Subgroups — #29688635

Order 15616.

1022 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839884	C₂	1	4	yes	maximal, complemented
2	#21454531	C₂	1	2	yes	maximal, complemented
3	#21446713	C₂	1	4	yes	maximal, complemented
4	#21446648	C₂	1	2	yes	maximal, complemented
5	#21445991	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14519687	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#3147	C₂	1	1	yes	maximal, characteristic
8	C₂×#14519784	C₂²	1	4	yes	—
9	#16839884	—	2	16	no	—
10	#16839840	—	2	4	no	—
11	D₄×#92418	—	2	4	no	—
12	C₂×#14519713	C₂²	1	4	yes	—
13	#16839786	C₂²	1	2	yes	—
14	C₂×#14519704	C₂²	1	8	yes	—
15	#16839729	C₂²	1	4	yes	—
16	C₂×#14519687	C₂²	1	2	yes	—
17	C₂×#14519687	C₂²	1	2	yes	—
18	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
19	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
20	C₂×#135×C₆₁	C₂²	1	4	yes	—
21	C₆₁×#532	C₂²	1	2	yes	—
22	C₆₁×#495	C₄	1	4	yes	complemented
23	C₆₁×#494	C₄	1	1	yes	characteristic, complemented
24	C₆₁×#493	C₄	1	1	yes	characteristic, complemented
25	C₆₁×#492	C₄	1	2	yes	complemented
26	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
27	#14519784	—	2	16	no	—
28	C₂×#11758194	—	2	4	no	—
29	C₂×#11758194	C₂³	1	4	yes	—
30	C₂²×#92418	—	2	4	no	—
31	C₂²×#92418	C₂³	1	1	yes	characteristic
32	C₂²×#92418	C₂³	1	1	yes	characteristic
33	C₂×C₄×Dic₆₁	—	2	2	no	—
34	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
35	#14519713	—	2	8	no	—
36	C₄⋊#92418	—	2	4	no	—
37	#14519704	—	2	16	no	—
38	C₂²⋊#92418	—	2	8	no	—
39	#14519687	—	2	4	no	—
40	#14519687	D₄	1	8	yes	complemented
41	C₄×#92418	D₄	1	4	yes	—
42	C₂×#41×C₆₁	C₂³	1	4	yes	—
43	#135×C₆₁	—	2	8	no	—
44	#135×C₆₁	—	2	8	no	—
45	C₂×Q₁₆×C₆₁	C₂×C₄	1	4	yes	—
46	C₂×SD₁₆×C₆₁	C₂×C₄	1	8	yes	—
47	C₂×D₈×C₆₁	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
49	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
50	#127×C₆₁	C₂×C₄	1	2	yes	—
51	#124×C₆₁	C₂×C₄	1	4	yes	—
52	#121×C₆₁	—	2	4	no	—
53	C₄×D₄×C₆₁	—	2	4	no	—
54	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
55	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
58	#41×C₆₁	—	2	8	no	—
59	#41×C₆₁	—	2	8	no	—
60	C₂×Q₈×C₆₁	C₂²×C₄	1	4	yes	—
61	C₂×D₄×C₆₁	—	2	4	no	—
62	C₂×D₄×C₆₁	—	2	4	no	—
63	C₂×D₄×C₆₁	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₆₁	—	2	4	no	—
65	C₂²×C₄×C₆₁	#41	1	2	yes	—
66	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₆₁	—	2	2	no	—
68	Q₁₆×C₆₁	—	2	8	no	—
69	SD₁₆×C₆₁	—	2	16	no	—
70	D₈×C₆₁	—	2	8	no	—
71	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
74	C₂×C₈×C₆₁	#41	1	4	yes	—
75	C₄⋊C₄×C₆₁	—	2	4	no	—
76	C₂²⋊C₄×C₆₁	—	2	8	no	—
77	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₆₁	C₂×D₄	1	2	yes	—
80	C₄×Dic₆₁	—	2	2	no	—
81	C₄×Dic₆₁	—	4	4	no	—
82	C₄×Dic₆₁	C₂×D₄	1	4	yes	—
83	#11758194	—	2	8	no	—
84	#11758194	—	4	8	no	—
85	C₂×#92418	—	2	2	no	—
86	C₂×#92418	—	2	4	no	—
87	C₂×#92418	—	4	8	no	—
88	C₂×#92418	C₂×D₄	1	2	yes	—
89	C₂×#92418	C₂×D₄	1	2	yes	—
90	C₂×#92418	C₂×D₄	1	4	yes	—
91	C₂³×C₆₁	—	2	4	no	—
92	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
93	Q₈×C₆₁	—	2	8	no	—
94	D₄×C₆₁	—	2	8	no	—
95	D₄×C₆₁	—	4	8	no	—
96	D₄×C₆₁	—	4	8	no	—
97	C₂×C₄×C₆₁	—	2	4	no	—
98	C₂×C₄×C₆₁	—	2	4	no	—
99	C₂×C₄×C₆₁	—	4	4	no	—
100	C₂×C₄×C₆₁	—	4	4	no	—
101	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
105	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
107	C₂×Dic₆₁	—	2	4	no	—
108	C₂×Dic₆₁	—	4	4	no	—
109	C₈×C₆₁	C₄×D₄	1	4	yes	—
110	C₈×C₆₁	C₄×D₄	1	4	yes	—
111	#92418	—	2	2	no	—
112	#92418	—	4	4	no	—
113	#92418	#127	1	4	yes	—
114	#21706	—	61	1	no	maximal, Sylow
115	C₂²×C₆₁	—	2	2	no	—
116	C₂²×C₆₁	—	4	4	no	—
117	C₂²×C₆₁	—	4	8	no	—
118	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂²×C₆₁	#532	1	2	yes	—
120	C₄×C₆₁	—	2	2	no	—
121	C₄×C₆₁	—	4	4	no	—
122	C₄×C₆₁	#444	1	4	yes	complemented
123	C₄×C₆₁	C₂×C₄×D₄	1	2	yes	—
124	C₄×C₆₁	C₂×#127	1	1	yes	characteristic
125	C₄×C₆₁	#532	1	1	yes	characteristic
126	Dic₆₁	—	4	4	no	—
127	#3147	D₆₁	1	1	yes	characteristic
128	C₂×#444	—	61	4	no	—
129	#2931	—	61	2	no	—
130	#1975	—	61	4	no	—
131	#1962	—	61	2	no	—
132	#1848	—	61	1	no	—
133	C₄×#97	—	61	1	no	—
134	C₂×C₆₁	—	2	2	no	—
135	C₂×C₆₁	—	8	4	no	—
136	C₂×C₆₁	C₄×#127	1	1	yes	characteristic
137	C₂×C₆₁	C₂×#444	1	2	yes	—
138	C₂×#135	C₂×D₆₁	1	4	yes	—
139	C₂×#131	—	61	4	no	—
140	#532	C₂×D₆₁	1	2	yes	—
141	#495	Dic₆₁	1	4	yes	complemented
142	#494	Dic₆₁	1	1	yes	characteristic, complemented
143	#493	Dic₆₁	1	1	yes	characteristic, complemented
144	#492	Dic₆₁	1	2	yes	complemented
145	#444	—	122	16	no	—
146	#436	—	122	4	no	—
147	C₈×D₄	—	122	4	no	—
148	C₂×#108	—	61	4	no	—
149	#423	—	61	2	no	—
150	C₂×#104	—	61	8	no	—
151	#408	—	61	4	no	—
152	C₂×#97	—	61	2	no	—
153	C₂×#97	—	61	2	no	—
154	C₂×C₄×C₈	—	61	1	no	—
155	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
156	C₄³	—	61	1	no	—
157	C₆₁	#21706	1	1	yes	characteristic, complemented, Sylow
158	C₂×#41	C₂²×D₆₁	1	4	yes	—
159	#135	—	2	8	no	—
160	#135	—	2	8	no	—
161	C₂×Q₁₆	C₂×Dic₆₁	1	4	yes	—
162	C₂×SD₁₆	C₂×Dic₆₁	1	8	yes	—
163	C₂×D₈	C₂×Dic₆₁	1	4	yes	—
164	#131	—	122	16	no	—
165	C₂×M₁₆	—	61	4	no	—
166	C₂×M₁₆	—	122	4	no	—
167	C₂²×C₈	—	61	1	no	—
168	C₂²×C₈	—	61	1	no	—
169	C₂²×C₈	—	122	4	no	—
170	C₂²×C₈	C₂²×D₆₁	1	2	yes	—
171	C₄⋊Q₈	C₂×Dic₆₁	1	2	yes	—
172	#127	C₂×Dic₆₁	1	2	yes	—
173	#124	C₂×Dic₆₁	1	4	yes	—
174	#121	—	2	4	no	—
175	C₄×D₄	—	2	4	no	—
176	C₂×C₄²	—	61	2	no	—
177	C₂×C₄²	—	122	2	no	—
178	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
179	#108	—	122	8	no	—
180	C₄⋊C₈	—	122	4	no	—
181	#104	—	122	16	no	—
182	C₂²⋊C₈	—	122	8	no	—
183	#97	—	61	8	no	—
184	#97	—	122	4	no	—
185	C₄×C₈	—	61	4	no	—
186	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
187	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
188	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
189	#41	—	2	8	no	—
190	#41	—	2	8	no	—
191	C₂×Q₈	C₂²×Dic₆₁	1	4	yes	—
192	C₂×D₄	—	2	4	no	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	C₂²×Dic₆₁	1	4	yes	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	122	2	no	—
197	C₂²×C₄	#11758224	1	2	yes	—
198	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
199	Q₁₆	—	2	8	no	—
200	SD₁₆	—	2	16	no	—
201	D₈	—	2	8	no	—
202	M₁₆	—	122	8	no	—
203	M₁₆	—	244	8	no	—
204	C₂×C₈	—	61	4	no	—
205	C₂×C₈	—	61	2	no	—
206	C₂×C₈	—	61	2	no	—
207	C₂×C₈	—	122	2	no	—
208	C₂×C₈	—	122	4	no	—
209	C₂×C₈	—	244	8	no	—
210	C₂×C₈	D₄×D₆₁	1	4	yes	—
211	C₂×C₈	#11758224	1	4	yes	—
212	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
213	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
214	C₄⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	—	2	8	no	—
216	C₄²	—	61	4	no	—
217	C₄²	—	122	2	no	—
218	C₄²	—	244	4	no	—
219	C₄²	D₄×D₆₁	1	2	yes	—
220	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
221	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
222	C₂³	—	2	4	no	—
223	C₂³	C₂×#11758224	1	1	yes	characteristic
224	Q₈	—	2	8	no	—
225	D₄	—	2	8	no	—
226	D₄	—	4	8	no	—
227	D₄	—	4	8	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	122	4	no	—
233	C₂×C₄	—	244	4	no	—
234	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
235	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
236	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
237	C₂×C₄	C₂×D₄×D₆₁	1	2	yes	—
238	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
239	C₂×C₄	C₂×#11758224	1	2	yes	—
240	C₈	—	61	4	no	—
241	C₈	—	122	2	no	—
242	C₈	—	244	4	no	—
243	C₈	D₄×Dic₆₁	1	4	yes	—
244	C₈	D₄×Dic₆₁	1	4	yes	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	8	no	—
247	C₂²	—	4	4	no	—
248	C₂²	C₂×D₄×Dic₆₁	1	1	yes	characteristic
249	C₂²	#16840310	1	2	yes	—
250	C₄	—	2	2	no	—
251	C₄	—	4	4	no	—
252	C₄	—	244	4	no	—
253	C₄	#16839884	1	4	yes	complemented
254	C₄	C₂×D₄×Dic₆₁	1	2	yes	—
255	C₄	#127×D₆₁	1	1	yes	characteristic
256	C₄	#16840310	1	1	yes	characteristic
257	C₂	—	2	2	no	—
258	C₂	—	8	4	no	—
259	C₂	#127×Dic₆₁	1	1	yes	characteristic
260	C₂	C₂×#16839884	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting, prime radical	C₆₁×#3147	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	#3147	D₆₁
hypocenter	C₆₁	#21706
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#16839884	C₂	semidirect
2. #21454531	C₂	semidirect
3. #21446713	C₂	semidirect
4. #21446648	C₂	semidirect
5. #21445991	C₂	semidirect
6. C₄×#14519687	C₂	semidirect
22. C₆₁×#495	C₄	semidirect
23. C₆₁×#494	C₄	semidirect
24. C₆₁×#493	C₄	semidirect
25. C₆₁×#492	C₄	semidirect
40. #14519687	D₄	semidirect
122. C₄×C₆₁	#444	semidirect
141. #495	Dic₆₁	semidirect
142. #494	Dic₆₁	semidirect
143. #493	Dic₆₁	semidirect
144. #492	Dic₆₁	semidirect
157. C₆₁	#21706	semidirect
253. C₄	#16839884	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839884	2
#21454531	2
#21446713	2
#21446648	2
#21445991	2
C₄×#14519687	2
C₆₁×#3147	2
#21706	61