Subgroups — C₂×C₆₇×#1839 #30700021

Order 17152.

1134 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	C₂×C₄³×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1839	C₂	1	8	yes	maximal, direct product factor
3	C₂×#336×C₆₇	C₂	1	6	yes	maximal
4	C₂³×C₈×C₆₇	C₂²	1	4	yes	—
5	C₂²×C₄²×C₆₇	C₄	1	3	yes	—
6	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
8	C₄³×C₆₇	C₄	1	4	yes	—
9	C₄³×C₆₇	C₂²	1	4	yes	—
10	#336×C₆₇	C₂²	1	24	yes	—
11	C₂³×C₄×C₆₇	C₂×C₄	1	3	yes	—
12	C₂³×C₄×C₆₇	C₂×C₄	1	3	yes	—
13	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
14	C₂²×C₈×C₆₇	—	2	24	no	—
15	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
16	C₂×C₄²×C₆₇	—	2	12	no	—
17	C₂×C₄²×C₆₇	—	2	12	no	—
18	C₂×C₄²×C₆₇	C₈	1	8	yes	complemented
19	C₂×C₄²×C₆₇	C₂×C₄	1	12	yes	—
20	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
21	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
23	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₆₇	—	2	6	no	—
25	C₂²×C₄×C₆₇	—	2	6	no	—
26	C₂²×C₄×C₆₇	—	2	6	no	—
27	C₂²×C₄×C₆₇	—	2	6	no	—
28	C₂²×C₄×C₆₇	C₂×C₈	1	12	yes	—
29	C₂²×C₄×C₆₇	M₁₆	1	12	yes	—
30	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
31	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
32	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₇	#41	1	12	yes	—
34	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
35	C₂×C₈×C₆₇	—	2	24	no	—
36	C₂×C₈×C₆₇	—	4	16	no	—
37	C₄²×C₆₇	—	2	12	no	—
38	C₄²×C₆₇	—	2	24	no	—
39	C₄²×C₆₇	—	2	12	no	—
40	C₄²×C₆₇	C₂×C₈	1	16	yes	complemented
41	C₂³×C₆₇	#117	1	6	yes	—
42	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
43	C₂³×C₆₇	C₂×M₁₆	1	6	yes	—
44	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₇	—	2	6	no	—
46	C₂×C₄×C₆₇	—	2	6	no	—
47	C₂×C₄×C₆₇	—	2	6	no	—
48	C₂×C₄×C₆₇	—	2	6	no	—
49	C₂×C₄×C₆₇	—	2	24	no	—
50	C₂×C₄×C₆₇	—	2	24	no	—
51	C₂×C₄×C₆₇	#117	1	6	yes	—
52	C₂×C₄×C₆₇	#126	1	16	yes	—
53	C₂×C₄×C₆₇	C₂²×C₈	1	12	yes	—
54	C₂×C₄×C₆₇	C₂×M₁₆	1	12	yes	—
55	C₂×C₄×C₆₇	C₂×#41	1	6	yes	—
56	C₈×C₆₇	—	4	8	no	—
57	C₂²×C₆₇	#383	1	4	yes	—
58	C₂²×C₆₇	#431	1	12	yes	—
59	C₂²×C₆₇	#432	1	12	yes	—
60	C₂²×C₆₇	C₂×#117	1	3	yes	—
61	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
62	C₂²×C₆₇	C₂²×M₁₆	1	3	yes	—
63	C₄×C₆₇	—	2	12	no	—
64	C₄×C₆₇	—	2	12	no	—
65	C₄×C₆₇	#383	1	4	yes	—
66	C₄×C₆₇	C₂×#126	1	4	yes	—
67	C₂×#1839	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₆₇	#1839	1	8	yes	direct product factor
69	C₂×C₆₇	C₂×#383	1	1	yes	characteristic
70	C₂×C₆₇	C₂×#431	1	3	yes	—
71	C₂×C₆₇	C₂×#432	1	3	yes	—
72	C₂×C₄³	C₂×C₆₇	1	1	yes	characteristic
73	#1839	C₂×C₆₇	1	8	yes	direct product factor
74	C₂×#336	C₂×C₆₇	1	6	yes	—
75	C₆₇	C₂×#1839	1	1	yes	characteristic, direct product factor, Sylow
76	C₂³×C₈	C₂²×C₆₇	1	4	yes	—
77	C₂²×C₄²	C₄×C₆₇	1	3	yes	—
78	C₂²×C₄²	C₄×C₆₇	1	1	yes	characteristic
79	C₂²×C₄²	C₂²×C₆₇	1	3	yes	—
80	C₄³	C₄×C₆₇	1	4	yes	—
81	C₄³	C₂²×C₆₇	1	4	yes	—
82	#336	C₂²×C₆₇	1	24	yes	—
83	C₂³×C₄	C₂×C₄×C₆₇	1	3	yes	—
84	C₂³×C₄	C₂×C₄×C₆₇	1	3	yes	—
85	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
86	C₂²×C₈	—	2	24	no	—
87	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
88	C₂×C₄²	—	2	12	no	—
89	C₂×C₄²	—	2	12	no	—
90	C₂×C₄²	C₈×C₆₇	1	8	yes	complemented
91	C₂×C₄²	C₂×C₄×C₆₇	1	6	yes	—
92	C₂×C₄²	C₂×C₄×C₆₇	1	12	yes	—
93	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
94	C₂×C₄²	C₂³×C₆₇	1	6	yes	—
95	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
96	C₂²×C₄	—	2	6	no	—
97	C₂²×C₄	—	2	6	no	—
98	C₂²×C₄	—	2	6	no	—
99	C₂²×C₄	—	2	6	no	—
100	C₂²×C₄	C₂×C₈×C₆₇	1	12	yes	—
101	C₂²×C₄	M₁₆×C₆₇	1	12	yes	—
102	C₂²×C₄	C₂²×C₄×C₆₇	1	6	yes	—
103	C₂²×C₄	C₂²×C₄×C₆₇	1	6	yes	—
104	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
105	C₂²×C₄	#41×C₆₇	1	12	yes	—
106	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
107	C₂×C₈	—	2	24	no	—
108	C₂×C₈	—	4	16	no	—
109	C₄²	—	2	12	no	—
110	C₄²	—	2	24	no	—
111	C₄²	—	2	12	no	—
112	C₄²	C₂×C₈×C₆₇	1	16	yes	complemented
113	C₂³	#117×C₆₇	1	6	yes	—
114	C₂³	C₂²×C₈×C₆₇	1	2	yes	—
115	C₂³	C₂×M₁₆×C₆₇	1	6	yes	—
116	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
117	C₂×C₄	—	2	6	no	—
118	C₂×C₄	—	2	6	no	—
119	C₂×C₄	—	2	6	no	—
120	C₂×C₄	—	2	6	no	—
121	C₂×C₄	—	2	24	no	—
122	C₂×C₄	—	2	24	no	—
123	C₂×C₄	#117×C₆₇	1	6	yes	—
124	C₂×C₄	#126×C₆₇	1	16	yes	—
125	C₂×C₄	C₂²×C₈×C₆₇	1	12	yes	—
126	C₂×C₄	C₂×M₁₆×C₆₇	1	12	yes	—
127	C₂×C₄	C₂×#41×C₆₇	1	6	yes	—
128	C₈	—	4	8	no	—
129	C₂²	#383×C₆₇	1	4	yes	—
130	C₂²	#431×C₆₇	1	12	yes	—
131	C₂²	#432×C₆₇	1	12	yes	—
132	C₂²	C₂×#117×C₆₇	1	3	yes	—
133	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
134	C₂²	C₂²×M₁₆×C₆₇	1	3	yes	—
135	C₄	—	2	12	no	—
136	C₄	—	2	12	no	—
137	C₄	#383×C₆₇	1	4	yes	—
138	C₄	C₂×#126×C₆₇	1	4	yes	—
139	C₂	C₆₇×#1839	1	8	yes	direct product factor
140	C₂	C₂×#383×C₆₇	1	1	yes	characteristic
141	C₂	C₂×#431×C₆₇	1	3	yes	—
142	C₂	C₂×#432×C₆₇	1	3	yes	—

	subgroup	G/H
center	C₂³×C₄×C₆₇	C₂³
commutator	C₂²	C₂³×C₈×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle, prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#1839	C₂	direct (N × H)
18. C₂×C₄²×C₆₇	C₈	semidirect
40. C₄²×C₆₇	C₂×C₈	semidirect
67. C₂×#1839	C₆₇	direct (N × H)
68. C₂×C₆₇	#1839	direct (N × H)
73. #1839	C₂×C₆₇	direct (N × H)
75. C₆₇	C₂×#1839	direct (N × H)
90. C₂×C₄²	C₈×C₆₇	semidirect
112. C₄²	C₂×C₈×C₆₇	semidirect
139. C₂	C₆₇×#1839	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×C₄³×C₆₇	2
C₆₇×#1839	2
C₂×#336×C₆₇	2
C₂×#1839	67

Subgroups — C2×C67×#1839 #30700021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1839 #30700021