Subgroups — C₂×C₆₇×#2844 #30715939

Order 17152.

1870 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2844	C₂	1	16	yes	maximal, direct product factor
2	C₂×#400×C₆₇	C₂	1	6	yes	maximal
3	C₂×#395×C₆₇	C₂	1	6	yes	maximal
4	C₂×#382×C₆₇	C₂	1	3	yes	maximal
5	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
6	C₂²×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
7	C₂²×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
8	C₂²×C₄⋊C₄×C₆₇	C₂²	1	6	yes	—
9	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
10	#400×C₆₇	C₂²	1	48	yes	—
11	#395×C₆₇	C₂²	1	48	yes	—
12	#382×C₆₇	C₂²	1	24	yes	—
13	C₂×#95×C₆₇	C₂²	1	12	yes	—
14	C₂×#95×C₆₇	C₂²	1	6	yes	—
15	C₂×#95×C₆₇	C₂²	1	2	yes	—
16	C₂²×Q₈×C₆₇	—	4	12	no	—
17	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
18	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
19	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
20	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
21	C₂×C₄⋊C₄×C₆₇	—	4	6	no	—
22	C₂×C₄⋊C₄×C₆₇	—	4	6	no	—
23	C₂×C₄⋊C₄×C₆₇	—	4	12	no	—
24	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
25	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
26	C₂×C₄⋊C₄×C₆₇	C₂³	1	24	yes	—
27	C₂×C₄²×C₆₇	Q₈	1	8	yes	complemented
28	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
29	#95×C₆₇	C₂³	1	48	yes	—
30	#95×C₆₇	C₂³	1	24	yes	—
31	#95×C₆₇	C₂³	1	8	yes	—
32	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
33	C₂×Q₈×C₆₇	—	4	24	no	—
34	C₂×Q₈×C₆₇	—	4	32	no	—
35	C₂²×C₄×C₆₇	—	2	6	no	—
36	C₂²×C₄×C₆₇	—	2	6	no	—
37	C₂²×C₄×C₆₇	—	2	12	no	—
38	C₂²×C₄×C₆₇	—	2	12	no	—
39	C₂²×C₄×C₆₇	—	2	24	no	—
40	C₂²×C₄×C₆₇	—	2	24	no	—
41	C₂²×C₄×C₆₇	C₂×Q₈	1	12	yes	—
42	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
43	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
44	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
45	C₄⋊C₄×C₆₇	—	4	12	no	—
46	C₄⋊C₄×C₆₇	—	4	12	no	—
47	C₄⋊C₄×C₆₇	—	4	24	no	—
48	C₄²×C₆₇	C₂×Q₈	1	16	yes	complemented
49	C₂³×C₆₇	C₂²×Q₈	1	2	yes	—
50	C₂³×C₆₇	H(5,𝔽₂)	1	6	yes	—
51	C₂³×C₆₇	2₋¹⁺⁴	1	6	yes	—
52	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
53	Q₈×C₆₇	—	4	32	no	—
54	C₂×C₄×C₆₇	—	2	24	no	—
55	C₂×C₄×C₆₇	—	2	6	no	—
56	C₂×C₄×C₆₇	—	2	6	no	—
57	C₂×C₄×C₆₇	—	2	12	no	—
58	C₂×C₄×C₆₇	—	2	12	no	—
59	C₂×C₄×C₆₇	—	2	24	no	—
60	C₂×C₄×C₆₇	—	2	24	no	—
61	C₂×C₄×C₆₇	—	4	24	no	—
62	C₂×C₄×C₆₇	—	4	24	no	—
63	C₂×C₄×C₆₇	C₂²×Q₈	1	12	yes	—
64	C₂²×C₆₇	#544	1	12	yes	—
65	C₂²×C₆₇	H(3,𝔽₄)	1	4	yes	—
66	C₂²×C₆₇	#563	1	12	yes	—
67	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
68	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	3	yes	—
69	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	3	yes	—
70	C₄×C₆₇	—	2	12	no	—
71	C₄×C₆₇	—	4	12	no	—
72	C₄×C₆₇	—	4	12	no	—
73	C₂×#2844	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂×C₆₇	#2844	1	8	yes	direct product factor
75	C₂×C₆₇	C₂×#544	1	3	yes	—
76	C₂×C₆₇	C₂×H(3,𝔽₄)	1	1	yes	characteristic
77	C₂×C₆₇	C₂×#563	1	3	yes	—
78	#2844	C₂×C₆₇	1	16	yes	direct product factor
79	C₂×#400	C₂×C₆₇	1	6	yes	—
80	C₂×#395	C₂×C₆₇	1	6	yes	—
81	C₂×#382	C₂×C₆₇	1	3	yes	—
82	C₆₇	C₂×#2844	1	1	yes	characteristic, direct product factor, Sylow
83	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
84	C₂²×C₄⋊C₄	C₂²×C₆₇	1	6	yes	—
85	C₂²×C₄⋊C₄	C₂²×C₆₇	1	3	yes	—
86	C₂²×C₄⋊C₄	C₂²×C₆₇	1	3	yes	—
87	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
88	#400	C₂²×C₆₇	1	48	yes	—
89	#395	C₂²×C₆₇	1	48	yes	—
90	#382	C₂²×C₆₇	1	24	yes	—
91	C₂×#95	C₂²×C₆₇	1	6	yes	—
92	C₂×#95	C₂²×C₆₇	1	12	yes	—
93	C₂×#95	C₂²×C₆₇	1	2	yes	—
94	C₂²×Q₈	—	4	12	no	—
95	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
96	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
97	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
98	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
99	C₂×C₄⋊C₄	—	4	6	no	—
100	C₂×C₄⋊C₄	—	4	6	no	—
101	C₂×C₄⋊C₄	—	4	12	no	—
102	C₂×C₄⋊C₄	C₂³×C₆₇	1	12	yes	—
103	C₂×C₄⋊C₄	C₂³×C₆₇	1	12	yes	—
104	C₂×C₄⋊C₄	C₂³×C₆₇	1	24	yes	—
105	C₂×C₄²	Q₈×C₆₇	1	8	yes	complemented
106	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
107	#95	C₂³×C₆₇	1	48	yes	—
108	#95	C₂³×C₆₇	1	24	yes	—
109	#95	C₂³×C₆₇	1	8	yes	—
110	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
111	C₂×Q₈	—	4	24	no	—
112	C₂×Q₈	—	4	32	no	—
113	C₂²×C₄	—	2	6	no	—
114	C₂²×C₄	—	2	6	no	—
115	C₂²×C₄	—	2	12	no	—
116	C₂²×C₄	—	2	12	no	—
117	C₂²×C₄	—	2	24	no	—
118	C₂²×C₄	—	2	24	no	—
119	C₂²×C₄	C₂×Q₈×C₆₇	1	12	yes	—
120	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
121	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
122	C₂²×C₄	C₂⁴×C₆₇	1	6	yes	—
123	C₄⋊C₄	—	4	12	no	—
124	C₄⋊C₄	—	4	12	no	—
125	C₄⋊C₄	—	4	24	no	—
126	C₄²	C₂×Q₈×C₆₇	1	16	yes	complemented
127	C₂³	C₂²×Q₈×C₆₇	1	2	yes	—
128	C₂³	H(5,𝔽₂)×C₆₇	1	6	yes	—
129	C₂³	2₋¹⁺⁴×C₆₇	1	6	yes	—
130	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
131	Q₈	—	4	32	no	—
132	C₂×C₄	—	2	24	no	—
133	C₂×C₄	—	2	6	no	—
134	C₂×C₄	—	2	6	no	—
135	C₂×C₄	—	2	12	no	—
136	C₂×C₄	—	2	12	no	—
137	C₂×C₄	—	2	24	no	—
138	C₂×C₄	—	2	24	no	—
139	C₂×C₄	—	4	24	no	—
140	C₂×C₄	—	4	24	no	—
141	C₂×C₄	C₂²×Q₈×C₆₇	1	12	yes	—
142	C₂²	#544×C₆₇	1	12	yes	—
143	C₂²	H(3,𝔽₄)×C₆₇	1	4	yes	—
144	C₂²	#563×C₆₇	1	12	yes	—
145	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
146	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	3	yes	—
147	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	3	yes	—
148	C₄	—	2	12	no	—
149	C₄	—	4	12	no	—
150	C₄	—	4	12	no	—
151	C₂	C₆₇×#2844	1	8	yes	direct product factor
152	C₂	C₂×#544×C₆₇	1	3	yes	—
153	C₂	C₂×H(3,𝔽₄)×C₆₇	1	1	yes	characteristic
154	C₂	C₂×#563×C₆₇	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2844	C₂	direct (N × H)
27. C₂×C₄²×C₆₇	Q₈	semidirect
48. C₄²×C₆₇	C₂×Q₈	semidirect
73. C₂×#2844	C₆₇	direct (N × H)
74. C₂×C₆₇	#2844	direct (N × H)
78. #2844	C₂×C₆₇	direct (N × H)
82. C₆₇	C₂×#2844	direct (N × H)
105. C₂×C₄²	Q₈×C₆₇	semidirect
126. C₄²	C₂×Q₈×C₆₇	semidirect
151. C₂	C₆₇×#2844	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2844	2
C₂×#400×C₆₇	2
C₂×#395×C₆₇	2
C₂×#382×C₆₇	2
C₂×#2844	67

Subgroups — C2×C67×#2844 #30715939

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2844 #30715939