Subgroups — #29835419

Order 15616.

834 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840194	C₂	1	4	yes	maximal, complemented
2	#21456247	C₂	1	2	yes	maximal, complemented
3	#21448080	C₂	1	4	yes	maximal, complemented
4	#21448075	C₂	1	2	yes	maximal, complemented
5	#21448072	C₂	1	1	yes	maximal, characteristic, complemented
6	#21448069	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14519799	—	2	2	no	—
9	C₂×#14519799	C₂²	1	4	yes	—
10	#16840312	—	2	2	no	—
11	#16840194	—	2	16	no	—
12	C₂×Dic₄₈₈	C₂²	1	4	yes	—
13	C₂×#14519722	C₂²	1	8	yes	—
14	C₂×D₉₇₆	C₂²	1	4	yes	—
15	#16840146	—	2	2	no	—
16	#16840146	C₂²	1	4	yes	—
17	#16840142	C₂²	1	2	yes	—
18	#16840139	C₂²	1	2	yes	—
19	#16840135	—	2	2	no	—
20	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
23	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
24	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
25	C₂×#11758222	—	2	4	no	—
26	#14519799	—	2	16	no	—
27	#14519799	—	4	8	no	—
28	C₂×Dic₂₄₄	—	2	2	no	—
29	C₂×Dic₂₄₄	C₂³	1	4	yes	—
30	C₂×#11758190	—	2	4	no	—
31	C₂×#11758190	—	2	4	no	—
32	C₂×D₄₈₈	—	2	2	no	—
33	C₂×D₄₈₈	C₂³	1	4	yes	—
34	#14519777	—	2	2	no	—
35	#14519774	—	2	2	no	—
36	#14519765	—	2	4	no	—
37	#14519751	—	4	4	no	—
38	C₄×D₂₄₄	—	4	4	no	—
39	Dic₄₈₈	—	2	8	no	—
40	#14519722	—	2	16	no	—
41	D₉₇₆	—	2	8	no	—
42	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
45	C₂×C₁₆×C₆₁	D₄	1	8	yes	complemented
46	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
47	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
48	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
49	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
50	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	D₈	1	2	yes	—
53	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₆₁	—	4	4	no	—
55	#11758222	—	4	8	no	—
56	#11758222	—	4	8	no	—
57	C₂×D₂₄₄	—	2	4	no	—
58	C₂×D₂₄₄	—	4	4	no	—
59	C₂×C₄×D₆₁	—	4	4	no	—
60	C₂×Dic₁₂₂	—	2	4	no	—
61	C₂×C₈×C₆₁	D₈	1	4	yes	—
62	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
64	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
67	C₄²×C₆₁	D₈	1	2	yes	—
68	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
70	#11758198	—	4	8	no	—
71	C₄⋊Dic₆₁	—	4	4	no	—
72	Dic₂₄₄	—	2	8	no	—
73	Dic₂₄₄	—	4	4	no	—
74	D₄₈₈	—	2	8	no	—
75	D₄₈₈	—	4	4	no	—
76	#11758190	—	4	8	no	—
77	#11758190	—	4	8	no	—
78	C₁₆×C₆₁	C₂×D₄	1	4	yes	—
79	C₁₆×C₆₁	C₂×D₄	1	4	yes	—
80	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
81	C₂²×D₆₁	—	4	4	no	—
82	C₂×C₄×C₆₁	—	2	4	no	—
83	C₂×C₄×C₆₁	#127	1	2	yes	—
84	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
85	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
89	C₂²⋊D₆₁	—	8	8	no	—
90	C₂×Dic₆₁	—	4	4	no	—
91	D₂₄₄	—	4	4	no	—
92	D₂₄₄	—	4	4	no	—
93	D₂₄₄	—	8	8	no	—
94	C₄×D₆₁	—	8	8	no	—
95	Dic₁₂₂	—	4	4	no	—
96	Dic₁₂₂	—	4	4	no	—
97	C₈×C₆₁	#127	1	2	yes	—
98	C₈×C₆₁	C₂×D₈	1	2	yes	—
99	C₈×C₆₁	C₂×D₈	1	2	yes	—
100	C₈×C₆₁	C₂²×D₄	1	2	yes	—
101	#33713	—	61	1	no	maximal, Sylow
102	C₂²×C₆₁	—	2	2	no	—
103	C₂²×C₆₁	#493	1	2	yes	—
104	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
105	C₂×D₆₁	—	8	4	no	—
106	C₂×D₆₁	—	8	8	no	—
107	C₄×C₆₁	—	2	2	no	—
108	C₄×C₆₁	#493	1	2	yes	—
109	C₄×C₆₁	#508	1	4	yes	complemented
110	C₄×C₆₁	C₂×#127	1	1	yes	characteristic
111	C₄×C₆₁	C₂²×D₈	1	1	yes	characteristic
112	Dic₆₁	—	8	4	no	—
113	C₂×#508	—	61	4	no	—
114	#3147	—	61	2	no	—
115	#2249	—	61	4	no	—
116	#2248	—	61	2	no	—
117	#2247	—	61	1	no	—
118	#2246	—	61	1	no	—
119	C₂×C₄×C₁₆	D₆₁	1	1	yes	characteristic, complemented
120	C₂×C₆₁	—	2	2	no	—
121	C₂×C₆₁	C₂×#493	1	1	yes	characteristic
122	C₂×C₆₁	C₂×#508	1	2	yes	—
123	D₆₁	—	16	4	no	—
124	C₂×#135	—	61	4	no	—
125	C₂×#135	—	122	2	no	—
126	#532	—	122	2	no	—
127	#508	—	122	16	no	—
128	C₂×Q₃₂	—	61	4	no	—
129	C₂×SD₃₂	—	61	8	no	—
130	C₂×D₁₆	—	61	4	no	—
131	C₂²×C₁₆	C₂×D₆₁	1	2	yes	—
132	#495	—	61	4	no	—
133	#495	—	122	2	no	—
134	#494	—	61	2	no	—
135	#493	—	61	2	no	—
136	#492	—	122	2	no	—
137	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
138	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
139	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
140	C₄×C₁₆	C₂×D₆₁	1	2	yes	—
141	C₆₁	#33713	1	1	yes	characteristic, complemented, Sylow
142	C₂×#41	—	122	4	no	—
143	#135	—	122	16	no	—
144	#135	—	244	8	no	—
145	C₂×Q₁₆	—	61	4	no	—
146	C₂×Q₁₆	—	122	2	no	—
147	C₂×SD₁₆	—	122	4	no	—
148	C₂×SD₁₆	—	122	4	no	—
149	C₂×D₈	—	61	4	no	—
150	C₂×D₈	—	122	2	no	—
151	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
153	C₄⋊Q₈	—	122	2	no	—
154	#127	—	122	2	no	—
155	#124	—	122	4	no	—
156	#121	—	244	4	no	—
157	C₄×D₄	—	244	4	no	—
158	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
159	Q₃₂	—	122	8	no	—
160	SD₃₂	—	122	16	no	—
161	D₁₆	—	122	8	no	—
162	C₂×C₁₆	D₂₄₄	1	8	yes	complemented
163	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
164	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
165	C₄×C₈	D₂₄₄	1	1	yes	characteristic
166	C₄×C₈	D₂₄₄	1	1	yes	characteristic
167	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
168	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
169	#41	—	244	8	no	—
170	#41	—	244	8	no	—
171	C₂×Q₈	—	122	4	no	—
172	C₂×D₄	—	122	4	no	—
173	C₂×D₄	—	244	4	no	—
174	C₂×D₄	—	244	4	no	—
175	C₂²×C₄	—	244	4	no	—
176	C₂²×C₄	D₄₈₈	1	2	yes	—
177	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
178	Q₁₆	—	122	8	no	—
179	Q₁₆	—	244	4	no	—
180	SD₁₆	—	244	8	no	—
181	SD₁₆	—	244	8	no	—
182	D₈	—	122	8	no	—
183	D₈	—	244	4	no	—
184	C₂×C₈	D₄₈₈	1	4	yes	—
185	C₂×C₈	C₂×D₂₄₄	1	4	yes	—
186	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
187	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
188	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
189	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
190	C₄⋊C₄	—	244	4	no	—
191	C₂²⋊C₄	—	244	8	no	—
192	C₄²	D₄₈₈	1	2	yes	—
193	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
194	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
195	C₁₆	C₂×D₂₄₄	1	4	yes	—
196	C₁₆	C₂×D₂₄₄	1	4	yes	—
197	C₂³	—	244	4	no	—
198	C₂³	C₂×D₄₈₈	1	1	yes	characteristic
199	Q₈	—	244	4	no	—
200	Q₈	—	244	4	no	—
201	D₄	—	488	8	no	—
202	D₄	—	488	8	no	—
203	D₄	—	244	4	no	—
204	D₄	—	244	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	488	8	no	—
207	C₂×C₄	—	244	4	no	—
208	C₂×C₄	#14519774	1	2	yes	—
209	C₂×C₄	C₂×D₄₈₈	1	2	yes	—
210	C₂×C₄	C₂×D₄₈₈	1	2	yes	—
211	C₂×C₄	C₂×D₄₈₈	1	2	yes	—
212	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
213	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
214	C₈	#14519774	1	2	yes	—
215	C₈	C₂×D₄₈₈	1	2	yes	—
216	C₈	C₂×D₄₈₈	1	2	yes	—
217	C₈	C₂²×D₂₄₄	1	2	yes	—
218	C₂²	—	2	2	no	—
219	C₂²	—	488	4	no	—
220	C₂²	—	488	8	no	—
221	C₂²	#16840139	1	2	yes	—
222	C₂²	C₂²×D₄₈₈	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	488	4	no	—
225	C₄	#16840139	1	2	yes	—
226	C₄	#16840194	1	4	yes	complemented
227	C₄	C₂×#14519774	1	1	yes	characteristic
228	C₄	C₂²×D₄₈₈	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	—	976	4	no	—
231	C₂	C₂×#16840139	1	1	yes	characteristic
232	C₂	C₂×#16840194	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₈₈
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂×C₄×C₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₈
hypercenter	C₂×C₄×C₁₆	D₆₁
hypocenter	C₆₁	#33713
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#16840194	C₂	semidirect
2. #21456247	C₂	semidirect
3. #21448080	C₂	semidirect
4. #21448075	C₂	semidirect
5. #21448072	C₂	semidirect
6. #21448069	C₂	semidirect
7. C₂×C₄×C₁₆×C₆₁	C₂	semidirect
45. C₂×C₁₆×C₆₁	D₄	semidirect
109. C₄×C₆₁	#508	semidirect
119. C₂×C₄×C₁₆	D₆₁	semidirect
141. C₆₁	#33713	semidirect
162. C₂×C₁₆	D₂₄₄	semidirect
226. C₄	#16840194	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840194	2
#21456247	2
#21448080	2
#21448075	2
#21448072	2
#21448069	2
C₂×C₄×C₁₆×C₆₁	2
#33713	61