Subgroups — #29571204

Order 15616.

438 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445674	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839513	C₂	1	1	yes	maximal, characteristic
4	#21443653	C₂	1	2	yes	maximal
5	C₂×C₆₁×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92418	C₂²	1	2	yes	—
7	C₂×#14519687	—	2	2	no	—
8	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₆₁	C₄	1	2	yes	—
10	C₂×C₈×Dic₆₁	C₄	1	2	yes	—
11	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
12	#16839516	—	2	2	no	—
13	#16839513	C₄	1	4	yes	complemented
14	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#334	C₄	1	4	yes	complemented
17	C₂²×#92418	—	2	2	no	—
18	C₂²×#92418	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
21	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
22	C₂×C₄×Dic₆₁	Q₈	1	2	yes	—
23	C₄⋊#92418	—	2	2	no	—
24	C₄⋊#92418	C₂×C₄	1	4	yes	—
25	#14519687	—	4	4	no	—
26	C₄×#92418	—	2	1	no	—
27	C₄×#92418	—	2	1	no	—
28	C₈×Dic₆₁	—	2	4	no	—
29	C₈×Dic₆₁	—	2	4	no	—
30	C₂²×C₈×C₆₁	—	2	2	no	—
31	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₆₁	—	2	2	no	—
35	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
36	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
41	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
42	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₆₁	—	2	2	no	—
44	C₂×C₈×C₆₁	—	2	4	no	—
45	C₂×C₈×C₆₁	—	4	2	no	—
46	C₂×C₈×C₆₁	—	4	2	no	—
47	C₂×C₈×C₆₁	C₄²	1	2	yes	—
48	C₂×C₈×C₆₁	C₄²	1	2	yes	—
49	C₂×C₈×C₆₁	M₁₆	1	4	yes	—
50	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₆₁	#41	1	1	yes	characteristic
53	C₄²×C₆₁	#41	1	1	yes	characteristic
54	C₄×Dic₆₁	—	2	2	no	—
55	C₄×Dic₆₁	—	2	2	no	—
56	C₄×Dic₆₁	—	2	2	no	—
57	C₄×Dic₆₁	—	2	4	no	—
58	C₄×Dic₆₁	SD₁₆	1	4	yes	—
59	C₂×#92418	—	2	2	no	—
60	C₂×#92418	—	2	2	no	—
61	C₂×#92418	—	2	4	no	—
62	C₂×#92418	—	4	2	no	—
63	C₂×#92418	—	4	2	no	—
64	C₂×#92418	M₁₆	1	4	yes	—
65	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₁	—	2	2	no	—
67	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
75	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₆₁	#135	1	2	yes	—
77	C₂×Dic₆₁	—	2	2	no	—
78	C₂×Dic₆₁	—	2	2	no	—
79	C₂×Dic₆₁	—	2	2	no	—
80	C₂×Dic₆₁	—	2	2	no	—
81	C₂×Dic₆₁	#106	1	4	yes	—
82	C₂×Dic₆₁	#108	1	4	yes	—
83	C₈×C₆₁	—	2	2	no	—
84	C₈×C₆₁	—	4	4	no	—
85	C₈×C₆₁	#97	1	4	yes	complemented
86	#92418	—	2	4	no	—
87	#92418	—	4	4	no	—
88	#8273	—	61	1	no	maximal, Sylow
89	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₆₁	#423	1	2	yes	—
91	C₂²×C₆₁	C₂×#106	1	1	yes	characteristic
92	C₂²×C₆₁	#427	1	1	yes	characteristic
93	C₂²×C₆₁	C₂×#108	1	2	yes	—
94	C₄×C₆₁	—	2	1	no	—
95	C₄×C₆₁	—	2	1	no	—
96	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
97	C₄×C₆₁	C₄×M₁₆	1	1	yes	characteristic
98	C₄×C₆₁	#432	1	2	yes	—
99	C₄×C₆₁	C₄×SD₁₆	1	2	yes	—
100	C₄×C₆₁	C₄×SD₁₆	1	2	yes	—
101	Dic₆₁	—	2	4	no	—
102	Dic₆₁	—	2	4	no	—
103	#1767	—	61	2	no	—
104	C₄²×C₈	—	61	1	no	—
105	C₂×#334	—	61	1	no	—
106	C₂×#334	D₆₁	1	1	yes	characteristic
107	#1277	—	61	2	no	—
108	C₂×C₆₁	#1566	1	2	yes	—
109	C₂×C₆₁	#1568	1	2	yes	—
110	C₂×C₆₁	#1767	1	1	yes	characteristic
111	C₂×C₆₁	C₄×#106	1	1	yes	characteristic
112	C₂×C₆₁	C₄×#108	1	1	yes	characteristic
113	C₂×C₄⋊C₈	—	61	2	no	—
114	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
115	C₂×#97	—	122	2	no	—
116	C₂×C₄×C₈	—	61	2	no	—
117	C₂×C₄×C₈	—	61	2	no	—
118	C₂×C₄×C₈	—	61	1	no	—
119	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
120	C₄³	—	61	1	no	—
121	#336	—	122	2	no	—
122	#334	—	61	4	no	—
123	#334	Dic₆₁	1	4	yes	complemented
124	C₆₁	#8273	1	1	yes	characteristic, complemented, Sylow
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	61	2	no	—
127	C₂²×C₈	—	122	2	no	—
128	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
129	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
130	C₂×C₄²	—	61	1	no	—
131	C₂×C₄²	—	61	2	no	—
132	C₂×C₄²	—	61	1	no	—
133	C₂×C₄²	—	61	2	no	—
134	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	61	4	no	—
137	C₄⋊C₈	—	122	2	no	—
138	C₄⋊C₈	C₂×Dic₆₁	1	4	yes	—
139	#97	—	244	4	no	—
140	C₄×C₈	—	122	4	no	—
141	C₄×C₈	—	122	4	no	—
142	C₄×C₈	—	122	1	no	—
143	C₄×C₈	—	122	1	no	—
144	C₄×C₈	C₄×D₆₁	1	2	yes	—
145	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
146	C₂²×C₄	—	61	2	no	—
147	C₂²×C₄	—	61	2	no	—
148	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
149	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
150	C₂²×C₄	Q₈×D₆₁	1	1	yes	characteristic
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	61	4	no	—
156	C₂×C₈	—	122	2	no	—
157	C₂×C₈	—	122	2	no	—
158	C₂×C₈	—	122	4	no	—
159	C₂×C₈	—	244	2	no	—
160	C₂×C₈	—	244	2	no	—
161	C₂×C₈	#11758189	1	4	yes	—
162	C₂×C₈	C₄×Dic₆₁	1	2	yes	—
163	C₂×C₈	C₄×Dic₆₁	1	2	yes	—
164	C₄²	—	61	4	no	—
165	C₄²	—	122	4	no	—
166	C₄²	—	122	2	no	—
167	C₄²	—	122	2	no	—
168	C₄²	—	122	2	no	—
169	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
170	C₄²	#11758224	1	1	yes	characteristic
171	C₄²	#11758226	1	1	yes	characteristic
172	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
173	C₂³	C₄⋊C₄×D₆₁	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	61	4	no	—
176	C₂×C₄	—	61	4	no	—
177	C₂×C₄	—	122	2	no	—
178	C₂×C₄	—	122	2	no	—
179	C₂×C₄	—	122	2	no	—
180	C₂×C₄	—	122	2	no	—
181	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
182	C₂×C₄	#14519735	1	1	yes	characteristic
183	C₂×C₄	#14519739	1	1	yes	characteristic
184	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
185	C₂×C₄	#14519742	1	1	yes	characteristic
186	C₂×C₄	Q₈×Dic₆₁	1	1	yes	characteristic
187	C₂×C₄	M₁₆×D₆₁	1	2	yes	—
188	C₂×C₄	C₂×#11758189	1	2	yes	—
189	C₂×C₄	SD₁₆×D₆₁	1	2	yes	—
190	C₂×C₄	#14519798	1	2	yes	—
191	C₈	—	2	2	no	—
192	C₈	—	4	4	no	—
193	C₈	—	122	4	no	—
194	C₈	—	244	4	no	—
195	C₈	#14519687	1	4	yes	complemented
196	C₂²	C₄⋊C₄×Dic₆₁	1	1	yes	characteristic
197	C₂²	#16839784	1	2	yes	—
198	C₂²	#106×D₆₁	1	1	yes	characteristic
199	C₂²	#16839797	1	1	yes	characteristic
200	C₂²	#108×D₆₁	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	122	4	no	—
204	C₄	—	122	4	no	—
205	C₄	C₂×#14519687	1	1	yes	characteristic
206	C₄	M₁₆×Dic₆₁	1	1	yes	characteristic
207	C₄	#16839818	1	2	yes	—
208	C₄	#16839849	1	2	yes	—
209	C₄	SD₁₆×Dic₆₁	1	2	yes	—
210	C₂	#21444573	1	2	yes	—
211	C₂	#21444579	1	2	yes	—
212	C₂	#21445674	1	1	yes	characteristic
213	C₂	#106×Dic₆₁	1	1	yes	characteristic
214	C₂	#108×Dic₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#334	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#334	D₆₁
hypocenter	C₆₁	#8273
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
13. #16839513	C₄	semidirect
16. C₆₁×#334	C₄	semidirect
85. C₈×C₆₁	#97	semidirect
123. #334	Dic₆₁	semidirect
124. C₆₁	#8273	semidirect
195. C₈	#14519687	semidirect

Maximal subgroups

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

subgroup	index
#21445674	2
C₄×C₈×Dic₆₁	2
C₂×#16839513	2
#21443653	2
C₂×C₆₁×#334	2
#8273	61