Subgroups — #21454565

Order 7808.

492 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14519785	C₂	1	2	yes	maximal, complemented
2	#16840312	C₂	1	1	yes	maximal, characteristic
3	#16839841	C₂	1	2	yes	maximal, complemented
4	#16839835	C₂	1	4	yes	maximal, complemented
5	C₈×D₂₄₄	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊#92418	C₂	1	1	yes	maximal, characteristic, complemented
7	#16839730	C₂	1	2	yes	maximal, complemented
8	C₄×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11758222	C₂²	1	2	yes	—
10	#14519785	—	2	8	no	—
11	C₂×#11758189	C₂²	1	4	yes	—
12	C₂²×#92418	C₂²	1	2	yes	—
13	C₂×C₈×D₆₁	C₂²	1	4	yes	—
14	#14519777	C₄	1	1	yes	characteristic
15	#14519774	C₄	1	1	yes	characteristic
16	#14519765	C₄	1	2	yes	—
17	#14519751	C₄	1	2	yes	—
18	#14519751	C₄	1	2	yes	—
19	C₄×D₂₄₄	C₂²	1	2	yes	—
20	C₄×D₂₄₄	C₂²	1	2	yes	—
21	C₄⋊#92418	C₂²	1	1	yes	characteristic
22	C₄⋊#92418	C₂²	1	2	yes	—
23	C₄⋊#92418	C₂²	1	1	yes	characteristic
24	#14519689	C₂²	1	4	yes	—
25	#14519689	C₂²	1	4	yes	—
26	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
29	#97×C₆₁	C₂²	1	2	yes	—
30	C₄×C₈×C₆₁	C₂²	1	2	yes	—
31	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊D₆₁	C₂×C₄	1	4	yes	—
35	#11758222	—	2	4	no	—
36	#11758222	—	2	4	no	—
37	C₂×D₂₄₄	C₂×C₄	1	2	yes	—
38	C₂×D₂₄₄	C₂×C₄	1	2	yes	—
39	C₂×D₂₄₄	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₆₁	C₂³	1	4	yes	—
41	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
42	M₁₆×C₆₁	—	2	2	no	—
43	M₁₆×C₆₁	D₄	1	4	yes	complemented
44	C₂×C₈×C₆₁	C₂³	1	2	yes	—
45	C₂×C₈×C₆₁	C₂³	1	2	yes	—
46	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₆₁	C₂³	1	1	yes	characteristic
49	C₄²×C₆₁	C₂³	1	1	yes	characteristic
50	#11758198	C₂×C₄	1	4	yes	—
51	#11758198	C₂×C₄	1	4	yes	—
52	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
53	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
54	C₂×#92418	—	2	4	no	—
55	C₂×#92418	C₂³	1	2	yes	—
56	C₂×#92418	C₂³	1	2	yes	—
57	#11758189	—	2	8	no	—
58	C₈×D₆₁	—	2	8	no	—
59	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×D₆₁	C₂²×C₄	1	4	yes	—
61	C₂×C₄×C₆₁	—	2	2	no	—
62	C₂×C₄×C₆₁	—	4	1	no	—
63	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₆₁	#41	1	2	yes	—
70	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
71	C₂²⋊D₆₁	—	2	8	no	—
72	C₂×Dic₆₁	C₂²×C₄	1	4	yes	—
73	D₂₄₄	—	2	4	no	—
74	D₂₄₄	—	2	4	no	—
75	D₂₄₄	—	4	2	no	—
76	C₄×D₆₁	—	2	4	no	—
77	C₄×D₆₁	—	2	4	no	—
78	Dic₁₂₂	—	2	4	no	—
79	C₈×C₆₁	—	2	2	no	—
80	C₈×C₆₁	C₂×D₄	1	4	yes	—
81	#92418	—	2	2	no	—
82	#92418	—	2	2	no	—
83	C₂²×C₆₁	—	2	1	no	—
84	C₂²×C₆₁	—	2	1	no	—
85	C₂²×C₆₁	C₄×D₄	1	2	yes	—
86	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
87	C₂×D₆₁	—	2	4	no	—
88	C₂×D₆₁	—	2	4	no	—
89	C₂×D₆₁	—	2	4	no	—
90	C₄×C₆₁	—	2	1	no	—
91	C₄×C₆₁	—	2	1	no	—
92	C₄×C₆₁	—	2	1	no	—
93	C₄×C₆₁	C₄×D₄	1	2	yes	—
94	C₄×C₆₁	#131	1	2	yes	complemented
95	C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
96	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
97	Dic₆₁	—	2	4	no	—
98	#2935	—	61	1	no	maximal, Sylow
99	C₂×C₆₁	—	2	2	no	—
100	C₂×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
101	C₂×C₆₁	C₂×#131	1	1	yes	characteristic
102	C₂×C₆₁	#568	1	1	yes	characteristic
103	D₆₁	—	4	4	no	—
104	C₂×#131	—	61	2	no	—
105	#532	—	61	1	no	—
106	#436	—	61	2	no	—
107	#435	—	61	4	no	—
108	C₈×D₄	—	61	2	no	—
109	C₂×C₄⋊C₈	—	61	1	no	—
110	#408	—	61	2	no	—
111	C₄×M₁₆	D₆₁	1	1	yes	characteristic, complemented
112	C₆₁	#2935	1	1	yes	characteristic, complemented, Sylow
113	C₂×#41	—	61	2	no	—
114	#131	—	122	8	no	—
115	C₂×M₁₆	—	61	4	no	—
116	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
117	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
118	C₂²×C₈	—	61	2	no	—
119	C₂²×C₈	—	61	4	no	—
120	C₄⋊Q₈	—	61	1	no	—
121	#127	—	61	1	no	—
122	#124	—	61	2	no	—
123	#121	—	61	2	no	—
124	#121	—	61	2	no	—
125	C₄×D₄	—	61	2	no	—
126	C₄×D₄	—	61	2	no	—
127	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
128	C₄⋊C₈	—	61	1	no	—
129	C₄⋊C₈	—	61	2	no	—
130	C₄⋊C₈	—	61	1	no	—
131	C₂²⋊C₈	—	61	4	no	—
132	C₂²⋊C₈	—	61	4	no	—
133	#97	C₂×D₆₁	1	2	yes	—
134	C₄×C₈	C₂×D₆₁	1	2	yes	—
135	#41	—	122	4	no	—
136	#41	—	122	4	no	—
137	C₂×Q₈	—	61	2	no	—
138	C₂×D₄	—	61	4	no	—
139	C₂×D₄	—	61	2	no	—
140	C₂×D₄	—	61	2	no	—
141	C₂×D₄	—	61	2	no	—
142	C₂²×C₄	—	61	4	no	—
143	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
144	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
146	M₁₆	—	2	2	no	—
147	M₁₆	—	122	8	no	—
148	M₁₆	D₂₄₄	1	4	yes	complemented
149	C₂×C₈	—	61	2	no	—
150	C₂×C₈	—	61	2	no	—
151	C₂×C₈	—	122	4	no	—
152	C₂×C₈	—	122	8	no	—
153	C₂×C₈	C₂²×D₆₁	1	2	yes	—
154	C₂×C₈	C₂²×D₆₁	1	2	yes	—
155	C₄⋊C₄	—	61	2	no	—
156	C₄⋊C₄	—	61	2	no	—
157	C₂²⋊C₄	—	61	4	no	—
158	C₂²⋊C₄	—	61	4	no	—
159	C₄²	C₄×D₆₁	1	1	yes	characteristic
160	C₄²	C₄×D₆₁	1	1	yes	characteristic
161	C₄²	C₂²×D₆₁	1	1	yes	characteristic
162	C₄²	C₂²×D₆₁	1	1	yes	characteristic
163	C₂³	—	61	4	no	—
164	C₂³	C₂×C₄×D₆₁	1	1	yes	characteristic
165	Q₈	—	122	4	no	—
166	D₄	—	122	8	no	—
167	D₄	—	122	4	no	—
168	D₄	—	122	4	no	—
169	D₄	—	244	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	61	4	no	—
173	C₂×C₄	—	122	4	no	—
174	C₂×C₄	—	122	4	no	—
175	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
176	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
180	C₂×C₄	C₂×D₂₄₄	1	2	yes	—
181	C₂×C₄	#11758222	1	2	yes	—
182	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	122	2	no	—
185	C₈	—	122	2	no	—
186	C₈	C₂×D₂₄₄	1	4	yes	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	122	4	no	—
190	C₂²	—	122	4	no	—
191	C₂²	—	122	4	no	—
192	C₂²	C₄×D₂₄₄	1	2	yes	—
193	C₂²	C₂²×C₄×D₆₁	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	122	4	no	—
198	C₄	C₄×D₂₄₄	1	2	yes	—
199	C₄	#14519785	1	2	yes	complemented
200	C₄	C₂²×D₂₄₄	1	1	yes	characteristic
201	C₄	C₂×#11758222	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	244	4	no	—
204	C₂	C₂×C₄×D₂₄₄	1	1	yes	characteristic
205	C₂	C₂×#14519785	1	1	yes	characteristic
206	C₂	#16840528	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₄×M₁₆×C₆₁	C₂
hypercenter	C₄×M₁₆	D₆₁
hypocenter	C₆₁	#2935
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#14519785	C₂	semidirect
3. #16839841	C₂	semidirect
4. #16839835	C₂	semidirect
5. C₈×D₂₄₄	C₂	semidirect
6. C₂×C₄⋊#92418	C₂	semidirect
7. #16839730	C₂	semidirect
8. C₄×M₁₆×C₆₁	C₂	semidirect
43. M₁₆×C₆₁	D₄	semidirect
94. C₄×C₆₁	#131	semidirect
111. C₄×M₁₆	D₆₁	semidirect
112. C₆₁	#2935	semidirect
148. M₁₆	D₂₄₄	semidirect
199. C₄	#14519785	semidirect