Subgroups — #29649250

Order 15616.

702 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454387	C₂	1	2	yes	maximal
2	#21444594	C₂	1	6	yes	maximal, complemented
3	#21444578	C₂	1	3	yes	maximal, complemented
4	C₂×#16839515	C₂	1	3	yes	maximal, complemented
5	C₈×#41×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519784	—	2	2	no	—
7	#16839789	—	2	6	no	—
8	#16839786	C₂²	1	6	yes	—
9	C₂×C₄⋊#92418	C₂²	1	6	yes	—
10	#16839515	C₂²	1	1	yes	characteristic
11	#16839515	C₂²	1	3	yes	—
12	#16839515	C₂²	1	6	yes	—
13	#16839515	C₂²	1	6	yes	—
14	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₆₁	C₄	1	1	yes	characteristic
16	C₈×Q₈×C₆₁	C₄	1	1	yes	characteristic
17	C₈×D₄×C₆₁	C₄	1	3	yes	—
18	C₈×D₄×C₆₁	C₄	1	3	yes	—
19	C₆₁×#431	C₂²	1	3	yes	—
20	C₂×C₄×C₈×C₆₁	C₂²	1	3	yes	—
21	#14519784	—	4	8	no	—
22	C₂×#11758194	—	2	6	no	—
23	C₂²×#92418	—	2	6	no	—
24	C₄⋊#92418	—	2	6	no	—
25	C₄⋊#92418	—	2	6	no	—
26	C₄⋊#92418	C₂³	1	2	yes	—
27	C₄⋊#92418	C₂³	1	6	yes	—
28	C₂×#41×C₆₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₁	C₂×C₄	1	3	yes	—
30	C₂²×C₈×C₆₁	C₂×C₄	1	3	yes	—
31	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₄×Q₈×C₆₁	Q₈	1	1	yes	characteristic
33	C₄×D₄×C₆₁	C₂×C₄	1	3	yes	—
34	C₄×D₄×C₆₁	Q₈	1	3	yes	—
35	#117×C₆₁	D₄	1	3	yes	—
36	C₂×C₄²×C₆₁	C₂³	1	3	yes	—
37	C₄⋊C₈×C₆₁	C₂×C₄	1	3	yes	—
38	C₄⋊C₈×C₆₁	C₂×C₄	1	3	yes	—
39	C₂²⋊C₈×C₆₁	C₂×C₄	1	6	yes	—
40	C₄×C₈×C₆₁	C₂×C₄	1	3	yes	—
41	C₄×C₈×C₆₁	C₂×C₄	1	3	yes	—
42	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
43	C₄×C₈×C₆₁	C₂³	1	3	yes	—
44	#41×C₆₁	D₈	1	4	yes	—
45	#41×C₆₁	Q₁₆	1	4	yes	—
46	C₂×Q₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
47	C₂×D₄×C₆₁	C₄⋊C₄	1	3	yes	—
48	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
49	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
50	C₂²×C₄×C₆₁	C₂×Q₈	1	3	yes	—
51	C₂×C₈×C₆₁	—	2	6	no	—
52	C₂×C₈×C₆₁	—	4	3	no	—
53	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
55	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
56	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₄⋊C₄×C₆₁	C₄⋊C₄	1	3	yes	—
58	C₄⋊C₄×C₆₁	C₄⋊C₄	1	3	yes	—
59	C₂²⋊C₄×C₆₁	C₄⋊C₄	1	6	yes	—
60	C₄²×C₆₁	C₂²×C₄	1	3	yes	—
61	C₄²×C₆₁	C₂×D₄	1	3	yes	—
62	C₄²×C₆₁	C₂×Q₈	1	3	yes	—
63	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
64	#11758194	—	4	12	no	—
65	C₂×#92418	—	2	2	no	—
66	C₂×#92418	—	2	6	no	—
67	C₂×#92418	—	4	12	no	—
68	C₂³×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
69	Q₈×C₆₁	#107	1	4	yes	—
70	D₄×C₆₁	#107	1	12	yes	—
71	C₂×C₄×C₆₁	—	2	3	no	—
72	C₂×C₄×C₆₁	—	2	3	no	—
73	C₂×C₄×C₆₁	—	4	3	no	—
74	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
76	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
77	C₂×C₄×C₆₁	C₂×D₈	1	6	yes	—
78	C₂×C₄×C₆₁	C₂×Q₁₆	1	6	yes	—
79	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
82	C₈×C₆₁	—	2	3	no	—
83	C₈×C₆₁	—	2	1	no	—
84	C₈×C₆₁	—	2	3	no	—
85	C₈×C₆₁	#131	1	2	yes	complemented
86	#92418	—	4	2	no	—
87	#92418	—	4	6	no	—
88	#17828	—	61	1	no	maximal, Sylow
89	C₂²×C₆₁	—	2	3	no	—
90	C₂²×C₆₁	—	2	3	no	—
91	C₂²×C₆₁	C₂×#107	1	6	yes	—
92	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
93	C₄×C₆₁	—	2	3	no	—
94	C₄×C₆₁	—	2	1	no	—
95	C₄×C₆₁	—	2	3	no	—
96	C₄×C₆₁	C₂×#107	1	6	yes	—
97	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
98	C₄×C₆₁	#568	1	1	yes	characteristic
99	C₄×C₆₁	C₂²×D₈	1	1	yes	characteristic
100	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
101	C₈×#41	D₆₁	1	1	yes	characteristic
102	#2906	—	61	2	no	—
103	#1571	—	61	6	no	—
104	#1567	—	61	3	no	—
105	C₂×#335	—	61	3	no	—
106	C₂×C₆₁	—	2	6	no	—
107	C₂×C₆₁	#2906	1	1	yes	characteristic
108	C₂×C₆₁	C₂²×#107	1	1	yes	characteristic
109	C₂×C₆₁	#2916	1	1	yes	characteristic
110	C₂×#131	—	122	2	no	—
111	C₄×#41	C₂×D₆₁	1	1	yes	characteristic
112	C₈×Q₈	Dic₆₁	1	1	yes	characteristic
113	C₈×Q₈	Dic₆₁	1	1	yes	characteristic
114	C₈×D₄	Dic₆₁	1	3	yes	—
115	C₈×D₄	Dic₆₁	1	3	yes	—
116	#431	C₂×D₆₁	1	3	yes	—
117	#424	—	122	6	no	—
118	#423	—	61	6	no	—
119	C₂×C₄⋊C₈	—	61	6	no	—
120	C₂×C₄×C₈	C₂×D₆₁	1	3	yes	—
121	#335	—	61	1	no	—
122	#335	—	61	3	no	—
123	#335	—	61	6	no	—
124	#335	—	61	6	no	—
125	C₆₁	#17828	1	1	yes	characteristic, complemented, Sylow
126	C₂×#41	D₂₄₄	1	1	yes	characteristic
127	#131	—	244	8	no	—
128	C₂×M₁₆	—	122	6	no	—
129	C₂²×C₈	—	122	6	no	—
130	C₂²×C₈	C₂×Dic₆₁	1	3	yes	—
131	C₂²×C₈	C₂×Dic₆₁	1	3	yes	—
132	C₄×Q₈	Dic₁₂₂	1	1	yes	characteristic
133	C₄×Q₈	C₂×Dic₆₁	1	1	yes	characteristic
134	C₄×D₄	Dic₁₂₂	1	3	yes	—
135	C₄×D₄	C₂×Dic₆₁	1	3	yes	—
136	#117	D₂₄₄	1	3	yes	—
137	C₂×C₄²	C₂²×D₆₁	1	3	yes	—
138	C₄⋊C₈	—	61	2	no	—
139	C₄⋊C₈	—	61	6	no	—
140	C₄⋊C₈	—	122	6	no	—
141	C₄⋊C₈	—	122	6	no	—
142	C₄⋊C₈	C₂×Dic₆₁	1	3	yes	—
143	C₄⋊C₈	C₂×Dic₆₁	1	3	yes	—
144	C₂²⋊C₈	C₂×Dic₆₁	1	6	yes	—
145	C₄×C₈	C₂×Dic₆₁	1	3	yes	—
146	C₄×C₈	C₂×Dic₆₁	1	3	yes	—
147	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
148	C₄×C₈	C₂²×D₆₁	1	3	yes	—
149	#41	D₄₈₈	1	4	yes	—
150	#41	Dic₂₄₄	1	4	yes	—
151	C₂×Q₈	C₄⋊Dic₆₁	1	1	yes	characteristic
152	C₂×D₄	C₄⋊Dic₆₁	1	3	yes	—
153	C₂²×C₄	C₂×Dic₁₂₂	1	3	yes	—
154	C₂²×C₄	C₂×D₂₄₄	1	3	yes	—
155	C₂²×C₄	C₂²×Dic₆₁	1	3	yes	—
156	M₁₆	—	244	12	no	—
157	C₂×C₈	—	2	6	no	—
158	C₂×C₈	—	4	3	no	—
159	C₂×C₈	—	122	2	no	—
160	C₂×C₈	—	122	6	no	—
161	C₂×C₈	—	244	12	no	—
162	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
163	C₂×C₈	C₂²×Dic₆₁	1	3	yes	—
164	C₂×C₈	C₂²×Dic₆₁	1	3	yes	—
165	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
166	C₄⋊C₄	C₄⋊Dic₆₁	1	3	yes	—
167	C₄⋊C₄	C₄⋊Dic₆₁	1	3	yes	—
168	C₂²⋊C₄	C₄⋊Dic₆₁	1	6	yes	—
169	C₄²	C₂×Dic₁₂₂	1	3	yes	—
170	C₄²	C₂×D₂₄₄	1	3	yes	—
171	C₄²	C₂²×Dic₆₁	1	3	yes	—
172	C₄²	C₂³×D₆₁	1	1	yes	characteristic
173	C₂³	C₂×C₄⋊Dic₆₁	1	3	yes	—
174	Q₈	#14519711	1	4	yes	—
175	D₄	#14519711	1	12	yes	—
176	C₂×C₄	—	2	3	no	—
177	C₂×C₄	—	2	3	no	—
178	C₂×C₄	—	4	3	no	—
179	C₂×C₄	C₂×C₄⋊Dic₆₁	1	3	yes	—
180	C₂×C₄	C₂×C₄⋊Dic₆₁	1	3	yes	—
181	C₂×C₄	C₂×C₄⋊Dic₆₁	1	3	yes	—
182	C₂×C₄	C₂×D₄₈₈	1	6	yes	—
183	C₂×C₄	C₂×Dic₂₄₄	1	6	yes	—
184	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
185	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
186	C₂×C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
187	C₈	—	2	3	no	—
188	C₈	—	2	1	no	—
189	C₈	—	2	3	no	—
190	C₈	—	244	2	no	—
191	C₈	—	244	6	no	—
192	C₈	#14519784	1	2	yes	complemented
193	C₂²	—	2	3	no	—
194	C₂²	—	2	3	no	—
195	C₂²	C₂×#14519711	1	6	yes	—
196	C₂²	C₂²×C₄⋊Dic₆₁	1	1	yes	characteristic
197	C₄	—	2	3	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	3	no	—
200	C₄	C₂×#14519711	1	6	yes	—
201	C₄	C₂×#14519784	1	1	yes	characteristic
202	C₄	#16840529	1	1	yes	characteristic
203	C₄	C₂²×D₄₈₈	1	1	yes	characteristic
204	C₄	C₂²×Dic₂₄₄	1	1	yes	characteristic
205	C₂	—	2	6	no	—
206	C₂	#21454387	1	1	yes	characteristic
207	C₂	C₂²×#14519711	1	1	yes	characteristic
208	C₂	#21454430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₄²	C₂³×D₆₁
Fitting	C₈×#41×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₆₁	D₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	C₈×#41	D₆₁
hypocenter	C₆₁	#17828
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. #21444594	C₂	semidirect
3. #21444578	C₂	semidirect
4. C₂×#16839515	C₂	semidirect
85. C₈×C₆₁	#131	semidirect
125. C₆₁	#17828	semidirect
192. C₈	#14519784	semidirect

Maximal subgroups

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

subgroup	index
#21454387	2
#21444594	2
#21444578	2
C₂×#16839515	2
C₈×#41×C₆₁	2
#17828	61