Subgroups — M₁₆×Dic₁₂₂ #21454772

Order 7808.

388 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×C₄×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
2	#16839892	C₂	1	2	yes	maximal, complemented
3	#16839890	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₁₂₂	C₂	1	2	yes	maximal, complemented
5	#16839786	C₂	1	1	yes	maximal, characteristic
6	#16839785	C₂	1	2	yes	maximal
7	M₁₆×Dic₆₁	C₂	1	2	yes	maximal
8	C₄×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₂₂	C₄	1	1	yes	characteristic
10	C₂×#11758194	C₂²	1	2	yes	—
11	C₄×Dic₁₂₂	C₄	1	1	yes	characteristic
12	C₄×Dic₁₂₂	C₄	1	2	yes	—
13	C₄×Dic₁₂₂	C₄	1	1	yes	characteristic
14	C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
15	C₄×Dic₁₂₂	C₂²	1	2	yes	—
16	C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₆₁	C₄	1	1	yes	characteristic
18	C₂×#11758196	C₄	1	2	yes	—
19	C₂×C₄×Dic₆₁	C₂²	1	2	yes	—
20	C₄⋊#92418	C₂²	1	2	yes	—
21	C₄⋊#92418	C₂²	1	2	yes	—
22	#14519706	C₂²	1	4	yes	—
23	#14519706	C₂²	1	4	yes	—
24	#14519687	C₂²	1	4	yes	—
25	C₈×Dic₆₁	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	C₂²×Dic₆₁	C₂×C₄	1	2	yes	—
35	C₂²×Dic₆₁	C₂×C₄	1	2	yes	—
36	C₂×Dic₁₂₂	—	2	4	no	—
37	C₂×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
38	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
39	C₂×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
40	M₁₆×C₆₁	—	2	2	no	—
41	M₁₆×C₆₁	Q₈	1	4	yes	complemented
42	C₂×C₈×C₆₁	C₂³	1	2	yes	—
43	C₂×C₈×C₆₁	C₂³	1	2	yes	—
44	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₆₁	C₂³	1	1	yes	characteristic
47	C₄²×C₆₁	C₂³	1	1	yes	characteristic
48	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
49	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
51	#11758196	C₂×C₄	1	2	yes	—
52	#11758196	C₂×C₄	1	2	yes	—
53	#11758196	C₂×C₄	1	2	yes	—
54	#11758196	C₂×C₄	1	2	yes	—
55	C₄×Dic₆₁	C₂×C₄	1	2	yes	—
56	C₄×Dic₆₁	C₂×C₄	1	2	yes	—
57	C₄×Dic₆₁	C₂³	1	2	yes	—
58	C₄×Dic₆₁	C₂³	1	2	yes	—
59	#11758194	—	2	4	no	—
60	C₂×#92418	C₂³	1	4	yes	—
61	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₁	—	2	2	no	—
63	C₂×C₄×C₆₁	—	4	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
70	C₂×C₄×C₆₁	#41	1	2	yes	—
71	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₆₁	—	2	4	no	—
73	C₂×Dic₆₁	—	4	2	no	—
74	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
75	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
76	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
77	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
78	Dic₁₂₂	—	2	2	no	—
79	Dic₁₂₂	—	2	4	no	—
80	Dic₁₂₂	M₁₆	1	4	yes	direct product factor
81	C₈×C₆₁	—	2	2	no	—
82	C₈×C₆₁	C₂×Q₈	1	4	yes	complemented
83	#92418	—	2	4	no	—
84	C₂²×C₆₁	—	2	1	no	—
85	C₂²×C₆₁	—	2	1	no	—
86	C₂²×C₆₁	C₄×Q₈	1	2	yes	—
87	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
88	C₄×C₆₁	—	2	1	no	—
89	C₄×C₆₁	—	2	1	no	—
90	C₄×C₆₁	—	2	1	no	—
91	C₄×C₆₁	C₄×Q₈	1	2	yes	—
92	C₄×C₆₁	C₂×M₁₆	1	2	yes	—
93	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
94	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
95	Dic₆₁	—	2	2	no	—
96	Dic₆₁	—	2	2	no	—
97	Dic₆₁	—	2	2	no	—
98	Dic₆₁	C₂×M₁₆	1	4	yes	—
99	Q₈×M₁₆	—	61	1	no	maximal, Sylow
100	C₂×C₆₁	—	2	2	no	—
101	C₂×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
102	C₂×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
103	C₂×C₆₁	#568	1	1	yes	characteristic
104	C₂×C₄×Q₈	—	61	1	no	—
105	#446	—	61	4	no	—
106	#446	—	61	2	no	—
107	C₈×Q₈	—	61	2	no	—
108	#423	—	61	2	no	—
109	#423	—	61	1	no	—
110	C₄×M₁₆	—	61	2	no	—
111	C₄×M₁₆	D₆₁	1	1	yes	characteristic
112	C₆₁	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₈	—	61	1	no	—
114	C₂×M₁₆	—	61	2	no	—
115	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
116	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
117	C₄×Q₈	—	61	1	no	—
118	C₄×Q₈	—	61	2	no	—
119	C₄×Q₈	—	61	1	no	—
120	C₄×Q₈	—	61	1	no	—
121	C₄×Q₈	—	61	2	no	—
122	C₄×Q₈	—	61	1	no	—
123	C₂×C₄⋊C₄	—	61	2	no	—
124	C₂×C₄⋊C₄	—	61	1	no	—
125	C₂×C₄²	—	61	2	no	—
126	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
127	C₄⋊C₈	—	61	4	no	—
128	C₄⋊C₈	—	61	4	no	—
129	C₄⋊C₈	—	61	2	no	—
130	C₄⋊C₈	—	61	2	no	—
131	#97	—	61	4	no	—
132	#97	C₂×D₆₁	1	2	yes	—
133	C₄×C₈	—	61	4	no	—
134	C₄×C₈	C₂×D₆₁	1	2	yes	—
135	C₂×Q₈	—	61	1	no	—
136	C₂×Q₈	—	61	2	no	—
137	C₂×Q₈	—	61	1	no	—
138	C₂×Q₈	—	122	4	no	—
139	C₂²×C₄	—	61	2	no	—
140	C₂²×C₄	—	61	2	no	—
141	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
142	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
143	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
144	M₁₆	—	2	2	no	—
145	M₁₆	—	122	4	no	—
146	M₁₆	Dic₁₂₂	1	4	yes	direct product factor
147	C₂×C₈	—	61	4	no	—
148	C₂×C₈	C₂²×D₆₁	1	2	yes	—
149	C₂×C₈	C₂²×D₆₁	1	2	yes	—
150	C₄⋊C₄	—	61	2	no	—
151	C₄⋊C₄	—	61	2	no	—
152	C₄⋊C₄	—	61	2	no	—
153	C₄⋊C₄	—	61	2	no	—
154	C₄⋊C₄	—	61	2	no	—
155	C₄⋊C₄	—	61	1	no	—
156	C₄⋊C₄	—	61	1	no	—
157	C₄²	—	61	2	no	—
158	C₄²	—	61	2	no	—
159	C₄²	—	61	2	no	—
160	C₄²	—	61	2	no	—
161	C₄²	C₄×D₆₁	1	1	yes	characteristic
162	C₄²	C₄×D₆₁	1	1	yes	characteristic
163	C₄²	C₂²×D₆₁	1	1	yes	characteristic
164	C₄²	C₂²×D₆₁	1	1	yes	characteristic
165	C₂³	C₂×C₄×D₆₁	1	1	yes	characteristic
166	Q₈	—	61	4	no	—
167	Q₈	—	122	4	no	—
168	Q₈	—	122	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	61	2	no	—
172	C₂×C₄	—	61	2	no	—
173	C₂×C₄	—	61	2	no	—
174	C₂×C₄	—	61	2	no	—
175	C₂×C₄	—	122	4	no	—
176	C₂×C₄	—	244	2	no	—
177	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
178	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
183	C₂×C₄	#11758222	1	2	yes	—
184	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	122	4	no	—
187	C₈	C₂×Dic₁₂₂	1	4	yes	complemented
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	C₄×Dic₁₂₂	1	2	yes	—
191	C₂²	C₂²×C₄×D₆₁	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	61	4	no	—
196	C₄	—	122	2	no	—
197	C₄	—	122	2	no	—
198	C₄	—	122	2	no	—
199	C₄	C₄×Dic₁₂₂	1	2	yes	—
200	C₄	M₁₆×D₆₁	1	2	yes	—
201	C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
202	C₄	C₂×#11758222	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	C₂×C₄×Dic₁₂₂	1	1	yes	characteristic
205	C₂	C₂×M₁₆×D₆₁	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	Q₈×M₁₆
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
2. #16839892	C₂	semidirect
3. #16839890	C₂	semidirect
4. C₈×Dic₁₂₂	C₂	semidirect
41. M₁₆×C₆₁	Q₈	semidirect
80. Dic₁₂₂	M₁₆	direct (N × H)
82. C₈×C₆₁	C₂×Q₈	semidirect
112. C₆₁	Q₈×M₁₆	semidirect
146. M₁₆	Dic₁₂₂	direct (N × H)
187. C₈	C₂×Dic₁₂₂	semidirect

Subgroups — M16×Dic122 #21454772

Conjugacy classes of subgroups

Complements

Subgroups — M₁₆×Dic₁₂₂ #21454772