Subgroups — #29667714

Order 15616.

922 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456398	C₂	1	4	yes	maximal, complemented
2	#21456247	C₂	1	2	yes	maximal, complemented
3	#21445437	C₂	1	1	yes	maximal, characteristic, complemented
4	#21445434	C₂	1	1	yes	maximal, characteristic, complemented
5	#21445425	C₂	1	2	yes	maximal, complemented
6	#21445411	C₂	1	4	yes	maximal, complemented
7	C₂×C₈²×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14519799	—	2	4	no	—
9	#16840312	C₂²	1	4	yes	—
10	#16840167	C₂²	1	4	yes	—
11	#16840146	—	2	4	no	—
12	#16840142	C₂²	1	2	yes	—
13	#16840139	C₂²	1	4	yes	—
14	#16840139	C₂²	1	2	yes	—
15	#16840135	C₂²	1	4	yes	—
16	#16840109	C₂²	1	8	yes	—
17	#16839998	—	2	8	no	—
18	C₄×D₄₈₈	—	2	8	no	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₈²×C₆₁	C₂²	1	2	yes	—
22	C₈²×C₆₁	C₂²	1	1	yes	characteristic
23	C₈²×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×#11758222	—	2	4	no	—
25	#14519799	—	4	16	no	—
26	C₂×Dic₂₄₄	—	2	4	no	—
27	C₂×#11758190	—	2	8	no	—
28	C₂×D₄₈₈	—	2	8	no	—
29	C₂×D₄₈₈	—	2	4	no	—
30	#14519777	C₂³	1	4	yes	—
31	#14519774	C₂³	1	4	yes	—
32	#14519765	—	2	4	no	—
33	#14519751	—	2	8	no	—
34	C₄×D₂₄₄	—	2	8	no	—
35	#14519711	—	2	8	no	—
36	#14519699	—	2	16	no	—
37	C₂²×C₈×C₆₁	D₄	1	4	yes	—
38	C₂²×C₈×C₆₁	D₄	1	2	yes	—
39	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₆₁	D₄	1	4	yes	—
41	C₄×C₈×C₆₁	D₄	1	2	yes	—
42	C₄×C₈×C₆₁	C₂³	1	2	yes	—
43	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
45	C₄×C₈×C₆₁	C₂³	1	2	yes	—
46	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₂²⋊D₆₁	—	4	4	no	—
49	#11758222	—	4	16	no	—
50	C₂×D₂₄₄	—	2	4	no	—
51	C₂×D₂₄₄	—	2	8	no	—
52	C₂×C₄×D₆₁	—	4	4	no	—
53	C₂×Dic₁₂₂	—	2	4	no	—
54	C₂×C₈×C₆₁	—	2	8	no	—
55	C₂×C₈×C₆₁	D₈	1	8	yes	complemented
56	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
57	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
60	C₄²×C₆₁	C₂×D₄	1	2	yes	—
61	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
63	#11758198	—	4	8	no	—
64	C₄⋊Dic₆₁	—	2	8	no	—
65	Dic₂₄₄	—	4	8	no	—
66	D₄₈₈	—	4	8	no	—
67	D₄₈₈	—	4	16	no	—
68	#11758190	—	4	16	no	—
69	C₂³×C₆₁	#127	1	1	yes	characteristic
70	C₂²×D₆₁	—	4	4	no	—
71	C₂×C₄×C₆₁	—	2	4	no	—
72	C₂×C₄×C₆₁	#127	1	2	yes	—
73	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂×D₈	1	4	yes	—
75	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
77	C₂²⋊D₆₁	—	8	8	no	—
78	C₂×Dic₆₁	—	4	4	no	—
79	D₂₄₄	—	4	8	no	—
80	D₂₄₄	—	4	16	no	—
81	C₄×D₆₁	—	8	8	no	—
82	Dic₁₂₂	—	4	8	no	—
83	C₈×C₆₁	—	2	4	no	—
84	C₈×C₆₁	C₂×D₈	1	4	yes	—
85	C₈×C₆₁	C₂×D₈	1	4	yes	—
86	C₈×C₆₁	#135	1	8	yes	complemented
87	#19603	—	61	1	no	maximal, Sylow
88	C₂²×C₆₁	—	2	2	no	—
89	C₂²×C₆₁	#493	1	2	yes	—
90	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
91	C₂×D₆₁	—	8	4	no	—
92	C₂×D₆₁	—	8	8	no	—
93	C₄×C₆₁	—	2	2	no	—
94	C₄×C₆₁	#493	1	2	yes	—
95	C₄×C₆₁	C₂²×D₈	1	2	yes	—
96	C₄×C₆₁	C₂×#135	1	4	yes	—
97	Dic₆₁	—	8	4	no	—
98	#3169	—	61	4	no	—
99	#3147	—	61	2	no	—
100	#1713	—	61	1	no	—
101	#1712	—	61	1	no	—
102	#1709	—	61	2	no	—
103	#1706	—	61	4	no	—
104	C₂×C₈²	D₆₁	1	1	yes	characteristic, complemented
105	C₂×C₆₁	—	2	2	no	—
106	C₂×C₆₁	C₂×#493	1	1	yes	characteristic
107	C₂×C₆₁	#3147	1	2	yes	—
108	D₆₁	—	16	4	no	—
109	C₂×#135	—	122	4	no	—
110	#532	—	61	4	no	—
111	#500	—	61	4	no	—
112	#495	—	122	4	no	—
113	#494	—	61	2	no	—
114	#493	—	61	2	no	—
115	#493	—	61	4	no	—
116	#492	—	61	4	no	—
117	#486	—	61	8	no	—
118	#466	—	122	8	no	—
119	C₄×D₈	—	122	8	no	—
120	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
122	C₈²	C₂×D₆₁	1	2	yes	—
123	C₈²	C₂×D₆₁	1	1	yes	characteristic
124	C₈²	C₂×D₆₁	1	1	yes	characteristic
125	C₆₁	#19603	1	1	yes	characteristic, complemented, Sylow
126	C₂×#41	—	122	4	no	—
127	#135	—	244	16	no	—
128	C₂×Q₁₆	—	122	4	no	—
129	C₂×SD₁₆	—	122	8	no	—
130	C₂×D₈	—	122	4	no	—
131	C₂×D₈	—	122	8	no	—
132	C₂²×C₈	D₂₄₄	1	4	yes	—
133	C₂²×C₈	D₂₄₄	1	2	yes	—
134	C₄⋊Q₈	—	61	4	no	—
135	#127	—	61	4	no	—
136	#124	—	122	4	no	—
137	#121	—	122	8	no	—
138	C₄×D₄	—	122	8	no	—
139	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
140	#107	—	122	8	no	—
141	#102	—	122	16	no	—
142	C₄×C₈	D₂₄₄	1	4	yes	—
143	C₄×C₈	D₂₄₄	1	2	yes	—
144	C₄×C₈	C₂²×D₆₁	1	2	yes	—
145	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
146	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
147	C₄×C₈	C₂²×D₆₁	1	2	yes	—
148	#41	—	244	16	no	—
149	C₂×Q₈	—	122	4	no	—
150	C₂×D₄	—	122	8	no	—
151	C₂×D₄	—	122	4	no	—
152	C₂×D₄	—	244	4	no	—
153	C₂²×C₄	—	244	4	no	—
154	C₂²×C₄	C₂×D₂₄₄	1	2	yes	—
155	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
156	Q₁₆	—	244	8	no	—
157	SD₁₆	—	244	16	no	—
158	D₈	—	244	8	no	—
159	D₈	—	244	16	no	—
160	C₂×C₈	—	2	8	no	—
161	C₂×C₈	D₄₈₈	1	8	yes	complemented
162	C₂×C₈	C₂×D₂₄₄	1	4	yes	—
163	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
164	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
165	C₂×C₈	C₂×D₂₄₄	1	4	yes	—
166	C₄⋊C₄	—	122	8	no	—
167	C₂²⋊C₄	—	244	8	no	—
168	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
169	C₄²	C₂×D₂₄₄	1	2	yes	—
170	C₄²	C₂³×D₆₁	1	1	yes	characteristic
171	C₂³	—	244	4	no	—
172	C₂³	#14519774	1	1	yes	characteristic
173	Q₈	—	244	8	no	—
174	D₄	—	488	8	no	—
175	D₄	—	244	8	no	—
176	D₄	—	244	16	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	488	8	no	—
179	C₂×C₄	—	244	4	no	—
180	C₂×C₄	#14519774	1	2	yes	—
181	C₂×C₄	#14519774	1	1	yes	characteristic
182	C₂×C₄	C₂×D₄₈₈	1	4	yes	—
183	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
184	C₂×C₄	C₂²×D₂₄₄	1	2	yes	—
185	C₈	—	2	4	no	—
186	C₈	C₂×D₄₈₈	1	4	yes	—
187	C₈	C₂×D₄₈₈	1	4	yes	—
188	C₈	#14519799	1	8	yes	complemented
189	C₂²	—	2	2	no	—
190	C₂²	—	488	4	no	—
191	C₂²	—	488	8	no	—
192	C₂²	#16840139	1	2	yes	—
193	C₂²	C₂×#14519774	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	488	4	no	—
196	C₄	#16840139	1	2	yes	—
197	C₄	C₂²×D₄₈₈	1	2	yes	—
198	C₄	C₂×#14519799	1	4	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	976	4	no	—
201	C₂	C₂×#16840139	1	1	yes	characteristic
202	C₂	#21456247	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14519774
commutator, cosocle	C₄²×C₆₁	C₂⁴
Frattini	C₄²	C₂³×D₆₁
Fitting	C₂×C₈²×C₆₁	C₂
socle	C₂²×C₆₁	C₂×#127
hypercenter	C₂×C₈²	D₆₁
hypocenter	C₆₁	#19603
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
1. #21456398	C₂	semidirect
2. #21456247	C₂	semidirect
3. #21445437	C₂	semidirect
4. #21445434	C₂	semidirect
5. #21445425	C₂	semidirect
6. #21445411	C₂	semidirect
7. C₂×C₈²×C₆₁	C₂	semidirect
55. C₂×C₈×C₆₁	D₈	semidirect
86. C₈×C₆₁	#135	semidirect
104. C₂×C₈²	D₆₁	semidirect
125. C₆₁	#19603	semidirect
161. C₂×C₈	D₄₈₈	semidirect
188. C₈	#14519799	semidirect

Maximal subgroups

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

subgroup	index
#21456398	2
#21456247	2
#21445437	2
#21445434	2
#21445425	2
#21445411	2
C₂×C₈²×C₆₁	2
#19603	61