Subgroups — C₂²×#347×C₆₇ #30712578

Order 17152.

782 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₆₇	C₂	1	2	yes	maximal
2	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#347×C₆₇	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₈×C₆₇	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₆₇	C₄	1	12	yes	complemented
8	C₂×M₃₂×C₆₇	C₂²	1	12	yes	—
9	C₂²×C₁₆×C₆₇	C₄	1	4	yes	complemented
10	C₂×#97×C₆₇	C₄	1	6	yes	—
11	C₂×#97×C₆₇	C₂²	1	6	yes	—
12	#347×C₆₇	C₂²	1	16	yes	direct product factor
13	C₂³×C₄×C₆₇	—	2	1	no	—
14	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₆₇	—	2	1	no	—
16	C₂²×C₈×C₆₇	—	2	3	no	—
17	C₂²×C₈×C₆₇	C₂×C₄	1	6	yes	—
18	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
19	C₂²×C₈×C₆₇	C₂×C₄	1	6	yes	—
20	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
22	C₂×C₄²×C₆₇	—	2	3	no	—
23	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
24	M₃₂×C₆₇	C₂×C₄	1	24	yes	complemented
25	M₃₂×C₆₇	C₂³	1	8	yes	—
26	C₂×C₁₆×C₆₇	C₂×C₄	1	24	yes	complemented
27	#97×C₆₇	C₂×C₄	1	12	yes	—
28	#97×C₆₇	C₂³	1	4	yes	—
29	C₂⁴×C₆₇	M₁₆	1	1	yes	characteristic
30	C₂²×C₄×C₆₇	—	2	6	no	—
31	C₂²×C₄×C₆₇	—	4	4	no	—
32	C₂²×C₄×C₆₇	C₄²	1	3	yes	—
33	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₆₇	M₁₆	1	3	yes	—
35	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	M₁₆	1	3	yes	—
37	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
38	C₂×C₈×C₆₇	—	2	6	no	—
39	C₂×C₈×C₆₇	—	2	6	no	—
40	C₂×C₈×C₆₇	C₄²	1	6	yes	—
41	C₂×C₈×C₆₇	C₄²	1	6	yes	—
42	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
43	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
44	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
45	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
46	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
47	C₄²×C₆₇	—	2	6	no	—
48	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
49	C₁₆×C₆₇	C₂²×C₄	1	16	yes	complemented
50	C₂³×C₆₇	—	2	4	no	—
51	C₂³×C₆₇	#97	1	3	yes	—
52	C₂³×C₆₇	#97	1	1	yes	characteristic
53	C₂³×C₆₇	C₂×M₁₆	1	3	yes	—
54	C₂×C₄×C₆₇	—	2	4	no	—
55	C₂×C₄×C₆₇	—	4	12	no	—
56	C₂×C₄×C₆₇	#97	1	6	yes	—
57	C₂×C₄×C₆₇	#97	1	3	yes	—
58	C₂×C₄×C₆₇	#97	1	3	yes	—
59	C₂×C₄×C₆₇	C₂×C₄²	1	3	yes	—
60	C₂×C₄×C₆₇	C₂×C₄²	1	3	yes	—
61	C₂×C₄×C₆₇	C₂×M₁₆	1	3	yes	—
62	C₂×C₄×C₆₇	C₂×M₁₆	1	3	yes	—
63	C₂×C₄×C₆₇	C₂×M₁₆	1	3	yes	—
64	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
65	C₈×C₆₇	—	2	4	no	—
66	C₈×C₆₇	C₂×C₄²	1	6	yes	—
67	C₈×C₆₇	C₂³×C₄	1	2	yes	—
68	C₂²×C₆₇	—	2	6	no	—
69	C₂²×C₆₇	—	2	6	no	—
70	C₂²×C₆₇	#347	1	4	yes	direct product factor
71	C₂²×C₆₇	C₂×#97	1	3	yes	—
72	C₂²×C₆₇	C₂×#97	1	3	yes	—
73	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
74	C₄×C₆₇	—	4	4	no	—
75	C₄×C₆₇	C₂×#97	1	3	yes	—
76	C₄×C₆₇	C₂×#97	1	3	yes	—
77	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
78	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
79	C₂²×#347	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂×C₆₇	—	2	1	no	—
81	C₂×C₆₇	—	2	3	no	—
82	C₂×C₆₇	C₂×#347	1	6	yes	direct product factor
83	C₂×C₆₇	C₂²×#97	1	1	yes	characteristic
84	C₂²×M₃₂	C₂×C₆₇	1	2	yes	—
85	C₂²×#97	C₂×C₆₇	1	1	yes	characteristic
86	C₂×#347	C₂×C₆₇	1	12	yes	direct product factor
87	C₆₇	C₂²×#347	1	1	yes	characteristic, direct product factor, Sylow
88	C₂³×C₈	C₄×C₆₇	1	1	yes	characteristic
89	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
90	C₂²×C₄²	C₄×C₆₇	1	1	yes	characteristic
91	C₂×M₃₂	C₄×C₆₇	1	12	yes	complemented
92	C₂×M₃₂	C₂²×C₆₇	1	12	yes	—
93	C₂²×C₁₆	C₄×C₆₇	1	4	yes	complemented
94	C₂×#97	C₄×C₆₇	1	6	yes	—
95	C₂×#97	C₂²×C₆₇	1	6	yes	—
96	#347	C₂²×C₆₇	1	16	yes	direct product factor
97	C₂³×C₄	—	2	1	no	—
98	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
99	C₂²×C₈	—	2	3	no	—
100	C₂²×C₈	—	2	1	no	—
101	C₂²×C₈	C₂×C₄×C₆₇	1	6	yes	—
102	C₂²×C₈	C₂×C₄×C₆₇	1	3	yes	—
103	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
104	C₂²×C₈	C₂×C₄×C₆₇	1	6	yes	—
105	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
106	C₂×C₄²	—	2	3	no	—
107	C₂×C₄²	C₂×C₄×C₆₇	1	6	yes	—
108	M₃₂	C₂×C₄×C₆₇	1	24	yes	complemented
109	M₃₂	C₂³×C₆₇	1	8	yes	—
110	C₂×C₁₆	C₂×C₄×C₆₇	1	24	yes	complemented
111	#97	C₂×C₄×C₆₇	1	12	yes	—
112	#97	C₂³×C₆₇	1	4	yes	—
113	C₂⁴	M₁₆×C₆₇	1	1	yes	characteristic
114	C₂²×C₄	—	2	6	no	—
115	C₂²×C₄	—	4	4	no	—
116	C₂²×C₄	C₄²×C₆₇	1	3	yes	—
117	C₂²×C₄	C₄²×C₆₇	1	1	yes	characteristic
118	C₂²×C₄	M₁₆×C₆₇	1	3	yes	—
119	C₂²×C₄	M₁₆×C₆₇	1	1	yes	characteristic
120	C₂²×C₄	M₁₆×C₆₇	1	3	yes	—
121	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
122	C₂×C₈	—	2	6	no	—
123	C₂×C₈	—	2	6	no	—
124	C₂×C₈	C₄²×C₆₇	1	6	yes	—
125	C₂×C₈	C₄²×C₆₇	1	6	yes	—
126	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
127	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
128	C₂×C₈	C₂²×C₄×C₆₇	1	6	yes	—
129	C₂×C₈	C₂²×C₄×C₆₇	1	6	yes	—
130	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
131	C₄²	—	2	6	no	—
132	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
133	C₁₆	C₂²×C₄×C₆₇	1	16	yes	complemented
134	C₂³	—	2	4	no	—
135	C₂³	#97×C₆₇	1	3	yes	—
136	C₂³	#97×C₆₇	1	1	yes	characteristic
137	C₂³	C₂×M₁₆×C₆₇	1	3	yes	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	4	12	no	—
140	C₂×C₄	#97×C₆₇	1	6	yes	—
141	C₂×C₄	#97×C₆₇	1	3	yes	—
142	C₂×C₄	#97×C₆₇	1	3	yes	—
143	C₂×C₄	C₂×C₄²×C₆₇	1	3	yes	—
144	C₂×C₄	C₂×C₄²×C₆₇	1	3	yes	—
145	C₂×C₄	C₂×M₁₆×C₆₇	1	3	yes	—
146	C₂×C₄	C₂×M₁₆×C₆₇	1	3	yes	—
147	C₂×C₄	C₂×M₁₆×C₆₇	1	3	yes	—
148	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
149	C₈	—	2	4	no	—
150	C₈	C₂×C₄²×C₆₇	1	6	yes	—
151	C₈	C₂³×C₄×C₆₇	1	2	yes	—
152	C₂²	—	2	6	no	—
153	C₂²	—	2	6	no	—
154	C₂²	#347×C₆₇	1	4	yes	direct product factor
155	C₂²	C₂×#97×C₆₇	1	3	yes	—
156	C₂²	C₂×#97×C₆₇	1	3	yes	—
157	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
158	C₄	—	4	4	no	—
159	C₄	C₂×#97×C₆₇	1	3	yes	—
160	C₄	C₂×#97×C₆₇	1	3	yes	—
161	C₄	C₂²×C₄²×C₆₇	1	1	yes	characteristic
162	C₄	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	3	no	—
165	C₂	C₂×#347×C₆₇	1	6	yes	direct product factor
166	C₂	C₂²×#97×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₄²
commutator	C₄	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂³×C₆₇	#97
prime radical	C₂⁴×C₆₇	M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#347×C₆₇	C₂	direct (N × H)
7. C₂×M₃₂×C₆₇	C₄	semidirect
9. C₂²×C₁₆×C₆₇	C₄	semidirect
12. #347×C₆₇	C₂²	direct (N × H)
24. M₃₂×C₆₇	C₂×C₄	semidirect
26. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
49. C₁₆×C₆₇	C₂²×C₄	semidirect
70. C₂²×C₆₇	#347	direct (N × H)
79. C₂²×#347	C₆₇	direct (N × H)
82. C₂×C₆₇	C₂×#347	direct (N × H)
86. C₂×#347	C₂×C₆₇	direct (N × H)
87. C₆₇	C₂²×#347	direct (N × H)
91. C₂×M₃₂	C₄×C₆₇	semidirect
93. C₂²×C₁₆	C₄×C₆₇	semidirect
96. #347	C₂²×C₆₇	direct (N × H)
108. M₃₂	C₂×C₄×C₆₇	semidirect
110. C₂×C₁₆	C₂×C₄×C₆₇	semidirect
133. C₁₆	C₂²×C₄×C₆₇	semidirect
154. C₂²	#347×C₆₇	direct (N × H)
165. C₂	C₂×#347×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
C₂²×M₃₂×C₆₇	2
C₂²×#97×C₆₇	2
C₂×#347×C₆₇	2
C₂²×#347	67

Subgroups — C22×#347×C67 #30712578

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#347×C₆₇ #30712578