Subgroups — C₂×C₆₇×#2247 #30712961

Order 17152.

814 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	C₂²×Q₃₂×C₆₇	C₂	1	4	yes	maximal
2	C₂×#494×C₆₇	C₂	1	2	yes	maximal
3	C₆₇×#2247	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₆₇	—	2	2	no	—
6	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	—
7	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
8	C₂×Q₃₂×C₆₇	—	2	16	no	—
9	C₂×Q₃₂×C₆₇	C₂²	1	16	yes	—
10	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
11	#494×C₆₇	—	2	4	no	—
12	#494×C₆₇	C₂²	1	8	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₆₇	C₂²	1	4	yes	—
15	C₂²×Q₈×C₆₇	—	2	4	no	—
16	C₂×Q₁₆×C₆₇	—	2	16	no	—
17	C₂×Q₁₆×C₆₇	—	2	4	no	—
18	C₂×Q₁₆×C₆₇	—	2	8	no	—
19	C₂×Q₁₆×C₆₇	—	4	8	no	—
20	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
21	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
23	C₄⋊Q₈×C₆₇	—	2	8	no	—
24	C₄⋊Q₈×C₆₇	—	4	4	no	—
25	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
26	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
27	Q₃₂×C₆₇	—	2	32	no	—
28	C₂×C₁₆×C₆₇	D₄	1	8	yes	—
29	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
30	C₄×C₈×C₆₇	D₄	1	2	yes	—
31	C₄×C₈×C₆₇	C₂³	1	2	yes	—
32	C₂×Q₈×C₆₇	—	2	8	no	—
33	C₂×Q₈×C₆₇	—	4	4	no	—
34	C₂×Q₈×C₆₇	—	4	8	no	—
35	C₂×Q₈×C₆₇	—	4	8	no	—
36	C₂²×C₄×C₆₇	—	4	4	no	—
37	C₂²×C₄×C₆₇	D₈	1	2	yes	—
38	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
39	Q₁₆×C₆₇	—	2	16	no	—
40	Q₁₆×C₆₇	—	4	8	no	—
41	Q₁₆×C₆₇	—	4	16	no	—
42	C₂×C₈×C₆₇	D₈	1	4	yes	—
43	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
45	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
47	C₄⋊C₄×C₆₇	—	4	16	no	—
48	C₄²×C₆₇	D₈	1	2	yes	—
49	C₄²×C₆₇	C₂×D₄	1	2	yes	—
50	C₁₆×C₆₇	C₂×D₄	1	8	yes	—
51	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
52	Q₈×C₆₇	—	4	8	no	—
53	Q₈×C₆₇	—	4	8	no	—
54	Q₈×C₆₇	—	8	8	no	—
55	C₂×C₄×C₆₇	—	4	8	no	—
56	C₂×C₄×C₆₇	—	8	8	no	—
57	C₂×C₄×C₆₇	Q₃₂	1	8	yes	complemented
58	C₂×C₄×C₆₇	#127	1	2	yes	—
59	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
60	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
62	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
63	C₈×C₆₇	#127	1	2	yes	—
64	C₈×C₆₇	C₂×D₈	1	4	yes	—
65	C₈×C₆₇	C₂²×D₄	1	2	yes	—
66	C₂²×C₆₇	#493	1	2	yes	—
67	C₂²×C₆₇	C₂×Q₃₂	1	4	yes	—
68	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
69	C₄×C₆₇	—	8	8	no	—
70	C₄×C₆₇	#493	1	2	yes	—
71	C₄×C₆₇	C₂×Q₃₂	1	8	yes	complemented
72	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
73	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
74	C₂×#2247	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂×C₆₇	#2247	1	4	yes	direct product factor
76	C₂×C₆₇	C₂×#493	1	1	yes	characteristic
77	C₂×C₆₇	C₂²×Q₃₂	1	2	yes	—
78	C₂²×Q₃₂	C₂×C₆₇	1	4	yes	—
79	C₂×#494	C₂×C₆₇	1	2	yes	—
80	#2247	C₂×C₆₇	1	8	yes	direct product factor
81	C₂×C₄×C₁₆	C₂×C₆₇	1	1	yes	characteristic
82	C₆₇	C₂×#2247	1	1	yes	characteristic, direct product factor, Sylow
83	C₂²×Q₁₆	—	2	2	no	—
84	C₂²×Q₁₆	C₂²×C₆₇	1	4	yes	—
85	C₂×C₄⋊Q₈	—	2	2	no	—
86	C₂×Q₃₂	—	2	16	no	—
87	C₂×Q₃₂	C₂²×C₆₇	1	16	yes	—
88	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	—
89	#494	—	2	4	no	—
90	#494	C₂²×C₆₇	1	8	yes	—
91	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
92	C₄×C₁₆	C₂²×C₆₇	1	4	yes	—
93	C₂²×Q₈	—	2	4	no	—
94	C₂×Q₁₆	—	2	16	no	—
95	C₂×Q₁₆	—	2	8	no	—
96	C₂×Q₁₆	—	2	4	no	—
97	C₂×Q₁₆	—	4	8	no	—
98	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
99	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
100	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
101	C₄⋊Q₈	—	2	8	no	—
102	C₄⋊Q₈	—	4	4	no	—
103	C₂×C₄⋊C₄	—	4	4	no	—
104	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
105	Q₃₂	—	2	32	no	—
106	C₂×C₁₆	D₄×C₆₇	1	8	yes	—
107	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
108	C₄×C₈	D₄×C₆₇	1	2	yes	—
109	C₄×C₈	C₂³×C₆₇	1	2	yes	—
110	C₂×Q₈	—	2	8	no	—
111	C₂×Q₈	—	4	4	no	—
112	C₂×Q₈	—	4	8	no	—
113	C₂×Q₈	—	4	8	no	—
114	C₂²×C₄	—	4	4	no	—
115	C₂²×C₄	D₈×C₆₇	1	2	yes	—
116	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
117	Q₁₆	—	2	16	no	—
118	Q₁₆	—	4	8	no	—
119	Q₁₆	—	4	16	no	—
120	C₂×C₈	D₈×C₆₇	1	4	yes	—
121	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
122	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
123	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
124	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
125	C₄⋊C₄	—	4	16	no	—
126	C₄²	D₈×C₆₇	1	2	yes	—
127	C₄²	C₂×D₄×C₆₇	1	2	yes	—
128	C₁₆	C₂×D₄×C₆₇	1	8	yes	—
129	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
130	Q₈	—	4	8	no	—
131	Q₈	—	4	8	no	—
132	Q₈	—	8	8	no	—
133	C₂×C₄	—	4	8	no	—
134	C₂×C₄	—	8	8	no	—
135	C₂×C₄	Q₃₂×C₆₇	1	8	yes	complemented
136	C₂×C₄	#127×C₆₇	1	2	yes	—
137	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
138	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
139	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
140	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
141	C₈	#127×C₆₇	1	2	yes	—
142	C₈	C₂×D₈×C₆₇	1	4	yes	—
143	C₈	C₂²×D₄×C₆₇	1	2	yes	—
144	C₂²	#493×C₆₇	1	2	yes	—
145	C₂²	C₂×Q₃₂×C₆₇	1	4	yes	—
146	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
147	C₄	—	8	8	no	—
148	C₄	#493×C₆₇	1	2	yes	—
149	C₄	C₂×Q₃₂×C₆₇	1	8	yes	complemented
150	C₄	C₂×#127×C₆₇	1	1	yes	characteristic
151	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
152	C₂	C₆₇×#2247	1	4	yes	direct product factor
153	C₂	C₂×#493×C₆₇	1	1	yes	characteristic
154	C₂	C₂²×Q₃₂×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2247	C₂	direct (N × H)
57. C₂×C₄×C₆₇	Q₃₂	semidirect
71. C₄×C₆₇	C₂×Q₃₂	semidirect
74. C₂×#2247	C₆₇	direct (N × H)
75. C₂×C₆₇	#2247	direct (N × H)
80. #2247	C₂×C₆₇	direct (N × H)
82. C₆₇	C₂×#2247	direct (N × H)
135. C₂×C₄	Q₃₂×C₆₇	semidirect
149. C₄	C₂×Q₃₂×C₆₇	semidirect
152. C₂	C₆₇×#2247	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Q₃₂×C₆₇	2
C₂×#494×C₆₇	2
C₆₇×#2247	2
C₂×C₄×C₁₆×C₆₇	2
C₂×#2247	67

Subgroups — C2×C67×#2247 #30712961

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2247 #30712961