yes	complemented
67	C₂×C₈×C₆₁	—	2	4	no	—
68	C₂×C₈×C₆₁	C₂×C₈	1	8	yes	complemented
69	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
71	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
72	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
73	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
74	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
76	C₄²×C₆₁	C₂×D₄	1	2	yes	—
77	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
79	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
80	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
81	C₂×#92418	—	2	8	no	—
82	C₂×#92418	—	4	8	no	—
83	C₂³×C₆₁	—	4	1	no	—
84	C₂³×C₆₁	—	4	1	no	—
85	C₂³×C₆₁	C₄⋊C₈	1	4	yes	—
86	C₂³×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
87	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₆₁	—	2	4	no	—
89	C₂×C₄×C₆₁	—	2	2	no	—
90	C₂×C₄×C₆₁	—	2	4	no	—
91	C₂×C₄×C₆₁	—	2	2	no	—
92	C₂×C₄×C₆₁	—	4	1	no	—
93	C₂×C₄×C₆₁	—	4	2	no	—
94	C₂×C₄×C₆₁	—	4	1	no	—
95	C₂×C₄×C₆₁	—	4	2	no	—
96	C₂×C₄×C₆₁	C₄⋊C₈	1	8	yes	—
97	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
98	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
106	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₆₁	C₂×M₁₆	1	4	yes	—
108	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
111	C₈×C₆₁	—	2	4	no	—
112	C₈×C₆₁	C₂²×C₈	1	8	yes	—
113	#92418	—	4	8	no	—
114	#17787	—	61	1	no	maximal, Sylow
115	C₂²×C₆₁	—	2	2	no	—
116	C₂²×C₆₁	—	2	1	no	—
117	C₂²×C₆₁	—	2	2	no	—
118	C₂²×C₆₁	—	2	1	no	—
119	C₂²×C₆₁	—	4	4	no	—
120	C₂²×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
121	C₂²×C₆₁	C₂×C₄⋊C₈	1	4	yes	—
122	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
123	C₄×C₆₁	—	2	2	no	—
124	C₄×C₆₁	—	2	1	no	—
125	C₄×C₆₁	—	2	2	no	—
126	C₄×C₆₁	—	2	1	no	—
127	C₄×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
128	C₄×C₆₁	C₂×C₄⋊C₈	1	4	yes	—
129	C₄×C₆₁	#430	1	2	yes	—
130	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
131	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
132	C₄×C₆₁	#575	1	2	yes	—
133	C₂×#431	—	61	2	no	—
134	C₂×C₄×M₁₆	D₆₁	1	1	yes	characteristic
135	#1565	—	61	8	no	—
136	#1564	—	61	4	no	—
137	C₂×C₆₁	—	2	4	no	—
138	C₂×C₆₁	—	2	2	no	—
139	C₂×C₆₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
140	C₂×C₆₁	#2911	1	1	yes	characteristic
141	C₂×C₆₁	C₂×#430	1	1	yes	characteristic
142	C₂²×M₁₆	Dic₆₁	1	1	yes	characteristic
143	C₂²×M₁₆	Dic₆₁	1	1	yes	characteristic
144	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic
145	#431	—	61	4	no	—
146	#431	—	61	4	no	—
147	#431	—	122	2	no	—
148	#431	—	122	2	no	—
149	C₂×C₄⋊C₈	—	61	4	no	—
150	C₂×C₂²⋊C₈	—	122	2	no	—
151	C₄×M₁₆	Dic₆₁	1	4	yes	—
152	C₄×M₁₆	C₂×D₆₁	1	4	yes	—
153	C₂×#97	Dic₆₁	1	2	yes	—
154	C₂×C₄×C₈	—	122	2	no	—
155	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
156	#335	—	61	8	no	—
157	#334	—	61	8	no	—
158	C₆₁	#17787	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₄	Dic₁₂₂	1	1	yes	characteristic
160	C₂³×C₄	D₂₄₄	1	1	yes	characteristic
161	C₂³×C₄	C₂×Dic₆₁	1	1	yes	characteristic
162	C₂×M₁₆	—	2	4	no	—
163	C₂×M₁₆	#92418	1	8	yes	complemented
164	C₂×M₁₆	C₂×Dic₆₁	1	4	yes	—
165	C₂×M₁₆	C₂×Dic₆₁	1	4	yes	—
166	C₂²×C₈	—	122	4	no	—
167	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
168	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
169	C₂×C₄²	Dic₁₂₂	1	1	yes	characteristic
170	C₂×C₄²	Dic₁₂₂	1	2	yes	—
171	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
172	C₂×C₄²	D₂₄₄	1	2	yes	—
173	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
174	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
175	C₂×C₄²	C₂²×D₆₁	1	2	yes	—
176	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
177	C₄⋊C₈	—	61	8	no	—
178	C₄⋊C₈	—	122	4	no	—
179	C₂²⋊C₈	—	122	4	no	—
180	C₂²⋊C₈	—	122	4	no	—
181	#97	C₂×Dic₆₁	1	8	yes	—
182	C₄×C₈	—	122	4	no	—
183	C₄×C₈	—	122	4	no	—
184	C₄×C₈	C₂×Dic₆₁	1	4	yes	—
185	C₄×C₈	C₂²×D₆₁	1	4	yes	—
186	C₂⁴	C₄⋊Dic₆₁	1	1	yes	characteristic
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	C₂×#92418	1	4	yes	—
190	C₂²×C₄	#11758194	1	4	yes	—
191	C₂²×C₄	C₄⋊Dic₆₁	1	2	yes	—
192	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
193	C₂²×C₄	C₄⋊Dic₆₁	1	2	yes	—
194	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
195	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
196	C₂²×C₄	C₄⋊Dic₆₁	1	2	yes	—
197	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
198	C₂²×C₄	C₂×Dic₁₂₂	1	2	yes	—
199	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₂₄₄	1	2	yes	—
201	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
202	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
203	M₁₆	—	2	8	no	—
204	M₁₆	C₂×#92418	1	16	yes	complemented
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	122	8	no	—
207	C₂×C₈	—	244	8	no	—
208	C₂×C₈	C₂×#92418	1	8	yes	complemented
209	C₂×C₈	C₂²×Dic₆₁	1	4	yes	—
210	C₂×C₈	C₂²×Dic₆₁	1	4	yes	—
211	C₄²	C₄⋊Dic₆₁	1	2	yes	—
212	C₄²	C₄⋊Dic₆₁	1	2	yes	—
213	C₄²	C₄⋊Dic₆₁	1	2	yes	—
214	C₄²	C₂×Dic₁₂₂	1	2	yes	—
215	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
216	C₄²	C₂×D₂₄₄	1	2	yes	—
217	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
218	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
219	C₄²	C₂²×Dic₆₁	1	2	yes	—
220	C₄²	C₂³×D₆₁	1	1	yes	characteristic
221	C₂³	—	4	1	no	—
222	C₂³	—	4	1	no	—
223	C₂³	C₄⋊#92418	1	4	yes	—
224	C₂³	C₂×C₄⋊Dic₆₁	1	2	yes	—
225	C₂³	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	C₄⋊#92418	1	4	yes	—
235	C₂×C₄	C₄⋊#92418	1	8	yes	—
236	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
237	C₂×C₄	C₂×C₄⋊Dic₆₁	1	2	yes	—
238	C₂×C₄	C₂×C₄⋊Dic₆₁	1	2	yes	—
239	C₂×C₄	C₂×C₄⋊Dic₆₁	1	2	yes	—
240	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
242	C₂×C₄	C₂²×#92418	1	2	yes	—
243	C₂×C₄	C₂²×#92418	1	4	yes	—
244	C₂×C₄	C₂×#11758194	1	2	yes	—
245	C₂×C₄	C₂×#11758194	1	4	yes	—
246	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
247	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
248	C₂×C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
249	C₈	—	2	4	no	—
250	C₈	—	244	8	no	—
251	C₈	C₂²×#92418	1	8	yes	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	4	4	no	—
257	C₂²	C₂×C₄⋊#92418	1	2	yes	—
258	C₂²	C₂×C₄⋊#92418	1	4	yes	—
259	C₂²	C₂²×C₄⋊Dic₆₁	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	2	no	—
264	C₄	C₂×C₄⋊#92418	1	4	yes	—
265	C₄	C₂×C₄⋊#92418	1	2	yes	—
266	C₄	#16839811	1	2	yes	—
267	C₄	C₂³×#92418	1	1	yes	characteristic
268	C₄	C₂²×#11758194	1	1	yes	characteristic
269	C₄	#16840562	1	2	yes	—
270	C₂	—	2	4	no	—
271	C₂	—	2	2	no	—
272	C₂	C₂²×C₄⋊#92418	1	1	yes	characteristic
273	C₂	#21454409	1	1	yes	characteristic
274	C₂	C₂×#16839811	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₂₄₄
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₄²	C₂³×D₆₁
Fitting	C₂×C₄×M₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₆₁	C₄⋊C₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₆₁
hypocenter	C₆₁	#17787
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #21444571	C₂	semidirect
3. #21444567	C₂	semidirect
12. #16839515	C₂²	semidirect
13. #16839513	C₂²	semidirect
32. C₂×M₁₆×C₆₁	C₈	semidirect
66. M₁₆×C₆₁	C₂×C₈	semidirect
68. C₂×C₈×C₆₁	C₂×C₈	semidirect
158. C₆₁	#17787	semidirect
163. C₂×M₁₆	#92418	semidirect
204. M₁₆	C₂×#92418	semidirect
208. 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
#21444571	2
#21444567	2
C₂×C₄×M₁₆×C₆₁	2
#17787	61