Subgroups — C₄²×#6122 #16796727

Order 3328.

1734 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#6122	C₂	1	6	yes	maximal
2	C₄×#88110	C₂	1	6	yes	maximal, complemented
3	C₄²×#1057	C₂	1	2	yes	maximal, complemented
4	C₂×C₄²×Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#6122	C₂²	1	4	yes	—
6	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
7	C₂²×C₄×Dic₁₃	C₄	1	3	yes	—
8	C₂²×C₄×Dic₁₃	C₂²	1	3	yes	—
9	C₄×#6122	C₄	1	48	yes	direct product factor
10	C₂×#88110	C₂²	1	12	yes	—
11	C₂×C₄×#1057	C₂²	1	12	yes	—
12	C₄²×Dic₁₃	C₄	1	1	yes	characteristic
13	C₄²×Dic₁₃	C₄	1	3	yes	—
14	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₄²×Dic₁₃	C₂²	1	3	yes	—
16	C₂³×C₄×C₁₃	C₂×C₄	1	3	yes	—
17	C₂³×Dic₁₃	C₂×C₄	1	3	yes	—
18	C₂³×Dic₁₃	C₂³	1	1	yes	characteristic
19	C₂×#6122	C₂×C₄	1	48	yes	—
20	C₂²×#1057	C₂³	1	8	yes	—
21	C₂×C₄²×C₁₃	—	2	4	no	—
22	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₃	—	2	12	no	—
25	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
26	C₂×C₄×Dic₁₃	C₂×C₄	1	6	yes	—
27	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
28	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
29	C₂×C₄×Dic₁₃	C₂×C₄	1	24	yes	—
30	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
31	C₂×C₄×Dic₁₃	C₂³	1	3	yes	—
32	C₂×C₄×Dic₁₃	C₂³	1	3	yes	—
33	#88110	C₂×C₄	1	48	yes	complemented
34	C₄×#1057	C₂×C₄	1	48	yes	complemented
35	C₄²×M₁₆	—	13	1	no	maximal, Sylow
36	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	—	2	6	no	—
38	C₂²×C₄×C₁₃	—	2	6	no	—
39	C₂²×C₄×C₁₃	C₄²	1	12	yes	—
40	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
41	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
42	C₂²×Dic₁₃	—	2	12	no	—
43	C₂²×Dic₁₃	C₄²	1	12	yes	—
44	C₂²×Dic₁₃	C₂²×C₄	1	3	yes	—
45	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₃	C₂²×C₄	1	12	yes	—
47	C₂²×Dic₁₃	C₂²×C₄	1	3	yes	—
48	C₂²×Dic₁₃	C₂⁴	1	1	yes	characteristic
49	#6122	C₄²	1	64	yes	direct product factor
50	C₂×#1057	C₂²×C₄	1	48	yes	—
51	C₄²×C₁₃	—	2	6	no	—
52	C₄²×C₁₃	M₁₆	1	4	yes	complemented
53	C₄×Dic₁₃	—	2	6	no	—
54	C₄×Dic₁₃	—	2	12	no	—
55	C₄×Dic₁₃	C₄²	1	12	yes	—
56	C₄×Dic₁₃	C₄²	1	12	yes	—
57	C₄×Dic₁₃	M₁₆	1	12	yes	complemented
58	C₄×Dic₁₃	C₂²×C₄	1	12	yes	—
59	C₄×Dic₁₃	C₂²×C₄	1	12	yes	—
60	C₂×C₄×M₁₆	—	13	6	no	—
61	C₂×C₄³	—	13	1	no	—
62	C₄×#97	—	13	6	no	—
63	C₄²×C₈	—	13	2	no	—
64	C₂³×C₁₃	—	2	3	no	—
65	C₂³×C₁₃	—	2	1	no	—
66	C₂³×C₁₃	C₂×C₄²	1	6	yes	—
67	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃	—	2	3	no	—
69	C₂×C₄×C₁₃	—	2	3	no	—
70	C₂×C₄×C₁₃	—	2	24	no	—
71	C₂×C₄×C₁₃	C₂×C₄²	1	6	yes	—
72	C₂×C₄×C₁₃	C₂×C₄²	1	6	yes	—
73	C₂×C₄×C₁₃	C₂×M₁₆	1	6	yes	—
74	C₂×C₄×C₁₃	C₂×M₁₆	1	6	yes	—
75	C₂×Dic₁₃	—	2	6	no	—
76	C₂×Dic₁₃	—	2	24	no	—
77	C₂×Dic₁₃	C₂×C₄²	1	6	yes	—
78	C₂×Dic₁₃	C₂×C₄²	1	6	yes	—
79	C₂×Dic₁₃	C₂×C₄²	1	16	yes	—
80	C₂×Dic₁₃	C₂×C₄²	1	6	yes	—
81	C₂×Dic₁₃	C₂×M₁₆	1	12	yes	—
82	C₂×Dic₁₃	C₂³×C₄	1	6	yes	—
83	#1057	C₂×C₄²	1	32	yes	complemented
84	C₂²×M₁₆	—	13	4	no	—
85	C₂²×C₄²	—	13	3	no	—
86	C₂²×C₄²	—	13	3	no	—
87	C₂²×C₄²	#261	1	1	yes	characteristic
88	C₄×M₁₆	—	13	48	no	—
89	C₂×#97	—	13	12	no	—
90	C₂×C₄×C₈	—	13	12	no	—
91	C₄³	—	13	3	no	—
92	C₄³	—	13	3	no	—
93	C₄³	—	13	1	no	—
94	C₄³	—	13	1	no	—
95	C₂²×C₁₃	—	2	6	no	—
96	C₂²×C₁₃	—	2	6	no	—
97	C₂²×C₁₃	C₄³	1	4	yes	—
98	C₂²×C₁₃	C₂²×C₄²	1	3	yes	—
99	C₂²×C₁₃	C₂²×M₁₆	1	3	yes	—
100	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
101	C₄×C₁₃	—	2	6	no	—
102	C₄×C₁₃	C₄×M₁₆	1	12	yes	complemented
103	Dic₁₃	—	2	6	no	—
104	Dic₁₃	C₄³	1	4	yes	—
105	Dic₁₃	C₄×M₁₆	1	12	yes	complemented
106	Dic₁₃	C₂²×C₄²	1	4	yes	—
107	C₂³×C₄	—	13	1	no	—
108	C₂³×C₄	—	13	3	no	—
109	C₂³×C₄	C₂×#261	1	3	yes	—
110	C₂×M₁₆	—	13	48	no	—
111	C₂²×C₈	—	13	8	no	—
112	C₂×C₄²	—	2	4	no	—
113	C₂×C₄²	—	13	24	no	—
114	C₂×C₄²	—	13	3	no	—
115	C₂×C₄²	—	13	3	no	—
116	C₂×C₄²	—	13	3	no	—
117	C₂×C₄²	—	13	3	no	—
118	C₂×C₄²	—	13	3	no	—
119	C₂×C₄²	—	13	6	no	—
120	C₂×C₄²	—	13	3	no	—
121	C₂×C₄²	—	26	12	no	—
122	C₂×C₄²	C₂×#261	1	3	yes	—
123	C₂×C₄²	C₂×#261	1	1	yes	characteristic
124	#97	—	13	48	no	—
125	C₄×C₈	—	13	48	no	—
126	C₂×C₁₃	—	2	4	no	—
127	C₂×C₁₃	C₂×C₄³	1	1	yes	characteristic
128	C₂×C₁₃	C₂×C₄×M₁₆	1	3	yes	—
129	C₂×C₁₃	C₂×C₄×M₁₆	1	3	yes	—
130	C₂⁴	C₂²×#261	1	1	yes	characteristic
131	C₂²×C₄	—	2	6	no	—
132	C₂²×C₄	—	2	6	no	—
133	C₂²×C₄	—	13	12	no	—
134	C₂²×C₄	—	13	1	no	—
135	C₂²×C₄	—	13	3	no	—
136	C₂²×C₄	—	13	1	no	—
137	C₂²×C₄	—	13	12	no	—
138	C₂²×C₄	—	13	3	no	—
139	C₂²×C₄	—	26	12	no	—
140	C₂²×C₄	C₄×#261	1	12	yes	—
141	C₂²×C₄	C₂²×#261	1	3	yes	—
142	C₂²×C₄	C₂²×#261	1	3	yes	—
143	M₁₆	—	13	64	no	—
144	C₂×C₈	—	13	48	no	—
145	C₄²	—	2	6	no	—
146	C₄²	—	13	12	no	—
147	C₄²	—	13	12	no	—
148	C₄²	—	13	12	no	—
149	C₄²	—	13	12	no	—
150	C₄²	—	13	12	no	—
151	C₄²	—	26	12	no	—
152	C₄²	—	26	6	no	—
153	C₄²	#6122	1	4	yes	direct product factor
154	C₁₃	C₄²×M₁₆	1	1	yes	characteristic, complemented, Sylow
155	C₂³	—	2	3	no	—
156	C₂³	—	2	1	no	—
157	C₂³	C₂×C₄×#261	1	6	yes	—
158	C₂³	C₂³×#261	1	1	yes	characteristic
159	C₂×C₄	—	2	3	no	—
160	C₂×C₄	—	2	3	no	—
161	C₂×C₄	—	2	24	no	—
162	C₂×C₄	—	13	16	no	—
163	C₂×C₄	—	13	12	no	—
164	C₂×C₄	—	13	6	no	—
165	C₂×C₄	—	13	6	no	—
166	C₂×C₄	—	13	6	no	—
167	C₂×C₄	—	13	6	no	—
168	C₂×C₄	—	26	24	no	—
169	C₂×C₄	—	26	6	no	—
170	C₂×C₄	D₁₃⋊M₁₆	1	6	yes	—
171	C₂×C₄	C₂×C₄×#261	1	6	yes	—
172	C₂×C₄	C₂×C₄×#261	1	6	yes	—
173	C₂×C₄	C₂×#6122	1	6	yes	—
174	C₈	—	13	32	no	—
175	C₂²	—	2	6	no	—
176	C₂²	—	2	6	no	—
177	C₂²	C₄²×#261	1	4	yes	—
178	C₂²	C₂×D₁₃⋊M₁₆	1	3	yes	—
179	C₂²	C₂²×C₄×#261	1	3	yes	—
180	C₂²	C₂²×#6122	1	1	yes	characteristic
181	C₄	—	2	6	no	—
182	C₄	—	13	12	no	—
183	C₄	—	13	4	no	—
184	C₄	—	13	4	no	—
185	C₄	—	26	6	no	—
186	C₄	C₄×#6122	1	12	yes	direct product factor
187	C₂	—	2	4	no	—
188	C₂	C₂×C₄²×#261	1	1	yes	characteristic
189	C₂	C₂×C₄×#6122	1	3	yes	—
190	C₂	C₄×D₁₃⋊M₁₆	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×#261
commutator	C₂×C₁₃	C₂×C₄³
Frattini	C₂³	C₂³×#261
Fitting	C₂²×C₄²×C₁₃	C₄
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×Dic₁₃	C₂⁴
hypercenter	C₂²×C₄²	#261
hypocenter	C₁₃	C₄²×M₁₆
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×#88110	C₂	semidirect
3. C₄²×#1057	C₂	semidirect
9. C₄×#6122	C₄	direct (N × H)
33. #88110	C₂×C₄	semidirect
34. C₄×#1057	C₂×C₄	semidirect
49. #6122	C₄²	direct (N × H)
52. C₄²×C₁₃	M₁₆	semidirect
57. C₄×Dic₁₃	M₁₆	semidirect
83. #1057	C₂×C₄²	semidirect
102. C₄×C₁₃	C₄×M₁₆	semidirect
105. Dic₁₃	C₄×M₁₆	semidirect
153. C₄²	#6122	direct (N × H)
154. C₁₃	C₄²×M₁₆	semidirect
186. C₄	C₄×#6122	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×#6122	2
C₄×#88110	2
C₄²×#1057	2
C₂×C₄²×Dic₁₃	2
C₄²×M₁₆	13

Subgroups — C42×#6122 #16796727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×#6122 #16796727