Subgroups — #29688893

Order 15616.

670 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454760	C₂	1	2	yes	maximal
2	#21446797	C₂	1	2	yes	maximal
3	#21446573	C₂	1	2	yes	maximal
4	#21446568	C₂	1	2	yes	maximal
5	#21445994	C₂	1	1	yes	maximal, characteristic
6	#21445705	C₂	1	2	yes	maximal
7	#21445700	C₂	1	2	yes	maximal
8	#97×Dic₆₁	C₂	1	1	yes	maximal, characteristic
9	C₆₁×#3162	C₂	1	1	yes	maximal, characteristic
10	#16839893	—	2	8	no	—
11	C₂×#14519713	C₂²	1	2	yes	—
12	C₂×#14519713	C₂²	1	2	yes	—
13	#16839789	C₂²	1	4	yes	—
14	#16839786	C₂²	1	2	yes	—
15	#16839718	C₂²	1	4	yes	—
16	C₂×#14519688	C₂²	1	1	yes	characteristic
17	C₂×#14519687	C₂²	1	1	yes	characteristic
18	C₂×#14519687	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
21	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
22	#16839521	C₂²	1	4	yes	—
23	#16839521	C₂²	1	4	yes	—
24	C₆₁×#533	C₂²	1	2	yes	—
25	C₆₁×#501	C₄	1	8	yes	complemented
26	C₆₁×#427	C₂²	1	2	yes	—
27	C₆₁×#427	C₂²	1	2	yes	—
28	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
29	C₂×#11758194	C₂³	1	4	yes	—
30	C₂²×#92418	C₂³	1	1	yes	characteristic
31	C₂²×#92418	C₂³	1	1	yes	characteristic
32	C₂×C₄×Dic₆₁	—	2	2	no	—
33	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
34	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
35	#14519713	—	2	4	no	—
36	#14519713	—	2	4	no	—
37	C₄⋊#92418	—	2	4	no	—
38	C₄⋊#92418	—	2	8	no	—
39	#14519688	—	2	4	no	—
40	#14519688	D₄	1	4	yes	—
41	#14519687	—	2	2	no	—
42	#14519687	Q₈	1	4	yes	complemented
43	C₄×#92418	—	2	4	no	—
44	C₈×Dic₆₁	—	2	2	no	—
45	C₈×Dic₆₁	Q₈	1	4	yes	complemented
46	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈×C₆₁	C₂×C₄	1	4	yes	—
49	#125×C₆₁	C₂×C₄	1	4	yes	—
50	C₂²⋊Q₈×C₆₁	—	2	4	no	—
51	C₄×Q₈×C₆₁	—	2	4	no	—
52	#117×C₆₁	C₂³	1	4	yes	—
53	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
54	#107×C₆₁	C₂×C₄	1	4	yes	—
55	#107×C₆₁	C₂×C₄	1	4	yes	—
56	#106×C₆₁	C₂×C₄	1	4	yes	—
57	#106×C₆₁	C₂×C₄	1	4	yes	—
58	#97×C₆₁	C₂×C₄	1	4	yes	—
59	C₂×Q₈×C₆₁	—	2	4	no	—
60	C₂²×C₄×C₆₁	#41	1	2	yes	—
61	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
62	C₂²×Dic₆₁	—	2	1	no	—
63	C₂²×Dic₆₁	—	2	1	no	—
64	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
67	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₆₁	#41	1	2	yes	—
69	C₂×C₈×C₆₁	#41	1	2	yes	—
70	C₄⋊C₄×C₆₁	—	2	8	no	—
71	C₄⋊C₄×C₆₁	—	2	4	no	—
72	C₄⋊C₄×C₆₁	C₂²×C₄	1	8	yes	—
73	C₂²⋊C₄×C₆₁	—	2	4	no	—
74	C₄²×C₆₁	—	2	4	no	—
75	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
76	C₄²×C₆₁	C₂×D₄	1	2	yes	—
77	C₄×Dic₆₁	—	2	1	no	—
78	C₄×Dic₆₁	—	2	1	no	—
79	C₄×Dic₆₁	—	4	4	no	—
80	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
81	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
82	#11758194	—	2	4	no	—
83	#11758194	—	2	4	no	—
84	C₂×#92418	—	2	1	no	—
85	C₂×#92418	—	2	1	no	—
86	C₂×#92418	—	2	4	no	—
87	C₂×#92418	C₂×D₄	1	2	yes	—
88	C₂×#92418	C₂×D₄	1	2	yes	—
89	C₂×#92418	C₂×Q₈	1	2	yes	—
90	C₂×#92418	C₂×Q₈	1	2	yes	—
91	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
92	Q₈×C₆₁	—	8	4	no	—
93	C₂×C₄×C₆₁	—	2	8	no	—
94	C₂×C₄×C₆₁	—	4	2	no	—
95	C₂×C₄×C₆₁	C₄×D₄	1	4	yes	—
96	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
102	C₂×Dic₆₁	—	2	4	no	—
103	C₂×Dic₆₁	—	2	2	no	—
104	C₂×Dic₆₁	—	2	2	no	—
105	C₂×Dic₆₁	—	4	2	no	—
106	C₈×C₆₁	C₄×Q₈	1	4	yes	—
107	C₈×C₆₁	C₄×Q₈	1	4	yes	—
108	#92418	—	2	2	no	—
109	#92418	—	4	4	no	—
110	#92418	C₄⋊Q₈	1	4	yes	—
111	#21736	—	61	1	no	maximal, Sylow
112	C₂²×C₆₁	—	2	2	no	—
113	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
114	C₂²×C₆₁	#533	1	1	yes	characteristic
115	C₂²×C₆₁	#533	1	1	yes	characteristic
116	C₄×C₆₁	—	2	4	no	—
117	C₄×C₆₁	—	4	8	no	—
118	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
119	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
120	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₄×C₆₁	#532	1	1	yes	characteristic
122	Dic₆₁	—	2	4	no	—
123	Dic₆₁	—	4	2	no	—
124	#3162	D₆₁	1	1	yes	characteristic
125	#2961	—	61	2	no	—
126	#1994	—	61	2	no	—
127	#1946	—	61	2	no	—
128	#1945	—	61	2	no	—
129	#1849	—	61	1	no	—
130	#1773	—	61	2	no	—
131	#1772	—	61	2	no	—
132	C₄×#97	—	61	1	no	—
133	C₂×C₆₁	—	2	1	no	—
134	C₂×C₆₁	—	2	1	no	—
135	C₂×C₆₁	C₄×C₄⋊Q₈	1	1	yes	characteristic
136	C₂×C₆₁	#2956	1	2	yes	—
137	#533	C₂×D₆₁	1	2	yes	—
138	#501	Dic₆₁	1	8	yes	complemented
139	#446	—	122	8	no	—
140	C₂×#108	—	61	2	no	—
141	C₂×#108	—	61	2	no	—
142	#427	C₂×D₆₁	1	2	yes	—
143	#427	C₂×D₆₁	1	2	yes	—
144	#424	—	61	4	no	—
145	#423	—	61	2	no	—
146	#405	—	61	4	no	—
147	C₂×#97	—	61	1	no	—
148	C₂×#97	—	61	1	no	—
149	C₂×#97	—	61	1	no	—
150	C₂×#97	C₂×D₆₁	1	1	yes	characteristic
151	C₂×C₄×C₈	—	61	1	no	—
152	C₂×C₄×C₈	—	61	1	no	—
153	C₄³	—	61	1	no	—
154	#339	—	61	4	no	—
155	#339	—	61	4	no	—
156	C₆₁	#21736	1	1	yes	characteristic, complemented, Sylow
157	C₂×M₁₆	—	61	4	no	—
158	C₂²×C₈	—	61	1	no	—
159	C₂²×C₈	—	61	1	no	—
160	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
162	C₄⋊Q₈	C₂×Dic₆₁	1	4	yes	—
163	#125	C₂×Dic₆₁	1	4	yes	—
164	C₂²⋊Q₈	—	2	4	no	—
165	C₄×Q₈	—	2	4	no	—
166	#117	C₂²×D₆₁	1	4	yes	—
167	C₂×C₄²	—	61	1	no	—
168	C₂×C₄²	—	61	1	no	—
169	C₂×C₄²	—	122	2	no	—
170	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
171	#108	—	122	4	no	—
172	#108	—	122	4	no	—
173	#107	C₂×Dic₆₁	1	4	yes	—
174	#107	C₂×Dic₆₁	1	4	yes	—
175	#106	C₂×Dic₆₁	1	4	yes	—
176	#106	C₂×Dic₆₁	1	4	yes	—
177	C₄⋊C₈	—	122	8	no	—
178	C₄⋊C₈	—	122	4	no	—
179	#97	—	61	4	no	—
180	#97	—	61	4	no	—
181	#97	—	122	4	no	—
182	#97	—	122	2	no	—
183	#97	C₂×Dic₆₁	1	4	yes	—
184	C₄×C₈	—	61	4	no	—
185	C₄×C₈	—	122	4	no	—
186	C₄×C₈	—	122	2	no	—
187	C₂×Q₈	—	2	4	no	—
188	C₂²×C₄	—	122	1	no	—
189	C₂²×C₄	—	122	1	no	—
190	C₂²×C₄	#11758224	1	2	yes	—
191	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
192	M₁₆	—	122	4	no	—
193	M₁₆	—	122	4	no	—
194	C₂×C₈	—	61	2	no	—
195	C₂×C₈	—	61	2	no	—
196	C₂×C₈	—	61	2	no	—
197	C₂×C₈	—	61	2	no	—
198	C₂×C₈	—	122	1	no	—
199	C₂×C₈	—	122	1	no	—
200	C₂×C₈	—	122	4	no	—
201	C₂×C₈	Q₈×D₆₁	1	2	yes	—
202	C₂×C₈	Q₈×D₆₁	1	2	yes	—
203	C₂×C₈	#11758226	1	2	yes	—
204	C₂×C₈	#11758226	1	2	yes	—
205	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
206	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
207	C₄⋊C₄	—	2	8	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	C₂²×Dic₆₁	1	8	yes	—
210	C₂²⋊C₄	—	2	4	no	—
211	C₄²	—	2	4	no	—
212	C₄²	—	61	2	no	—
213	C₄²	—	61	2	no	—
214	C₄²	—	122	1	no	—
215	C₄²	—	122	1	no	—
216	C₄²	—	244	4	no	—
217	C₄²	D₄×D₆₁	1	2	yes	—
218	C₄²	C₂²×Dic₆₁	1	2	yes	—
219	C₂³	C₂×#11758224	1	1	yes	characteristic
220	Q₈	—	8	4	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	122	4	no	—
224	C₂×C₄	—	122	2	no	—
225	C₂×C₄	—	122	2	no	—
226	C₂×C₄	—	244	2	no	—
227	C₂×C₄	D₄×Dic₆₁	1	4	yes	—
228	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
229	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
230	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
231	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
232	C₂×C₄	C₂×#11758226	1	1	yes	characteristic
233	C₂×C₄	C₂×#11758226	1	1	yes	characteristic
234	C₈	—	61	4	no	—
235	C₈	—	122	2	no	—
236	C₈	—	244	4	no	—
237	C₈	Q₈×Dic₆₁	1	4	yes	—
238	C₈	Q₈×Dic₆₁	1	4	yes	—
239	C₂²	—	2	2	no	—
240	C₂²	C₂×D₄×Dic₆₁	1	1	yes	characteristic
241	C₂²	#16840317	1	1	yes	characteristic
242	C₂²	#16840317	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	—	4	8	no	—
245	C₄	—	122	4	no	—
246	C₄	—	244	2	no	—
247	C₄	C₂×Q₈×Dic₆₁	1	1	yes	characteristic
248	C₄	C₂×Q₈×Dic₆₁	1	1	yes	characteristic
249	C₄	C₄⋊Q₈×D₆₁	1	1	yes	characteristic
250	C₄	#16840313	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	C₄⋊Q₈×Dic₆₁	1	1	yes	characteristic
254	C₂	#21454729	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₆₁×#3162	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂³×C₆₁	C₂×#41
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	#3162	D₆₁
hypocenter	C₆₁	#21736
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3162

Complements

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

N	H ≅ G/N	type
25. C₆₁×#501	C₄	semidirect
42. #14519687	Q₈	semidirect
45. C₈×Dic₆₁	Q₈	semidirect
138. #501	Dic₆₁	semidirect
156. C₆₁	#21736	semidirect

Maximal subgroups

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

subgroup	index
#21454760	2
#21446797	2
#21446573	2
#21446568	2
#21445994	2
#21445705	2
#21445700	2
#97×Dic₆₁	2
C₆₁×#3162	2
#21736	61