Subgroups — C₆₇×#11405 #30690656

Order 17152.

282 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#346×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1450	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1313	C₂	1	4	yes	maximal, complemented
5	C₂×M₃₂×C₆₇	C₄	1	2	yes	—
6	C₂²×C₁₆×C₆₇	C₄	1	2	yes	—
7	#431×C₆₇	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
9	#346×C₆₇	C₄	1	2	yes	—
10	#346×C₆₇	C₂²	1	2	yes	—
11	#346×C₆₇	C₂²	1	2	yes	—
12	C₄×C₁₆×C₆₇	C₄	1	2	yes	—
13	#322×C₆₇	C₂²	1	2	yes	—
14	C₈²×C₆₇	C₄	1	2	yes	—
15	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
18	M₃₂×C₆₇	—	2	4	no	—
19	C₂×C₁₆×C₆₇	C₈	1	8	yes	complemented
20	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
21	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
22	C₄⋊C₈×C₆₇	—	2	2	no	—
23	C₂²⋊C₈×C₆₇	—	2	2	no	—
24	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
27	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
30	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
31	C₂×C₈×C₆₇	—	2	4	no	—
32	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
33	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
35	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
36	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₆₇	C₄²	1	1	yes	characteristic
39	C₄²×C₆₇	M₁₆	1	2	yes	—
40	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₁₆×C₆₇	—	2	4	no	—
42	C₁₆×C₆₇	C₂×C₈	1	8	yes	—
43	C₂³×C₆₇	#97	1	1	yes	characteristic
44	C₂×C₄×C₆₇	—	4	2	no	—
45	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₆₇	#97	1	2	yes	—
47	C₂×C₄×C₆₇	#97	1	1	yes	characteristic
48	C₂×C₄×C₆₇	#97	1	2	yes	—
49	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
51	C₈×C₆₇	—	4	4	no	—
52	C₈×C₆₇	C₄×C₈	1	2	yes	—
53	C₈×C₆₇	#97	1	2	yes	—
54	C₈×C₆₇	C₂²×C₈	1	2	yes	—
55	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
56	C₂²×C₆₇	—	2	2	no	—
57	C₂²×C₆₇	#322	1	2	yes	—
58	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
59	C₄×C₆₇	—	2	2	no	—
60	C₄×C₆₇	—	2	2	no	—
61	C₄×C₆₇	#322	1	2	yes	—
62	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
63	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
64	#11405	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂×C₆₇	—	2	2	no	—
66	C₂×C₆₇	C₂×#322	1	1	yes	characteristic
67	C₂×C₆₇	#2110	1	2	yes	—
68	C₄×M₃₂	C₂×C₆₇	1	1	yes	characteristic
69	C₂×#346	C₂×C₆₇	1	1	yes	characteristic
70	#1450	C₂×C₆₇	1	1	yes	characteristic
71	#1313	C₂×C₆₇	1	4	yes	complemented
72	C₆₇	#11405	1	1	yes	characteristic, direct product factor, Sylow
73	C₂×M₃₂	C₄×C₆₇	1	2	yes	—
74	C₂²×C₁₆	C₄×C₆₇	1	2	yes	—
75	#431	C₄×C₆₇	1	2	yes	—
76	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
77	#346	C₄×C₆₇	1	2	yes	—
78	#346	C₂²×C₆₇	1	2	yes	—
79	#346	C₂²×C₆₇	1	2	yes	—
80	C₄×C₁₆	C₄×C₆₇	1	2	yes	—
81	#322	C₂²×C₆₇	1	2	yes	—
82	C₈²	C₄×C₆₇	1	2	yes	—
83	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
84	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
85	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
86	M₃₂	—	2	4	no	—
87	C₂×C₁₆	C₈×C₆₇	1	8	yes	complemented
88	C₂×C₁₆	C₂×C₄×C₆₇	1	4	yes	—
89	C₂×C₁₆	C₂×C₄×C₆₇	1	4	yes	—
90	C₄⋊C₈	—	2	2	no	—
91	C₂²⋊C₈	—	2	2	no	—
92	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
93	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
94	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	—
95	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
96	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
97	C₂²×C₄	C₄²×C₆₇	1	1	yes	characteristic
98	C₂²×C₄	M₁₆×C₆₇	1	2	yes	—
99	C₂×C₈	—	2	4	no	—
100	C₂×C₈	C₄²×C₆₇	1	1	yes	characteristic
101	C₂×C₈	C₄²×C₆₇	1	1	yes	characteristic
102	C₂×C₈	C₂×C₈×C₆₇	1	4	yes	—
103	C₂×C₈	M₁₆×C₆₇	1	4	yes	—
104	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
105	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
106	C₄²	C₄²×C₆₇	1	1	yes	characteristic
107	C₄²	M₁₆×C₆₇	1	2	yes	—
108	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
109	C₁₆	—	2	4	no	—
110	C₁₆	C₂×C₈×C₆₇	1	8	yes	—
111	C₂³	#97×C₆₇	1	1	yes	characteristic
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₄×C₈×C₆₇	1	2	yes	—
114	C₂×C₄	#97×C₆₇	1	2	yes	—
115	C₂×C₄	#97×C₆₇	1	1	yes	characteristic
116	C₂×C₄	#97×C₆₇	1	2	yes	—
117	C₂×C₄	C₂×C₄²×C₆₇	1	1	yes	characteristic
118	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
119	C₈	—	4	4	no	—
120	C₈	C₄×C₈×C₆₇	1	2	yes	—
121	C₈	#97×C₆₇	1	2	yes	—
122	C₈	C₂²×C₈×C₆₇	1	2	yes	—
123	C₈	C₂×M₁₆×C₆₇	1	2	yes	—
124	C₂²	—	2	2	no	—
125	C₂²	#322×C₆₇	1	2	yes	—
126	C₂²	C₂×#97×C₆₇	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	2	no	—
129	C₄	#322×C₆₇	1	2	yes	—
130	C₄	C₂×C₄×C₈×C₆₇	1	1	yes	characteristic
131	C₄	C₂×#97×C₆₇	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×#322×C₆₇	1	1	yes	characteristic
134	C₂	C₆₇×#2110	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₆₇	C₄²
commutator	C₄	C₂×C₄×C₈×C₆₇
Frattini, cosocle	C₄×C₈	C₂³×C₆₇
socle	C₂²×C₆₇	C₂×#97
prime radical	C₂³×C₆₇	#97
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1313	C₂	semidirect
19. C₂×C₁₆×C₆₇	C₈	semidirect
64. #11405	C₆₇	direct (N × H)
71. #1313	C₂×C₆₇	semidirect
72. C₆₇	#11405	direct (N × H)
87. C₂×C₁₆	C₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₃₂×C₆₇	2
C₂×#346×C₆₇	2
C₆₇×#1450	2
C₆₇×#1313	2
#11405	67

Subgroups — C67×#11405 #30690656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#11405 #30690656