Subgroups — C₄×#504×C₆₇ #30712576

Order 17152.

834 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#504×C₆₇	C₂	1	2	yes	maximal
2	C₄×#131×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2108	C₂	1	6	yes	maximal, complemented
4	C₄×M₃₂×C₆₇	C₂	1	3	yes	maximal, complemented
5	C₂×C₄×C₁₆×C₆₇	C₂	1	3	yes	maximal, complemented
6	C₂×#131×C₆₇	C₄	1	1	yes	characteristic
7	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
8	C₄×#41×C₆₇	C₄	1	1	yes	characteristic
9	#504×C₆₇	C₄	1	16	yes	direct product factor
10	C₂×M₃₂×C₆₇	C₂²	1	6	yes	—
11	C₂²×C₁₆×C₆₇	C₂²	1	6	yes	—
12	#405×C₆₇	C₄	1	3	yes	—
13	#405×C₆₇	C₂²	1	3	yes	—
14	C₄×M₁₆×C₆₇	C₄	1	3	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
16	#346×C₆₇	C₂²	1	6	yes	—
17	C₄×C₁₆×C₆₇	C₂²	1	3	yes	—
18	C₄×C₁₆×C₆₇	C₂²	1	6	yes	—
19	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
21	#131×C₆₇	C₈	1	8	yes	—
22	#131×C₆₇	C₂×C₄	1	8	yes	—
23	C₂×M₁₆×C₆₇	C₂×C₄	1	3	yes	—
24	C₂×M₁₆×C₆₇	C₂×C₄	1	3	yes	—
25	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
26	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
27	C₄×Q₈×C₆₇	C₈	1	2	yes	—
28	C₄×D₄×C₆₇	C₈	1	6	yes	—
29	#117×C₆₇	C₂×C₄	1	3	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
31	M₃₂×C₆₇	C₂×C₄	1	24	yes	complemented
32	C₂×C₁₆×C₆₇	C₂×C₄	1	24	yes	complemented
33	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
34	C₂×C₁₆×C₆₇	C₂³	1	6	yes	—
35	#97×C₆₇	C₂×C₄	1	6	yes	—
36	C₄×C₈×C₆₇	C₂×C₄	1	3	yes	—
37	C₄×C₈×C₆₇	C₂×C₄	1	3	yes	—
38	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₄×C₈×C₆₇	C₂³	1	3	yes	—
40	#41×C₆₇	C₄²	1	4	yes	—
41	#41×C₆₇	C₂×C₈	1	4	yes	—
42	C₂×Q₈×C₆₇	C₂×C₈	1	1	yes	characteristic
43	C₂×D₄×C₆₇	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₆₇	C₂×C₈	1	6	yes	—
45	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
46	M₁₆×C₆₇	C₄²	1	12	yes	—
47	M₁₆×C₆₇	C₂×C₈	1	12	yes	—
48	C₂×C₈×C₆₇	C₂×C₈	1	12	yes	—
49	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
50	C₂×C₈×C₆₇	C₂²×C₄	1	12	yes	—
51	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
53	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
54	C₄⋊C₄×C₆₇	C₂×C₈	1	6	yes	—
55	C₂²⋊C₄×C₆₇	C₂×C₈	1	6	yes	—
56	C₄²×C₆₇	C₂×C₈	1	6	yes	—
57	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₆₇	C₂²×C₄	1	3	yes	—
59	C₁₆×C₆₇	C₂²×C₄	1	4	yes	—
60	C₁₆×C₆₇	C₂²×C₄	1	12	yes	—
61	C₂³×C₆₇	C₂²×C₈	1	3	yes	—
62	Q₈×C₆₇	C₄×C₈	1	4	yes	—
63	D₄×C₆₇	C₄×C₈	1	12	yes	—
64	C₂×C₄×C₆₇	—	2	12	no	—
65	C₂×C₄×C₆₇	C₂×C₄²	1	6	yes	—
66	C₂×C₄×C₆₇	C₂²×C₈	1	6	yes	—
67	C₂×C₄×C₆₇	C₂²×C₈	1	6	yes	—
68	C₂×C₄×C₆₇	C₂²×C₈	1	3	yes	—
69	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
71	C₈×C₆₇	C₂×C₄²	1	6	yes	—
72	C₈×C₆₇	C₂²×C₈	1	2	yes	—
73	C₈×C₆₇	C₂²×C₈	1	6	yes	—
74	C₈×C₆₇	C₂³×C₄	1	2	yes	—
75	C₂²×C₆₇	—	2	6	no	—
76	C₂²×C₆₇	C₂×C₄×C₈	1	6	yes	—
77	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
78	C₄×C₆₇	—	2	6	no	—
79	C₄×C₆₇	C₂×C₄×C₈	1	6	yes	—
80	C₄×C₆₇	#504	1	4	yes	direct product factor
81	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
82	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
83	C₄×#504	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂×C₆₇	—	2	6	no	—
85	C₂×C₆₇	C₂²×C₄×C₈	1	1	yes	characteristic
86	C₂×C₆₇	C₂×#504	1	2	yes	—
87	C₂×#504	C₂×C₆₇	1	2	yes	—
88	C₄×#131	C₂×C₆₇	1	1	yes	characteristic
89	#2108	C₂×C₆₇	1	6	yes	complemented
90	C₄×M₃₂	C₂×C₆₇	1	3	yes	complemented
91	C₂×C₄×C₁₆	C₂×C₆₇	1	3	yes	complemented
92	C₆₇	C₄×#504	1	1	yes	characteristic, direct product factor, Sylow
93	C₂×#131	C₄×C₆₇	1	1	yes	characteristic
94	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
95	C₄×#41	C₄×C₆₇	1	1	yes	characteristic
96	#504	C₄×C₆₇	1	16	yes	direct product factor
97	C₂×M₃₂	C₂²×C₆₇	1	6	yes	—
98	C₂²×C₁₆	C₂²×C₆₇	1	6	yes	—
99	#405	C₄×C₆₇	1	3	yes	—
100	#405	C₂²×C₆₇	1	3	yes	—
101	C₄×M₁₆	C₄×C₆₇	1	3	yes	—
102	C₂×C₄×C₈	C₂²×C₆₇	1	3	yes	—
103	#346	C₂²×C₆₇	1	6	yes	—
104	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
105	C₄×C₁₆	C₂²×C₆₇	1	3	yes	—
106	C₄×C₁₆	C₂²×C₆₇	1	6	yes	—
107	C₂×#41	C₂×C₄×C₆₇	1	1	yes	characteristic
108	#131	C₈×C₆₇	1	8	yes	—
109	#131	C₂×C₄×C₆₇	1	8	yes	—
110	C₂×M₁₆	C₂×C₄×C₆₇	1	3	yes	—
111	C₂×M₁₆	C₂×C₄×C₆₇	1	3	yes	—
112	C₂²×C₈	C₂×C₄×C₆₇	1	3	yes	—
113	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
114	C₄×Q₈	C₈×C₆₇	1	2	yes	—
115	C₄×D₄	C₈×C₆₇	1	6	yes	—
116	#117	C₂×C₄×C₆₇	1	3	yes	—
117	C₂×C₄²	C₂×C₄×C₆₇	1	3	yes	—
118	M₃₂	C₂×C₄×C₆₇	1	24	yes	complemented
119	C₂×C₁₆	C₂×C₄×C₆₇	1	24	yes	complemented
120	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
121	C₂×C₁₆	C₂³×C₆₇	1	6	yes	—
122	#97	C₂×C₄×C₆₇	1	6	yes	—
123	C₄×C₈	C₂×C₄×C₆₇	1	3	yes	—
124	C₄×C₈	C₂×C₄×C₆₇	1	3	yes	—
125	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
126	C₄×C₈	C₂³×C₆₇	1	3	yes	—
127	#41	C₄²×C₆₇	1	4	yes	—
128	#41	C₂×C₈×C₆₇	1	4	yes	—
129	C₂×Q₈	C₂×C₈×C₆₇	1	1	yes	characteristic
130	C₂×D₄	C₂×C₈×C₆₇	1	3	yes	—
131	C₂²×C₄	C₂×C₈×C₆₇	1	6	yes	—
132	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
133	M₁₆	C₄²×C₆₇	1	12	yes	—
134	M₁₆	C₂×C₈×C₆₇	1	12	yes	—
135	C₂×C₈	C₂×C₈×C₆₇	1	12	yes	—
136	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
137	C₂×C₈	C₂²×C₄×C₆₇	1	12	yes	—
138	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
139	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
140	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
141	C₄⋊C₄	C₂×C₈×C₆₇	1	6	yes	—
142	C₂²⋊C₄	C₂×C₈×C₆₇	1	6	yes	—
143	C₄²	C₂×C₈×C₆₇	1	6	yes	—
144	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
145	C₄²	C₂²×C₄×C₆₇	1	3	yes	—
146	C₁₆	C₂²×C₄×C₆₇	1	4	yes	—
147	C₁₆	C₂²×C₄×C₆₇	1	12	yes	—
148	C₂³	C₂²×C₈×C₆₇	1	3	yes	—
149	Q₈	C₄×C₈×C₆₇	1	4	yes	—
150	D₄	C₄×C₈×C₆₇	1	12	yes	—
151	C₂×C₄	—	2	12	no	—
152	C₂×C₄	C₂×C₄²×C₆₇	1	6	yes	—
153	C₂×C₄	C₂²×C₈×C₆₇	1	6	yes	—
154	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
155	C₂×C₄	C₂²×C₈×C₆₇	1	6	yes	—
156	C₂×C₄	C₂²×C₈×C₆₇	1	3	yes	—
157	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
158	C₈	C₂×C₄²×C₆₇	1	6	yes	—
159	C₈	C₂²×C₈×C₆₇	1	2	yes	—
160	C₈	C₂²×C₈×C₆₇	1	6	yes	—
161	C₈	C₂³×C₄×C₆₇	1	2	yes	—
162	C₂²	—	2	6	no	—
163	C₂²	C₂×C₄×C₈×C₆₇	1	6	yes	—
164	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
165	C₄	—	2	6	no	—
166	C₄	C₂×C₄×C₈×C₆₇	1	6	yes	—
167	C₄	#504×C₆₇	1	4	yes	direct product factor
168	C₄	C₂²×C₄²×C₆₇	1	1	yes	characteristic
169	C₄	C₂³×C₈×C₆₇	1	1	yes	characteristic
170	C₂	—	2	6	no	—
171	C₂	C₂²×C₄×C₈×C₆₇	1	1	yes	characteristic
172	C₂	C₂×#504×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₄×C₁₆×C₆₇	C₂²
commutator	C₂	C₂²×C₄×C₈×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂³×C₈
prime radical	C₂×#41×C₆₇	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2108	C₂	semidirect
4. C₄×M₃₂×C₆₇	C₂	semidirect
5. C₂×C₄×C₁₆×C₆₇	C₂	semidirect
9. #504×C₆₇	C₄	direct (N × H)
31. M₃₂×C₆₇	C₂×C₄	semidirect
32. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
80. C₄×C₆₇	#504	direct (N × H)
83. C₄×#504	C₆₇	direct (N × H)
89. #2108	C₂×C₆₇	semidirect
90. C₄×M₃₂	C₂×C₆₇	semidirect
91. C₂×C₄×C₁₆	C₂×C₆₇	semidirect
92. C₆₇	C₄×#504	direct (N × H)
96. #504	C₄×C₆₇	direct (N × H)
118. M₃₂	C₂×C₄×C₆₇	semidirect
119. C₂×C₁₆	C₂×C₄×C₆₇	semidirect
167. C₄	#504×C₆₇	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×#504×C₆₇	2
C₄×#131×C₆₇	2
C₆₇×#2108	2
C₄×M₃₂×C₆₇	2
C₂×C₄×C₁₆×C₆₇	2
C₄×#504	67

Subgroups — C4×#504×C67 #30712576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#504×C₆₇ #30712576