Subgroups — C₂×#41×Dic₁₀₁ #27184314

Order 12928.

1728 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×Dic₁₀₁	C₂	1	16	yes	maximal, direct product factor
2	C₂×Q₈×Dic₁₀₁	C₂	1	2	yes	maximal, complemented
3	C₂×D₄×Dic₁₀₁	C₂	1	6	yes	maximal, complemented
4	C₂×#15692214	C₂	1	3	yes	maximal, complemented
5	C₂²×C₄×Dic₁₀₁	C₂	1	3	yes	maximal, complemented
6	C₂²×#41×C₁₀₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₀₁	C₂²	1	6	yes	—
8	Q₈×Dic₁₀₁	C₂²	1	16	yes	complemented
9	D₄×Dic₁₀₁	C₂²	1	48	yes	complemented
10	#15692214	C₂²	1	24	yes	complemented
11	C₂×C₄⋊Dic₁₀₁	C₂²	1	6	yes	—
12	C₂×C₂²⋊Dic₁₀₁	C₂²	1	6	yes	—
13	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₁₀₁	C₂²	1	3	yes	—
15	C₂×C₄×Dic₁₀₁	C₂²	1	6	yes	—
16	C₂×C₄×Dic₁₀₁	C₂²	1	24	yes	complemented
17	C₂×#41×C₁₀₁	C₄	1	16	yes	complemented
18	C₂×#41×C₁₀₁	C₂²	1	8	yes	—
19	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
20	C₂²×D₄×C₁₀₁	C₂²	1	3	yes	—
21	C₂³×C₄×C₁₀₁	C₂²	1	3	yes	—
22	C₂²×Dic₁₀₁	—	2	24	no	—
23	C₂²×Dic₁₀₁	C₂³	1	6	yes	—
24	C₂²×Dic₁₀₁	C₂³	1	24	yes	—
25	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
26	C₄⋊Dic₁₀₁	C₂³	1	24	yes	—
27	C₂²⋊Dic₁₀₁	C₂³	1	24	yes	—
28	C₄×Dic₁₀₁	C₂³	1	4	yes	—
29	C₄×Dic₁₀₁	C₂³	1	12	yes	—
30	C₄×Dic₁₀₁	C₂³	1	24	yes	—
31	C₂⁴×C₁₀₁	C₂³	1	3	yes	—
32	#41×C₁₀₁	C₂×C₄	1	64	yes	complemented
33	C₂×Q₈×C₁₀₁	C₂×C₄	1	8	yes	complemented
34	C₂×Q₈×C₁₀₁	C₂³	1	4	yes	—
35	C₂×D₄×C₁₀₁	C₂×C₄	1	24	yes	complemented
36	C₂×D₄×C₁₀₁	C₂³	1	12	yes	—
37	C₂²×C₄×C₁₀₁	C₂×C₄	1	24	yes	complemented
38	C₂²×C₄×C₁₀₁	C₂³	1	3	yes	—
39	C₂²×C₄×C₁₀₁	C₂³	1	12	yes	—
40	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
41	C₂³×C₁₀₁	—	2	12	no	—
42	C₂³×C₁₀₁	C₂²×C₄	1	12	yes	—
43	C₂³×C₁₀₁	C₂⁴	1	6	yes	—
44	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
45	Q₈×C₁₀₁	C₂²×C₄	1	16	yes	complemented
46	D₄×C₁₀₁	C₂²×C₄	1	48	yes	complemented
47	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₁₀₁	C₂²×C₄	1	12	yes	—
49	C₂×C₄×C₁₀₁	C₂²×C₄	1	48	yes	complemented
50	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
51	C₂×C₄×C₁₀₁	C₂⁴	1	6	yes	—
52	C₂×Dic₁₀₁	—	2	12	no	—
53	C₂×Dic₁₀₁	—	2	48	no	—
54	C₂×Dic₁₀₁	#41	1	8	yes	direct product factor
55	C₂×Dic₁₀₁	C₂⁴	1	12	yes	—
56	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
57	C₂²×C₁₀₁	—	2	24	no	—
58	C₂²×C₁₀₁	—	2	12	no	—
59	C₂²×C₁₀₁	C₂³×C₄	1	12	yes	—
60	C₂²×C₁₀₁	C₂³×C₄	1	2	yes	—
61	C₂²×C₁₀₁	C₂×#41	1	4	yes	—
62	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
63	C₄×C₁₀₁	C₂³×C₄	1	4	yes	—
64	C₄×C₁₀₁	C₂³×C₄	1	12	yes	—
65	Dic₁₀₁	—	2	12	no	—
66	Dic₁₀₁	C₂×#41	1	8	yes	direct product factor
67	C₂×C₁₀₁	—	2	12	no	—
68	C₂×C₁₀₁	C₄×#41	1	4	yes	complemented
69	C₂×C₁₀₁	C₂⁴×C₄	1	1	yes	characteristic
70	C₂×C₁₀₁	C₂²×#41	1	2	yes	—
71	C₂×C₄×#41	—	101	1	no	maximal, Sylow
72	C₁₀₁	C₂×C₄×#41	1	1	yes	characteristic, complemented, Sylow
73	C₂²×#41	D₁₀₁	1	1	yes	characteristic
74	C₄×#41	—	101	16	no	—
75	C₂×C₄×Q₈	—	101	2	no	—
76	C₂×C₄×D₄	—	101	6	no	—
77	C₂×#117	—	101	3	no	—
78	C₂²×C₄²	—	101	3	no	—
79	C₂×#41	Dic₁₀₁	1	16	yes	direct product factor
80	C₂×#41	C₂×D₁₀₁	1	8	yes	—
81	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
82	C₂²×D₄	C₂×D₁₀₁	1	3	yes	—
83	C₂³×C₄	—	101	6	no	—
84	C₂³×C₄	C₂×D₁₀₁	1	3	yes	—
85	C₄×Q₈	—	101	16	no	—
86	C₄×D₄	—	101	48	no	—
87	#117	—	101	24	no	—
88	C₂×C₄⋊C₄	—	101	6	no	—
89	C₂×C₂²⋊C₄	—	101	6	no	—
90	C₂×C₄²	—	101	6	no	—
91	C₂×C₄²	—	101	24	no	—
92	C₂×C₄²	—	101	3	no	—
93	C₂×C₄²	—	101	1	no	—
94	C₂⁴	C₂²×D₁₀₁	1	3	yes	—
95	#41	C₂×Dic₁₀₁	1	64	yes	direct product factor
96	C₂×Q₈	C₂×Dic₁₀₁	1	8	yes	complemented
97	C₂×Q₈	C₂²×D₁₀₁	1	4	yes	—
98	C₂×D₄	C₂×Dic₁₀₁	1	24	yes	complemented
99	C₂×D₄	C₂²×D₁₀₁	1	12	yes	—
100	C₂²×C₄	—	101	2	no	—
101	C₂²×C₄	—	101	6	no	—
102	C₂²×C₄	—	101	24	no	—
103	C₂²×C₄	—	202	24	no	—
104	C₂²×C₄	C₂×Dic₁₀₁	1	24	yes	complemented
105	C₂²×C₄	C₂²×D₁₀₁	1	3	yes	—
106	C₂²×C₄	C₂²×D₁₀₁	1	12	yes	—
107	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
108	C₄⋊C₄	—	101	24	no	—
109	C₂²⋊C₄	—	101	24	no	—
110	C₄²	—	101	4	no	—
111	C₄²	—	101	12	no	—
112	C₄²	—	101	24	no	—
113	C₂³	—	2	12	no	—
114	C₂³	C₂²×Dic₁₀₁	1	12	yes	—
115	C₂³	C₂³×D₁₀₁	1	6	yes	—
116	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
117	Q₈	C₂²×Dic₁₀₁	1	16	yes	complemented
118	D₄	C₂²×Dic₁₀₁	1	48	yes	complemented
119	C₂×C₄	—	101	8	no	—
120	C₂×C₄	—	101	4	no	—
121	C₂×C₄	—	101	12	no	—
122	C₂×C₄	—	202	48	no	—
123	C₂×C₄	—	202	12	no	—
124	C₂×C₄	C₂²×Dic₁₀₁	1	4	yes	—
125	C₂×C₄	C₂²×Dic₁₀₁	1	12	yes	—
126	C₂×C₄	C₂²×Dic₁₀₁	1	48	yes	complemented
127	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
128	C₂×C₄	C₂³×D₁₀₁	1	6	yes	—
129	C₂²	—	2	12	no	—
130	C₂²	—	2	24	no	—
131	C₂²	C₂³×Dic₁₀₁	1	12	yes	—
132	C₂²	C₂³×Dic₁₀₁	1	2	yes	—
133	C₂²	#41×D₁₀₁	1	4	yes	—
134	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
135	C₄	—	101	8	no	—
136	C₄	—	202	12	no	—
137	C₄	C₂³×Dic₁₀₁	1	4	yes	—
138	C₄	C₂³×Dic₁₀₁	1	12	yes	—
139	C₂	—	2	12	no	—
140	C₂	#41×Dic₁₀₁	1	4	yes	direct product factor
141	C₂	C₂⁴×Dic₁₀₁	1	1	yes	characteristic
142	C₂	C₂×#41×D₁₀₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₁
commutator	C₂×C₁₀₁	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting, prime radical	C₂²×#41×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
cosocle	C₂²×C₁₀₁	C₂⁵
hypercenter	C₂²×#41	D₁₀₁
hypocenter	C₁₀₁	C₂×C₄×#41
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. #41×Dic₁₀₁	C₂	direct (N × H)
2. C₂×Q₈×Dic₁₀₁	C₂	semidirect
3. C₂×D₄×Dic₁₀₁	C₂	semidirect
4. C₂×#15692214	C₂	semidirect
5. C₂²×C₄×Dic₁₀₁	C₂	semidirect
8. Q₈×Dic₁₀₁	C₂²	semidirect
9. D₄×Dic₁₀₁	C₂²	semidirect
10. #15692214	C₂²	semidirect
16. C₂×C₄×Dic₁₀₁	C₂²	semidirect
17. C₂×#41×C₁₀₁	C₄	semidirect
32. #41×C₁₀₁	C₂×C₄	semidirect
33. C₂×Q₈×C₁₀₁	C₂×C₄	semidirect
35. C₂×D₄×C₁₀₁	C₂×C₄	semidirect
37. C₂²×C₄×C₁₀₁	C₂×C₄	semidirect
45. Q₈×C₁₀₁	C₂²×C₄	semidirect
46. D₄×C₁₀₁	C₂²×C₄	semidirect
49. C₂×C₄×C₁₀₁	C₂²×C₄	semidirect
54. C₂×Dic₁₀₁	#41	direct (N × H)
66. Dic₁₀₁	C₂×#41	direct (N × H)
68. C₂×C₁₀₁	C₄×#41	semidirect
72. C₁₀₁	C₂×C₄×#41	semidirect
79. C₂×#41	Dic₁₀₁	direct (N × H)
95. #41	C₂×Dic₁₀₁	direct (N × H)
96. C₂×Q₈	C₂×Dic₁₀₁	semidirect
98. C₂×D₄	C₂×Dic₁₀₁	semidirect
104. C₂²×C₄	C₂×Dic₁₀₁	semidirect
117. Q₈	C₂²×Dic₁₀₁	semidirect
118. D₄	C₂²×Dic₁₀₁	semidirect
126. C₂×C₄	C₂²×Dic₁₀₁	semidirect
140. C₂	#41×Dic₁₀₁	direct (N × H)

Maximal subgroups

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

subgroup	index
#41×Dic₁₀₁	2
C₂×Q₈×Dic₁₀₁	2
C₂×D₄×Dic₁₀₁	2
C₂×#15692214	2
C₂²×C₄×Dic₁₀₁	2
C₂²×#41×C₁₀₁	2
C₂×C₄×#41	101

Subgroups — C2×#41×Dic101 #27184314

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#41×Dic₁₀₁ #27184314