Subgroups — Q₈×M₁₆×C₁₀₁ #27168317

Order 12928.

388 conjugacy classes of subgroups.

The subgroup lattice has 134 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	#446×C₁₀₁	C₂	1	6	yes	maximal, complemented
3	C₈×Q₈×C₁₀₁	C₂	1	2	yes	maximal, complemented
4	#423×C₁₀₁	C₂	1	3	yes	maximal
5	C₄×M₁₆×C₁₀₁	C₂	1	3	yes	maximal
6	C₂²×Q₈×C₁₀₁	C₄	1	1	yes	characteristic
7	C₂×M₁₆×C₁₀₁	C₂²	1	3	yes	—
8	C₂×M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
9	C₄×Q₈×C₁₀₁	C₄	1	3	yes	—
10	C₄×Q₈×C₁₀₁	C₄	1	1	yes	characteristic
11	C₄×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
12	C₄×Q₈×C₁₀₁	C₂²	1	3	yes	—
13	C₂×C₄⋊C₄×C₁₀₁	C₄	1	3	yes	—
14	C₂×C₄²×C₁₀₁	C₂²	1	3	yes	—
15	C₄⋊C₈×C₁₀₁	C₂²	1	6	yes	—
16	C₄⋊C₈×C₁₀₁	C₂²	1	6	yes	—
17	#97×C₁₀₁	C₂²	1	6	yes	—
18	C₄×C₈×C₁₀₁	C₂²	1	6	yes	—
19	C₂×Q₈×C₁₀₁	—	2	4	no	—
20	C₂×Q₈×C₁₀₁	C₂×C₄	1	3	yes	—
21	C₂×Q₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₁	C₂×C₄	1	3	yes	—
23	C₂²×C₄×C₁₀₁	C₂×C₄	1	3	yes	—
24	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
25	M₁₆×C₁₀₁	—	2	6	no	—
26	M₁₆×C₁₀₁	Q₈	1	4	yes	direct product factor
27	C₂×C₈×C₁₀₁	C₂³	1	6	yes	—
28	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
29	C₄⋊C₄×C₁₀₁	C₂×C₄	1	6	yes	—
30	C₄⋊C₄×C₁₀₁	C₂×C₄	1	3	yes	—
31	C₄⋊C₄×C₁₀₁	C₂×C₄	1	3	yes	—
32	C₄²×C₁₀₁	C₂×C₄	1	3	yes	—
33	C₄²×C₁₀₁	C₂×C₄	1	3	yes	—
34	C₄²×C₁₀₁	C₂³	1	3	yes	—
35	C₄²×C₁₀₁	C₂³	1	3	yes	—
36	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₁₀₁	—	2	6	no	—
38	Q₈×C₁₀₁	M₁₆	1	4	yes	direct product factor
39	C₂×C₄×C₁₀₁	—	2	6	no	—
40	C₂×C₄×C₁₀₁	—	4	3	no	—
41	C₂×C₄×C₁₀₁	C₂²×C₄	1	3	yes	—
42	C₂×C₄×C₁₀₁	C₂²×C₄	1	3	yes	—
43	C₂×C₄×C₁₀₁	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₁	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₁₀₁	#41	1	2	yes	—
48	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
49	C₈×C₁₀₁	—	2	6	no	—
50	C₈×C₁₀₁	C₂×Q₈	1	4	yes	complemented
51	C₂²×C₁₀₁	—	2	1	no	—
52	C₂²×C₁₀₁	—	2	1	no	—
53	C₂²×C₁₀₁	C₄×Q₈	1	2	yes	—
54	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
55	C₄×C₁₀₁	—	2	3	no	—
56	C₄×C₁₀₁	—	2	3	no	—
57	C₄×C₁₀₁	—	2	3	no	—
58	C₄×C₁₀₁	C₄×Q₈	1	2	yes	—
59	C₄×C₁₀₁	C₂×M₁₆	1	6	yes	—
60	C₄×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
61	C₄×C₁₀₁	C₂×#41	1	1	yes	characteristic
62	C₂×C₁₀₁	—	2	2	no	—
63	C₂×C₁₀₁	C₂×C₄×Q₈	1	1	yes	characteristic
64	C₂×C₁₀₁	C₂²×M₁₆	1	1	yes	characteristic
65	C₂×C₁₀₁	#568	1	1	yes	characteristic
66	Q₈×M₁₆	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₁₀₁	Q₈×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
68	C₂×C₄×Q₈	C₂×C₁₀₁	1	1	yes	characteristic
69	#446	C₂×C₁₀₁	1	6	yes	complemented
70	C₈×Q₈	C₂×C₁₀₁	1	2	yes	complemented
71	#423	C₂×C₁₀₁	1	3	yes	—
72	C₄×M₁₆	C₂×C₁₀₁	1	3	yes	—
73	C₂²×Q₈	C₄×C₁₀₁	1	1	yes	characteristic
74	C₂×M₁₆	C₂²×C₁₀₁	1	3	yes	—
75	C₂×M₁₆	C₂²×C₁₀₁	1	1	yes	characteristic
76	C₄×Q₈	C₄×C₁₀₁	1	3	yes	—
77	C₄×Q₈	C₄×C₁₀₁	1	1	yes	characteristic
78	C₄×Q₈	C₂²×C₁₀₁	1	1	yes	characteristic
79	C₄×Q₈	C₂²×C₁₀₁	1	3	yes	—
80	C₂×C₄⋊C₄	C₄×C₁₀₁	1	3	yes	—
81	C₂×C₄²	C₂²×C₁₀₁	1	3	yes	—
82	C₄⋊C₈	C₂²×C₁₀₁	1	6	yes	—
83	C₄⋊C₈	C₂²×C₁₀₁	1	6	yes	—
84	#97	C₂²×C₁₀₁	1	6	yes	—
85	C₄×C₈	C₂²×C₁₀₁	1	6	yes	—
86	C₂×Q₈	—	2	4	no	—
87	C₂×Q₈	C₂×C₄×C₁₀₁	1	3	yes	—
88	C₂×Q₈	C₂×C₄×C₁₀₁	1	1	yes	characteristic
89	C₂²×C₄	C₂×C₄×C₁₀₁	1	3	yes	—
90	C₂²×C₄	C₂×C₄×C₁₀₁	1	3	yes	—
91	C₂²×C₄	C₂³×C₁₀₁	1	1	yes	characteristic
92	M₁₆	—	2	6	no	—
93	M₁₆	Q₈×C₁₀₁	1	4	yes	direct product factor
94	C₂×C₈	C₂³×C₁₀₁	1	6	yes	—
95	C₂×C₈	C₂³×C₁₀₁	1	2	yes	—
96	C₄⋊C₄	C₂×C₄×C₁₀₁	1	6	yes	—
97	C₄⋊C₄	C₂×C₄×C₁₀₁	1	3	yes	—
98	C₄⋊C₄	C₂×C₄×C₁₀₁	1	3	yes	—
99	C₄²	C₂×C₄×C₁₀₁	1	3	yes	—
100	C₄²	C₂×C₄×C₁₀₁	1	3	yes	—
101	C₄²	C₂³×C₁₀₁	1	3	yes	—
102	C₄²	C₂³×C₁₀₁	1	3	yes	—
103	C₂³	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
104	Q₈	—	2	6	no	—
105	Q₈	M₁₆×C₁₀₁	1	4	yes	direct product factor
106	C₂×C₄	—	2	6	no	—
107	C₂×C₄	—	4	3	no	—
108	C₂×C₄	C₂²×C₄×C₁₀₁	1	3	yes	—
109	C₂×C₄	C₂²×C₄×C₁₀₁	1	3	yes	—
110	C₂×C₄	C₂²×C₄×C₁₀₁	1	3	yes	—
111	C₂×C₄	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
112	C₂×C₄	C₂²×C₄×C₁₀₁	1	3	yes	—
113	C₂×C₄	C₂×Q₈×C₁₀₁	1	2	yes	—
114	C₂×C₄	#41×C₁₀₁	1	2	yes	—
115	C₂×C₄	C₂⁴×C₁₀₁	1	1	yes	characteristic
116	C₈	—	2	6	no	—
117	C₈	C₂×Q₈×C₁₀₁	1	4	yes	complemented
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₄×Q₈×C₁₀₁	1	2	yes	—
121	C₂²	C₂³×C₄×C₁₀₁	1	1	yes	characteristic
122	C₄	—	2	3	no	—
123	C₄	—	2	3	no	—
124	C₄	—	2	3	no	—
125	C₄	C₄×Q₈×C₁₀₁	1	2	yes	—
126	C₄	C₂×M₁₆×C₁₀₁	1	6	yes	—
127	C₄	C₂²×Q₈×C₁₀₁	1	1	yes	characteristic
128	C₄	C₂×#41×C₁₀₁	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	C₂×C₄×Q₈×C₁₀₁	1	1	yes	characteristic
131	C₂	C₂²×M₁₆×C₁₀₁	1	1	yes	characteristic
132	C₂	#568×C₁₀₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₁₀₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₁
socle	C₂²×C₁₀₁	C₂³×C₄
prime radical	C₂³×C₁₀₁	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #446×C₁₀₁	C₂	semidirect
3. C₈×Q₈×C₁₀₁	C₂	semidirect
26. M₁₆×C₁₀₁	Q₈	direct (N × H)
38. Q₈×C₁₀₁	M₁₆	direct (N × H)
50. C₈×C₁₀₁	C₂×Q₈	semidirect
66. Q₈×M₁₆	C₁₀₁	direct (N × H)
67. C₁₀₁	Q₈×M₁₆	direct (N × H)
69. #446	C₂×C₁₀₁	semidirect
70. C₈×Q₈	C₂×C₁₀₁	semidirect
93. M₁₆	Q₈×C₁₀₁	direct (N × H)
105. Q₈	M₁₆×C₁₀₁	direct (N × H)
117. C₈	C₂×Q₈×C₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₁₀₁	2
#446×C₁₀₁	2
C₈×Q₈×C₁₀₁	2
#423×C₁₀₁	2
C₄×M₁₆×C₁₀₁	2
Q₈×M₁₆	101

Subgroups — Q8×M16×C101 #27168317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×M₁₆×C₁₀₁ #27168317